Require terminaison. Require Relations. Require term. Require List. Require equational_theory. Require rpo_extension. Require equational_extension. Require closure_extension. Require term_extension. Require dp. Require Inclusion. Require or_ext_generated. Require ZArith. Require ring_extention. Require Zwf. Require Inverse_Image. Require matrix. Require more_list_extention. Import List. Import ZArith. Set Implicit Arguments. Module algebra. Module F <:term.Signature. Inductive symb : Set := (* id_a____ *) | id_a____ : symb (* id_U31 *) | id_U31 : symb (* id_a__U56 *) | id_a__U56 : symb (* id_a__U26 *) | id_a__U26 : symb (* id_U61 *) | id_U61 : symb (* id_e *) | id_e : symb (* id_a__U13 *) | id_a__U13 : symb (* id_U45 *) | id_U45 : symb (* id_a__U74 *) | id_a__U74 : symb (* id_a__U44 *) | id_a__U44 : symb (* id_U81 *) | id_U81 : symb (* id_isNeList *) | id_isNeList : symb (* id_a__U11 *) | id_a__U11 : symb (* id_U41 *) | id_U41 : symb (* id_a__U71 *) | id_a__U71 : symb (* id_a__isQid *) | id_a__isQid : symb (* id_U72 *) | id_U72 : symb (* id_U11 *) | id_U11 : symb (* id_a__U23 *) | id_a__U23 : symb (* id_U53 *) | id_U53 : symb (* id_a__isNePal *) | id_a__isNePal : symb (* id_a__U52 *) | id_a__U52 : symb (* id_U91 *) | id_U91 : symb (* id_U24 *) | id_U24 : symb (* id_mark *) | id_mark : symb (* id_U33 *) | id_U33 : symb (* id_a__U62 *) | id_a__U62 : symb (* id_a__U32 *) | id_a__U32 : symb (* id_U63 *) | id_U63 : symb (* id_o *) | id_o : symb (* id_a__U21 *) | id_a__U21 : symb (* id_U51 *) | id_U51 : symb (* id_a__U82 *) | id_a__U82 : symb (* id_a__U46 *) | id_a__U46 : symb (* id_U83 *) | id_U83 : symb (* id_U22 *) | id_U22 : symb (* id_a__U12 *) | id_a__U12 : symb (* id_U43 *) | id_U43 : symb (* id_a__U73 *) | id_a__U73 : symb (* id_a__U42 *) | id_a__U42 : symb (* id_isPal *) | id_isPal : symb (* id_isPalListKind *) | id_isPalListKind : symb (* id_a__U25 *) | id_a__U25 : symb (* id_U55 *) | id_U55 : symb (* id_a__U92 *) | id_a__U92 : symb (* id_a__U54 *) | id_a__U54 : symb (* id_isList *) | id_isList : symb (* id___ *) | id___ : symb (* id_U32 *) | id_U32 : symb (* id_a__U61 *) | id_a__U61 : symb (* id_a__U31 *) | id_a__U31 : symb (* id_U62 *) | id_U62 : symb (* id_i *) | id_i : symb (* id_a__isNeList *) | id_a__isNeList : symb (* id_U46 *) | id_U46 : symb (* id_a__U81 *) | id_a__U81 : symb (* id_a__U45 *) | id_a__U45 : symb (* id_U82 *) | id_U82 : symb (* id_U21 *) | id_U21 : symb (* id_tt *) | id_tt : symb (* id_U42 *) | id_U42 : symb (* id_a__U72 *) | id_a__U72 : symb (* id_a__U41 *) | id_a__U41 : symb (* id_U73 *) | id_U73 : symb (* id_U12 *) | id_U12 : symb (* id_a__U24 *) | id_a__U24 : symb (* id_U54 *) | id_U54 : symb (* id_a__U91 *) | id_a__U91 : symb (* id_a__U53 *) | id_a__U53 : symb (* id_U92 *) | id_U92 : symb (* id_U25 *) | id_U25 : symb (* id_nil *) | id_nil : symb (* id_isQid *) | id_isQid : symb (* id_a__U63 *) | id_a__U63 : symb (* id_a__U33 *) | id_a__U33 : symb (* id_U71 *) | id_U71 : symb (* id_u *) | id_u : symb (* id_a__U22 *) | id_a__U22 : symb (* id_U52 *) | id_U52 : symb (* id_a__U83 *) | id_a__U83 : symb (* id_a__U51 *) | id_a__U51 : symb (* id_isNePal *) | id_isNePal : symb (* id_U23 *) | id_U23 : symb (* id_a__isPalListKind *) | id_a__isPalListKind : symb (* id_U44 *) | id_U44 : symb (* id_a__isPal *) | id_a__isPal : symb (* id_a__U43 *) | id_a__U43 : symb (* id_U74 *) | id_U74 : symb (* id_U13 *) | id_U13 : symb (* id_a__isList *) | id_a__isList : symb (* id_U56 *) | id_U56 : symb (* id_a *) | id_a : symb (* id_a__U55 *) | id_a__U55 : symb (* id_U26 *) | id_U26 : symb . Definition symb_eq_bool (f1 f2:symb) : bool := match f1,f2 with | id_a____,id_a____ => true | id_U31,id_U31 => true | id_a__U56,id_a__U56 => true | id_a__U26,id_a__U26 => true | id_U61,id_U61 => true | id_e,id_e => true | id_a__U13,id_a__U13 => true | id_U45,id_U45 => true | id_a__U74,id_a__U74 => true | id_a__U44,id_a__U44 => true | id_U81,id_U81 => true | id_isNeList,id_isNeList => true | id_a__U11,id_a__U11 => true | id_U41,id_U41 => true | id_a__U71,id_a__U71 => true | id_a__isQid,id_a__isQid => true | id_U72,id_U72 => true | id_U11,id_U11 => true | id_a__U23,id_a__U23 => true | id_U53,id_U53 => true | id_a__isNePal,id_a__isNePal => true | id_a__U52,id_a__U52 => true | id_U91,id_U91 => true | id_U24,id_U24 => true | id_mark,id_mark => true | id_U33,id_U33 => true | id_a__U62,id_a__U62 => true | id_a__U32,id_a__U32 => true | id_U63,id_U63 => true | id_o,id_o => true | id_a__U21,id_a__U21 => true | id_U51,id_U51 => true | id_a__U82,id_a__U82 => true | id_a__U46,id_a__U46 => true | id_U83,id_U83 => true | id_U22,id_U22 => true | id_a__U12,id_a__U12 => true | id_U43,id_U43 => true | id_a__U73,id_a__U73 => true | id_a__U42,id_a__U42 => true | id_isPal,id_isPal => true | id_isPalListKind,id_isPalListKind => true | id_a__U25,id_a__U25 => true | id_U55,id_U55 => true | id_a__U92,id_a__U92 => true | id_a__U54,id_a__U54 => true | id_isList,id_isList => true | id___,id___ => true | id_U32,id_U32 => true | id_a__U61,id_a__U61 => true | id_a__U31,id_a__U31 => true | id_U62,id_U62 => true | id_i,id_i => true | id_a__isNeList,id_a__isNeList => true | id_U46,id_U46 => true | id_a__U81,id_a__U81 => true | id_a__U45,id_a__U45 => true | id_U82,id_U82 => true | id_U21,id_U21 => true | id_tt,id_tt => true | id_U42,id_U42 => true | id_a__U72,id_a__U72 => true | id_a__U41,id_a__U41 => true | id_U73,id_U73 => true | id_U12,id_U12 => true | id_a__U24,id_a__U24 => true | id_U54,id_U54 => true | id_a__U91,id_a__U91 => true | id_a__U53,id_a__U53 => true | id_U92,id_U92 => true | id_U25,id_U25 => true | id_nil,id_nil => true | id_isQid,id_isQid => true | id_a__U63,id_a__U63 => true | id_a__U33,id_a__U33 => true | id_U71,id_U71 => true | id_u,id_u => true | id_a__U22,id_a__U22 => true | id_U52,id_U52 => true | id_a__U83,id_a__U83 => true | id_a__U51,id_a__U51 => true | id_isNePal,id_isNePal => true | id_U23,id_U23 => true | id_a__isPalListKind,id_a__isPalListKind => true | id_U44,id_U44 => true | id_a__isPal,id_a__isPal => true | id_a__U43,id_a__U43 => true | id_U74,id_U74 => true | id_U13,id_U13 => true | id_a__isList,id_a__isList => true | id_U56,id_U56 => true | id_a,id_a => true | id_a__U55,id_a__U55 => true | id_U26,id_U26 => true | _,_ => false end. (* Proof of decidability of equality over symb *) Definition symb_eq_bool_ok(f1 f2:symb) : match symb_eq_bool f1 f2 with | true => f1 = f2 | false => f1 <> f2 end. Proof. intros f1 f2. refine match f1 as u1,f2 as u2 return match symb_eq_bool u1 u2 return Prop with | true => u1 = u2 | false => u1 <> u2 end with | id_a____,id_a____ => refl_equal _ | id_U31,id_U31 => refl_equal _ | id_a__U56,id_a__U56 => refl_equal _ | id_a__U26,id_a__U26 => refl_equal _ | id_U61,id_U61 => refl_equal _ | id_e,id_e => refl_equal _ | id_a__U13,id_a__U13 => refl_equal _ | id_U45,id_U45 => refl_equal _ | id_a__U74,id_a__U74 => refl_equal _ | id_a__U44,id_a__U44 => refl_equal _ | id_U81,id_U81 => refl_equal _ | id_isNeList,id_isNeList => refl_equal _ | id_a__U11,id_a__U11 => refl_equal _ | id_U41,id_U41 => refl_equal _ | id_a__U71,id_a__U71 => refl_equal _ | id_a__isQid,id_a__isQid => refl_equal _ | id_U72,id_U72 => refl_equal _ | id_U11,id_U11 => refl_equal _ | id_a__U23,id_a__U23 => refl_equal _ | id_U53,id_U53 => refl_equal _ | id_a__isNePal,id_a__isNePal => refl_equal _ | id_a__U52,id_a__U52 => refl_equal _ | id_U91,id_U91 => refl_equal _ | id_U24,id_U24 => refl_equal _ | id_mark,id_mark => refl_equal _ | id_U33,id_U33 => refl_equal _ | id_a__U62,id_a__U62 => refl_equal _ | id_a__U32,id_a__U32 => refl_equal _ | id_U63,id_U63 => refl_equal _ | id_o,id_o => refl_equal _ | id_a__U21,id_a__U21 => refl_equal _ | id_U51,id_U51 => refl_equal _ | id_a__U82,id_a__U82 => refl_equal _ | id_a__U46,id_a__U46 => refl_equal _ | id_U83,id_U83 => refl_equal _ | id_U22,id_U22 => refl_equal _ | id_a__U12,id_a__U12 => refl_equal _ | id_U43,id_U43 => refl_equal _ | id_a__U73,id_a__U73 => refl_equal _ | id_a__U42,id_a__U42 => refl_equal _ | id_isPal,id_isPal => refl_equal _ | id_isPalListKind,id_isPalListKind => refl_equal _ | id_a__U25,id_a__U25 => refl_equal _ | id_U55,id_U55 => refl_equal _ | id_a__U92,id_a__U92 => refl_equal _ | id_a__U54,id_a__U54 => refl_equal _ | id_isList,id_isList => refl_equal _ | id___,id___ => refl_equal _ | id_U32,id_U32 => refl_equal _ | id_a__U61,id_a__U61 => refl_equal _ | id_a__U31,id_a__U31 => refl_equal _ | id_U62,id_U62 => refl_equal _ | id_i,id_i => refl_equal _ | id_a__isNeList,id_a__isNeList => refl_equal _ | id_U46,id_U46 => refl_equal _ | id_a__U81,id_a__U81 => refl_equal _ | id_a__U45,id_a__U45 => refl_equal _ | id_U82,id_U82 => refl_equal _ | id_U21,id_U21 => refl_equal _ | id_tt,id_tt => refl_equal _ | id_U42,id_U42 => refl_equal _ | id_a__U72,id_a__U72 => refl_equal _ | id_a__U41,id_a__U41 => refl_equal _ | id_U73,id_U73 => refl_equal _ | id_U12,id_U12 => refl_equal _ | id_a__U24,id_a__U24 => refl_equal _ | id_U54,id_U54 => refl_equal _ | id_a__U91,id_a__U91 => refl_equal _ | id_a__U53,id_a__U53 => refl_equal _ | id_U92,id_U92 => refl_equal _ | id_U25,id_U25 => refl_equal _ | id_nil,id_nil => refl_equal _ | id_isQid,id_isQid => refl_equal _ | id_a__U63,id_a__U63 => refl_equal _ | id_a__U33,id_a__U33 => refl_equal _ | id_U71,id_U71 => refl_equal _ | id_u,id_u => refl_equal _ | id_a__U22,id_a__U22 => refl_equal _ | id_U52,id_U52 => refl_equal _ | id_a__U83,id_a__U83 => refl_equal _ | id_a__U51,id_a__U51 => refl_equal _ | id_isNePal,id_isNePal => refl_equal _ | id_U23,id_U23 => refl_equal _ | id_a__isPalListKind,id_a__isPalListKind => refl_equal _ | id_U44,id_U44 => refl_equal _ | id_a__isPal,id_a__isPal => refl_equal _ | id_a__U43,id_a__U43 => refl_equal _ | id_U74,id_U74 => refl_equal _ | id_U13,id_U13 => refl_equal _ | id_a__isList,id_a__isList => refl_equal _ | id_U56,id_U56 => refl_equal _ | id_a,id_a => refl_equal _ | id_a__U55,id_a__U55 => refl_equal _ | id_U26,id_U26 => refl_equal _ | _,_ => _ end;intros abs;discriminate. Defined. Definition arity (f:symb) := match f with | id_a____ => term.Free 2 | id_U31 => term.Free 2 | id_a__U56 => term.Free 1 | id_a__U26 => term.Free 1 | id_U61 => term.Free 2 | id_e => term.Free 0 | id_a__U13 => term.Free 1 | id_U45 => term.Free 2 | id_a__U74 => term.Free 1 | id_a__U44 => term.Free 3 | id_U81 => term.Free 2 | id_isNeList => term.Free 1 | id_a__U11 => term.Free 2 | id_U41 => term.Free 3 | id_a__U71 => term.Free 3 | id_a__isQid => term.Free 1 | id_U72 => term.Free 2 | id_U11 => term.Free 2 | id_a__U23 => term.Free 3 | id_U53 => term.Free 3 | id_a__isNePal => term.Free 1 | id_a__U52 => term.Free 3 | id_U91 => term.Free 2 | id_U24 => term.Free 3 | id_mark => term.Free 1 | id_U33 => term.Free 1 | id_a__U62 => term.Free 2 | id_a__U32 => term.Free 2 | id_U63 => term.Free 1 | id_o => term.Free 0 | id_a__U21 => term.Free 3 | id_U51 => term.Free 3 | id_a__U82 => term.Free 2 | id_a__U46 => term.Free 1 | id_U83 => term.Free 1 | id_U22 => term.Free 3 | id_a__U12 => term.Free 2 | id_U43 => term.Free 3 | id_a__U73 => term.Free 2 | id_a__U42 => term.Free 3 | id_isPal => term.Free 1 | id_isPalListKind => term.Free 1 | id_a__U25 => term.Free 2 | id_U55 => term.Free 2 | id_a__U92 => term.Free 1 | id_a__U54 => term.Free 3 | id_isList => term.Free 1 | id___ => term.Free 2 | id_U32 => term.Free 2 | id_a__U61 => term.Free 2 | id_a__U31 => term.Free 2 | id_U62 => term.Free 2 | id_i => term.Free 0 | id_a__isNeList => term.Free 1 | id_U46 => term.Free 1 | id_a__U81 => term.Free 2 | id_a__U45 => term.Free 2 | id_U82 => term.Free 2 | id_U21 => term.Free 3 | id_tt => term.Free 0 | id_U42 => term.Free 3 | id_a__U72 => term.Free 2 | id_a__U41 => term.Free 3 | id_U73 => term.Free 2 | id_U12 => term.Free 2 | id_a__U24 => term.Free 3 | id_U54 => term.Free 3 | id_a__U91 => term.Free 2 | id_a__U53 => term.Free 3 | id_U92 => term.Free 1 | id_U25 => term.Free 2 | id_nil => term.Free 0 | id_isQid => term.Free 1 | id_a__U63 => term.Free 1 | id_a__U33 => term.Free 1 | id_U71 => term.Free 3 | id_u => term.Free 0 | id_a__U22 => term.Free 3 | id_U52 => term.Free 3 | id_a__U83 => term.Free 1 | id_a__U51 => term.Free 3 | id_isNePal => term.Free 1 | id_U23 => term.Free 3 | id_a__isPalListKind => term.Free 1 | id_U44 => term.Free 3 | id_a__isPal => term.Free 1 | id_a__U43 => term.Free 3 | id_U74 => term.Free 1 | id_U13 => term.Free 1 | id_a__isList => term.Free 1 | id_U56 => term.Free 1 | id_a => term.Free 0 | id_a__U55 => term.Free 2 | id_U26 => term.Free 1 end. Definition symb_order (f1 f2:symb) : bool := match f1,f2 with | id_a____,id_a____ => true | id_a____,id_U31 => false | id_a____,id_a__U56 => false | id_a____,id_a__U26 => false | id_a____,id_U61 => false | id_a____,id_e => false | id_a____,id_a__U13 => false | id_a____,id_U45 => false | id_a____,id_a__U74 => false | id_a____,id_a__U44 => false | id_a____,id_U81 => false | id_a____,id_isNeList => false | id_a____,id_a__U11 => false | id_a____,id_U41 => false | id_a____,id_a__U71 => false | id_a____,id_a__isQid => false | id_a____,id_U72 => false | id_a____,id_U11 => false | id_a____,id_a__U23 => false | id_a____,id_U53 => false | id_a____,id_a__isNePal => false | id_a____,id_a__U52 => false | id_a____,id_U91 => false | id_a____,id_U24 => false | id_a____,id_mark => false | id_a____,id_U33 => false | id_a____,id_a__U62 => false | id_a____,id_a__U32 => false | id_a____,id_U63 => false | id_a____,id_o => false | id_a____,id_a__U21 => false | id_a____,id_U51 => false | id_a____,id_a__U82 => false | id_a____,id_a__U46 => false | id_a____,id_U83 => false | id_a____,id_U22 => false | id_a____,id_a__U12 => false | id_a____,id_U43 => false | id_a____,id_a__U73 => false | id_a____,id_a__U42 => false | id_a____,id_isPal => false | id_a____,id_isPalListKind => false | id_a____,id_a__U25 => false | id_a____,id_U55 => false | id_a____,id_a__U92 => false | id_a____,id_a__U54 => false | id_a____,id_isList => false | id_a____,id___ => false | id_a____,id_U32 => false | id_a____,id_a__U61 => false | id_a____,id_a__U31 => false | id_a____,id_U62 => false | id_a____,id_i => false | id_a____,id_a__isNeList => false | id_a____,id_U46 => false | id_a____,id_a__U81 => false | id_a____,id_a__U45 => false | id_a____,id_U82 => false | id_a____,id_U21 => false | id_a____,id_tt => false | id_a____,id_U42 => false | id_a____,id_a__U72 => false | id_a____,id_a__U41 => false | id_a____,id_U73 => false | id_a____,id_U12 => false | id_a____,id_a__U24 => false | id_a____,id_U54 => false | id_a____,id_a__U91 => false | id_a____,id_a__U53 => false | id_a____,id_U92 => false | id_a____,id_U25 => false | id_a____,id_nil => false | id_a____,id_isQid => false | id_a____,id_a__U63 => false | id_a____,id_a__U33 => false | id_a____,id_U71 => false | id_a____,id_u => false | id_a____,id_a__U22 => false | id_a____,id_U52 => false | id_a____,id_a__U83 => false | id_a____,id_a__U51 => false | id_a____,id_isNePal => false | id_a____,id_U23 => false | id_a____,id_a__isPalListKind => false | id_a____,id_U44 => false | id_a____,id_a__isPal => false | id_a____,id_a__U43 => false | id_a____,id_U74 => false | id_a____,id_U13 => false | id_a____,id_a__isList => false | id_a____,id_U56 => false | id_a____,id_a => false | id_a____,id_a__U55 => false | id_a____,id_U26 => false | id_U31,id_a____ => true | id_U31,id_U31 => true | id_U31,id_a__U56 => false | id_U31,id_a__U26 => false | id_U31,id_U61 => false | id_U31,id_e => false | id_U31,id_a__U13 => false | id_U31,id_U45 => false | 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true | id_a__U55,id_a__isNeList => true | id_a__U55,id_U46 => true | id_a__U55,id_a__U81 => true | id_a__U55,id_a__U45 => true | id_a__U55,id_U82 => true | id_a__U55,id_U21 => true | id_a__U55,id_tt => true | id_a__U55,id_U42 => true | id_a__U55,id_a__U72 => true | id_a__U55,id_a__U41 => true | id_a__U55,id_U73 => true | id_a__U55,id_U12 => true | id_a__U55,id_a__U24 => true | id_a__U55,id_U54 => true | id_a__U55,id_a__U91 => true | id_a__U55,id_a__U53 => true | id_a__U55,id_U92 => true | id_a__U55,id_U25 => true | id_a__U55,id_nil => true | id_a__U55,id_isQid => true | id_a__U55,id_a__U63 => true | id_a__U55,id_a__U33 => true | id_a__U55,id_U71 => true | id_a__U55,id_u => true | id_a__U55,id_a__U22 => true | id_a__U55,id_U52 => true | id_a__U55,id_a__U83 => true | id_a__U55,id_a__U51 => true | id_a__U55,id_isNePal => true | id_a__U55,id_U23 => true | id_a__U55,id_a__isPalListKind => true | id_a__U55,id_U44 => true | id_a__U55,id_a__isPal => true | id_a__U55,id_a__U43 => true | id_a__U55,id_U74 => true | id_a__U55,id_U13 => true | id_a__U55,id_a__isList => true | id_a__U55,id_U56 => true | id_a__U55,id_a => true | id_a__U55,id_a__U55 => true | id_a__U55,id_U26 => false | id_U26,id_a____ => true | id_U26,id_U31 => true | id_U26,id_a__U56 => true | id_U26,id_a__U26 => true | id_U26,id_U61 => true | id_U26,id_e => true | id_U26,id_a__U13 => true | id_U26,id_U45 => true | id_U26,id_a__U74 => true | id_U26,id_a__U44 => true | id_U26,id_U81 => true | id_U26,id_isNeList => true | id_U26,id_a__U11 => true | id_U26,id_U41 => true | id_U26,id_a__U71 => true | id_U26,id_a__isQid => true | id_U26,id_U72 => true | id_U26,id_U11 => true | id_U26,id_a__U23 => true | id_U26,id_U53 => true | id_U26,id_a__isNePal => true | id_U26,id_a__U52 => true | id_U26,id_U91 => true | id_U26,id_U24 => true | id_U26,id_mark => true | id_U26,id_U33 => true | id_U26,id_a__U62 => true | id_U26,id_a__U32 => true | id_U26,id_U63 => true | id_U26,id_o => true | id_U26,id_a__U21 => true | id_U26,id_U51 => true | id_U26,id_a__U82 => true | id_U26,id_a__U46 => true | id_U26,id_U83 => true | id_U26,id_U22 => true | id_U26,id_a__U12 => true | id_U26,id_U43 => true | id_U26,id_a__U73 => true | id_U26,id_a__U42 => true | id_U26,id_isPal => true | id_U26,id_isPalListKind => true | id_U26,id_a__U25 => true | id_U26,id_U55 => true | id_U26,id_a__U92 => true | id_U26,id_a__U54 => true | id_U26,id_isList => true | id_U26,id___ => true | id_U26,id_U32 => true | id_U26,id_a__U61 => true | id_U26,id_a__U31 => true | id_U26,id_U62 => true | id_U26,id_i => true | id_U26,id_a__isNeList => true | id_U26,id_U46 => true | id_U26,id_a__U81 => true | id_U26,id_a__U45 => true | id_U26,id_U82 => true | id_U26,id_U21 => true | id_U26,id_tt => true | id_U26,id_U42 => true | id_U26,id_a__U72 => true | id_U26,id_a__U41 => true | id_U26,id_U73 => true | id_U26,id_U12 => true | id_U26,id_a__U24 => true | id_U26,id_U54 => true | id_U26,id_a__U91 => true | id_U26,id_a__U53 => true | id_U26,id_U92 => true | id_U26,id_U25 => true | id_U26,id_nil => true | id_U26,id_isQid => true | id_U26,id_a__U63 => true | id_U26,id_a__U33 => true | id_U26,id_U71 => true | id_U26,id_u => true | id_U26,id_a__U22 => true | id_U26,id_U52 => true | id_U26,id_a__U83 => true | id_U26,id_a__U51 => true | id_U26,id_isNePal => true | id_U26,id_U23 => true | id_U26,id_a__isPalListKind => true | id_U26,id_U44 => true | id_U26,id_a__isPal => true | id_U26,id_a__U43 => true | id_U26,id_U74 => true | id_U26,id_U13 => true | id_U26,id_a__isList => true | id_U26,id_U56 => true | id_U26,id_a => true | id_U26,id_a__U55 => true | id_U26,id_U26 => true end. Module Symb. Definition A := symb. Definition eq_A := @eq A. Definition eq_proof : equivalence A eq_A. Proof. constructor. red ;reflexivity . red ;intros ;transitivity y ;assumption. red ;intros ;symmetry ;assumption. Defined. Add Relation A eq_A reflexivity proved by (@equiv_refl _ _ eq_proof) symmetry proved by (@equiv_sym _ _ eq_proof) transitivity proved by (@equiv_trans _ _ eq_proof) as EQA . Definition eq_bool := symb_eq_bool. Definition eq_bool_ok := symb_eq_bool_ok. End Symb. Export Symb. End F. Module Alg := term.Make'(F)(term_extension.IntVars). Module Alg_ext := term_extension.Make(Alg). Module EQT := equational_theory.Make(Alg). Module EQT_ext := equational_extension.Make(EQT). End algebra. Module R_xml_0_deep_rew. Inductive R_xml_0_rules : algebra.Alg.term ->algebra.Alg.term ->Prop := (* a____(__(X_,Y_),Z_) -> a____(mark(X_),a____(mark(Y_),mark(Z_))) *) | R_xml_0_rule_0 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil)):: (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 2)::nil)):: (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 3)::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil)):: (algebra.Alg.Var 3)::nil)) (* a____(X_,nil) -> mark(X_) *) | R_xml_0_rule_1 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil)) (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Var 1):: (algebra.Alg.Term algebra.F.id_nil nil)::nil)) (* a____(nil,X_) -> mark(X_) *) | R_xml_0_rule_2 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil)) (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Var 1)::nil)) (* a__U11(tt,V_) -> a__U12(a__isPalListKind(V_),V_) *) | R_xml_0_rule_3 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 4)::nil))::(algebra.Alg.Var 4)::nil)) (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil)) (* a__U12(tt,V_) -> a__U13(a__isNeList(V_)) *) | R_xml_0_rule_4 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 4)::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil)) (* a__U13(tt) -> tt *) | R_xml_0_rule_5 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (* a__U21(tt,V1_,V2_) -> a__U22(a__isPalListKind(V1_),V1_,V2_) *) | R_xml_0_rule_6 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 5)::nil))::(algebra.Alg.Var 5):: (algebra.Alg.Var 6)::nil)) (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)) (* a__U22(tt,V1_,V2_) -> a__U23(a__isPalListKind(V2_),V1_,V2_) *) | R_xml_0_rule_7 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 5):: (algebra.Alg.Var 6)::nil)) (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)) (* a__U23(tt,V1_,V2_) -> a__U24(a__isPalListKind(V2_),V1_,V2_) *) | R_xml_0_rule_8 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U24 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 5):: (algebra.Alg.Var 6)::nil)) (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)) (* a__U24(tt,V1_,V2_) -> a__U25(a__isList(V1_),V2_) *) | R_xml_0_rule_9 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U25 ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 6)::nil)) (algebra.Alg.Term algebra.F.id_a__U24 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)) (* a__U25(tt,V2_) -> a__U26(a__isList(V2_)) *) | R_xml_0_rule_10 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U26 ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 6)::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U25 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 6)::nil)) (* a__U26(tt) -> tt *) | R_xml_0_rule_11 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) (algebra.Alg.Term algebra.F.id_a__U26 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (* a__U31(tt,V_) -> a__U32(a__isPalListKind(V_),V_) *) | R_xml_0_rule_12 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 4)::nil))::(algebra.Alg.Var 4)::nil)) (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil)) (* a__U32(tt,V_) -> a__U33(a__isQid(V_)) *) | R_xml_0_rule_13 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Var 4)::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil)) (* a__U33(tt) -> tt *) | R_xml_0_rule_14 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (* a__U41(tt,V1_,V2_) -> a__U42(a__isPalListKind(V1_),V1_,V2_) *) | R_xml_0_rule_15 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 5)::nil))::(algebra.Alg.Var 5):: (algebra.Alg.Var 6)::nil)) (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)) (* a__U42(tt,V1_,V2_) -> a__U43(a__isPalListKind(V2_),V1_,V2_) *) | R_xml_0_rule_16 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 5):: (algebra.Alg.Var 6)::nil)) (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)) (* a__U43(tt,V1_,V2_) -> a__U44(a__isPalListKind(V2_),V1_,V2_) *) | R_xml_0_rule_17 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 5):: (algebra.Alg.Var 6)::nil)) (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)) (* a__U44(tt,V1_,V2_) -> a__U45(a__isList(V1_),V2_) *) | R_xml_0_rule_18 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 6)::nil)) (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)) (* a__U45(tt,V2_) -> a__U46(a__isNeList(V2_)) *) | R_xml_0_rule_19 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 6)::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 6)::nil)) (* a__U46(tt) -> tt *) | R_xml_0_rule_20 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (* a__U51(tt,V1_,V2_) -> a__U52(a__isPalListKind(V1_),V1_,V2_) *) | R_xml_0_rule_21 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 5)::nil))::(algebra.Alg.Var 5):: (algebra.Alg.Var 6)::nil)) (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)) (* a__U52(tt,V1_,V2_) -> a__U53(a__isPalListKind(V2_),V1_,V2_) *) | R_xml_0_rule_22 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 5):: (algebra.Alg.Var 6)::nil)) (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)) (* a__U53(tt,V1_,V2_) -> a__U54(a__isPalListKind(V2_),V1_,V2_) *) | R_xml_0_rule_23 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U54 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 5):: (algebra.Alg.Var 6)::nil)) (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)) (* a__U54(tt,V1_,V2_) -> a__U55(a__isNeList(V1_),V2_) *) | R_xml_0_rule_24 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U55 ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 6)::nil)) (algebra.Alg.Term algebra.F.id_a__U54 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)) (* a__U55(tt,V2_) -> a__U56(a__isList(V2_)) *) | R_xml_0_rule_25 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U56 ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 6)::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U55 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 6)::nil)) (* a__U56(tt) -> tt *) | R_xml_0_rule_26 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_tt nil) (algebra.Alg.Term algebra.F.id_a__U56 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)) (* a__U61(tt,V_) -> a__U62(a__isPalListKind(V_),V_) *) | R_xml_0_rule_27 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_a__U62 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 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algebra.F.id_a__U63 ((algebra.Alg.Var 1)::nil)) (* a__U71(X1_,X2_,X3_) -> U71(X1_,X2_,X3_) *) | R_xml_0_rule_143 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_U71 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)) (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)) (* a__U72(X1_,X2_) -> U72(X1_,X2_) *) | R_xml_0_rule_144 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_U72 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)) (algebra.Alg.Term algebra.F.id_a__U72 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)) (* a__U73(X1_,X2_) -> U73(X1_,X2_) *) | R_xml_0_rule_145 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_U73 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)) (algebra.Alg.Term algebra.F.id_a__U73 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)) (* a__isPal(X_) -> isPal(X_) *) | R_xml_0_rule_146 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_isPal ((algebra.Alg.Var 1)::nil)) (algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Var 1)::nil)) (* a__U74(X_) -> U74(X_) *) | R_xml_0_rule_147 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_U74 ((algebra.Alg.Var 1)::nil)) (algebra.Alg.Term algebra.F.id_a__U74 ((algebra.Alg.Var 1)::nil)) (* a__U81(X1_,X2_) -> U81(X1_,X2_) *) | R_xml_0_rule_148 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_U81 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)) (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)) (* a__U82(X1_,X2_) -> U82(X1_,X2_) *) | R_xml_0_rule_149 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_U82 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)) (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)) (* a__U83(X_) -> U83(X_) *) | R_xml_0_rule_150 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_U83 ((algebra.Alg.Var 1)::nil)) (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Var 1)::nil)) (* a__isNePal(X_) -> isNePal(X_) *) | R_xml_0_rule_151 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_isNePal ((algebra.Alg.Var 1)::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 1)::nil)) (* a__U91(X1_,X2_) -> U91(X1_,X2_) *) | R_xml_0_rule_152 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_U91 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)) (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)) (* a__U92(X_) -> U92(X_) *) | R_xml_0_rule_153 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_U92 ((algebra.Alg.Var 1)::nil)) (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Var 1)::nil)) . Definition R_xml_0_rule_as_list_0 := ((algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil)):: (algebra.Alg.Var 3)::nil)), (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::(algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 2)::nil))::(algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 3)::nil))::nil))::nil)))::nil. Definition R_xml_0_rule_as_list_1 := ((algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Var 1):: (algebra.Alg.Term algebra.F.id_nil nil)::nil)), (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_0. Definition R_xml_0_rule_as_list_2 := ((algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_1. Definition R_xml_0_rule_as_list_3 := ((algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil)), (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 4)::nil)):: (algebra.Alg.Var 4)::nil)))::R_xml_0_rule_as_list_2. Definition R_xml_0_rule_as_list_4 := ((algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil)), (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 4)::nil))::nil))):: R_xml_0_rule_as_list_3. Definition R_xml_0_rule_as_list_5 := ((algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_4. Definition R_xml_0_rule_as_list_6 := ((algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)))::R_xml_0_rule_as_list_5 . Definition R_xml_0_rule_as_list_7 := ((algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil)):: (algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)))::R_xml_0_rule_as_list_6 . Definition R_xml_0_rule_as_list_8 := ((algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U24 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil)):: (algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)))::R_xml_0_rule_as_list_7 . Definition R_xml_0_rule_as_list_9 := ((algebra.Alg.Term algebra.F.id_a__U24 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U25 ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 6)::nil)))::R_xml_0_rule_as_list_8. Definition R_xml_0_rule_as_list_10 := ((algebra.Alg.Term algebra.F.id_a__U25 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U26 ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 6)::nil))::nil))):: R_xml_0_rule_as_list_9. Definition R_xml_0_rule_as_list_11 := ((algebra.Alg.Term algebra.F.id_a__U26 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_10. Definition R_xml_0_rule_as_list_12 := ((algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil)), (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 4)::nil)):: (algebra.Alg.Var 4)::nil)))::R_xml_0_rule_as_list_11. Definition R_xml_0_rule_as_list_13 := ((algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil)), (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Var 4)::nil))::nil))):: R_xml_0_rule_as_list_12. Definition R_xml_0_rule_as_list_14 := ((algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_13. Definition R_xml_0_rule_as_list_15 := ((algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))):: R_xml_0_rule_as_list_14. Definition R_xml_0_rule_as_list_16 := ((algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil)):: (algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))):: R_xml_0_rule_as_list_15. Definition R_xml_0_rule_as_list_17 := ((algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil)):: (algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))):: R_xml_0_rule_as_list_16. Definition R_xml_0_rule_as_list_18 := ((algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 6)::nil)))::R_xml_0_rule_as_list_17. Definition R_xml_0_rule_as_list_19 := ((algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 6)::nil))::nil))):: R_xml_0_rule_as_list_18. Definition R_xml_0_rule_as_list_20 := ((algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_19. Definition R_xml_0_rule_as_list_21 := ((algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))):: R_xml_0_rule_as_list_20. Definition R_xml_0_rule_as_list_22 := ((algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil)):: (algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))):: R_xml_0_rule_as_list_21. Definition R_xml_0_rule_as_list_23 := ((algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U54 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil)):: (algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))):: R_xml_0_rule_as_list_22. Definition R_xml_0_rule_as_list_24 := ((algebra.Alg.Term algebra.F.id_a__U54 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U55 ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 6)::nil)))::R_xml_0_rule_as_list_23. Definition R_xml_0_rule_as_list_25 := ((algebra.Alg.Term algebra.F.id_a__U55 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U56 ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 6)::nil))::nil))):: R_xml_0_rule_as_list_24. Definition R_xml_0_rule_as_list_26 := ((algebra.Alg.Term algebra.F.id_a__U56 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_25. Definition R_xml_0_rule_as_list_27 := ((algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil)), (algebra.Alg.Term algebra.F.id_a__U62 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 4)::nil)):: (algebra.Alg.Var 4)::nil)))::R_xml_0_rule_as_list_26. Definition R_xml_0_rule_as_list_28 := ((algebra.Alg.Term algebra.F.id_a__U62 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil)), (algebra.Alg.Term algebra.F.id_a__U63 ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Var 4)::nil))::nil))):: R_xml_0_rule_as_list_27. Definition R_xml_0_rule_as_list_29 := ((algebra.Alg.Term algebra.F.id_a__U63 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_28. Definition R_xml_0_rule_as_list_30 := ((algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 7)::(algebra.Alg.Var 8)::nil)), (algebra.Alg.Term algebra.F.id_a__U72 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 7)::nil)):: (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_29. Definition R_xml_0_rule_as_list_31 := ((algebra.Alg.Term algebra.F.id_a__U72 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 8)::nil)), (algebra.Alg.Term algebra.F.id_a__U73 ((algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_30. Definition R_xml_0_rule_as_list_32 := ((algebra.Alg.Term algebra.F.id_a__U73 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 8)::nil)), (algebra.Alg.Term algebra.F.id_a__U74 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 8)::nil))::nil))):: R_xml_0_rule_as_list_31. Definition R_xml_0_rule_as_list_33 := ((algebra.Alg.Term algebra.F.id_a__U74 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_32. Definition R_xml_0_rule_as_list_34 := ((algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil)), (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 4)::nil)):: (algebra.Alg.Var 4)::nil)))::R_xml_0_rule_as_list_33. Definition R_xml_0_rule_as_list_35 := ((algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 4)::nil)), (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 4)::nil))::nil))):: R_xml_0_rule_as_list_34. Definition R_xml_0_rule_as_list_36 := ((algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_35. Definition R_xml_0_rule_as_list_37 := ((algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term algebra.F.id_tt nil)::(algebra.Alg.Var 6)::nil)), (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 6)::nil))::nil))):: R_xml_0_rule_as_list_36. Definition R_xml_0_rule_as_list_38 := ((algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term algebra.F.id_tt nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_37. Definition R_xml_0_rule_as_list_39 := ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 4)::nil)), (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 4)::nil)):: (algebra.Alg.Var 4)::nil)))::R_xml_0_rule_as_list_38. Definition R_xml_0_rule_as_list_40 := ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Term algebra.F.id_nil nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_39. Definition R_xml_0_rule_as_list_41 := ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))):: R_xml_0_rule_as_list_40. Definition R_xml_0_rule_as_list_42 := ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 4)::nil)), (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 4)::nil)):: (algebra.Alg.Var 4)::nil)))::R_xml_0_rule_as_list_41. Definition R_xml_0_rule_as_list_43 := ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))):: R_xml_0_rule_as_list_42. Definition R_xml_0_rule_as_list_44 := ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))):: R_xml_0_rule_as_list_43. Definition R_xml_0_rule_as_list_45 := ((algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 4)::nil)), (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 4)::nil)):: (algebra.Alg.Var 4)::nil)))::R_xml_0_rule_as_list_44. Definition R_xml_0_rule_as_list_46 := ((algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 7)::(algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 8)::(algebra.Alg.Var 7)::nil))::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 7):: (algebra.Alg.Var 8)::nil)))::R_xml_0_rule_as_list_45. Definition R_xml_0_rule_as_list_47 := ((algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Var 4)::nil)), (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 4)::nil)):: (algebra.Alg.Var 4)::nil)))::R_xml_0_rule_as_list_46. Definition R_xml_0_rule_as_list_48 := ((algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Term algebra.F.id_nil nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_47. Definition R_xml_0_rule_as_list_49 := ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Term algebra.F.id_a nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_48. Definition R_xml_0_rule_as_list_50 := ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Term algebra.F.id_e nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_49. Definition R_xml_0_rule_as_list_51 := ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Term algebra.F.id_i nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_50. Definition R_xml_0_rule_as_list_52 := ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Term algebra.F.id_nil nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_51. Definition R_xml_0_rule_as_list_53 := ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Term algebra.F.id_o nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_52. Definition R_xml_0_rule_as_list_54 := ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Term algebra.F.id_u nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_53. Definition R_xml_0_rule_as_list_55 := ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 5)::(algebra.Alg.Var 6)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 5)::nil)):: (algebra.Alg.Var 6)::nil)))::R_xml_0_rule_as_list_54. Definition R_xml_0_rule_as_list_56 := ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term algebra.F.id_a nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_55. Definition R_xml_0_rule_as_list_57 := ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term algebra.F.id_e nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_56. Definition R_xml_0_rule_as_list_58 := ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term algebra.F.id_i nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_57. Definition R_xml_0_rule_as_list_59 := ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term algebra.F.id_o nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_58. Definition R_xml_0_rule_as_list_60 := ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Term algebra.F.id_u nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_59. Definition R_xml_0_rule_as_list_61 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 10)::nil))::nil))):: R_xml_0_rule_as_list_60. Definition R_xml_0_rule_as_list_62 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_61. Definition R_xml_0_rule_as_list_63 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U12 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_62. Definition R_xml_0_rule_as_list_64 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_isPalListKind ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 1)::nil)))::R_xml_0_rule_as_list_63. Definition R_xml_0_rule_as_list_65 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U13 ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil))):: R_xml_0_rule_as_list_64. Definition R_xml_0_rule_as_list_66 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_isNeList ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_65. Definition R_xml_0_rule_as_list_67 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U21 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_66. Definition R_xml_0_rule_as_list_68 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U22 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_67. Definition R_xml_0_rule_as_list_69 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U23 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_68. Definition R_xml_0_rule_as_list_70 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U24 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U24 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_69. Definition R_xml_0_rule_as_list_71 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U25 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U25 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_70. Definition R_xml_0_rule_as_list_72 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_isList ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_71. Definition R_xml_0_rule_as_list_73 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U26 ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U26 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil))):: R_xml_0_rule_as_list_72. Definition R_xml_0_rule_as_list_74 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U31 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_73. Definition R_xml_0_rule_as_list_75 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U32 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_74. Definition R_xml_0_rule_as_list_76 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U33 ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil))):: R_xml_0_rule_as_list_75. Definition R_xml_0_rule_as_list_77 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_isQid ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_76. Definition R_xml_0_rule_as_list_78 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U41 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_77. Definition R_xml_0_rule_as_list_79 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U42 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_78. Definition R_xml_0_rule_as_list_80 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U43 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_79. Definition R_xml_0_rule_as_list_81 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U44 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_80. Definition R_xml_0_rule_as_list_82 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U45 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_81. Definition R_xml_0_rule_as_list_83 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U46 ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil))):: R_xml_0_rule_as_list_82. Definition R_xml_0_rule_as_list_84 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U51 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_83. Definition R_xml_0_rule_as_list_85 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U52 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_84. Definition R_xml_0_rule_as_list_86 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U53 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_85. Definition R_xml_0_rule_as_list_87 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U54 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U54 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_86. Definition R_xml_0_rule_as_list_88 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U55 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U55 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_87. Definition R_xml_0_rule_as_list_89 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U56 ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U56 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil))):: R_xml_0_rule_as_list_88. Definition R_xml_0_rule_as_list_90 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U61 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_89. Definition R_xml_0_rule_as_list_91 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U62 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U62 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_90. Definition R_xml_0_rule_as_list_92 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U63 ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U63 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil))):: R_xml_0_rule_as_list_91. Definition R_xml_0_rule_as_list_93 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U71 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10):: (algebra.Alg.Var 11)::nil)))::R_xml_0_rule_as_list_92. Definition R_xml_0_rule_as_list_94 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U72 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U72 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_93. Definition R_xml_0_rule_as_list_95 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U73 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U73 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_94. Definition R_xml_0_rule_as_list_96 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_isPal ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_95. Definition R_xml_0_rule_as_list_97 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U74 ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U74 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil))):: R_xml_0_rule_as_list_96. Definition R_xml_0_rule_as_list_98 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U81 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_97. Definition R_xml_0_rule_as_list_99 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U82 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_98. Definition R_xml_0_rule_as_list_100 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U83 ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil))):: R_xml_0_rule_as_list_99. Definition R_xml_0_rule_as_list_101 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_isNePal ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_100. Definition R_xml_0_rule_as_list_102 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U91 ((algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_101. Definition R_xml_0_rule_as_list_103 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_U92 ((algebra.Alg.Var 1)::nil))::nil)), (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Var 1)::nil))::nil))):: R_xml_0_rule_as_list_102. Definition R_xml_0_rule_as_list_104 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_nil nil)::nil)),(algebra.Alg.Term algebra.F.id_nil nil)):: R_xml_0_rule_as_list_103. Definition R_xml_0_rule_as_list_105 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_tt nil)::nil)),(algebra.Alg.Term algebra.F.id_tt nil)):: R_xml_0_rule_as_list_104. Definition R_xml_0_rule_as_list_106 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_a nil)::nil)),(algebra.Alg.Term algebra.F.id_a nil)):: R_xml_0_rule_as_list_105. Definition R_xml_0_rule_as_list_107 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_e nil)::nil)),(algebra.Alg.Term algebra.F.id_e nil)):: R_xml_0_rule_as_list_106. Definition R_xml_0_rule_as_list_108 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_i nil)::nil)),(algebra.Alg.Term algebra.F.id_i nil)):: R_xml_0_rule_as_list_107. Definition R_xml_0_rule_as_list_109 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_o nil)::nil)),(algebra.Alg.Term algebra.F.id_o nil)):: R_xml_0_rule_as_list_108. Definition R_xml_0_rule_as_list_110 := ((algebra.Alg.Term algebra.F.id_mark ((algebra.Alg.Term algebra.F.id_u nil)::nil)),(algebra.Alg.Term algebra.F.id_u nil)):: R_xml_0_rule_as_list_109. Definition R_xml_0_rule_as_list_111 := ((algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id___ ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_110. Definition R_xml_0_rule_as_list_112 := ((algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U11 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_111. Definition R_xml_0_rule_as_list_113 := ((algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U12 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_112. Definition R_xml_0_rule_as_list_114 := ((algebra.Alg.Term algebra.F.id_a__isPalListKind ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_isPalListKind ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_113. Definition R_xml_0_rule_as_list_115 := ((algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_U13 ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_114. Definition R_xml_0_rule_as_list_116 := ((algebra.Alg.Term algebra.F.id_a__isNeList ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_isNeList ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_115. Definition R_xml_0_rule_as_list_117 := ((algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U21 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_116. Definition R_xml_0_rule_as_list_118 := ((algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U22 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_117. Definition R_xml_0_rule_as_list_119 := ((algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U23 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_118. Definition R_xml_0_rule_as_list_120 := ((algebra.Alg.Term algebra.F.id_a__U24 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U24 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_119. Definition R_xml_0_rule_as_list_121 := ((algebra.Alg.Term algebra.F.id_a__U25 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U25 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_120. Definition R_xml_0_rule_as_list_122 := ((algebra.Alg.Term algebra.F.id_a__isList ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_isList ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_121. Definition R_xml_0_rule_as_list_123 := ((algebra.Alg.Term algebra.F.id_a__U26 ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_U26 ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_122. Definition R_xml_0_rule_as_list_124 := ((algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U31 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_123. Definition R_xml_0_rule_as_list_125 := ((algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U32 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_124. Definition R_xml_0_rule_as_list_126 := ((algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_U33 ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_125. Definition R_xml_0_rule_as_list_127 := ((algebra.Alg.Term algebra.F.id_a__isQid ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_isQid ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_126. Definition R_xml_0_rule_as_list_128 := ((algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U41 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_127. Definition R_xml_0_rule_as_list_129 := ((algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U42 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_128. Definition R_xml_0_rule_as_list_130 := ((algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U43 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_129. Definition R_xml_0_rule_as_list_131 := ((algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U44 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_130. Definition R_xml_0_rule_as_list_132 := ((algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U45 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_131. Definition R_xml_0_rule_as_list_133 := ((algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_U46 ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_132. Definition R_xml_0_rule_as_list_134 := ((algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U51 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_133. Definition R_xml_0_rule_as_list_135 := ((algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U52 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_134. Definition R_xml_0_rule_as_list_136 := ((algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U53 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_135. Definition R_xml_0_rule_as_list_137 := ((algebra.Alg.Term algebra.F.id_a__U54 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U54 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_136. Definition R_xml_0_rule_as_list_138 := ((algebra.Alg.Term algebra.F.id_a__U55 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U55 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_137. Definition R_xml_0_rule_as_list_139 := ((algebra.Alg.Term algebra.F.id_a__U56 ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_U56 ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_138. Definition R_xml_0_rule_as_list_140 := ((algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U61 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_139. Definition R_xml_0_rule_as_list_141 := ((algebra.Alg.Term algebra.F.id_a__U62 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U62 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_140. Definition R_xml_0_rule_as_list_142 := ((algebra.Alg.Term algebra.F.id_a__U63 ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_U63 ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_141. Definition R_xml_0_rule_as_list_143 := ((algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil)), (algebra.Alg.Term algebra.F.id_U71 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::(algebra.Alg.Var 11)::nil))):: R_xml_0_rule_as_list_142. Definition R_xml_0_rule_as_list_144 := ((algebra.Alg.Term algebra.F.id_a__U72 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U72 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_143. Definition R_xml_0_rule_as_list_145 := ((algebra.Alg.Term algebra.F.id_a__U73 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U73 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_144. Definition R_xml_0_rule_as_list_146 := ((algebra.Alg.Term algebra.F.id_a__isPal ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_isPal ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_145. Definition R_xml_0_rule_as_list_147 := ((algebra.Alg.Term algebra.F.id_a__U74 ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_U74 ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_146. Definition R_xml_0_rule_as_list_148 := ((algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U81 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_147. Definition R_xml_0_rule_as_list_149 := ((algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U82 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_148. Definition R_xml_0_rule_as_list_150 := ((algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_U83 ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_149. Definition R_xml_0_rule_as_list_151 := ((algebra.Alg.Term algebra.F.id_a__isNePal ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_isNePal ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_150. Definition R_xml_0_rule_as_list_152 := ((algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_U91 ((algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)))::R_xml_0_rule_as_list_151. Definition R_xml_0_rule_as_list_153 := ((algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Var 1)::nil)), (algebra.Alg.Term algebra.F.id_U92 ((algebra.Alg.Var 1)::nil))):: R_xml_0_rule_as_list_152. Definition R_xml_0_rule_as_list := R_xml_0_rule_as_list_153. Lemma R_xml_0_rules_included : forall l r, R_xml_0_rules r l <-> In (l,r) R_xml_0_rule_as_list. Proof. intros l r. constructor. intros H. case H;clear H; (apply (more_list.mem_impl_in (@eq (algebra.Alg.term*algebra.Alg.term))); [tauto|idtac]); match goal with | |- _ _ _ ?t ?l => let u := fresh "u" in (generalize (more_list.mem_bool_ok _ _ algebra.Alg_ext.eq_term_term_bool_ok t l); set (u:=more_list.mem_bool algebra.Alg_ext.eq_term_term_bool t l) in *; vm_compute in u|-;unfold u in *;clear u;intros H;refine H) end . intros H. vm_compute in H|-. rewrite <- or_ext_generated.or25_equiv in H|-. case H;clear H;intros H. injection H;intros ;subst;constructor 154. injection H;intros ;subst;constructor 153. injection H;intros ;subst;constructor 152. injection H;intros ;subst;constructor 151. injection H;intros ;subst;constructor 150. injection H;intros ;subst;constructor 149. injection H;intros ;subst;constructor 148. injection H;intros ;subst;constructor 147. injection H;intros ;subst;constructor 146. injection H;intros ;subst;constructor 145. injection H;intros ;subst;constructor 144. injection H;intros ;subst;constructor 143. injection H;intros ;subst;constructor 142. injection H;intros ;subst;constructor 141. injection H;intros ;subst;constructor 140. injection H;intros ;subst;constructor 139. injection H;intros ;subst;constructor 138. injection H;intros ;subst;constructor 137. injection H;intros ;subst;constructor 136. injection H;intros ;subst;constructor 135. injection H;intros ;subst;constructor 134. injection H;intros ;subst;constructor 133. injection H;intros ;subst;constructor 132. injection H;intros ;subst;constructor 131. rewrite <- or_ext_generated.or25_equiv in H|-. case H;clear H;intros H. injection H;intros ;subst;constructor 130. injection H;intros ;subst;constructor 129. injection H;intros ;subst;constructor 128. injection H;intros ;subst;constructor 127. injection H;intros ;subst;constructor 126. injection H;intros ;subst;constructor 125. injection H;intros ;subst;constructor 124. injection H;intros ;subst;constructor 123. injection H;intros ;subst;constructor 122. injection H;intros ;subst;constructor 121. injection H;intros ;subst;constructor 120. injection H;intros ;subst;constructor 119. injection H;intros ;subst;constructor 118. injection H;intros ;subst;constructor 117. injection H;intros ;subst;constructor 116. injection H;intros ;subst;constructor 115. injection H;intros ;subst;constructor 114. injection H;intros ;subst;constructor 113. injection H;intros ;subst;constructor 112. injection H;intros ;subst;constructor 111. injection H;intros ;subst;constructor 110. injection H;intros ;subst;constructor 109. injection H;intros ;subst;constructor 108. injection H;intros ;subst;constructor 107. rewrite <- or_ext_generated.or25_equiv in H|-. case H;clear H;intros H. injection H;intros ;subst;constructor 106. injection H;intros ;subst;constructor 105. injection H;intros ;subst;constructor 104. injection H;intros ;subst;constructor 103. injection H;intros ;subst;constructor 102. injection H;intros ;subst;constructor 101. injection H;intros ;subst;constructor 100. injection H;intros ;subst;constructor 99. injection H;intros ;subst;constructor 98. injection H;intros ;subst;constructor 97. injection H;intros ;subst;constructor 96. injection H;intros ;subst;constructor 95. injection H;intros ;subst;constructor 94. injection H;intros ;subst;constructor 93. injection H;intros ;subst;constructor 92. injection H;intros ;subst;constructor 91. injection H;intros ;subst;constructor 90. injection H;intros ;subst;constructor 89. injection H;intros ;subst;constructor 88. injection H;intros ;subst;constructor 87. injection H;intros ;subst;constructor 86. injection H;intros ;subst;constructor 85. injection H;intros ;subst;constructor 84. injection H;intros ;subst;constructor 83. rewrite <- or_ext_generated.or25_equiv in H|-. case H;clear H;intros H. injection H;intros ;subst;constructor 82. injection H;intros ;subst;constructor 81. injection H;intros ;subst;constructor 80. injection H;intros ;subst;constructor 79. injection H;intros ;subst;constructor 78. injection H;intros ;subst;constructor 77. injection H;intros ;subst;constructor 76. injection H;intros ;subst;constructor 75. injection H;intros ;subst;constructor 74. injection H;intros ;subst;constructor 73. injection H;intros ;subst;constructor 72. injection H;intros ;subst;constructor 71. injection H;intros ;subst;constructor 70. injection H;intros ;subst;constructor 69. injection H;intros ;subst;constructor 68. injection H;intros ;subst;constructor 67. injection H;intros ;subst;constructor 66. injection H;intros ;subst;constructor 65. injection H;intros ;subst;constructor 64. injection H;intros ;subst;constructor 63. injection H;intros ;subst;constructor 62. injection H;intros ;subst;constructor 61. injection H;intros ;subst;constructor 60. injection H;intros ;subst;constructor 59. rewrite <- or_ext_generated.or25_equiv in H|-. case H;clear H;intros H. injection H;intros ;subst;constructor 58. injection H;intros ;subst;constructor 57. injection H;intros ;subst;constructor 56. injection H;intros ;subst;constructor 55. injection H;intros ;subst;constructor 54. injection H;intros ;subst;constructor 53. injection H;intros ;subst;constructor 52. injection H;intros ;subst;constructor 51. injection H;intros ;subst;constructor 50. injection H;intros ;subst;constructor 49. injection H;intros ;subst;constructor 48. injection H;intros ;subst;constructor 47. injection H;intros ;subst;constructor 46. injection H;intros ;subst;constructor 45. injection H;intros ;subst;constructor 44. injection H;intros ;subst;constructor 43. injection H;intros ;subst;constructor 42. injection H;intros ;subst;constructor 41. injection H;intros ;subst;constructor 40. injection H;intros ;subst;constructor 39. injection H;intros ;subst;constructor 38. injection H;intros ;subst;constructor 37. injection H;intros ;subst;constructor 36. injection H;intros ;subst;constructor 35. rewrite <- or_ext_generated.or25_equiv in H|-. case H;clear H;intros H. injection H;intros ;subst;constructor 34. injection H;intros ;subst;constructor 33. injection H;intros ;subst;constructor 32. injection H;intros ;subst;constructor 31. injection H;intros ;subst;constructor 30. injection H;intros ;subst;constructor 29. injection H;intros ;subst;constructor 28. injection H;intros ;subst;constructor 27. injection H;intros ;subst;constructor 26. injection H;intros ;subst;constructor 25. injection H;intros ;subst;constructor 24. injection H;intros ;subst;constructor 23. injection H;intros ;subst;constructor 22. injection H;intros ;subst;constructor 21. injection H;intros ;subst;constructor 20. injection H;intros ;subst;constructor 19. injection H;intros ;subst;constructor 18. injection H;intros ;subst;constructor 17. injection H;intros ;subst;constructor 16. injection H;intros ;subst;constructor 15. injection H;intros ;subst;constructor 14. injection H;intros ;subst;constructor 13. injection H;intros ;subst;constructor 12. injection H;intros ;subst;constructor 11. rewrite <- or_ext_generated.or11_equiv in H|-. case H;clear H;intros H. injection H;intros ;subst;constructor 10. injection H;intros ;subst;constructor 9. injection H;intros ;subst;constructor 8. injection H;intros ;subst;constructor 7. injection H;intros ;subst;constructor 6. injection H;intros ;subst;constructor 5. injection H;intros ;subst;constructor 4. injection H;intros ;subst;constructor 3. injection H;intros ;subst;constructor 2. injection H;intros ;subst;constructor 1. elim H. Qed. Lemma R_xml_0_non_var : forall x t, ~R_xml_0_rules t (algebra.EQT.T.Var x). Proof. intros x t H. inversion H. Qed. Lemma R_xml_0_reg : forall s t, (R_xml_0_rules s t) -> forall x, In x (algebra.Alg.var_list s) ->In x (algebra.Alg.var_list t). Proof. intros s t H. inversion H;intros x Hx; (apply (more_list.mem_impl_in (@eq algebra.Alg.variable));[tauto|idtac]); apply (more_list.in_impl_mem (@eq algebra.Alg.variable)) in Hx; vm_compute in Hx|-*;tauto. Qed. Inductive and_50 (x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30 x31 x32 x33 x34 x35 x36 x37 x38 x39 x40 x41 x42 x43 x44 x45 x46 x47 x48 x49 x50 x51 x52 x53 x54 x55 x56 x57 x58 x59 x60 x61 x62:Prop) : Prop := | conj_50 : x13->x14->x15->x16->x17->x18->x19->x20->x21->x22->x23->x24->x25-> x26->x27->x28->x29->x30->x31->x32->x33->x34->x35->x36->x37->x38-> x39->x40->x41->x42->x43->x44->x45->x46->x47->x48->x49->x50->x51-> x52->x53->x54->x55->x56->x57->x58->x59->x60->x61->x62-> and_50 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30 x31 x32 x33 x34 x35 x36 x37 x38 x39 x40 x41 x42 x43 x44 x45 x46 x47 x48 x49 x50 x51 x52 x53 x54 x55 x56 x57 x58 x59 x60 x61 x62 . Lemma are_constuctors_of_R_xml_0 : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> and_50 (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U31 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U31 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U61 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U61 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (t = (algebra.Alg.Term algebra.F.id_e nil) -> t' = (algebra.Alg.Term algebra.F.id_e nil)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U45 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U45 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U81 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U81 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (forall x14, t = (algebra.Alg.Term algebra.F.id_isNeList (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_isNeList (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U41 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U41 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U72 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U72 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U11 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U11 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U53 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U53 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U91 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U91 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U24 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U24 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (forall x14, t = (algebra.Alg.Term algebra.F.id_U33 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U33 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (forall x14, t = (algebra.Alg.Term algebra.F.id_U63 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U63 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (t = (algebra.Alg.Term algebra.F.id_o nil) -> t' = (algebra.Alg.Term algebra.F.id_o nil)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U51 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U51 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (forall x14, t = (algebra.Alg.Term algebra.F.id_U83 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U83 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U22 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U22 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U43 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U43 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (forall x14, t = (algebra.Alg.Term algebra.F.id_isPal (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_isPal (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (forall x14, t = (algebra.Alg.Term algebra.F.id_isPalListKind (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U55 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U55 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (forall x14, t = (algebra.Alg.Term algebra.F.id_isList (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_isList (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id___ (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id___ (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U32 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U32 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U62 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U62 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (t = (algebra.Alg.Term algebra.F.id_i nil) -> t' = (algebra.Alg.Term algebra.F.id_i nil)) (forall x14, t = (algebra.Alg.Term algebra.F.id_U46 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U46 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U82 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U82 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U21 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U21 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (t = (algebra.Alg.Term algebra.F.id_tt nil) -> t' = (algebra.Alg.Term algebra.F.id_tt nil)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U42 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U42 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U73 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U73 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U12 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U12 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U54 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U54 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (forall x14, t = (algebra.Alg.Term algebra.F.id_U92 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U92 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U25 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U25 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)) (t = (algebra.Alg.Term algebra.F.id_nil nil) -> t' = (algebra.Alg.Term algebra.F.id_nil nil)) (forall x14, t = (algebra.Alg.Term algebra.F.id_isQid (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_isQid (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U71 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U71 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (t = (algebra.Alg.Term algebra.F.id_u nil) -> t' = (algebra.Alg.Term algebra.F.id_u nil)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U52 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U52 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (forall x14, t = (algebra.Alg.Term algebra.F.id_isNePal (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_isNePal (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U23 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U23 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U44 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U44 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18)) (forall x14, t = (algebra.Alg.Term algebra.F.id_U74 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U74 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (forall x14, t = (algebra.Alg.Term algebra.F.id_U13 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U13 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (forall x14, t = (algebra.Alg.Term algebra.F.id_U56 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U56 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)) (t = (algebra.Alg.Term algebra.F.id_a nil) -> t' = (algebra.Alg.Term algebra.F.id_a nil)) (forall x14, t = (algebra.Alg.Term algebra.F.id_U26 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U26 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)). Proof. intros t t' H. induction H as [|y IH z z_to_y] using closure_extension.refl_trans_clos_ind2. constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros H;intuition;constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros H;intuition;constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros H;intuition;constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros H;intuition;constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros x14 x16 H;exists x14;exists x16;intuition;constructor 1. intros H;intuition;constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros H;intuition;constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros x14 x16 x18 H;exists x14;exists x16;exists x18;intuition; constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros x14 H;exists x14;intuition;constructor 1. intros H;intuition;constructor 1. intros x14 H;exists x14;intuition;constructor 1. inversion z_to_y as [t1 t2 H H0 H1|f l1 l2 H0 H H2];clear z_to_y;subst. inversion H as [t1 t2 sigma H2 H1 H0];clear H IH;subst;inversion H2; clear ;constructor;try (intros until 0 );clear ;intros abs; discriminate abs. destruct IH as [H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26]. constructor. clear H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U31 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U31 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U61 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U61 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros H;injection H;clear H; intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). clear H_id_U31 H_id_U61 H_id_e H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U45 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U45 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U81 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U81 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_isNeList y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U41 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U41 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U41 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U72 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U72 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U11 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U11 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U53 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U53 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U53 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U91 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U91 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U24 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U24 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U24 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U33 y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U63 y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros H;injection H;clear H; intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U51 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U51 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U51 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U83 y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U22 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U22 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U22 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U43 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U43 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U43 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_isPal y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H;clear H;intros ; subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_isPalListKind y (refl_equal _)) as [x13];intros ; intuition;exists x13;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U55 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U55 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_isList y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id___ y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id___ x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U32 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U32 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U62 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U62 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros H;injection H;clear H; intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U46 y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U82 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U82 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U21 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U21 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U21 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros H;injection H;clear H; intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U42 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U42 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U42 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U73 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U73 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U12 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U12 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U54 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U54 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U54 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U92 y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U25 y x16 (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U25 x14 y (refl_equal _)) as [x13 [x15]];intros ; intuition;exists x13;exists x15;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros H;injection H;clear H; intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H;clear H; intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_isQid y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U71 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U71 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U71 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros H;injection H;clear H; intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U52 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U52 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U52 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H;clear H; intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_isNePal y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U23 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U23 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U23 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U74 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 x16 x18 H; injection H;clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U44 y x16 x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x16 |- _ => destruct (H_id_U44 x14 y x18 (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x18 |- _ => destruct (H_id_U44 x14 x16 y (refl_equal _)) as [x13 [x15 [x17]]]; intros ;intuition;exists x13;exists x15;exists x17;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U13 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U74 y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U56 H_id_a H_id_U26;intros x14 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U13 y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_a H_id_U26;intros x14 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U56 y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_U26;intros H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). clear H_id_U31 H_id_U61 H_id_e H_id_U45 H_id_U81 H_id_isNeList H_id_U41 H_id_U72 H_id_U11 H_id_U53 H_id_U91 H_id_U24 H_id_U33 H_id_U63 H_id_o H_id_U51 H_id_U83 H_id_U22 H_id_U43 H_id_isPal H_id_isPalListKind H_id_U55 H_id_isList H_id___ H_id_U32 H_id_U62 H_id_i H_id_U46 H_id_U82 H_id_U21 H_id_tt H_id_U42 H_id_U73 H_id_U12 H_id_U54 H_id_U92 H_id_U25 H_id_nil H_id_isQid H_id_U71 H_id_u H_id_U52 H_id_isNePal H_id_U23 H_id_U44 H_id_U74 H_id_U13 H_id_U56 H_id_a;intros x14 H;injection H; clear H;intros ;subst; repeat ( match goal with | H:closure.one_step_list (algebra.EQT.one_step _) _ _ |- _ => inversion H;clear H;subst end ). match goal with | H:algebra.EQT.one_step _ ?y x14 |- _ => destruct (H_id_U26 y (refl_equal _)) as [x13];intros ;intuition; exists x13;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . Qed. Lemma id_U31_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U31 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U31 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U61_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U61 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U61 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_e_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> t = (algebra.Alg.Term algebra.F.id_e nil) -> t' = (algebra.Alg.Term algebra.F.id_e nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U45_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U45 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U45 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U81_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U81 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U81 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_isNeList_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_isNeList (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_isNeList (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U41_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U41 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U41 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U72_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U72 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U72 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U11_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U11 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U11 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U53_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U53 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U53 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U91_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U91 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U91 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U24_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U24 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U24 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U33_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_U33 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U33 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U63_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_U63 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U63 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_o_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> t = (algebra.Alg.Term algebra.F.id_o nil) -> t' = (algebra.Alg.Term algebra.F.id_o nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U51_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U51 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U51 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U83_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_U83 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U83 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U22_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U22 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U22 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U43_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U43 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U43 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_isPal_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_isPal (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_isPal (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_isPalListKind_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_isPalListKind (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U55_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U55 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U55 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_isList_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_isList (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_isList (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id____is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id___ (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id___ (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U32_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U32 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U32 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U62_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U62 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U62 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_i_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> t = (algebra.Alg.Term algebra.F.id_i nil) -> t' = (algebra.Alg.Term algebra.F.id_i nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U46_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_U46 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U46 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U82_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U82 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U82 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U21_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U21 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U21 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_tt_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> t = (algebra.Alg.Term algebra.F.id_tt nil) -> t' = (algebra.Alg.Term algebra.F.id_tt nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U42_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U42 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U42 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U73_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U73 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U73 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U12_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U12 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U12 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U54_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U54 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U54 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U92_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_U92 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U92 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U25_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16, t = (algebra.Alg.Term algebra.F.id_U25 (x14::x16::nil)) -> exists x13, exists x15, t' = (algebra.Alg.Term algebra.F.id_U25 (x13::x15::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_nil_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> t = (algebra.Alg.Term algebra.F.id_nil nil) -> t' = (algebra.Alg.Term algebra.F.id_nil nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_isQid_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_isQid (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_isQid (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U71_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U71 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U71 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_u_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> t = (algebra.Alg.Term algebra.F.id_u nil) -> t' = (algebra.Alg.Term algebra.F.id_u nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U52_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U52 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U52 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_isNePal_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_isNePal (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_isNePal (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U23_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U23 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U23 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U44_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14 x16 x18, t = (algebra.Alg.Term algebra.F.id_U44 (x14::x16::x18::nil)) -> exists x13, exists x15, exists x17, t' = (algebra.Alg.Term algebra.F.id_U44 (x13::x15::x17::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x15 x16)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x17 x18). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U74_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_U74 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U74 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U13_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_U13 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U13 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U56_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_U56 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U56 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_a_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> t = (algebra.Alg.Term algebra.F.id_a nil) -> t' = (algebra.Alg.Term algebra.F.id_a nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_U26_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x14, t = (algebra.Alg.Term algebra.F.id_U26 (x14::nil)) -> exists x13, t' = (algebra.Alg.Term algebra.F.id_U26 (x13::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x13 x14). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Ltac impossible_star_reduction_R_xml_0 := match goal with | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U31 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U31_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U61 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U61_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_e nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_e_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; impossible_star_reduction_R_xml_0 )) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U45 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U45_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U81 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U81_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_isNeList (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isNeList_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U41 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U41_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U72 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U72_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U11 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U11_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U53 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U53_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U91 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U91_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U24 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U24_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U33 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U33_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U63 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U63_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_o nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_o_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; impossible_star_reduction_R_xml_0 )) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U51 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U51_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U83 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U83_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U22 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U22_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U43 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U43_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_isPal (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isPal_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_isPalListKind (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isPalListKind_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U55 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U55_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_isList (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isList_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id___ (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id____is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U32 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U32_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U62 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U62_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_i nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_i_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; impossible_star_reduction_R_xml_0 )) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U46 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U46_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U82 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U82_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U21 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U21_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_tt nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_tt_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; impossible_star_reduction_R_xml_0 )) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U42 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U42_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U73 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U73_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U12 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U12_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U54 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U54_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U92 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U92_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U25 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U25_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_nil nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_nil_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; impossible_star_reduction_R_xml_0 )) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_isQid (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isQid_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U71 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U71_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_u nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_u_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; impossible_star_reduction_R_xml_0 )) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U52 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U52_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_isNePal (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isNePal_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U23 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U23_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U44 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U44_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U74 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U74_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U13 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U13_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U56 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U56_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_a nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_a_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; impossible_star_reduction_R_xml_0 )) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U26 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U26_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; impossible_star_reduction_R_xml_0 )))) end . Ltac simplify_star_reduction_R_xml_0 := match goal with | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U31 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U31_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U61 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U61_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_e nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_e_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; try (simplify_star_reduction_R_xml_0 ))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U45 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U45_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U81 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U81_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_isNeList (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isNeList_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U41 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U41_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U72 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U72_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U11 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U11_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U53 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U53_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U91 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U91_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U24 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U24_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U33 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U33_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U63 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U63_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_o nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_o_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; try (simplify_star_reduction_R_xml_0 ))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U51 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U51_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U83 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U83_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U22 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U22_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U43 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U43_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_isPal (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isPal_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_isPalListKind (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isPalListKind_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U55 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U55_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_isList (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isList_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id___ (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id____is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U32 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U32_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U62 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U62_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_i nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_i_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; try (simplify_star_reduction_R_xml_0 ))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U46 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U46_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U82 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U82_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U21 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U21_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_tt nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_tt_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; try (simplify_star_reduction_R_xml_0 ))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U42 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U42_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U73 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U73_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U12 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U12_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U54 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U54_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U92 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U92_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U25 (?x14::?x13::nil)) |- _ => let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_U25_is_R_xml_0_constructor H (refl_equal _)) as [x14 [x13 [Heq [Hred2 Hred1]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_nil nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_nil_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; try (simplify_star_reduction_R_xml_0 ))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_isQid (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isQid_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U71 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U71_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_u nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_u_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; try (simplify_star_reduction_R_xml_0 ))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U52 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U52_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_isNePal (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_isNePal_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U23 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U23_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U44 (?x15::?x14::?x13::nil)) |- _ => let x15 := fresh "x" in (let x14 := fresh "x" in (let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (let Hred3 := fresh "Hred" in (destruct (id_U44_is_R_xml_0_constructor H (refl_equal _)) as [x15 [x14 [x13 [Heq [Hred3 [Hred2 Hred1]]]]]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U74 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U74_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U13 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U13_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U56 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U56_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_a nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_a_is_R_xml_0_constructor H (refl_equal _)) in *; (discriminate Heq)|| (clearbody Heq;try (subst);try (clear Heq);clear H; try (simplify_star_reduction_R_xml_0 ))) | H:closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) _ (algebra.Alg.Term algebra.F.id_U26 (?x13::nil)) |- _ => let x13 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_U26_is_R_xml_0_constructor H (refl_equal _)) as [x13 [Heq Hred1]]; (discriminate Heq)|| (injection Heq;intros ;subst;clear Heq;clear H; try (simplify_star_reduction_R_xml_0 ))))) end . End R_xml_0_deep_rew. Module InterpGen := interp.Interp(algebra.EQT). Module ddp := dp.MakeDP(algebra.EQT). Module SymbType. Definition A := algebra.Alg.F.Symb.A. End SymbType. Module Symb_more_list := more_list_extention.Make(SymbType)(algebra.Alg.F.Symb). Module SymbSet := list_set.Make(algebra.F.Symb). Module Interp. Section S. Require Import interp. Hypothesis A : Type. Hypothesis Ale Alt Aeq : A -> A -> Prop. Hypothesis Aop : interp.ordering_pair Aeq Alt Ale. Hypothesis A0 : A. Notation Local "a <= b" := (Ale a b). Hypothesis P_id_a____ : A ->A ->A. Hypothesis P_id_U31 : A ->A ->A. Hypothesis P_id_a__U56 : A ->A. Hypothesis P_id_a__U26 : A ->A. Hypothesis P_id_U61 : A ->A ->A. Hypothesis P_id_e : A. Hypothesis P_id_a__U13 : A ->A. Hypothesis P_id_U45 : A ->A ->A. Hypothesis P_id_a__U74 : A ->A. Hypothesis P_id_a__U44 : A ->A ->A ->A. Hypothesis P_id_U81 : A ->A ->A. Hypothesis P_id_isNeList : A ->A. Hypothesis P_id_a__U11 : A ->A ->A. Hypothesis P_id_U41 : A ->A ->A ->A. Hypothesis P_id_a__U71 : A ->A ->A ->A. Hypothesis P_id_a__isQid : A ->A. Hypothesis P_id_U72 : A ->A ->A. Hypothesis P_id_U11 : A ->A ->A. Hypothesis P_id_a__U23 : A ->A ->A ->A. Hypothesis P_id_U53 : A ->A ->A ->A. Hypothesis P_id_a__isNePal : A ->A. Hypothesis P_id_a__U52 : A ->A ->A ->A. Hypothesis P_id_U91 : A ->A ->A. Hypothesis P_id_U24 : A ->A ->A ->A. Hypothesis P_id_mark : A ->A. Hypothesis P_id_U33 : A ->A. Hypothesis P_id_a__U62 : A ->A ->A. Hypothesis P_id_a__U32 : A ->A ->A. Hypothesis P_id_U63 : A ->A. Hypothesis P_id_o : A. Hypothesis P_id_a__U21 : A ->A ->A ->A. Hypothesis P_id_U51 : A ->A ->A ->A. Hypothesis P_id_a__U82 : A ->A ->A. Hypothesis P_id_a__U46 : A ->A. Hypothesis P_id_U83 : A ->A. Hypothesis P_id_U22 : A ->A ->A ->A. Hypothesis P_id_a__U12 : A ->A ->A. Hypothesis P_id_U43 : A ->A ->A ->A. Hypothesis P_id_a__U73 : A ->A ->A. Hypothesis P_id_a__U42 : A ->A ->A ->A. Hypothesis P_id_isPal : A ->A. Hypothesis P_id_isPalListKind : A ->A. Hypothesis P_id_a__U25 : A ->A ->A. Hypothesis P_id_U55 : A ->A ->A. Hypothesis P_id_a__U92 : A ->A. Hypothesis P_id_a__U54 : A ->A ->A ->A. Hypothesis P_id_isList : A ->A. Hypothesis P_id___ : A ->A ->A. Hypothesis P_id_U32 : A ->A ->A. Hypothesis P_id_a__U61 : A ->A ->A. Hypothesis P_id_a__U31 : A ->A ->A. Hypothesis P_id_U62 : A ->A ->A. Hypothesis P_id_i : A. Hypothesis P_id_a__isNeList : A ->A. Hypothesis P_id_U46 : A ->A. Hypothesis P_id_a__U81 : A ->A ->A. Hypothesis P_id_a__U45 : A ->A ->A. Hypothesis P_id_U82 : A ->A ->A. Hypothesis P_id_U21 : A ->A ->A ->A. Hypothesis P_id_tt : A. Hypothesis P_id_U42 : A ->A ->A ->A. Hypothesis P_id_a__U72 : A ->A ->A. Hypothesis P_id_a__U41 : A ->A ->A ->A. Hypothesis P_id_U73 : A ->A ->A. Hypothesis P_id_U12 : A ->A ->A. Hypothesis P_id_a__U24 : A ->A ->A ->A. Hypothesis P_id_U54 : A ->A ->A ->A. Hypothesis P_id_a__U91 : A ->A ->A. Hypothesis P_id_a__U53 : A ->A ->A ->A. Hypothesis P_id_U92 : A ->A. Hypothesis P_id_U25 : A ->A ->A. Hypothesis P_id_nil : A. Hypothesis P_id_isQid : A ->A. Hypothesis P_id_a__U63 : A ->A. Hypothesis P_id_a__U33 : A ->A. Hypothesis P_id_U71 : A ->A ->A ->A. Hypothesis P_id_u : A. Hypothesis P_id_a__U22 : A ->A ->A ->A. Hypothesis P_id_U52 : A ->A ->A ->A. Hypothesis P_id_a__U83 : A ->A. Hypothesis P_id_a__U51 : A ->A ->A ->A. Hypothesis P_id_isNePal : A ->A. Hypothesis P_id_U23 : A ->A ->A ->A. Hypothesis P_id_a__isPalListKind : A ->A. Hypothesis P_id_U44 : A ->A ->A ->A. Hypothesis P_id_a__isPal : A ->A. Hypothesis P_id_a__U43 : A ->A ->A ->A. Hypothesis P_id_U74 : A ->A. Hypothesis P_id_U13 : A ->A. Hypothesis P_id_a__isList : A ->A. Hypothesis P_id_U56 : A ->A. Hypothesis P_id_a : A. Hypothesis P_id_a__U55 : A ->A ->A. Hypothesis P_id_U26 : A ->A. Hypothesis P_id_a_____monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a____ x14 x16 <= P_id_a____ x13 x15. Hypothesis P_id_U31_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U31 x14 x16 <= P_id_U31 x13 x15. Hypothesis P_id_a__U56_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Hypothesis P_id_a__U26_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Hypothesis P_id_U61_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U61 x14 x16 <= P_id_U61 x13 x15. Hypothesis P_id_a__U13_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Hypothesis P_id_U45_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U45 x14 x16 <= P_id_U45 x13 x15. Hypothesis P_id_a__U74_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Hypothesis P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Hypothesis P_id_U81_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U81 x14 x16 <= P_id_U81 x13 x15. Hypothesis P_id_isNeList_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_isNeList x14 <= P_id_isNeList x13. Hypothesis P_id_a__U11_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Hypothesis P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Hypothesis P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Hypothesis P_id_a__isQid_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__isQid x14 <= P_id_a__isQid x13. Hypothesis P_id_U72_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U72 x14 x16 <= P_id_U72 x13 x15. Hypothesis P_id_U11_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U11 x14 x16 <= P_id_U11 x13 x15. Hypothesis P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Hypothesis P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Hypothesis P_id_a__isNePal_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__isNePal x14 <= P_id_a__isNePal x13. Hypothesis P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Hypothesis P_id_U91_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U91 x14 x16 <= P_id_U91 x13 x15. Hypothesis P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Hypothesis P_id_mark_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Hypothesis P_id_U33_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Hypothesis P_id_a__U62_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Hypothesis P_id_a__U32_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Hypothesis P_id_U63_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Hypothesis P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Hypothesis P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Hypothesis P_id_a__U82_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Hypothesis P_id_a__U46_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Hypothesis P_id_U83_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Hypothesis P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Hypothesis P_id_a__U12_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Hypothesis P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Hypothesis P_id_a__U73_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Hypothesis P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Hypothesis P_id_isPal_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Hypothesis P_id_isPalListKind_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Hypothesis P_id_a__U25_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Hypothesis P_id_U55_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U55 x14 x16 <= P_id_U55 x13 x15. Hypothesis P_id_a__U92_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Hypothesis P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Hypothesis P_id_isList_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Hypothesis P_id____monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Hypothesis P_id_U32_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U32 x14 x16 <= P_id_U32 x13 x15. Hypothesis P_id_a__U61_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Hypothesis P_id_a__U31_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Hypothesis P_id_U62_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U62 x14 x16 <= P_id_U62 x13 x15. Hypothesis P_id_a__isNeList_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Hypothesis P_id_U46_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Hypothesis P_id_a__U81_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Hypothesis P_id_a__U45_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Hypothesis P_id_U82_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U82 x14 x16 <= P_id_U82 x13 x15. Hypothesis P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Hypothesis P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Hypothesis P_id_a__U72_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Hypothesis P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Hypothesis P_id_U73_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U73 x14 x16 <= P_id_U73 x13 x15. Hypothesis P_id_U12_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U12 x14 x16 <= P_id_U12 x13 x15. Hypothesis P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Hypothesis P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Hypothesis P_id_a__U91_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Hypothesis P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Hypothesis P_id_U92_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Hypothesis P_id_U25_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_U25 x14 x16 <= P_id_U25 x13 x15. Hypothesis P_id_isQid_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Hypothesis P_id_a__U63_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Hypothesis P_id_a__U33_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Hypothesis P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Hypothesis P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Hypothesis P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Hypothesis P_id_a__U83_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Hypothesis P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Hypothesis P_id_isNePal_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Hypothesis P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Hypothesis P_id_a__isPalListKind_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Hypothesis P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Hypothesis P_id_a__isPal_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__isPal x14 <= P_id_a__isPal x13. Hypothesis P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Hypothesis P_id_U74_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Hypothesis P_id_U13_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Hypothesis P_id_a__isList_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_a__isList x14 <= P_id_a__isList x13. Hypothesis P_id_U56_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Hypothesis P_id_a__U55_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Hypothesis P_id_U26_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Hypothesis P_id_a_____bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a____ x14 x13. Hypothesis P_id_U31_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U31 x14 x13. Hypothesis P_id_a__U56_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__U56 x13. Hypothesis P_id_a__U26_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__U26 x13. Hypothesis P_id_U61_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U61 x14 x13. Hypothesis P_id_e_bounded : A0 <= P_id_e . Hypothesis P_id_a__U13_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__U13 x13. Hypothesis P_id_U45_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U45 x14 x13. Hypothesis P_id_a__U74_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__U74 x13. Hypothesis P_id_a__U44_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U44 x15 x14 x13. Hypothesis P_id_U81_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U81 x14 x13. Hypothesis P_id_isNeList_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_isNeList x13. Hypothesis P_id_a__U11_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U11 x14 x13. Hypothesis P_id_U41_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U41 x15 x14 x13. Hypothesis P_id_a__U71_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U71 x15 x14 x13. Hypothesis P_id_a__isQid_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__isQid x13. Hypothesis P_id_U72_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U72 x14 x13. Hypothesis P_id_U11_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U11 x14 x13. Hypothesis P_id_a__U23_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U23 x15 x14 x13. Hypothesis P_id_U53_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U53 x15 x14 x13. Hypothesis P_id_a__isNePal_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__isNePal x13. Hypothesis P_id_a__U52_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U52 x15 x14 x13. Hypothesis P_id_U91_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U91 x14 x13. Hypothesis P_id_U24_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U24 x15 x14 x13. Hypothesis P_id_mark_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_mark x13. Hypothesis P_id_U33_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_U33 x13. Hypothesis P_id_a__U62_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U62 x14 x13. Hypothesis P_id_a__U32_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U32 x14 x13. Hypothesis P_id_U63_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_U63 x13. Hypothesis P_id_o_bounded : A0 <= P_id_o . Hypothesis P_id_a__U21_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U21 x15 x14 x13. Hypothesis P_id_U51_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U51 x15 x14 x13. Hypothesis P_id_a__U82_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U82 x14 x13. Hypothesis P_id_a__U46_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__U46 x13. Hypothesis P_id_U83_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_U83 x13. Hypothesis P_id_U22_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U22 x15 x14 x13. Hypothesis P_id_a__U12_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U12 x14 x13. Hypothesis P_id_U43_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U43 x15 x14 x13. Hypothesis P_id_a__U73_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U73 x14 x13. Hypothesis P_id_a__U42_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U42 x15 x14 x13. Hypothesis P_id_isPal_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_isPal x13. Hypothesis P_id_isPalListKind_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_isPalListKind x13. Hypothesis P_id_a__U25_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U25 x14 x13. Hypothesis P_id_U55_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U55 x14 x13. Hypothesis P_id_a__U92_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__U92 x13. Hypothesis P_id_a__U54_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U54 x15 x14 x13. Hypothesis P_id_isList_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_isList x13. Hypothesis P_id____bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id___ x14 x13. Hypothesis P_id_U32_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U32 x14 x13. Hypothesis P_id_a__U61_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U61 x14 x13. Hypothesis P_id_a__U31_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U31 x14 x13. Hypothesis P_id_U62_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U62 x14 x13. Hypothesis P_id_i_bounded : A0 <= P_id_i . Hypothesis P_id_a__isNeList_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__isNeList x13. Hypothesis P_id_U46_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_U46 x13. Hypothesis P_id_a__U81_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U81 x14 x13. Hypothesis P_id_a__U45_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U45 x14 x13. Hypothesis P_id_U82_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U82 x14 x13. Hypothesis P_id_U21_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U21 x15 x14 x13. Hypothesis P_id_tt_bounded : A0 <= P_id_tt . Hypothesis P_id_U42_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U42 x15 x14 x13. Hypothesis P_id_a__U72_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U72 x14 x13. Hypothesis P_id_a__U41_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U41 x15 x14 x13. Hypothesis P_id_U73_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U73 x14 x13. Hypothesis P_id_U12_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U12 x14 x13. Hypothesis P_id_a__U24_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U24 x15 x14 x13. Hypothesis P_id_U54_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U54 x15 x14 x13. Hypothesis P_id_a__U91_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U91 x14 x13. Hypothesis P_id_a__U53_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U53 x15 x14 x13. Hypothesis P_id_U92_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_U92 x13. Hypothesis P_id_U25_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_U25 x14 x13. Hypothesis P_id_nil_bounded : A0 <= P_id_nil . Hypothesis P_id_isQid_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_isQid x13. Hypothesis P_id_a__U63_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__U63 x13. Hypothesis P_id_a__U33_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__U33 x13. Hypothesis P_id_U71_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U71 x15 x14 x13. Hypothesis P_id_u_bounded : A0 <= P_id_u . Hypothesis P_id_a__U22_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U22 x15 x14 x13. Hypothesis P_id_U52_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U52 x15 x14 x13. Hypothesis P_id_a__U83_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__U83 x13. Hypothesis P_id_a__U51_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U51 x15 x14 x13. Hypothesis P_id_isNePal_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_isNePal x13. Hypothesis P_id_U23_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U23 x15 x14 x13. Hypothesis P_id_a__isPalListKind_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__isPalListKind x13. Hypothesis P_id_U44_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_U44 x15 x14 x13. Hypothesis P_id_a__isPal_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__isPal x13. Hypothesis P_id_a__U43_bounded : forall x14 x13 x15, (A0 <= x13) ->(A0 <= x14) ->(A0 <= x15) ->A0 <= P_id_a__U43 x15 x14 x13. Hypothesis P_id_U74_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_U74 x13. Hypothesis P_id_U13_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_U13 x13. Hypothesis P_id_a__isList_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_a__isList x13. Hypothesis P_id_U56_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_U56 x13. Hypothesis P_id_a_bounded : A0 <= P_id_a . Hypothesis P_id_a__U55_bounded : forall x14 x13, (A0 <= x13) ->(A0 <= x14) ->A0 <= P_id_a__U55 x14 x13. Hypothesis P_id_U26_bounded : forall x13, (A0 <= x13) ->A0 <= P_id_U26 x13. Fixpoint measure t { struct t } := match t with | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14::x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14::x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14::x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14::x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14::x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14::x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14::x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14::x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14::x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14::x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14::x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14::x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14::x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14::x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14::x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => A0 end. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14::x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14::x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14::x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14::x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14::x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14::x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14::x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14::x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14::x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14::x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14::x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14::x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14::x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14::x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14::x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => A0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Definition Pols f : InterpGen.Pol_type A (InterpGen.get_arity f) := match f with | algebra.F.id_a____ => P_id_a____ | algebra.F.id_U31 => P_id_U31 | algebra.F.id_a__U56 => P_id_a__U56 | algebra.F.id_a__U26 => P_id_a__U26 | algebra.F.id_U61 => P_id_U61 | algebra.F.id_e => P_id_e | algebra.F.id_a__U13 => P_id_a__U13 | algebra.F.id_U45 => P_id_U45 | algebra.F.id_a__U74 => P_id_a__U74 | algebra.F.id_a__U44 => P_id_a__U44 | algebra.F.id_U81 => P_id_U81 | algebra.F.id_isNeList => P_id_isNeList | algebra.F.id_a__U11 => P_id_a__U11 | algebra.F.id_U41 => P_id_U41 | algebra.F.id_a__U71 => P_id_a__U71 | algebra.F.id_a__isQid => P_id_a__isQid | algebra.F.id_U72 => P_id_U72 | algebra.F.id_U11 => P_id_U11 | algebra.F.id_a__U23 => P_id_a__U23 | algebra.F.id_U53 => P_id_U53 | algebra.F.id_a__isNePal => P_id_a__isNePal | algebra.F.id_a__U52 => P_id_a__U52 | algebra.F.id_U91 => P_id_U91 | algebra.F.id_U24 => P_id_U24 | algebra.F.id_mark => P_id_mark | algebra.F.id_U33 => P_id_U33 | algebra.F.id_a__U62 => P_id_a__U62 | algebra.F.id_a__U32 => P_id_a__U32 | algebra.F.id_U63 => P_id_U63 | algebra.F.id_o => P_id_o | algebra.F.id_a__U21 => P_id_a__U21 | algebra.F.id_U51 => P_id_U51 | algebra.F.id_a__U82 => P_id_a__U82 | algebra.F.id_a__U46 => P_id_a__U46 | algebra.F.id_U83 => P_id_U83 | algebra.F.id_U22 => P_id_U22 | algebra.F.id_a__U12 => P_id_a__U12 | algebra.F.id_U43 => P_id_U43 | algebra.F.id_a__U73 => P_id_a__U73 | algebra.F.id_a__U42 => P_id_a__U42 | algebra.F.id_isPal => P_id_isPal | algebra.F.id_isPalListKind => P_id_isPalListKind | algebra.F.id_a__U25 => P_id_a__U25 | algebra.F.id_U55 => P_id_U55 | algebra.F.id_a__U92 => P_id_a__U92 | algebra.F.id_a__U54 => P_id_a__U54 | algebra.F.id_isList => P_id_isList | algebra.F.id___ => P_id___ | algebra.F.id_U32 => P_id_U32 | algebra.F.id_a__U61 => P_id_a__U61 | algebra.F.id_a__U31 => P_id_a__U31 | algebra.F.id_U62 => P_id_U62 | algebra.F.id_i => P_id_i | algebra.F.id_a__isNeList => P_id_a__isNeList | algebra.F.id_U46 => P_id_U46 | algebra.F.id_a__U81 => P_id_a__U81 | algebra.F.id_a__U45 => P_id_a__U45 | algebra.F.id_U82 => P_id_U82 | algebra.F.id_U21 => P_id_U21 | algebra.F.id_tt => P_id_tt | algebra.F.id_U42 => P_id_U42 | algebra.F.id_a__U72 => P_id_a__U72 | algebra.F.id_a__U41 => P_id_a__U41 | algebra.F.id_U73 => P_id_U73 | algebra.F.id_U12 => P_id_U12 | algebra.F.id_a__U24 => P_id_a__U24 | algebra.F.id_U54 => P_id_U54 | algebra.F.id_a__U91 => P_id_a__U91 | algebra.F.id_a__U53 => P_id_a__U53 | algebra.F.id_U92 => P_id_U92 | algebra.F.id_U25 => P_id_U25 | algebra.F.id_nil => P_id_nil | algebra.F.id_isQid => P_id_isQid | algebra.F.id_a__U63 => P_id_a__U63 | algebra.F.id_a__U33 => P_id_a__U33 | algebra.F.id_U71 => P_id_U71 | algebra.F.id_u => P_id_u | algebra.F.id_a__U22 => P_id_a__U22 | algebra.F.id_U52 => P_id_U52 | algebra.F.id_a__U83 => P_id_a__U83 | algebra.F.id_a__U51 => P_id_a__U51 | algebra.F.id_isNePal => P_id_isNePal | algebra.F.id_U23 => P_id_U23 | algebra.F.id_a__isPalListKind => P_id_a__isPalListKind | algebra.F.id_U44 => P_id_U44 | algebra.F.id_a__isPal => P_id_a__isPal | algebra.F.id_a__U43 => P_id_a__U43 | algebra.F.id_U74 => P_id_U74 | algebra.F.id_U13 => P_id_U13 | algebra.F.id_a__isList => P_id_a__isList | algebra.F.id_U56 => P_id_U56 | algebra.F.id_a => P_id_a | algebra.F.id_a__U55 => P_id_a__U55 | algebra.F.id_U26 => P_id_U26 end. Lemma same_measure : forall t, measure t = InterpGen.measure A0 Pols t. Proof. fix 1 . intros [a| f l]. simpl in |-*. unfold eq_rect_r, eq_rect, sym_eq in |-*. reflexivity . refine match f with | algebra.F.id_a____ => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U31 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U56 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U26 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U61 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_e => match l with | nil => _ | _::_ => _ end | algebra.F.id_a__U13 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U45 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U74 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U44 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U81 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_isNeList => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U11 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U41 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U71 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__isQid => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U72 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U11 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U23 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U53 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__isNePal => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U52 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U91 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U24 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_mark => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U33 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U62 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U32 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U63 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_o => match l with | nil => _ | _::_ => _ end | algebra.F.id_a__U21 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U51 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U82 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U46 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U83 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U22 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U12 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U43 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U73 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U42 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_isPal => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_isPalListKind => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U25 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U55 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U92 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U54 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_isList => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id___ => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U32 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U61 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U31 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U62 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_i => match l with | nil => _ | _::_ => _ end | algebra.F.id_a__isNeList => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U46 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U81 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U45 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U82 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U21 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_tt => match l with | nil => _ | _::_ => _ end | algebra.F.id_U42 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U72 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U41 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U73 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U12 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U24 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U54 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U91 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U53 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U92 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U25 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_nil => match l with | nil => _ | _::_ => _ end | algebra.F.id_isQid => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U63 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U33 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U71 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_u => match l with | nil => _ | _::_ => _ end | algebra.F.id_a__U22 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U52 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U83 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U51 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_isNePal => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U23 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__isPalListKind => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U44 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__isPal => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U43 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U74 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U13 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__isList => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U56 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a => match l with | nil => _ | _::_ => _ end | algebra.F.id_a__U55 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U26 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end end;simpl in |-*;unfold eq_rect_r, eq_rect, sym_eq in |-*; try (reflexivity );f_equal ;auto. Qed. Lemma measure_bounded : forall t, A0 <= measure t. Proof. intros t. rewrite same_measure in |-*. apply (InterpGen.measure_bounded Aop). intros f. case f. vm_compute in |-*;intros ;apply P_id_a_____bounded;assumption. vm_compute in |-*;intros ;apply P_id_U31_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U56_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U26_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U61_bounded;assumption. vm_compute in |-*;intros ;apply P_id_e_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U13_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U45_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U74_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U44_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U81_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isNeList_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U11_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U41_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U71_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isQid_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U72_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U23_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U53_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isNePal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U52_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U91_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U24_bounded;assumption. vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U33_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U62_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U32_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U63_bounded;assumption. vm_compute in |-*;intros ;apply P_id_o_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U21_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U51_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U82_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U46_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U83_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U22_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U12_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U43_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U73_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U42_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isPal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isPalListKind_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U25_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U55_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U92_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U54_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isList_bounded;assumption. vm_compute in |-*;intros ;apply P_id____bounded;assumption. vm_compute in |-*;intros ;apply P_id_U32_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U61_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U31_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U62_bounded;assumption. vm_compute in |-*;intros ;apply P_id_i_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isNeList_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U46_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U81_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U45_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U82_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U21_bounded;assumption. vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U42_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U72_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U41_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U73_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U12_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U24_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U54_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U91_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U53_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U92_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U25_bounded;assumption. vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isQid_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U63_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U33_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U71_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U22_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U52_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U83_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U51_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isNePal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U23_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isPalListKind_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U44_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isPal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U43_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U74_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U13_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isList_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U56_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U55_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U26_bounded;assumption. Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Hypothesis rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. intros . do 2 (rewrite same_measure in |-*). apply InterpGen.measure_star_monotonic with (1:=Aop) (Pols:=Pols) (rules:=R_xml_0_deep_rew.R_xml_0_rules). intros f. case f. vm_compute in |-*;intros ;apply P_id_a_____monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U31_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U56_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U26_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U61_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_a__U13_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U45_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U74_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U44_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U81_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_isNeList_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U11_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U41_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U71_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__isQid_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U72_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U11_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U23_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U53_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__isNePal_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U52_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U91_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U24_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_mark_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U33_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U62_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U32_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U63_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_a__U21_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U51_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U82_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U46_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U83_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U22_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U12_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U43_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U73_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U42_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_isPal_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_isPalListKind_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U25_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U55_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U92_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U54_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_isList_monotonic;assumption. vm_compute in |-*;intros ;apply P_id____monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U32_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U61_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U31_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U62_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_a__isNeList_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U46_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U81_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U45_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U82_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U21_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_U42_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U72_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U41_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U73_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U12_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U24_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U54_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U91_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U53_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U92_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U25_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_isQid_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U63_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U33_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U71_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_a__U22_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U52_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U83_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U51_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_isNePal_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U23_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__isPalListKind_monotonic; assumption. vm_compute in |-*;intros ;apply P_id_U44_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__isPal_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U43_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U74_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U13_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__isList_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U56_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_a__U55_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U26_monotonic;assumption. intros f. case f. vm_compute in |-*;intros ;apply P_id_a_____bounded;assumption. vm_compute in |-*;intros ;apply P_id_U31_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U56_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U26_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U61_bounded;assumption. vm_compute in |-*;intros ;apply P_id_e_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U13_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U45_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U74_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U44_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U81_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isNeList_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U11_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U41_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U71_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isQid_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U72_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U23_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U53_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isNePal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U52_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U91_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U24_bounded;assumption. vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U33_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U62_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U32_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U63_bounded;assumption. vm_compute in |-*;intros ;apply P_id_o_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U21_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U51_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U82_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U46_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U83_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U22_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U12_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U43_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U73_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U42_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isPal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isPalListKind_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U25_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U55_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U92_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U54_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isList_bounded;assumption. vm_compute in |-*;intros ;apply P_id____bounded;assumption. vm_compute in |-*;intros ;apply P_id_U32_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U61_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U31_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U62_bounded;assumption. vm_compute in |-*;intros ;apply P_id_i_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isNeList_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U46_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U81_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U45_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U82_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U21_bounded;assumption. vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U42_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U72_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U41_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U73_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U12_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U24_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U54_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U91_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U53_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U92_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U25_bounded;assumption. vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isQid_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U63_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U33_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U71_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U22_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U52_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U83_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U51_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isNePal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U23_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isPalListKind_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U44_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isPal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U43_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U74_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U13_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isList_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U56_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U55_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U26_bounded;assumption. intros . do 2 (rewrite <- same_measure in |-*). apply rules_monotonic;assumption. assumption. Qed. Hypothesis P_id_A__U72 : A ->A ->A. Hypothesis P_id_A__U11 : A ->A ->A. Hypothesis P_id_A__U41 : A ->A ->A ->A. Hypothesis P_id_A__U91 : A ->A ->A. Hypothesis P_id_A__U24 : A ->A ->A ->A. Hypothesis P_id_A__U53 : A ->A ->A ->A. Hypothesis P_id_A__U81 : A ->A ->A. Hypothesis P_id_A__ISNELIST : A ->A. Hypothesis P_id_A__U45 : A ->A ->A. Hypothesis P_id_A__U31 : A ->A ->A. Hypothesis P_id_A__U61 : A ->A ->A. Hypothesis P_id_A__ISPAL : A ->A. Hypothesis P_id_A__ISPALLISTKIND : A ->A. Hypothesis P_id_A__U43 : A ->A ->A ->A. Hypothesis P_id_A__ISLIST : A ->A. Hypothesis P_id_A__U55 : A ->A ->A. Hypothesis P_id_A__U83 : A ->A. Hypothesis P_id_A__U22 : A ->A ->A ->A. Hypothesis P_id_A__U51 : A ->A ->A ->A. Hypothesis P_id_A__U33 : A ->A. Hypothesis P_id_A____ : A ->A ->A. Hypothesis P_id_A__U63 : A ->A. Hypothesis P_id_A__U73 : A ->A ->A. Hypothesis P_id_A__U12 : A ->A ->A. Hypothesis P_id_A__U42 : A ->A ->A ->A. Hypothesis P_id_A__U92 : A ->A. Hypothesis P_id_A__U25 : A ->A ->A. Hypothesis P_id_A__U54 : A ->A ->A ->A. Hypothesis P_id_A__U82 : A ->A ->A. Hypothesis P_id_A__U21 : A ->A ->A ->A. Hypothesis P_id_A__U46 : A ->A. Hypothesis P_id_A__U32 : A ->A ->A. Hypothesis P_id_A__U62 : A ->A ->A. Hypothesis P_id_A__U74 : A ->A. Hypothesis P_id_A__U13 : A ->A. Hypothesis P_id_A__U44 : A ->A ->A ->A. Hypothesis P_id_A__U26 : A ->A. Hypothesis P_id_A__U56 : A ->A. Hypothesis P_id_A__ISNEPAL : A ->A. Hypothesis P_id_A__U23 : A ->A ->A ->A. Hypothesis P_id_A__U52 : A ->A ->A ->A. Hypothesis P_id_A__ISQID : A ->A. Hypothesis P_id_MARK : A ->A. Hypothesis P_id_A__U71 : A ->A ->A ->A. Hypothesis P_id_A__U72_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Hypothesis P_id_A__U11_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Hypothesis P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Hypothesis P_id_A__U91_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Hypothesis P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Hypothesis P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Hypothesis P_id_A__U81_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Hypothesis P_id_A__ISNELIST_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Hypothesis P_id_A__U45_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Hypothesis P_id_A__U31_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Hypothesis P_id_A__U61_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Hypothesis P_id_A__ISPAL_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Hypothesis P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Hypothesis P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Hypothesis P_id_A__ISLIST_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Hypothesis P_id_A__U55_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Hypothesis P_id_A__U83_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Hypothesis P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Hypothesis P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Hypothesis P_id_A__U33_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Hypothesis P_id_A_____monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A____ x14 x16 <= P_id_A____ x13 x15. Hypothesis P_id_A__U63_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Hypothesis P_id_A__U73_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Hypothesis P_id_A__U12_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Hypothesis P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Hypothesis P_id_A__U92_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Hypothesis P_id_A__U25_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Hypothesis P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Hypothesis P_id_A__U82_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Hypothesis P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Hypothesis P_id_A__U46_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Hypothesis P_id_A__U32_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Hypothesis P_id_A__U62_monotonic : forall x16 x14 x13 x15, (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Hypothesis P_id_A__U74_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Hypothesis P_id_A__U13_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Hypothesis P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Hypothesis P_id_A__U26_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Hypothesis P_id_A__U56_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Hypothesis P_id_A__ISNEPAL_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Hypothesis P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Hypothesis P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Hypothesis P_id_A__ISQID_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_A__ISQID x14 <= P_id_A__ISQID x13. Hypothesis P_id_MARK_monotonic : forall x14 x13, (A0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Hypothesis P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (A0 <= x18)/\ (x18 <= x17) -> (A0 <= x16)/\ (x16 <= x15) -> (A0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Definition marked_measure t := match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Definition Marked_pols : forall f, (algebra.EQT.defined R_xml_0_deep_rew.R_xml_0_rules f) -> InterpGen.Pol_type A (InterpGen.get_arity f). Proof. intros f H. apply ddp.defined_list_complete with (1:=R_xml_0_deep_rew.R_xml_0_rules_included) in H . apply (Symb_more_list.change_in algebra.F.symb_order) in H . set (u := (Symb_more_list.qs algebra.F.symb_order (Symb_more_list.XSet.remove_red (ddp.defined_list R_xml_0_deep_rew.R_xml_0_rule_as_list)))) in * . vm_compute in u . unfold u in * . clear u . unfold more_list.mem_bool in H . match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U55 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__ISLIST x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U43 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__ISPAL x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__ISPALLISTKIND x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U51 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__U83 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U22 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__U33 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__U63 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U53 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U91 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U24 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U41 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U72 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U45 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U81 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__ISNELIST x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U31 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U61 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U54 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__U92 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U25 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U42 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U73 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U12 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__U46 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U82 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U21 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U32 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U62 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_MARK x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U52 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__ISNEPAL x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U23 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__ISQID x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U71 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A__U11 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x15 x14 x13;apply (P_id_A__U44 x15 x14 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__U74 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__U13 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__U26 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x13;apply (P_id_A__U56 x13). match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros x14 x13;apply (P_id_A____ x14 x13). discriminate H. Defined. Lemma same_marked_measure : forall t, marked_measure t = InterpGen.marked_measure A0 Pols Marked_pols (ddp.defined_dec _ _ R_xml_0_deep_rew.R_xml_0_rules_included) t. Proof. intros [a| f l]. simpl in |-*. unfold eq_rect_r, eq_rect, sym_eq in |-*. reflexivity . refine match f with | algebra.F.id_a____ => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U31 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U56 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U26 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U61 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_e => match l with | nil => _ | _::_ => _ end | algebra.F.id_a__U13 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U45 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U74 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U44 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U81 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_isNeList => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U11 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U41 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U71 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__isQid => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U72 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U11 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U23 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U53 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__isNePal => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U52 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U91 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U24 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_mark => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U33 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U62 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U32 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U63 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_o => match l with | nil => _ | _::_ => _ end | algebra.F.id_a__U21 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U51 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U82 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U46 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U83 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U22 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U12 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U43 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U73 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U42 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_isPal => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_isPalListKind => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U25 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U55 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U92 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U54 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_isList => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id___ => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U32 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U61 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U31 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U62 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_i => match l with | nil => _ | _::_ => _ end | algebra.F.id_a__isNeList => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U46 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U81 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U45 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U82 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U21 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_tt => match l with | nil => _ | _::_ => _ end | algebra.F.id_U42 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U72 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U41 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U73 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U12 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U24 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U54 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U91 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_a__U53 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U92 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U25 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_nil => match l with | nil => _ | _::_ => _ end | algebra.F.id_isQid => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U63 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U33 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U71 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_u => match l with | nil => _ | _::_ => _ end | algebra.F.id_a__U22 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U52 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__U83 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U51 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_isNePal => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U23 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__isPalListKind => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U44 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_a__isPal => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__U43 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::_ => _ end | algebra.F.id_U74 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U13 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a__isList => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_U56 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_a => match l with | nil => _ | _::_ => _ end | algebra.F.id_a__U55 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_U26 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end end. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . vm_compute in |-*;reflexivity . lazy - [measure InterpGen.measure Pols] in |-* ;f_equal ; apply same_measure. vm_compute in |-*;reflexivity . Qed. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. intros . do 2 (rewrite same_marked_measure in |-*). apply InterpGen.marked_measure_star_monotonic with (1:=Aop) (Pols:= Pols) (rules:=R_xml_0_deep_rew.R_xml_0_rules). clear f. intros f. case f. vm_compute in |-*;intros ;apply P_id_a_____monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U31_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U56_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U26_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U61_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_a__U13_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U45_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U74_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U44_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U81_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_isNeList_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U11_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U41_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U71_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__isQid_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U72_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U11_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U23_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U53_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__isNePal_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U52_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U91_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U24_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_mark_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U33_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U62_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U32_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U63_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_a__U21_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U51_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U82_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U46_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U83_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U22_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U12_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U43_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U73_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U42_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_isPal_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_isPalListKind_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U25_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U55_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U92_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U54_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_isList_monotonic;assumption. vm_compute in |-*;intros ;apply P_id____monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U32_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U61_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U31_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U62_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_a__isNeList_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U46_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U81_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U45_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U82_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U21_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_U42_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U72_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U41_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U73_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U12_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U24_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U54_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U91_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U53_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U92_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U25_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_isQid_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U63_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U33_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U71_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_a__U22_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U52_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U83_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U51_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_isNePal_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U23_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__isPalListKind_monotonic; assumption. vm_compute in |-*;intros ;apply P_id_U44_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__isPal_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__U43_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U74_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U13_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_a__isList_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U56_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_a__U55_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_U26_monotonic;assumption. clear f. intros f. case f. vm_compute in |-*;intros ;apply P_id_a_____bounded;assumption. vm_compute in |-*;intros ;apply P_id_U31_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U56_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U26_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U61_bounded;assumption. vm_compute in |-*;intros ;apply P_id_e_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U13_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U45_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U74_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U44_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U81_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isNeList_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U11_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U41_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U71_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isQid_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U72_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U11_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U23_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U53_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isNePal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U52_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U91_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U24_bounded;assumption. vm_compute in |-*;intros ;apply P_id_mark_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U33_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U62_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U32_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U63_bounded;assumption. vm_compute in |-*;intros ;apply P_id_o_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U21_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U51_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U82_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U46_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U83_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U22_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U12_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U43_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U73_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U42_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isPal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isPalListKind_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U25_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U55_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U92_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U54_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isList_bounded;assumption. vm_compute in |-*;intros ;apply P_id____bounded;assumption. vm_compute in |-*;intros ;apply P_id_U32_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U61_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U31_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U62_bounded;assumption. vm_compute in |-*;intros ;apply P_id_i_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isNeList_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U46_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U81_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U45_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U82_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U21_bounded;assumption. vm_compute in |-*;intros ;apply P_id_tt_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U42_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U72_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U41_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U73_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U12_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U24_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U54_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U91_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U53_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U92_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U25_bounded;assumption. vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isQid_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U63_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U33_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U71_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U22_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U52_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U83_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U51_bounded;assumption. vm_compute in |-*;intros ;apply P_id_isNePal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U23_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isPalListKind_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U44_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isPal_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U43_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U74_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U13_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__isList_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U56_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a_bounded;assumption. vm_compute in |-*;intros ;apply P_id_a__U55_bounded;assumption. vm_compute in |-*;intros ;apply P_id_U26_bounded;assumption. intros . do 2 (rewrite <- same_measure in |-*). apply rules_monotonic;assumption. clear f. intros f. clear H. intros H. generalize H. apply ddp.defined_list_complete with (1:=R_xml_0_deep_rew.R_xml_0_rules_included) in H . apply (Symb_more_list.change_in algebra.F.symb_order) in H . set (u := (Symb_more_list.qs algebra.F.symb_order (Symb_more_list.XSet.remove_red (ddp.defined_list R_xml_0_deep_rew.R_xml_0_rule_as_list)))) in * . vm_compute in u . unfold u in * . clear u . unfold more_list.mem_bool in H . match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U55_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__ISLIST_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U43_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__ISPAL_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__ISPALLISTKIND_monotonic; assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U51_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U83_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U22_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U33_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U63_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U53_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U91_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U24_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U41_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U72_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U45_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U81_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__ISNELIST_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U31_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U61_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U54_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U92_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U25_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U42_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U73_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U12_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U46_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U82_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U21_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U32_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U62_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_MARK_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U52_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__ISNEPAL_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U23_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__ISQID_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U71_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U11_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U44_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U74_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U13_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U26_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A__U56_monotonic;assumption. match type of H with | orb ?a ?b = true => assert (H':{a = true}+{b = true});[ revert H;case a;[left;reflexivity|simpl;intros h;right;exact h]| clear H;destruct H' as [H|H]] end . match type of H with | _ _ ?t = true => generalize (algebra.F.symb_eq_bool_ok f t); unfold algebra.Alg.eq_symb_bool in H; rewrite H;clear H;intros ;subst end . vm_compute in |-*;intros ;apply P_id_A_____monotonic;assumption. discriminate H. assumption. Qed. End S. End Interp. Module InterpZ. Section S. Open Scope Z_scope. Hypothesis min_value : Z. Import ring_extention. Notation Local "'Alt'" := (Zwf.Zwf min_value). Notation Local "'Ale'" := Zle. Notation Local "'Aeq'" := (@eq Z). Notation Local "a <= b" := (Ale a b). Notation Local "a < b" := (Alt a b). Hypothesis P_id_a____ : Z ->Z ->Z. Hypothesis P_id_U31 : Z ->Z ->Z. Hypothesis P_id_a__U56 : Z ->Z. Hypothesis P_id_a__U26 : Z ->Z. Hypothesis P_id_U61 : Z ->Z ->Z. Hypothesis P_id_e : Z. Hypothesis P_id_a__U13 : Z ->Z. Hypothesis P_id_U45 : Z ->Z ->Z. Hypothesis P_id_a__U74 : Z ->Z. Hypothesis P_id_a__U44 : Z ->Z ->Z ->Z. Hypothesis P_id_U81 : Z ->Z ->Z. Hypothesis P_id_isNeList : Z ->Z. Hypothesis P_id_a__U11 : Z ->Z ->Z. Hypothesis P_id_U41 : Z ->Z ->Z ->Z. Hypothesis P_id_a__U71 : Z ->Z ->Z ->Z. Hypothesis P_id_a__isQid : Z ->Z. Hypothesis P_id_U72 : Z ->Z ->Z. Hypothesis P_id_U11 : Z ->Z ->Z. Hypothesis P_id_a__U23 : Z ->Z ->Z ->Z. Hypothesis P_id_U53 : Z ->Z ->Z ->Z. Hypothesis P_id_a__isNePal : Z ->Z. Hypothesis P_id_a__U52 : Z ->Z ->Z ->Z. Hypothesis P_id_U91 : Z ->Z ->Z. Hypothesis P_id_U24 : Z ->Z ->Z ->Z. Hypothesis P_id_mark : Z ->Z. Hypothesis P_id_U33 : Z ->Z. Hypothesis P_id_a__U62 : Z ->Z ->Z. Hypothesis P_id_a__U32 : Z ->Z ->Z. Hypothesis P_id_U63 : Z ->Z. Hypothesis P_id_o : Z. Hypothesis P_id_a__U21 : Z ->Z ->Z ->Z. Hypothesis P_id_U51 : Z ->Z ->Z ->Z. Hypothesis P_id_a__U82 : Z ->Z ->Z. Hypothesis P_id_a__U46 : Z ->Z. Hypothesis P_id_U83 : Z ->Z. Hypothesis P_id_U22 : Z ->Z ->Z ->Z. Hypothesis P_id_a__U12 : Z ->Z ->Z. Hypothesis P_id_U43 : Z ->Z ->Z ->Z. Hypothesis P_id_a__U73 : Z ->Z ->Z. Hypothesis P_id_a__U42 : Z ->Z ->Z ->Z. Hypothesis P_id_isPal : Z ->Z. Hypothesis P_id_isPalListKind : Z ->Z. Hypothesis P_id_a__U25 : Z ->Z ->Z. Hypothesis P_id_U55 : Z ->Z ->Z. Hypothesis P_id_a__U92 : Z ->Z. Hypothesis P_id_a__U54 : Z ->Z ->Z ->Z. Hypothesis P_id_isList : Z ->Z. Hypothesis P_id___ : Z ->Z ->Z. Hypothesis P_id_U32 : Z ->Z ->Z. Hypothesis P_id_a__U61 : Z ->Z ->Z. Hypothesis P_id_a__U31 : Z ->Z ->Z. Hypothesis P_id_U62 : Z ->Z ->Z. Hypothesis P_id_i : Z. Hypothesis P_id_a__isNeList : Z ->Z. Hypothesis P_id_U46 : Z ->Z. Hypothesis P_id_a__U81 : Z ->Z ->Z. Hypothesis P_id_a__U45 : Z ->Z ->Z. Hypothesis P_id_U82 : Z ->Z ->Z. Hypothesis P_id_U21 : Z ->Z ->Z ->Z. Hypothesis P_id_tt : Z. Hypothesis P_id_U42 : Z ->Z ->Z ->Z. Hypothesis P_id_a__U72 : Z ->Z ->Z. Hypothesis P_id_a__U41 : Z ->Z ->Z ->Z. Hypothesis P_id_U73 : Z ->Z ->Z. Hypothesis P_id_U12 : Z ->Z ->Z. Hypothesis P_id_a__U24 : Z ->Z ->Z ->Z. Hypothesis P_id_U54 : Z ->Z ->Z ->Z. Hypothesis P_id_a__U91 : Z ->Z ->Z. Hypothesis P_id_a__U53 : Z ->Z ->Z ->Z. Hypothesis P_id_U92 : Z ->Z. Hypothesis P_id_U25 : Z ->Z ->Z. Hypothesis P_id_nil : Z. Hypothesis P_id_isQid : Z ->Z. Hypothesis P_id_a__U63 : Z ->Z. Hypothesis P_id_a__U33 : Z ->Z. Hypothesis P_id_U71 : Z ->Z ->Z ->Z. Hypothesis P_id_u : Z. Hypothesis P_id_a__U22 : Z ->Z ->Z ->Z. Hypothesis P_id_U52 : Z ->Z ->Z ->Z. Hypothesis P_id_a__U83 : Z ->Z. Hypothesis P_id_a__U51 : Z ->Z ->Z ->Z. Hypothesis P_id_isNePal : Z ->Z. Hypothesis P_id_U23 : Z ->Z ->Z ->Z. Hypothesis P_id_a__isPalListKind : Z ->Z. Hypothesis P_id_U44 : Z ->Z ->Z ->Z. Hypothesis P_id_a__isPal : Z ->Z. Hypothesis P_id_a__U43 : Z ->Z ->Z ->Z. Hypothesis P_id_U74 : Z ->Z. Hypothesis P_id_U13 : Z ->Z. Hypothesis P_id_a__isList : Z ->Z. Hypothesis P_id_U56 : Z ->Z. Hypothesis P_id_a : Z. Hypothesis P_id_a__U55 : Z ->Z ->Z. Hypothesis P_id_U26 : Z ->Z. Hypothesis P_id_a_____monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Hypothesis P_id_U31_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U31 x14 x16 <= P_id_U31 x13 x15. Hypothesis P_id_a__U56_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Hypothesis P_id_a__U26_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Hypothesis P_id_U61_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U61 x14 x16 <= P_id_U61 x13 x15. Hypothesis P_id_a__U13_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Hypothesis P_id_U45_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U45 x14 x16 <= P_id_U45 x13 x15. Hypothesis P_id_a__U74_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Hypothesis P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Hypothesis P_id_U81_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U81 x14 x16 <= P_id_U81 x13 x15. Hypothesis P_id_isNeList_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_isNeList x14 <= P_id_isNeList x13. Hypothesis P_id_a__U11_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Hypothesis P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Hypothesis P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Hypothesis P_id_a__isQid_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_a__isQid x14 <= P_id_a__isQid x13. Hypothesis P_id_U72_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U72 x14 x16 <= P_id_U72 x13 x15. Hypothesis P_id_U11_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U11 x14 x16 <= P_id_U11 x13 x15. Hypothesis P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Hypothesis P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Hypothesis P_id_a__isNePal_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Hypothesis P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Hypothesis P_id_U91_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U91 x14 x16 <= P_id_U91 x13 x15. Hypothesis P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Hypothesis P_id_mark_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Hypothesis P_id_U33_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Hypothesis P_id_a__U62_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Hypothesis P_id_a__U32_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Hypothesis P_id_U63_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Hypothesis P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Hypothesis P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Hypothesis P_id_a__U82_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Hypothesis P_id_a__U46_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Hypothesis P_id_U83_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Hypothesis P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Hypothesis P_id_a__U12_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Hypothesis P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Hypothesis P_id_a__U73_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Hypothesis P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Hypothesis P_id_isPal_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Hypothesis P_id_isPalListKind_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Hypothesis P_id_a__U25_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Hypothesis P_id_U55_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U55 x14 x16 <= P_id_U55 x13 x15. Hypothesis P_id_a__U92_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Hypothesis P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Hypothesis P_id_isList_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Hypothesis P_id____monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Hypothesis P_id_U32_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U32 x14 x16 <= P_id_U32 x13 x15. Hypothesis P_id_a__U61_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Hypothesis P_id_a__U31_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Hypothesis P_id_U62_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U62 x14 x16 <= P_id_U62 x13 x15. Hypothesis P_id_a__isNeList_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Hypothesis P_id_U46_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Hypothesis P_id_a__U81_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Hypothesis P_id_a__U45_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Hypothesis P_id_U82_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U82 x14 x16 <= P_id_U82 x13 x15. Hypothesis P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Hypothesis P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Hypothesis P_id_a__U72_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Hypothesis P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Hypothesis P_id_U73_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U73 x14 x16 <= P_id_U73 x13 x15. Hypothesis P_id_U12_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U12 x14 x16 <= P_id_U12 x13 x15. Hypothesis P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Hypothesis P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Hypothesis P_id_a__U91_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Hypothesis P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Hypothesis P_id_U92_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Hypothesis P_id_U25_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U25 x14 x16 <= P_id_U25 x13 x15. Hypothesis P_id_isQid_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Hypothesis P_id_a__U63_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Hypothesis P_id_a__U33_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Hypothesis P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Hypothesis P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Hypothesis P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Hypothesis P_id_a__U83_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Hypothesis P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Hypothesis P_id_isNePal_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Hypothesis P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Hypothesis P_id_a__isPalListKind_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Hypothesis P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Hypothesis P_id_a__isPal_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_a__isPal x14 <= P_id_a__isPal x13. Hypothesis P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Hypothesis P_id_U74_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Hypothesis P_id_U13_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Hypothesis P_id_a__isList_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Hypothesis P_id_U56_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Hypothesis P_id_a__U55_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Hypothesis P_id_U26_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Hypothesis P_id_a_____bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a____ x14 x13. Hypothesis P_id_U31_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U31 x14 x13. Hypothesis P_id_a__U56_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__U56 x13. Hypothesis P_id_a__U26_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__U26 x13. Hypothesis P_id_U61_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U61 x14 x13. Hypothesis P_id_e_bounded : min_value <= P_id_e . Hypothesis P_id_a__U13_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__U13 x13. Hypothesis P_id_U45_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U45 x14 x13. Hypothesis P_id_a__U74_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__U74 x13. Hypothesis P_id_a__U44_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U44 x15 x14 x13. Hypothesis P_id_U81_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U81 x14 x13. Hypothesis P_id_isNeList_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_isNeList x13. Hypothesis P_id_a__U11_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U11 x14 x13. Hypothesis P_id_U41_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U41 x15 x14 x13. Hypothesis P_id_a__U71_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U71 x15 x14 x13. Hypothesis P_id_a__isQid_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__isQid x13. Hypothesis P_id_U72_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U72 x14 x13. Hypothesis P_id_U11_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U11 x14 x13. Hypothesis P_id_a__U23_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U23 x15 x14 x13. Hypothesis P_id_U53_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U53 x15 x14 x13. Hypothesis P_id_a__isNePal_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__isNePal x13. Hypothesis P_id_a__U52_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U52 x15 x14 x13. Hypothesis P_id_U91_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U91 x14 x13. Hypothesis P_id_U24_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U24 x15 x14 x13. Hypothesis P_id_mark_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_mark x13. Hypothesis P_id_U33_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_U33 x13. Hypothesis P_id_a__U62_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U62 x14 x13. Hypothesis P_id_a__U32_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U32 x14 x13. Hypothesis P_id_U63_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_U63 x13. Hypothesis P_id_o_bounded : min_value <= P_id_o . Hypothesis P_id_a__U21_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U21 x15 x14 x13. Hypothesis P_id_U51_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U51 x15 x14 x13. Hypothesis P_id_a__U82_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U82 x14 x13. Hypothesis P_id_a__U46_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__U46 x13. Hypothesis P_id_U83_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_U83 x13. Hypothesis P_id_U22_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U22 x15 x14 x13. Hypothesis P_id_a__U12_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U12 x14 x13. Hypothesis P_id_U43_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U43 x15 x14 x13. Hypothesis P_id_a__U73_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U73 x14 x13. Hypothesis P_id_a__U42_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U42 x15 x14 x13. Hypothesis P_id_isPal_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_isPal x13. Hypothesis P_id_isPalListKind_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_isPalListKind x13. Hypothesis P_id_a__U25_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U25 x14 x13. Hypothesis P_id_U55_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U55 x14 x13. Hypothesis P_id_a__U92_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__U92 x13. Hypothesis P_id_a__U54_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U54 x15 x14 x13. Hypothesis P_id_isList_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_isList x13. Hypothesis P_id____bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id___ x14 x13. Hypothesis P_id_U32_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U32 x14 x13. Hypothesis P_id_a__U61_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U61 x14 x13. Hypothesis P_id_a__U31_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U31 x14 x13. Hypothesis P_id_U62_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U62 x14 x13. Hypothesis P_id_i_bounded : min_value <= P_id_i . Hypothesis P_id_a__isNeList_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__isNeList x13. Hypothesis P_id_U46_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_U46 x13. Hypothesis P_id_a__U81_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U81 x14 x13. Hypothesis P_id_a__U45_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U45 x14 x13. Hypothesis P_id_U82_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U82 x14 x13. Hypothesis P_id_U21_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U21 x15 x14 x13. Hypothesis P_id_tt_bounded : min_value <= P_id_tt . Hypothesis P_id_U42_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U42 x15 x14 x13. Hypothesis P_id_a__U72_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U72 x14 x13. Hypothesis P_id_a__U41_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U41 x15 x14 x13. Hypothesis P_id_U73_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U73 x14 x13. Hypothesis P_id_U12_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U12 x14 x13. Hypothesis P_id_a__U24_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U24 x15 x14 x13. Hypothesis P_id_U54_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U54 x15 x14 x13. Hypothesis P_id_a__U91_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U91 x14 x13. Hypothesis P_id_a__U53_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U53 x15 x14 x13. Hypothesis P_id_U92_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_U92 x13. Hypothesis P_id_U25_bounded : forall x14 x13, (min_value <= x13) ->(min_value <= x14) ->min_value <= P_id_U25 x14 x13. Hypothesis P_id_nil_bounded : min_value <= P_id_nil . Hypothesis P_id_isQid_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_isQid x13. Hypothesis P_id_a__U63_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__U63 x13. Hypothesis P_id_a__U33_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__U33 x13. Hypothesis P_id_U71_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U71 x15 x14 x13. Hypothesis P_id_u_bounded : min_value <= P_id_u . Hypothesis P_id_a__U22_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U22 x15 x14 x13. Hypothesis P_id_U52_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U52 x15 x14 x13. Hypothesis P_id_a__U83_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__U83 x13. Hypothesis P_id_a__U51_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U51 x15 x14 x13. Hypothesis P_id_isNePal_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_isNePal x13. Hypothesis P_id_U23_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U23 x15 x14 x13. Hypothesis P_id_a__isPalListKind_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__isPalListKind x13. Hypothesis P_id_U44_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_U44 x15 x14 x13. Hypothesis P_id_a__isPal_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__isPal x13. Hypothesis P_id_a__U43_bounded : forall x14 x13 x15, (min_value <= x13) -> (min_value <= x14) -> (min_value <= x15) ->min_value <= P_id_a__U43 x15 x14 x13. Hypothesis P_id_U74_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_U74 x13. Hypothesis P_id_U13_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_U13 x13. Hypothesis P_id_a__isList_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_a__isList x13. Hypothesis P_id_U56_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_U56 x13. Hypothesis P_id_a_bounded : min_value <= P_id_a . Hypothesis P_id_a__U55_bounded : forall x14 x13, (min_value <= x13) -> (min_value <= x14) ->min_value <= P_id_a__U55 x14 x13. Hypothesis P_id_U26_bounded : forall x13, (min_value <= x13) ->min_value <= P_id_U26 x13. Definition measure := Interp.measure min_value P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14::x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14::x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14::x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14::x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14::x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14::x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14::x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14::x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14::x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14::x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14::x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14::x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14::x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14::x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14::x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => min_value end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, min_value <= measure t. Proof. unfold measure in |-*. apply Interp.measure_bounded with Alt Aeq; (apply interp.o_Z)|| (cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;auto with zarith). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Hypothesis rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply Interp.measure_star_monotonic with Alt Aeq. (apply interp.o_Z)|| (cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;auto with zarith). intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Hypothesis P_id_A__U72 : Z ->Z ->Z. Hypothesis P_id_A__U11 : Z ->Z ->Z. Hypothesis P_id_A__U41 : Z ->Z ->Z ->Z. Hypothesis P_id_A__U91 : Z ->Z ->Z. Hypothesis P_id_A__U24 : Z ->Z ->Z ->Z. Hypothesis P_id_A__U53 : Z ->Z ->Z ->Z. Hypothesis P_id_A__U81 : Z ->Z ->Z. Hypothesis P_id_A__ISNELIST : Z ->Z. Hypothesis P_id_A__U45 : Z ->Z ->Z. Hypothesis P_id_A__U31 : Z ->Z ->Z. Hypothesis P_id_A__U61 : Z ->Z ->Z. Hypothesis P_id_A__ISPAL : Z ->Z. Hypothesis P_id_A__ISPALLISTKIND : Z ->Z. Hypothesis P_id_A__U43 : Z ->Z ->Z ->Z. Hypothesis P_id_A__ISLIST : Z ->Z. Hypothesis P_id_A__U55 : Z ->Z ->Z. Hypothesis P_id_A__U83 : Z ->Z. Hypothesis P_id_A__U22 : Z ->Z ->Z ->Z. Hypothesis P_id_A__U51 : Z ->Z ->Z ->Z. Hypothesis P_id_A__U33 : Z ->Z. Hypothesis P_id_A____ : Z ->Z ->Z. Hypothesis P_id_A__U63 : Z ->Z. Hypothesis P_id_A__U73 : Z ->Z ->Z. Hypothesis P_id_A__U12 : Z ->Z ->Z. Hypothesis P_id_A__U42 : Z ->Z ->Z ->Z. Hypothesis P_id_A__U92 : Z ->Z. Hypothesis P_id_A__U25 : Z ->Z ->Z. Hypothesis P_id_A__U54 : Z ->Z ->Z ->Z. Hypothesis P_id_A__U82 : Z ->Z ->Z. Hypothesis P_id_A__U21 : Z ->Z ->Z ->Z. Hypothesis P_id_A__U46 : Z ->Z. Hypothesis P_id_A__U32 : Z ->Z ->Z. Hypothesis P_id_A__U62 : Z ->Z ->Z. Hypothesis P_id_A__U74 : Z ->Z. Hypothesis P_id_A__U13 : Z ->Z. Hypothesis P_id_A__U44 : Z ->Z ->Z ->Z. Hypothesis P_id_A__U26 : Z ->Z. Hypothesis P_id_A__U56 : Z ->Z. Hypothesis P_id_A__ISNEPAL : Z ->Z. Hypothesis P_id_A__U23 : Z ->Z ->Z ->Z. Hypothesis P_id_A__U52 : Z ->Z ->Z ->Z. Hypothesis P_id_A__ISQID : Z ->Z. Hypothesis P_id_MARK : Z ->Z. Hypothesis P_id_A__U71 : Z ->Z ->Z ->Z. Hypothesis P_id_A__U72_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Hypothesis P_id_A__U11_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Hypothesis P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Hypothesis P_id_A__U91_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Hypothesis P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Hypothesis P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Hypothesis P_id_A__U81_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Hypothesis P_id_A__ISNELIST_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Hypothesis P_id_A__U45_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Hypothesis P_id_A__U31_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Hypothesis P_id_A__U61_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Hypothesis P_id_A__ISPAL_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Hypothesis P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Hypothesis P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Hypothesis P_id_A__ISLIST_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Hypothesis P_id_A__U55_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Hypothesis P_id_A__U83_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Hypothesis P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Hypothesis P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Hypothesis P_id_A__U33_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Hypothesis P_id_A_____monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Hypothesis P_id_A__U63_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Hypothesis P_id_A__U73_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Hypothesis P_id_A__U12_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Hypothesis P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Hypothesis P_id_A__U92_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Hypothesis P_id_A__U25_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Hypothesis P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Hypothesis P_id_A__U82_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Hypothesis P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Hypothesis P_id_A__U46_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Hypothesis P_id_A__U32_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Hypothesis P_id_A__U62_monotonic : forall x16 x14 x13 x15, (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Hypothesis P_id_A__U74_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Hypothesis P_id_A__U13_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Hypothesis P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Hypothesis P_id_A__U26_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Hypothesis P_id_A__U56_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Hypothesis P_id_A__ISNEPAL_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Hypothesis P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Hypothesis P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Hypothesis P_id_A__ISQID_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) -> P_id_A__ISQID x14 <= P_id_A__ISQID x13. Hypothesis P_id_MARK_monotonic : forall x14 x13, (min_value <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Hypothesis P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (min_value <= x18)/\ (x18 <= x17) -> (min_value <= x16)/\ (x16 <= x15) -> (min_value <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Definition marked_measure := Interp.marked_measure min_value P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply Interp.marked_measure_star_monotonic with Alt Aeq. (apply interp.o_Z)|| (cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;auto with zarith). intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. End S. End InterpZ. Module WF_R_xml_0_deep_rew. Inductive DP_R_xml_0 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_0 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::(algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil))::(algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_1 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_2 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil))::(algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_3 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x2::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_4 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x3::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_5 : forall x14 x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_nil nil) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_6 : forall x14 x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_nil nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_7 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) (* *) | DP_R_xml_0_8 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) (* *) | DP_R_xml_0_9 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term algebra.F.id_a__isNeList (x4::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) (* *) | DP_R_xml_0_10 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNeList (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) (* *) | DP_R_xml_0_11 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_12 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_13 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_14 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_15 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U24 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_16 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_17 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U25 ((algebra.Alg.Term algebra.F.id_a__isList (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_18 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isList (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_19 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U26 ((algebra.Alg.Term algebra.F.id_a__isList (x6::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) (* *) | DP_R_xml_0_20 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isList (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) (* *) | DP_R_xml_0_21 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) (* *) | DP_R_xml_0_22 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) (* *) | DP_R_xml_0_23 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term algebra.F.id_a__isQid (x4::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) (* *) | DP_R_xml_0_24 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isQid (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) (* *) | DP_R_xml_0_25 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_26 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_27 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_28 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_29 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_30 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_31 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term algebra.F.id_a__isList (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_32 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isList (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_33 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term algebra.F.id_a__isNeList (x6::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) (* *) | DP_R_xml_0_34 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNeList (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) (* *) | DP_R_xml_0_35 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_36 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_37 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_38 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_39 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U54 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_40 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_41 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U55 ((algebra.Alg.Term algebra.F.id_a__isNeList (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_42 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNeList (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_43 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U56 ((algebra.Alg.Term algebra.F.id_a__isList (x6::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) (* *) | DP_R_xml_0_44 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isList (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) (* *) | DP_R_xml_0_45 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U62 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) (* *) | DP_R_xml_0_46 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) (* *) | DP_R_xml_0_47 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U63 ((algebra.Alg.Term algebra.F.id_a__isQid (x4::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) (* *) | DP_R_xml_0_48 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isQid (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) (* *) | DP_R_xml_0_49 : forall x8 x14 x13 x7 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x7 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U72 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x7::nil))::x8::nil)) (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_50 : forall x8 x14 x13 x7 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x7 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x7::nil)) (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_51 : forall x8 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U73 ((algebra.Alg.Term algebra.F.id_a__isPal (x8::nil))::x8::nil)) (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) (* *) | DP_R_xml_0_52 : forall x8 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPal (x8::nil)) (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) (* *) | DP_R_xml_0_53 : forall x8 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U74 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x8::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) (* *) | DP_R_xml_0_54 : forall x8 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x8::nil)) (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) (* *) | DP_R_xml_0_55 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) (* *) | DP_R_xml_0_56 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) (* *) | DP_R_xml_0_57 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term algebra.F.id_a__isNePal (x4::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) (* *) | DP_R_xml_0_58 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNePal (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) (* *) | DP_R_xml_0_59 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) (* *) | DP_R_xml_0_60 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) (* *) | DP_R_xml_0_61 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) (* *) | DP_R_xml_0_62 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) (* *) | DP_R_xml_0_63 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) (* *) | DP_R_xml_0_64 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) (* *) | DP_R_xml_0_65 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) (* *) | DP_R_xml_0_66 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) (* *) | DP_R_xml_0_67 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) (* *) | DP_R_xml_0_68 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) (* *) | DP_R_xml_0_69 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) (* *) | DP_R_xml_0_70 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) (* *) | DP_R_xml_0_71 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) (* *) | DP_R_xml_0_72 : forall x8 x13 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x7::(algebra.Alg.Term algebra.F.id___ (x8::x7::nil))::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term algebra.F.id_a__isQid (x7::nil))::x7::x8::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) (* *) | DP_R_xml_0_73 : forall x8 x13 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x7::(algebra.Alg.Term algebra.F.id___ (x8::x7::nil))::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isQid (x7::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) (* *) | DP_R_xml_0_74 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) (* *) | DP_R_xml_0_75 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) (* *) | DP_R_xml_0_76 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) (* *) | DP_R_xml_0_77 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) (* *) | DP_R_xml_0_78 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::(algebra.Alg.Term algebra.F.id_mark (x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_79 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_80 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_81 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_82 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_83 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_84 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_85 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isPalListKind (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_86 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_87 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_88 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isNeList (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNeList (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_89 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U21 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_90 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U21 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_91 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U22 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_92 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U22 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_93 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U23 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_94 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U23 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_95 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U24 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U24 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_96 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U24 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_97 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U25 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U25 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_98 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U25 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_99 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isList (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isList (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_100 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U26 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U26 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_101 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U26 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_102 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_103 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_104 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_105 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_106 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_107 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_108 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isQid (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isQid (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_109 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U41 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_110 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U41 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_111 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U42 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_112 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U42 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_113 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U43 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_114 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U43 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_115 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U44 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_116 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U44 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_117 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U45 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_118 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U45 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_119 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U46 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_120 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U46 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_121 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_122 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_123 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_124 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_125 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_126 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_127 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U54 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_128 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_129 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U55 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_130 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_131 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U56 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_132 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_133 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U61 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_134 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U61 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_135 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U62 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U62 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_136 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U62 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_137 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U63 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U63 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_138 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U63 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_139 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U71 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_140 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U71 (x9::x10::x11::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_141 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U72 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U72 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_142 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U72 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_143 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U73 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U73 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_144 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U73 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_145 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isPal (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isPal (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_146 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U74 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U74 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_147 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U74 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_148 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U81 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_149 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U81 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_150 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U82 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_151 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U82 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_152 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U83 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_153 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U83 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_154 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__isNePal (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_155 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_156 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_157 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_158 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module ddp := dp.MakeDP(algebra.EQT). Lemma R_xml_0_dp_step_spec : forall x y, (ddp.dp_step R_xml_0_deep_rew.R_xml_0_rules x y) -> exists f, exists l1, exists l2, y = algebra.Alg.Term f l2/\ (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2)/\ (ddp.dp R_xml_0_deep_rew.R_xml_0_rules x (algebra.Alg.Term f l1)). Proof. intros x y H. induction H. inversion H. subst. destruct t0. refine ((False_ind) _ _). refine (R_xml_0_deep_rew.R_xml_0_non_var H0). simpl in H|-*. exists a. exists ((List.map) (algebra.Alg.apply_subst sigma) l). exists ((List.map) (algebra.Alg.apply_subst sigma) l). repeat (constructor). assumption. exists f. exists l2. exists l1. constructor. constructor. constructor. constructor. rewrite <- closure.rwr_list_trans_clos_one_step_list. assumption. assumption. Qed. Ltac included_dp_tac H := injection H;clear H;intros;subst; repeat (match goal with | H: closure.refl_trans_clos (closure.one_step_list _) (_::_) _ |- _=> let x := fresh "x" in let l := fresh "l" in let h1 := fresh "h" in let h2 := fresh "h" in let h3 := fresh "h" in destruct (@algebra.EQT_ext.one_step_list_star_decompose_cons _ _ _ _ H) as [x [l[h1[h2 h3]]]];clear H;subst | H: closure.refl_trans_clos (closure.one_step_list _) nil _ |- _ => rewrite (@algebra.EQT_ext.one_step_list_star_decompose_nil _ _ H) in *;clear H end );simpl; econstructor eassumption . Ltac dp_concl_tac h2 h cont_tac t := match t with | False => let h' := fresh "a" in (set (h':=t) in *;cont_tac h'; repeat ( let e := type of h in (match e with | ?t => unfold t in h|-; (case h; [abstract (clear h;intros h;injection h; clear h;intros ;subst; included_dp_tac h2)| clear h;intros h;clear t]) | ?t => unfold t in h|-;elim h end ) )) | or ?a ?b => let cont_tac h' := let h'' := fresh "a" in (set (h'':=or a h') in *;cont_tac h'') in (dp_concl_tac h2 h cont_tac b) end . Module WF_DP_R_xml_0. Inductive DP_R_xml_0_non_scc_1 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_1_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_1 : forall x y, (DP_R_xml_0_non_scc_1 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_2 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_2_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U83 (x1::nil)) x13) -> DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_2 : forall x y, (DP_R_xml_0_non_scc_2 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_3 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_3_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U74 (x1::nil)) x13) -> DP_R_xml_0_non_scc_3 (algebra.Alg.Term algebra.F.id_a__U74 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_3 : forall x y, (DP_R_xml_0_non_scc_3 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_4 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_4_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U63 (x1::nil)) x13) -> DP_R_xml_0_non_scc_4 (algebra.Alg.Term algebra.F.id_a__U63 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_4 : forall x y, (DP_R_xml_0_non_scc_4 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_5 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_5_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0_non_scc_5 (algebra.Alg.Term algebra.F.id_a__U56 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_5 : forall x y, (DP_R_xml_0_non_scc_5 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_6 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_6_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U46 (x1::nil)) x13) -> DP_R_xml_0_non_scc_6 (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_6 : forall x y, (DP_R_xml_0_non_scc_6 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_7 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_7_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isQid (x1::nil)) x13) -> DP_R_xml_0_non_scc_7 (algebra.Alg.Term algebra.F.id_a__isQid (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_7 : forall x y, (DP_R_xml_0_non_scc_7 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_8 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_8_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_non_scc_8 (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_8 : forall x y, (DP_R_xml_0_non_scc_8 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_9 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_9_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U26 (x1::nil)) x13) -> DP_R_xml_0_non_scc_9 (algebra.Alg.Term algebra.F.id_a__U26 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_9 : forall x y, (DP_R_xml_0_non_scc_9 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_10 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_10_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_non_scc_10 (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_10 : forall x y, (DP_R_xml_0_non_scc_10 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_11 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_11_0 : forall x8 x13 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x7::(algebra.Alg.Term algebra.F.id___ (x8::x7::nil))::nil)) x13) -> DP_R_xml_0_non_scc_11 (algebra.Alg.Term algebra.F.id_a__isQid (x7::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_11 : forall x y, (DP_R_xml_0_non_scc_11 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_12 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_12_0 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_12 (algebra.Alg.Term algebra.F.id_a__U92 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_12 : forall x y, (DP_R_xml_0_non_scc_12 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_scc_13 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_13_0 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_13 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_13_1 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_scc_13 (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) (* *) | DP_R_xml_0_scc_13_2 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_scc_13 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) . Module WF_DP_R_xml_0_scc_13. Inductive DP_R_xml_0_scc_13_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_13_large_0 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_13_large (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) . Inductive DP_R_xml_0_scc_13_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_13_strict_0 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_scc_13_strict (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)):: x6::nil)) (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) (* *) | DP_R_xml_0_scc_13_strict_1 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_scc_13_strict (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) . Module WF_DP_R_xml_0_scc_13_large. Inductive DP_R_xml_0_scc_13_large_non_scc_1 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_13_large_non_scc_1_0 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_13_large_non_scc_1 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_13_large_non_scc_1 : forall x y, (DP_R_xml_0_scc_13_large_non_scc_1 x y) -> Acc WF_DP_R_xml_0_scc_13.DP_R_xml_0_scc_13_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Lemma wf : well_founded WF_DP_R_xml_0_scc_13.DP_R_xml_0_scc_13_large. Proof. constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13_large_non_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_13_large_non_scc_0; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail))). Qed. End WF_DP_R_xml_0_scc_13_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 2 + 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 0. Definition P_id_a__U56 (x13:Z) := 0. Definition P_id_a__U26 (x13:Z) := 0. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 1. Definition P_id_a__U13 (x13:Z) := 0. Definition P_id_U45 (x13:Z) (x14:Z) := 0. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U81 (x13:Z) (x14:Z) := 0. Definition P_id_isNeList (x13:Z) := 2* x13. Definition P_id_a__U11 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 1* x13. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 2* x15. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isNePal (x13:Z) := 0. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U91 (x13:Z) (x14:Z) := 0. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 0. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 0. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 2 + 2* x15. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 0. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 2* x15. Definition P_id_a__U12 (x13:Z) (x14:Z) := 0. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 2* x13. Definition P_id_a__U25 (x13:Z) (x14:Z) := 0. Definition P_id_U55 (x13:Z) (x14:Z) := 0. Definition P_id_a__U92 (x13:Z) := 0. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isList (x13:Z) := 2* x13. Definition P_id___ (x13:Z) (x14:Z) := 2 + 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 0. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 0. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 1. Definition P_id_a__isNeList (x13:Z) := 2* x13. Definition P_id_U46 (x13:Z) := 0. Definition P_id_a__U81 (x13:Z) (x14:Z) := 0. Definition P_id_a__U45 (x13:Z) (x14:Z) := 0. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 2 + 2* x15. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 0. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U91 (x13:Z) (x14:Z) := 0. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U92 (x13:Z) := 0. Definition P_id_U25 (x13:Z) (x14:Z) := 0. Definition P_id_nil := 2. Definition P_id_isQid (x13:Z) := 1* x13. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 0. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 2* x15. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isNePal (x13:Z) := 0. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 2* x15. Definition P_id_a__isPalListKind (x13:Z) := 2* x13. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 0. Definition P_id_a__isList (x13:Z) := 2* x13. Definition P_id_U56 (x13:Z) := 0. Definition P_id_a := 2. Definition P_id_a__U55 (x13:Z) (x14:Z) := 0. Definition P_id_U26 (x13:Z) := 0. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14::x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14::x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14::x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14::x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14::x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14::x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14::x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14::x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14::x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14::x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14::x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14::x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14::x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14::x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14::x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 2* x13 + 2* x14. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 2* x13. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 0. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_13_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_13_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_13_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_13_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_13_large := WF_DP_R_xml_0_scc_13_large.wf. Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_13. Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_13_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_13_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_13_large_in_le;econstructor eassumption])). apply wf_DP_R_xml_0_scc_13_large. Qed. End WF_DP_R_xml_0_scc_13. Definition wf_DP_R_xml_0_scc_13 := WF_DP_R_xml_0_scc_13.wf. Lemma acc_DP_R_xml_0_scc_13 : forall x y, (DP_R_xml_0_scc_13 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_13). intros x' _ Hrec y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply Hrec;econstructor eassumption)|| ((eapply acc_DP_R_xml_0_non_scc_12; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). apply wf_DP_R_xml_0_scc_13. Qed. Inductive DP_R_xml_0_non_scc_14 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_14_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isPalListKind (x1::nil)) x13) -> DP_R_xml_0_non_scc_14 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_14 : forall x y, (DP_R_xml_0_non_scc_14 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_15 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_15_0 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_15 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_15 : forall x y, (DP_R_xml_0_non_scc_15 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_16 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_16_0 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_16 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_16 : forall x y, (DP_R_xml_0_non_scc_16 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_17 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_17_0 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_non_scc_17 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_17 : forall x y, (DP_R_xml_0_non_scc_17 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_18 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_18_0 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_18 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_18 : forall x y, (DP_R_xml_0_non_scc_18 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_19 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_19_0 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_non_scc_19 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_19 : forall x y, (DP_R_xml_0_non_scc_19 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_20 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_20_0 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_20 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_20 : forall x y, (DP_R_xml_0_non_scc_20 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_21 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_21_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_21 (algebra.Alg.Term algebra.F.id_a__U91 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_21 : forall x y, (DP_R_xml_0_non_scc_21 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_12; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_22 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_22_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_22 (algebra.Alg.Term algebra.F.id_a__U83 ((algebra.Alg.Term algebra.F.id_a__isNePal (x4::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_22 : forall x y, (DP_R_xml_0_non_scc_22 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_23 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_23_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_23 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_23 : forall x y, (DP_R_xml_0_non_scc_23 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_24 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_24_0 : forall x8 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0_non_scc_24 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x8::nil)) (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_24 : forall x y, (DP_R_xml_0_non_scc_24 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_25 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_25_0 : forall x8 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0_non_scc_25 (algebra.Alg.Term algebra.F.id_a__U74 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x8::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_25 : forall x y, (DP_R_xml_0_non_scc_25 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_26 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_26_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U73 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_26 (algebra.Alg.Term algebra.F.id_a__U73 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_26 : forall x y, (DP_R_xml_0_non_scc_26 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_non_scc_25; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_24; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_27 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_27_0 : forall x8 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0_non_scc_27 (algebra.Alg.Term algebra.F.id_a__U73 ((algebra.Alg.Term algebra.F.id_a__isPal (x8::nil))::x8::nil)) (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_27 : forall x y, (DP_R_xml_0_non_scc_27 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_non_scc_25; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_24; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_28 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_28_0 : forall x8 x14 x13 x7 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x7 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0_non_scc_28 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x7::nil)) (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_28 : forall x y, (DP_R_xml_0_non_scc_28 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_29 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_29_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_29 (algebra.Alg.Term algebra.F.id_a__isQid (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_29 : forall x y, (DP_R_xml_0_non_scc_29 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_30 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_30_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_30 (algebra.Alg.Term algebra.F.id_a__U63 ((algebra.Alg.Term algebra.F.id_a__isQid (x4::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_30 : forall x y, (DP_R_xml_0_non_scc_30 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_31 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_31_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U62 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_31 (algebra.Alg.Term algebra.F.id_a__U62 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_31 : forall x y, (DP_R_xml_0_non_scc_31 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_non_scc_30; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_29; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_32 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_32_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_32 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_32 : forall x y, (DP_R_xml_0_non_scc_32 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_33 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_33_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_33 (algebra.Alg.Term algebra.F.id_a__U62 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)):: x4::nil)) (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_33 : forall x y, (DP_R_xml_0_non_scc_33 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_non_scc_30; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_29; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_34 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_34_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U61 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_34 (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_34 : forall x y, (DP_R_xml_0_non_scc_34 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_non_scc_33; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_32; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_35 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_35_0 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_35 (algebra.Alg.Term algebra.F.id_a__U61 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)):: x4::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_35 : forall x y, (DP_R_xml_0_non_scc_35 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_non_scc_33; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_32; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_scc_36 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_36_0 : forall x8 x13 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x7::(algebra.Alg.Term algebra.F.id___ (x8::x7::nil))::nil)) x13) -> DP_R_xml_0_scc_36 (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term algebra.F.id_a__isQid (x7::nil))::x7::x8::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) (* *) | DP_R_xml_0_scc_36_1 : forall x8 x14 x13 x7 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x7 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0_scc_36 (algebra.Alg.Term algebra.F.id_a__U72 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x7::nil))::x8::nil)) (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_36_2 : forall x8 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0_scc_36 (algebra.Alg.Term algebra.F.id_a__isPal (x8::nil)) (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_36_3 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_36 (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) (* *) | DP_R_xml_0_scc_36_4 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_36 (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_36_5 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_36 (algebra.Alg.Term algebra.F.id_a__isNePal (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) . Module WF_DP_R_xml_0_scc_36. Inductive DP_R_xml_0_scc_36_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_36_large_0 : forall x8 x14 x13 x7 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x7 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0_scc_36_large (algebra.Alg.Term algebra.F.id_a__U72 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x7::nil)):: x8::nil)) (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_36_large_1 : forall x8 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0_scc_36_large (algebra.Alg.Term algebra.F.id_a__isPal (x8::nil)) (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_36_large_2 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_36_large (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)):: x4::nil)) (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) (* *) | DP_R_xml_0_scc_36_large_3 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_36_large (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)):: x4::nil)) (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_36_large_4 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_36_large (algebra.Alg.Term algebra.F.id_a__isNePal (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) . Inductive DP_R_xml_0_scc_36_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_36_strict_0 : forall x8 x13 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x7::(algebra.Alg.Term algebra.F.id___ (x8::x7::nil))::nil)) x13) -> DP_R_xml_0_scc_36_strict (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term algebra.F.id_a__isQid (x7::nil))::x7::x8::nil)) (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) . Module WF_DP_R_xml_0_scc_36_large. Inductive DP_R_xml_0_scc_36_large_non_scc_1 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_36_large_non_scc_1_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_36_large_non_scc_1 (algebra.Alg.Term algebra.F.id_a__isNePal (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_36_large_non_scc_1 : forall x y, (DP_R_xml_0_scc_36_large_non_scc_1 x y) -> Acc WF_DP_R_xml_0_scc_36.DP_R_xml_0_scc_36_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_scc_36_large_non_scc_2 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_36_large_non_scc_2_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_36_large_non_scc_2 (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_36_large_non_scc_2 : forall x y, (DP_R_xml_0_scc_36_large_non_scc_2 x y) -> Acc WF_DP_R_xml_0_scc_36.DP_R_xml_0_scc_36_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_36_large_non_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_36_large_non_scc_3 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_36_large_non_scc_3_0 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_36_large_non_scc_3 (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) . Lemma acc_DP_R_xml_0_scc_36_large_non_scc_3 : forall x y, (DP_R_xml_0_scc_36_large_non_scc_3 x y) -> Acc WF_DP_R_xml_0_scc_36.DP_R_xml_0_scc_36_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_36_large_non_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_36_large_non_scc_4 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_36_large_non_scc_4_0 : forall x8 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0_scc_36_large_non_scc_4 (algebra.Alg.Term algebra.F.id_a__isPal (x8::nil)) (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_36_large_non_scc_4 : forall x y, (DP_R_xml_0_scc_36_large_non_scc_4 x y) -> Acc WF_DP_R_xml_0_scc_36.DP_R_xml_0_scc_36_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_36_large_non_scc_3; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_36_large_non_scc_5 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_36_large_non_scc_5_0 : forall x8 x14 x13 x7 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x7 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x13) -> DP_R_xml_0_scc_36_large_non_scc_5 (algebra.Alg.Term algebra.F.id_a__U72 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x7::nil))::x8::nil)) (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_36_large_non_scc_5 : forall x y, (DP_R_xml_0_scc_36_large_non_scc_5 x y) -> Acc WF_DP_R_xml_0_scc_36.DP_R_xml_0_scc_36_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_36_large_non_scc_4; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Lemma wf : well_founded WF_DP_R_xml_0_scc_36.DP_R_xml_0_scc_36_large. Proof. constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_36_large_non_scc_5; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_36_large_non_scc_4; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_36_large_non_scc_3; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_36_large_non_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_36_large_non_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_36_large_non_scc_0; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail))))))). Qed. End WF_DP_R_xml_0_scc_36_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 2 + 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 0. Definition P_id_a__U56 (x13:Z) := 0. Definition P_id_a__U26 (x13:Z) := 0. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 0. Definition P_id_U45 (x13:Z) (x14:Z) := 0. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U81 (x13:Z) (x14:Z) := 0. Definition P_id_isNeList (x13:Z) := 0. Definition P_id_a__U11 (x13:Z) (x14:Z) := 0. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 0. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isNePal (x13:Z) := 0. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U91 (x13:Z) (x14:Z) := 1. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 0. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 0. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 0. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U12 (x13:Z) (x14:Z) := 0. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 1* x13. Definition P_id_a__U25 (x13:Z) (x14:Z) := 0. Definition P_id_U55 (x13:Z) (x14:Z) := 0. Definition P_id_a__U92 (x13:Z) := 0. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isList (x13:Z) := 0. Definition P_id___ (x13:Z) (x14:Z) := 2 + 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 0. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 0. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 0. Definition P_id_U46 (x13:Z) := 0. Definition P_id_a__U81 (x13:Z) (x14:Z) := 0. Definition P_id_a__U45 (x13:Z) (x14:Z) := 0. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 0. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U92 (x13:Z) := 0. Definition P_id_U25 (x13:Z) (x14:Z) := 0. Definition P_id_nil := 2. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 0. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isNePal (x13:Z) := 0. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPalListKind (x13:Z) := 1* x13. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 0. Definition P_id_a__isList (x13:Z) := 0. Definition P_id_U56 (x13:Z) := 0. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 0. Definition P_id_U26 (x13:Z) := 0. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14::x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14::x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14::x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14::x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14::x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14::x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14::x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14::x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14::x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14::x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14::x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14::x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14::x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14::x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14::x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 1* x14. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 1* x14. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 1* x13. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 1* x14. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 1* x13. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 0. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 1* x15. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_36_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_36_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_36_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_36_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_36_large := WF_DP_R_xml_0_scc_36_large.wf. Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_36. Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_36_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_36_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_36_large_in_le;econstructor eassumption])). apply wf_DP_R_xml_0_scc_36_large. Qed. End WF_DP_R_xml_0_scc_36. Definition wf_DP_R_xml_0_scc_36 := WF_DP_R_xml_0_scc_36.wf. Lemma acc_DP_R_xml_0_scc_36 : forall x y, (DP_R_xml_0_scc_36 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_36). intros x' _ Hrec y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply Hrec;econstructor eassumption)|| ((eapply acc_DP_R_xml_0_non_scc_35; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_28; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_27; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_23; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_22; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_16; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_15; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_11; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))))))))). apply wf_DP_R_xml_0_scc_36. Qed. Inductive DP_R_xml_0_non_scc_37 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_37_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U82 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_37 (algebra.Alg.Term algebra.F.id_a__U82 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_37 : forall x y, (DP_R_xml_0_non_scc_37 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_36; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_22; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_38 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_38_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U81 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_38 (algebra.Alg.Term algebra.F.id_a__U81 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_38 : forall x y, (DP_R_xml_0_non_scc_38 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_36; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_23; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_39 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_39_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isPal (x1::nil)) x13) -> DP_R_xml_0_non_scc_39 (algebra.Alg.Term algebra.F.id_a__isPal (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_39 : forall x y, (DP_R_xml_0_non_scc_39 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_36; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_15; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_40 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_40_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U72 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_40 (algebra.Alg.Term algebra.F.id_a__U72 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_40 : forall x y, (DP_R_xml_0_non_scc_40 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_36; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_27; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_41 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_41_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U71 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_41 (algebra.Alg.Term algebra.F.id_a__U71 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_41 : forall x y, (DP_R_xml_0_non_scc_41 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_36; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_28; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_42 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_42_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isNePal (x1::nil)) x13) -> DP_R_xml_0_non_scc_42 (algebra.Alg.Term algebra.F.id_a__isNePal (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_42 : forall x y, (DP_R_xml_0_non_scc_42 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_36; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_35; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_16; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_11; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))). Qed. Inductive DP_R_xml_0_non_scc_43 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_43_0 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_43 (algebra.Alg.Term algebra.F.id_a__U56 ((algebra.Alg.Term algebra.F.id_a__isList (x6::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_43 : forall x y, (DP_R_xml_0_non_scc_43 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_44 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_44_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_44 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_44 : forall x y, (DP_R_xml_0_non_scc_44 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_45 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_45_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_45 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_45 : forall x y, (DP_R_xml_0_non_scc_45 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_46 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_46_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_46 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_46 : forall x y, (DP_R_xml_0_non_scc_46 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_47 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_47_0 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_47 (algebra.Alg.Term algebra.F.id_a__U46 ((algebra.Alg.Term algebra.F.id_a__isNeList (x6::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_47 : forall x y, (DP_R_xml_0_non_scc_47 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_48 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_48_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_48 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_48 : forall x y, (DP_R_xml_0_non_scc_48 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_49 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_49_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_49 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_49 : forall x y, (DP_R_xml_0_non_scc_49 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_50 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_50_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_50 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_50 : forall x y, (DP_R_xml_0_non_scc_50 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_51 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_51_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_51 (algebra.Alg.Term algebra.F.id_a__isQid (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_51 : forall x y, (DP_R_xml_0_non_scc_51 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_52 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_52_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_52 (algebra.Alg.Term algebra.F.id_a__U33 ((algebra.Alg.Term algebra.F.id_a__isQid (x4::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_52 : forall x y, (DP_R_xml_0_non_scc_52 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_53 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_53_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_53 (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_53 : forall x y, (DP_R_xml_0_non_scc_53 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_non_scc_52; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_51; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_54 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_54_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_54 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_54 : forall x y, (DP_R_xml_0_non_scc_54 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_55 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_55_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_55 (algebra.Alg.Term algebra.F.id_a__U32 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)):: x4::nil)) (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_55 : forall x y, (DP_R_xml_0_non_scc_55 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_non_scc_52; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_51; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_56 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_56_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_56 (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_56 : forall x y, (DP_R_xml_0_non_scc_56 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_non_scc_55; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_54; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_57 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_57_0 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_57 (algebra.Alg.Term algebra.F.id_a__U31 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)):: x4::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_57 : forall x y, (DP_R_xml_0_non_scc_57 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_non_scc_55; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_54; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_58 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_58_0 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_58 (algebra.Alg.Term algebra.F.id_a__U26 ((algebra.Alg.Term algebra.F.id_a__isList (x6::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_58 : forall x y, (DP_R_xml_0_non_scc_58 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_59 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_59_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_59 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_59 : forall x y, (DP_R_xml_0_non_scc_59 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_60 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_60_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_60 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_60 : forall x y, (DP_R_xml_0_non_scc_60 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_61 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_61_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_non_scc_61 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_61 : forall x y, (DP_R_xml_0_non_scc_61 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_62 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_62_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_62 (algebra.Alg.Term algebra.F.id_a__U13 ((algebra.Alg.Term algebra.F.id_a__isNeList (x4::nil))::nil)) (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_62 : forall x y, (DP_R_xml_0_non_scc_62 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_non_scc_63 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_63_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_non_scc_63 (algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_63 : forall x y, (DP_R_xml_0_non_scc_63 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__isNeList (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_1 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) (* *) | DP_R_xml_0_scc_64_2 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_3 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_4 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_5 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term algebra.F.id_a__isList (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_6 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__isNeList (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_7 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) (* *) | DP_R_xml_0_scc_64_8 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_9 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_10 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U54 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_11 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U55 ((algebra.Alg.Term algebra.F.id_a__isNeList (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_12 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__isList (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_13 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) (* *) | DP_R_xml_0_scc_64_14 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_15 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) (* *) | DP_R_xml_0_scc_64_16 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_17 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_18 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U24 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_19 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__U25 ((algebra.Alg.Term algebra.F.id_a__isList (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_20 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__isList (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_21 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__isList (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_22 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__isNeList (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_23 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64 (algebra.Alg.Term algebra.F.id_a__isList (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) . Module WF_DP_R_xml_0_scc_64. Inductive DP_R_xml_0_scc_64_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__isNeList (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_1 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)):: x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) (* *) | DP_R_xml_0_scc_64_large_2 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)):: x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_3 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)):: x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_4 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term algebra.F.id_a__isList (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_5 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__isNeList (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_6 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)):: x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_7 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)):: x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_8 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U54 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)):: x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_9 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U55 ((algebra.Alg.Term algebra.F.id_a__isNeList (x5::nil)):: x6::nil)) (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_10 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__isList (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_11 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)):: x4::nil)) (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) (* *) | DP_R_xml_0_scc_64_large_12 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil)):: x4::nil)) (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_13 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)):: x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_14 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)):: x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_15 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U24 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil)):: x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_16 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__U25 ((algebra.Alg.Term algebra.F.id_a__isList (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_17 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__isList (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_18 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__isList (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_19 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__isNeList (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_large_20 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large (algebra.Alg.Term algebra.F.id_a__isList (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) . Inductive DP_R_xml_0_scc_64_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_strict_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_strict (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) (* *) | DP_R_xml_0_scc_64_strict_1 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_scc_64_strict (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)):: x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) (* *) | DP_R_xml_0_scc_64_strict_2 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_scc_64_strict (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil)):: x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) . Module WF_DP_R_xml_0_scc_64_large. Inductive DP_R_xml_0_scc_64_large_non_scc_1 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_1_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_1 (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_1 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_1 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_2 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_2_0 : forall x6 x5 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x5::x6::nil)) x13) -> DP_R_xml_0_scc_64_large_non_scc_2 (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_2 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_2 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_3 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_3_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_3 (algebra.Alg.Term algebra.F.id_a__isNeList (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_3 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_3 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_4 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_4_0 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_4 (algebra.Alg.Term algebra.F.id_a__isNeList (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_4 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_4 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_5 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_5_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_5 (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term algebra.F.id_a__isList (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_5 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_5 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_4; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_6 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_6_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_64_large_non_scc_6 (algebra.Alg.Term algebra.F.id_a__isNeList (x4::nil)) (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_6 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_6 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_7 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_7_0 : forall x4 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_64_large_non_scc_7 (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_7 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_7 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_6; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_8 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_8_0 : forall x4 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x13) -> DP_R_xml_0_scc_64_large_non_scc_8 (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x4::nil))::x4::nil)) (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_8 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_8 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_7; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_9 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_9_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_9 (algebra.Alg.Term algebra.F.id_a__isList (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_9 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_9 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_8; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_10 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_10_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_10 (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_10 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_10 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_9; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_5; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_11 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_11_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_11 (algebra.Alg.Term algebra.F.id_a__isList (x5::nil)) (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_11 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_11 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_8; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_12 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_12_0 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_12 (algebra.Alg.Term algebra.F.id_a__isList (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_12 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_12 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_8; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_13 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_13_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_13 (algebra.Alg.Term algebra.F.id_a__U25 ((algebra.Alg.Term algebra.F.id_a__isList (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_13 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_13 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_12; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_14 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_14_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_14 (algebra.Alg.Term algebra.F.id_a__U24 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_14 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_14 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_11; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_15 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_15_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_15 (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_15 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_15 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_14; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_16 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_16_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_16 (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_16 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_16 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_15; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_17 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_17_0 : forall x6 x14 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_17 (algebra.Alg.Term algebra.F.id_a__isList (x6::nil)) (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_17 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_17 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_8; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_18 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_18_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_18 (algebra.Alg.Term algebra.F.id_a__U55 ((algebra.Alg.Term algebra.F.id_a__isNeList (x5::nil))::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_18 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_18 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_17; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_19 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_19_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_19 (algebra.Alg.Term algebra.F.id_a__U54 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_19 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_19 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_18; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_3; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_20 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_20_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_20 (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x6::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_20 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_20 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_19; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_64_large_non_scc_21 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_64_large_non_scc_21_0 : forall x6 x14 x5 x13 x15, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_tt nil) x15) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x6 x13) -> DP_R_xml_0_scc_64_large_non_scc_21 (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term algebra.F.id_a__isPalListKind (x5::nil))::x5::x6::nil)) (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_64_large_non_scc_21 : forall x y, (DP_R_xml_0_scc_64_large_non_scc_21 x y) -> Acc WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_20; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Lemma wf : well_founded WF_DP_R_xml_0_scc_64.DP_R_xml_0_scc_64_large. Proof. constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64_large_non_scc_21; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_20; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_19; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_18; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_17; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_16; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_15; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_14; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_13; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_12; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_11; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_10; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_9; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_8; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_7; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_6; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_5; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_4; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_3; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_64_large_non_scc_0; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (fail))))))))))))))))))))))). Qed. End WF_DP_R_xml_0_scc_64_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 1 + 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 0. Definition P_id_a__U56 (x13:Z) := 0. Definition P_id_a__U26 (x13:Z) := 0. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 0. Definition P_id_U45 (x13:Z) (x14:Z) := 0. Definition P_id_a__U74 (x13:Z) := 3* x13. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U81 (x13:Z) (x14:Z) := 0. Definition P_id_isNeList (x13:Z) := 0. Definition P_id_a__U11 (x13:Z) (x14:Z) := 0. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 0. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 2. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__isNePal (x13:Z) := 0. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U91 (x13:Z) (x14:Z) := 0. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 2. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 0. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 0. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 2 + 1* x15. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 0. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 2. Definition P_id_a__U12 (x13:Z) (x14:Z) := 0. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U73 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 0. Definition P_id_U55 (x13:Z) (x14:Z) := 0. Definition P_id_a__U92 (x13:Z) := 0. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isList (x13:Z) := 2* x13. Definition P_id___ (x13:Z) (x14:Z) := 1 + 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 0. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 0. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 1. Definition P_id_a__isNeList (x13:Z) := 0. Definition P_id_U46 (x13:Z) := 0. Definition P_id_a__U81 (x13:Z) (x14:Z) := 0. Definition P_id_a__U45 (x13:Z) (x14:Z) := 0. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 2 + 1* x15. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U73 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U12 (x13:Z) (x14:Z) := 0. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 2. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U91 (x13:Z) (x14:Z) := 0. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U92 (x13:Z) := 0. Definition P_id_U25 (x13:Z) (x14:Z) := 0. Definition P_id_nil := 1. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 0. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 2. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isNePal (x13:Z) := 0. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 2. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U74 (x13:Z) := 3* x13. Definition P_id_U13 (x13:Z) := 0. Definition P_id_a__isList (x13:Z) := 2* x13. Definition P_id_U56 (x13:Z) := 0. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 0. Definition P_id_U26 (x13:Z) := 0. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14::x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14::x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14::x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14::x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14::x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14::x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14::x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14::x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14::x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14::x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14::x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14::x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14::x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14::x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14::x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 1* x14. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x14 + 1* x15. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 1* x13. Definition P_id_A__U45 (x13:Z) (x14:Z) := 1* x14. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_A__ISLIST (x13:Z) := 1* x13. Definition P_id_A__U55 (x13:Z) (x14:Z) := 1* x14. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 1* x14. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x14 + 1* x15. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 1* x14. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 0. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_64_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_64_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_64_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_64_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_64_large := WF_DP_R_xml_0_scc_64_large.wf. Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_64. Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_64_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_64_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_64_large_in_le;econstructor eassumption])). apply wf_DP_R_xml_0_scc_64_large. Qed. End WF_DP_R_xml_0_scc_64. Definition wf_DP_R_xml_0_scc_64 := WF_DP_R_xml_0_scc_64.wf. Lemma acc_DP_R_xml_0_scc_64 : forall x y, (DP_R_xml_0_scc_64 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_64). intros x' _ Hrec y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply Hrec;econstructor eassumption)|| ((eapply acc_DP_R_xml_0_non_scc_63; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_62; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_61; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_60; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_59; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_58; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_57; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_50; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_49; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_48; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_47; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_46; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_45; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_44; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_43; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_20; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_19; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_18; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_17; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))))))))))))))))))). apply wf_DP_R_xml_0_scc_64. Qed. Inductive DP_R_xml_0_non_scc_65 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_65_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_65 (algebra.Alg.Term algebra.F.id_a__U11 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_65 : forall x y, (DP_R_xml_0_non_scc_65 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_63; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_66 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_66_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U23 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_66 (algebra.Alg.Term algebra.F.id_a__U23 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_66 : forall x y, (DP_R_xml_0_non_scc_66 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_59; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_67 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_67_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U22 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_67 (algebra.Alg.Term algebra.F.id_a__U22 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_67 : forall x y, (DP_R_xml_0_non_scc_67 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_60; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_68 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_68_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U21 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_68 (algebra.Alg.Term algebra.F.id_a__U21 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_68 : forall x y, (DP_R_xml_0_non_scc_68 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_61; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_69 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_69_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isList (x1::nil)) x13) -> DP_R_xml_0_non_scc_69 (algebra.Alg.Term algebra.F.id_a__isList (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_69 : forall x y, (DP_R_xml_0_non_scc_69 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_20; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_19; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))))). Qed. Inductive DP_R_xml_0_non_scc_70 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_70_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_70 (algebra.Alg.Term algebra.F.id_a__U55 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_70 : forall x y, (DP_R_xml_0_non_scc_70 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_43; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_71 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_71_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U43 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_71 (algebra.Alg.Term algebra.F.id_a__U43 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_71 : forall x y, (DP_R_xml_0_non_scc_71 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_48; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_72 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_72_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U42 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_72 (algebra.Alg.Term algebra.F.id_a__U42 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_72 : forall x y, (DP_R_xml_0_non_scc_72 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_49; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_73 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_73_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U41 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_73 (algebra.Alg.Term algebra.F.id_a__U41 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_73 : forall x y, (DP_R_xml_0_non_scc_73 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_50; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_74 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_74_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_74 (algebra.Alg.Term algebra.F.id_a__U54 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_74 : forall x y, (DP_R_xml_0_non_scc_74 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_75 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_75_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_75 (algebra.Alg.Term algebra.F.id_a__U53 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_75 : forall x y, (DP_R_xml_0_non_scc_75 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_44; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_76 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_76_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_76 (algebra.Alg.Term algebra.F.id_a__U52 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_76 : forall x y, (DP_R_xml_0_non_scc_76 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_45; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_77 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_77_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_77 (algebra.Alg.Term algebra.F.id_a__U51 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_77 : forall x y, (DP_R_xml_0_non_scc_77 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_46; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_78 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_78_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_isNeList (x1::nil)) x13) -> DP_R_xml_0_non_scc_78 (algebra.Alg.Term algebra.F.id_a__isNeList (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_78 : forall x y, (DP_R_xml_0_non_scc_78 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_57; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_18; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_17; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))))). Qed. Inductive DP_R_xml_0_non_scc_79 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_79_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U45 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_79 (algebra.Alg.Term algebra.F.id_a__U45 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_79 : forall x y, (DP_R_xml_0_non_scc_79 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_47; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_80 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_80_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U44 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_80 (algebra.Alg.Term algebra.F.id_a__U44 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_80 : forall x y, (DP_R_xml_0_non_scc_80 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_non_scc_81 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_81_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_81 (algebra.Alg.Term algebra.F.id_a__U12 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_81 : forall x y, (DP_R_xml_0_non_scc_81 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_62; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_82 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_82_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U25 (x9::x10::nil)) x13) -> DP_R_xml_0_non_scc_82 (algebra.Alg.Term algebra.F.id_a__U25 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_82 : forall x y, (DP_R_xml_0_non_scc_82 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_58; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' )))). Qed. Inductive DP_R_xml_0_non_scc_83 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_83_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U24 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_non_scc_83 (algebra.Alg.Term algebra.F.id_a__U24 ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::x10::x11::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Lemma acc_DP_R_xml_0_non_scc_83 : forall x y, (DP_R_xml_0_non_scc_83 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec;econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_84 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_0 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_scc_84_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x9::nil)):: (algebra.Alg.Term algebra.F.id_mark (x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_2 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x1::nil)):: (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil)):: (algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_scc_84_3 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil)):: (algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_scc_84_4 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x2::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_scc_84_5 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_6 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_7 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_8 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_9 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_10 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U21 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_11 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U22 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_12 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U23 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_13 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U24 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_14 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U25 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_15 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U26 (x1::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_16 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_17 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_18 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_19 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U41 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_20 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U42 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_21 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U43 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_22 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U44 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_23 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U45 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_24 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U46 (x1::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_25 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_26 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_27 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_28 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_29 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_30 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_31 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U61 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_32 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U62 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_33 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U63 (x1::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_34 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U71 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_35 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U72 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_36 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U73 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_37 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U74 (x1::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_38 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U81 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_39 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U82 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_40 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U83 (x1::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_41 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_42 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_43 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x3::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_scc_84_44 : forall x14 x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_nil nil) x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_scc_84_45 : forall x14 x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_nil nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x13) -> DP_R_xml_0_scc_84 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) . Module WF_DP_R_xml_0_scc_84. Inductive DP_R_xml_0_scc_84_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_3 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U21 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_4 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U22 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_5 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U23 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_6 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U24 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_7 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U25 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_8 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U26 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_9 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_10 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_11 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_12 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U41 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_13 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U42 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_14 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U43 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_15 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U44 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_16 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U45 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_17 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U46 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_18 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_19 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_20 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_21 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_22 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_23 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_24 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U61 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_25 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U62 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_26 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U63 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_27 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U71 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_28 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U72 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_29 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U73 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_30 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U74 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_31 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U81 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_32 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U82 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_33 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U83 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_34 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_35 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_36 : forall x14 x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_nil nil) x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_scc_84_large_37 : forall x14 x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_nil nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x13) -> DP_R_xml_0_scc_84_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) . Inductive DP_R_xml_0_scc_84_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_strict_0 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0_scc_84_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_scc_84_strict_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_strict (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x9::nil))::(algebra.Alg.Term algebra.F.id_mark (x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_strict_2 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0_scc_84_strict (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x1::nil))::(algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil)):: (algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_scc_84_strict_3 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0_scc_84_strict (algebra.Alg.Term algebra.F.id_a____ ((algebra.Alg.Term algebra.F.id_mark (x2::nil))::(algebra.Alg.Term algebra.F.id_mark (x3::nil))::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_scc_84_strict_4 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0_scc_84_strict (algebra.Alg.Term algebra.F.id_mark (x2::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) (* *) | DP_R_xml_0_scc_84_strict_5 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_strict_6 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_strict (algebra.Alg.Term algebra.F.id_mark (x10::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_strict_7 : forall x2 x14 x1 x13 x3, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id___ (x1::x2::nil)) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x3 x13) -> DP_R_xml_0_scc_84_strict (algebra.Alg.Term algebra.F.id_mark (x3::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) . Module WF_DP_R_xml_0_scc_84_large. Inductive DP_R_xml_0_scc_84_large_scc_1 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_3 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U21 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_4 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U22 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_5 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U23 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_6 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U24 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_7 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U25 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_8 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U26 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_9 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_10 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_11 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_12 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U41 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_13 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U42 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_14 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U43 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_15 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U44 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_16 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U45 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_17 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U46 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_18 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_19 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_20 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_21 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_22 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_23 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_24 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U61 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_25 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U62 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_26 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U63 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_27 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U71 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_28 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U72 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_29 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U73 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_30 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U74 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_31 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U81 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_32 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U82 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_33 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U83 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_34 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_35 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1. Inductive DP_R_xml_0_scc_84_large_scc_1_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_3 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U21 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_4 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U22 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_5 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U23 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_6 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U24 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_7 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U25 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_8 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U26 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_9 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_10 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_11 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_12 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U41 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_13 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U42 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_14 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U43 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_15 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U44 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_16 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U45 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_17 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U46 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_18 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_19 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_20 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_21 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_22 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_23 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_24 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U73 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_25 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U74 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_26 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U83 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_27 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_28 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_strict_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U61 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_strict_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U62 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_strict_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U63 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_strict_3 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U71 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_strict_4 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U72 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_strict_5 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U81 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_strict_6 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U82 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_3 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U21 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_4 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U22 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_5 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U23 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_6 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U24 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_7 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U25 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_8 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U26 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_9 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_10 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_11 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_12 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U41 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_13 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U42 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_14 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U43 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_15 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U44 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_16 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U45 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_17 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U46 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_18 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_19 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_20 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_21 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_22 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_23 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_24 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U83 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_25 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_26 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_strict_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U73 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_strict_1 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U74 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_3 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U21 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_4 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U22 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_5 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U23 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_6 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U24 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_7 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U25 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_8 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U26 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_9 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_10 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_11 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_12 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U41 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_13 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U42 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_14 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U43 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_15 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U44 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_16 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U45 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_17 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U46 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_18 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_19 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_20 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_21 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_22 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_23 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_24 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_25 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_strict_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U83 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_3 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U21 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_4 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U22 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_5 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U23 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_6 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U24 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_7 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U25 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_8 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U26 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_9 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_10 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_11 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_12 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U42 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_13 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U43 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_14 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U44 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_15 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U45 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_16 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U46 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_17 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_18 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_19 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_20 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_21 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_22 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_23 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_24 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_strict_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U41 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_3 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U22 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_4 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U23 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_5 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U24 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_6 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U25 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_7 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U26 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_8 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_9 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_10 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_11 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U42 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_12 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U43 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_13 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U44 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_14 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U45 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_15 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U46 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_16 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_17 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_18 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_19 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_20 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_21 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_22 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_23 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_strict_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U21 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_3 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_4 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_5 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_6 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_7 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_8 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_9 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_10 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_11 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_12 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_13 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U22 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict_1 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U23 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict_2 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U24 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict_3 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U25 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict_4 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U26 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict_5 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U42 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict_6 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U43 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict_7 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U44 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict_8 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U45 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict_9 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U46 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_3 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_4 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_5 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_6 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_7 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_8 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_9 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_10 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_11 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_12 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_strict_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U51 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_3 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_4 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_5 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_6 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_7 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict_0 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U52 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict_1 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U53 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict_2 : forall x10 x9 x13 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U54 (x9::x10::x11::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict_3 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U55 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict_4 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U56 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_1 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_2 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_3 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_4 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_5 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_6 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_strict_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U11 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_3 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_4 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_strict_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U12 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_strict_1 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U13 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_1 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_2 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_3 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_strict_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U31 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_1 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_2 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_strict_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U32 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_1 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U33 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large. Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_0 : forall x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U92 (x1::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Inductive DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_strict_0 : forall x10 x9 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_U91 (x9::x10::nil)) x13) -> DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_strict (algebra.Alg.Term algebra.F.id_mark (x9::nil)) (algebra.Alg.Term algebra.F.id_mark (x13::nil)) . Module WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 1 + 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 1 + 2* x14. Definition P_id_a__U56 (x13:Z) := 0. Definition P_id_a__U26 (x13:Z) := 0. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 0. Definition P_id_U45 (x13:Z) (x14:Z) := 0. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U81 (x13:Z) (x14:Z) := 0. Definition P_id_isNeList (x13:Z) := 1 + 2* x13. Definition P_id_a__U11 (x13:Z) (x14:Z) := 0. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 0. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 1* x14. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isNePal (x13:Z) := 0. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U91 (x13:Z) (x14:Z) := 1 + 2* x14. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 1* x14. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 0. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 1* x14. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 1* x14. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 0. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 1* x14. Definition P_id_a__U12 (x13:Z) (x14:Z) := 0. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isPal (x13:Z) := 2. Definition P_id_isPalListKind (x13:Z) := 2* x13. Definition P_id_a__U25 (x13:Z) (x14:Z) := 0. Definition P_id_U55 (x13:Z) (x14:Z) := 0. Definition P_id_a__U92 (x13:Z) := 1 + 1* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isList (x13:Z) := 2* x13. Definition P_id___ (x13:Z) (x14:Z) := 1 + 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 1* x14. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 1 + 2* x14. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 1 + 2* x13. Definition P_id_U46 (x13:Z) := 0. Definition P_id_a__U81 (x13:Z) (x14:Z) := 0. Definition P_id_a__U45 (x13:Z) (x14:Z) := 0. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 1* x14. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 0. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 1* x14. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1 + 2* x14. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U92 (x13:Z) := 1 + 1* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 0. Definition P_id_nil := 2. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 0. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 1* x14. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isNePal (x13:Z) := 0. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 1* x14. Definition P_id_a__isPalListKind (x13:Z) := 2* x13. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPal (x13:Z) := 2. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 0. Definition P_id_a__isList (x13:Z) := 2* x13. Definition P_id_U56 (x13:Z) := 0. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 0. Definition P_id_U26 (x13:Z) := 0. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14::x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14::x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14::x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14::x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14::x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14::x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14::x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14::x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14:: x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14::x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14::x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14::x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14::x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14::x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14::x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *; set (v:=measure x) in *;clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *; set (v:=measure x) in *;clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0; subst;simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 2* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large . Proof. intros x. apply well_founded_ind with (R:=fun x y => (Zwf.Zwf 0) (marked_measure x) (marked_measure y)). apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. clear x. intros x IHx. repeat ( constructor;inversion 1;subst; full_prove_ineq algebra.Alg.Term ltac:(algebra.Alg_ext.find_replacement ) algebra.EQT_ext.one_step_list_refl_trans_clos marked_measure marked_measure_star_monotonic (Zwf.Zwf 0) (interp.o_Z 0) ltac:(fun _ => R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 ) ltac:(fun _ => rewrite_and_unfold ) ltac:(fun _ => generate_pos_hyp ) ltac:(fun _ => cbv beta iota zeta delta - [Zplus Zmult Zle Zlt] in * ; try (constructor)) IHx ). Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 1 + 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 0. Definition P_id_a__U56 (x13:Z) := 0. Definition P_id_a__U26 (x13:Z) := 0. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 0. Definition P_id_U45 (x13:Z) (x14:Z) := 0. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U81 (x13:Z) (x14:Z) := 0. Definition P_id_isNeList (x13:Z) := 2. Definition P_id_a__U11 (x13:Z) (x14:Z) := 0. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 2* x13. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 0. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 1. Definition P_id_a__isNePal (x13:Z) := 0. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 1. Definition P_id_U91 (x13:Z) (x14:Z) := 1 + 1* x13 + 2* x14. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 0. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 0. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 2. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 1. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 0. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U12 (x13:Z) (x14:Z) := 0. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 2* x13. Definition P_id_a__U25 (x13:Z) (x14:Z) := 0. Definition P_id_U55 (x13:Z) (x14:Z) := 0. Definition P_id_a__U92 (x13:Z) := 1* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 1. Definition P_id_isList (x13:Z) := 0. Definition P_id___ (x13:Z) (x14:Z) := 1 + 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 0. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 0. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 2. Definition P_id_a__isNeList (x13:Z) := 2. Definition P_id_U46 (x13:Z) := 0. Definition P_id_a__U81 (x13:Z) (x14:Z) := 0. Definition P_id_a__U45 (x13:Z) (x14:Z) := 0. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 0. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 1. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1 + 1* x13 + 2* x14. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 1. Definition P_id_U92 (x13:Z) := 1* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 0. Definition P_id_nil := 0. Definition P_id_isQid (x13:Z) := 2* x13. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 0. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 1. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 1. Definition P_id_isNePal (x13:Z) := 0. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPalListKind (x13:Z) := 2* x13. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 0. Definition P_id_a__isList (x13:Z) := 0. Definition P_id_U56 (x13:Z) := 0. Definition P_id_a := 1. Definition P_id_a__U55 (x13:Z) (x14:Z) := 0. Definition P_id_U26 (x13:Z) := 0. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14:: x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14:: x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14:: x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14:: x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15:: x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14:: x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14:: x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15:: x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14:: x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15:: x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15:: x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15:: x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15:: x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14:: x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14:: x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14:: x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14:: x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14:: x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15:: x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15:: x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14:: x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14:: x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15:: x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14:: x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15:: x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15:: x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15:: x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15:: x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *; set (v:=measure x) in *;clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *; set (v:=measure x) in *;clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 2* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[ idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos );(assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[ idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos );(assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large.wf . Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large) . clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx'; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx'; apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 1. Definition P_id_a__U56 (x13:Z) := 0. Definition P_id_a__U26 (x13:Z) := 0. Definition P_id_U61 (x13:Z) (x14:Z) := 2* x14. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 0. Definition P_id_U45 (x13:Z) (x14:Z) := 0. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U81 (x13:Z) (x14:Z) := 0. Definition P_id_isNeList (x13:Z) := 2. Definition P_id_a__U11 (x13:Z) (x14:Z) := 2* x14. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 1. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 2* x14. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isNePal (x13:Z) := 3* x13. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 1 + 2* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 1* x14. Definition P_id_a__U32 (x13:Z) (x14:Z) := 1. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 2* x15. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 0. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U12 (x13:Z) (x14:Z) := 2* x14. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 0. Definition P_id_U55 (x13:Z) (x14:Z) := 0. Definition P_id_a__U92 (x13:Z) := 2* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isList (x13:Z) := 2* x13. Definition P_id___ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 1. Definition P_id_a__U61 (x13:Z) (x14:Z) := 2* x14. Definition P_id_a__U31 (x13:Z) (x14:Z) := 1. Definition P_id_U62 (x13:Z) (x14:Z) := 1* x14. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 2. Definition P_id_U46 (x13:Z) := 0. Definition P_id_a__U81 (x13:Z) (x14:Z) := 0. Definition P_id_a__U45 (x13:Z) (x14:Z) := 0. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 2* x15. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 1. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 2* x14. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U92 (x13:Z) := 2* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 0. Definition P_id_nil := 0. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 1 + 2* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isNePal (x13:Z) := 3* x13. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 0. Definition P_id_a__isList (x13:Z) := 2* x13. Definition P_id_U56 (x13:Z) := 0. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 0. Definition P_id_U26 (x13:Z) := 0. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14:: x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14:: x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14:: x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14:: x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15:: x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14:: x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14:: x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15:: x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14:: x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15:: x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15:: x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15:: x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15:: x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14:: x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14:: x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14:: x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14:: x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14:: x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15:: x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15:: x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14:: x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14:: x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15:: x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14:: x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15:: x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15:: x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15:: x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15:: x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *; set (v:=measure x) in *;clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *; set (v:=measure x) in *;clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 3* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[ idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos );(assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[ idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos );(assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large.wf . Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large) . clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx'; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx'; apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 1. Definition P_id_a__U56 (x13:Z) := 0. Definition P_id_a__U26 (x13:Z) := 0. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 0. Definition P_id_U45 (x13:Z) (x14:Z) := 0. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U81 (x13:Z) (x14:Z) := 2* x13. Definition P_id_isNeList (x13:Z) := 2. Definition P_id_a__U11 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 1. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isNePal (x13:Z) := 0. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U91 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 1* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 1 + 2* x13. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 0. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U12 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 0. Definition P_id_U55 (x13:Z) (x14:Z) := 0. Definition P_id_a__U92 (x13:Z) := 2* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isList (x13:Z) := 0. Definition P_id___ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 1 + 2* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 1. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 2. Definition P_id_U46 (x13:Z) := 0. Definition P_id_a__U81 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U45 (x13:Z) (x14:Z) := 0. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U91 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U92 (x13:Z) := 2* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 0. Definition P_id_nil := 0. Definition P_id_isQid (x13:Z) := 1. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 1* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isNePal (x13:Z) := 0. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 0. Definition P_id_a__isList (x13:Z) := 0. Definition P_id_U56 (x13:Z) := 0. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 0. Definition P_id_U26 (x13:Z) := 0. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14:: x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14:: x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14:: x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14:: x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15:: x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14:: x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14:: x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15:: x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14:: x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15:: x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15:: x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15:: x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15:: x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14:: x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14:: x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14:: x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14:: x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14:: x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15:: x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15:: x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14:: x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14:: x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15:: x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14:: x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15:: x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15:: x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15:: x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15:: x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *; set (v:=measure x) in *;clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *; set (v:=measure x) in *;clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 1* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[ idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos );(assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[ idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos );(assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)|| (repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large.wf . Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large) . clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx'; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx'; apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 2 + 2* x13. Definition P_id_a__U56 (x13:Z) := 0. Definition P_id_a__U26 (x13:Z) := 0. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 0. Definition P_id_U45 (x13:Z) (x14:Z) := 0. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U81 (x13:Z) (x14:Z) := 2* x13. Definition P_id_isNeList (x13:Z) := 2. Definition P_id_a__U11 (x13:Z) (x14:Z) := 2* x14. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 2* x14. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isNePal (x13:Z) := 0. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 2* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 1. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 0. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U12 (x13:Z) (x14:Z) := 0. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isPal (x13:Z) := 1* x13. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 0. Definition P_id_U55 (x13:Z) (x14:Z) := 0. Definition P_id_a__U92 (x13:Z) := 2* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isList (x13:Z) := 2* x13. Definition P_id___ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 2 + 2* x13. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 2. Definition P_id_U46 (x13:Z) := 0. Definition P_id_a__U81 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U45 (x13:Z) (x14:Z) := 0. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 0. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U92 (x13:Z) := 2* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 0. Definition P_id_nil := 0. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 2* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 1. Definition P_id_isNePal (x13:Z) := 0. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPal (x13:Z) := 1* x13. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 0. Definition P_id_a__isList (x13:Z) := 2* x13. Definition P_id_U56 (x13:Z) := 0. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 0. Definition P_id_U26 (x13:Z) := 0. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14:: x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14:: x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14:: x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15:: x14::x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14:: x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15:: x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15:: x14::x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14:: x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14:: x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15:: x14::x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15:: x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15:: x14::x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14:: x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15:: x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15:: x14::x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15:: x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15:: x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15:: x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15:: x14::x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14:: x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15:: x14::x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14:: x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14:: x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14:: x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14:: x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15:: x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15:: x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15:: x14::x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14:: x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14:: x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15:: x14::x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15:: x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15:: x14::x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14:: x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15:: x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15:: x14::x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15:: x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15:: x14::x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15:: x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15:: x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15:: x14::x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *; set (v:=measure x) in *;clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *; set (v:=measure x) in *;clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 2* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71 . Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ; intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large.wf . Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large) . clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx'; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx'; apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U56 (x13:Z) := 0. Definition P_id_a__U26 (x13:Z) := 0. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 1 + 2* x13. Definition P_id_U45 (x13:Z) (x14:Z) := 0. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U81 (x13:Z) (x14:Z) := 0. Definition P_id_isNeList (x13:Z) := 0. Definition P_id_a__U11 (x13:Z) (x14:Z) := 1. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 1. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isNePal (x13:Z) := 0. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 2* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 0. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U12 (x13:Z) (x14:Z) := 1 + 1* x13. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 0. Definition P_id_U55 (x13:Z) (x14:Z) := 0. Definition P_id_a__U92 (x13:Z) := 1* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isList (x13:Z) := 2. Definition P_id___ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 0. Definition P_id_U46 (x13:Z) := 0. Definition P_id_a__U81 (x13:Z) (x14:Z) := 0. Definition P_id_a__U45 (x13:Z) (x14:Z) := 0. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 1 + 1* x13. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U92 (x13:Z) := 1* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 0. Definition P_id_nil := 2. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 2* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isNePal (x13:Z) := 0. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 1 + 2* x13. Definition P_id_a__isList (x13:Z) := 2. Definition P_id_U56 (x13:Z) := 0. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 0. Definition P_id_U26 (x13:Z) := 0. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14:: x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14:: x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14:: x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15:: x14::x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14:: x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15:: x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15:: x14::x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14:: x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14:: x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15:: x14::x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15:: x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15:: x14::x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14:: x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15:: x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15:: x14::x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15:: x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15:: x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15:: x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15:: x14::x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14:: x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15:: x14::x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14:: x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14:: x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14:: x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14:: x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15:: x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15:: x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15:: x14::x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14:: x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14:: x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15:: x14::x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15:: x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15:: x14::x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14:: x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15:: x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15:: x14::x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15:: x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15:: x14::x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15:: x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15:: x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15:: x14::x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *; set (v:=measure x) in *;clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *; set (v:=measure x) in *;clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 1* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large.wf . Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large.DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large) . clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx'; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx'; apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U56 (x13:Z) := 0. Definition P_id_a__U26 (x13:Z) := 0. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 1* x13. Definition P_id_U45 (x13:Z) (x14:Z) := 0. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U81 (x13:Z) (x14:Z) := 1* x14. Definition P_id_isNeList (x13:Z) := 0. Definition P_id_a__U11 (x13:Z) (x14:Z) := 1 + 2* x13. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 1 + 2* x13. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isNePal (x13:Z) := 0. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 1* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U63 (x13:Z) := 2* x13. Definition P_id_o := 1. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 0. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U12 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_isPal (x13:Z) := 2* x13. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 0. Definition P_id_U55 (x13:Z) (x14:Z) := 0. Definition P_id_a__U92 (x13:Z) := 2* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isList (x13:Z) := 2. Definition P_id___ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 0. Definition P_id_U46 (x13:Z) := 0. Definition P_id_a__U81 (x13:Z) (x14:Z) := 1* x14. Definition P_id_a__U45 (x13:Z) (x14:Z) := 0. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U92 (x13:Z) := 2* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 0. Definition P_id_nil := 0. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 2* x13. Definition P_id_a__U33 (x13:Z) := 1* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 2. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isNePal (x13:Z) := 0. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPal (x13:Z) := 2* x13. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 1* x13. Definition P_id_a__isList (x13:Z) := 2. Definition P_id_U56 (x13:Z) := 0. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 0. Definition P_id_U26 (x13:Z) := 0. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14:: x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14:: x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14:: x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15:: x14::x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14:: x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14:: x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15:: x14::x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14:: x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14:: x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15:: x14::x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14:: x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15:: x14::x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14:: x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14:: x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15:: x14::x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14:: x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14:: x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14:: x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15:: x14::x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14:: x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15:: x14::x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14:: x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14:: x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14:: x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14:: x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14:: x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14:: x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15:: x14::x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14:: x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14:: x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15:: x14::x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14:: x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15:: x14::x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14:: x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14:: x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15:: x14::x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14:: x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15:: x14::x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14:: x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14:: x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15:: x14::x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 2* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large.wf . Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large.DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large) . clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx'; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 1 + 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U56 (x13:Z) := 1 + 1* x13. Definition P_id_a__U26 (x13:Z) := 0. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 1* x13. Definition P_id_U45 (x13:Z) (x14:Z) := 0. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U81 (x13:Z) (x14:Z) := 0. Definition P_id_isNeList (x13:Z) := 1* x13. Definition P_id_a__U11 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 1 + 2* x13 + 2* x14 + 1* x15. Definition P_id_a__isNePal (x13:Z) := 1. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 2* x14 + 1* x15. Definition P_id_U91 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 1* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 1 + 2* x14 + 1* x15. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 0. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U12 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 3* x13. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 0. Definition P_id_U55 (x13:Z) (x14:Z) := 1 + 2* x13 + 1* x14. Definition P_id_a__U92 (x13:Z) := 1* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 2* x14 + 1* x15. Definition P_id_isList (x13:Z) := 1* x13. Definition P_id___ (x13:Z) (x14:Z) := 1 + 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 1* x13. Definition P_id_U46 (x13:Z) := 0. Definition P_id_a__U81 (x13:Z) (x14:Z) := 0. Definition P_id_a__U45 (x13:Z) (x14:Z) := 0. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 2* x14 + 1* x15. Definition P_id_a__U91 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 1 + 2* x13 + 2* x14 + 1* x15. Definition P_id_U92 (x13:Z) := 1* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 0. Definition P_id_nil := 1. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 1* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 2. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 2* x14 + 1* x15. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 1 + 2* x14 + 1* x15. Definition P_id_isNePal (x13:Z) := 1. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 3* x13. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 1* x13. Definition P_id_a__isList (x13:Z) := 1* x13. Definition P_id_U56 (x13:Z) := 1 + 1* x13. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 1 + 2* x13 + 1* x14. Definition P_id_U26 (x13:Z) := 0. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14:: x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14:: x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14:: x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15:: x14::x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14:: x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14:: x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15:: x14::x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14:: x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14:: x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15:: x14::x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14:: x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15:: x14::x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14:: x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14:: x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15:: x14::x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14:: x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14:: x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14:: x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15:: x14::x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14:: x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15:: x14::x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14:: x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14:: x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14:: x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14:: x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14:: x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14:: x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15:: x14::x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14:: x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14:: x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15:: x14::x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14:: x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15:: x14::x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14:: x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14:: x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15:: x14::x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14:: x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15:: x14::x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14:: x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14:: x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15:: x14::x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 1* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large.wf . Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large.DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large) . clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx'; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 3 + 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U56 (x13:Z) := 1* x13. Definition P_id_a__U26 (x13:Z) := 0. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 1* x13. Definition P_id_U45 (x13:Z) (x14:Z) := 0. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U81 (x13:Z) (x14:Z) := 0. Definition P_id_isNeList (x13:Z) := 2* x13. Definition P_id_a__U11 (x13:Z) (x14:Z) := 2* x13 + 2* x14. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 2. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 2* x13 + 2* x14. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 2* x15. Definition P_id_a__isNePal (x13:Z) := 1 + 2* x13. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 2* x15. Definition P_id_U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 1* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 2* x14. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 2* x14 + 2* x15. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 0. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 2* x14. Definition P_id_a__U12 (x13:Z) (x14:Z) := 2* x13 + 2* x14. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 0. Definition P_id_U55 (x13:Z) (x14:Z) := 1* x13 + 2* x14. Definition P_id_a__U92 (x13:Z) := 2* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 2* x15. Definition P_id_isList (x13:Z) := 2* x13. Definition P_id___ (x13:Z) (x14:Z) := 3 + 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 2* x13. Definition P_id_U46 (x13:Z) := 0. Definition P_id_a__U81 (x13:Z) (x14:Z) := 0. Definition P_id_a__U45 (x13:Z) (x14:Z) := 0. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 2* x14. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 2. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 2* x13 + 2* x14. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 2* x15. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 2* x15. Definition P_id_U92 (x13:Z) := 2* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 0. Definition P_id_nil := 1. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 1* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 2* x14. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 2* x15. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 2* x14 + 2* x15. Definition P_id_isNePal (x13:Z) := 1 + 2* x13. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 1* x13. Definition P_id_a__isList (x13:Z) := 2* x13. Definition P_id_U56 (x13:Z) := 1* x13. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 1* x13 + 2* x14. Definition P_id_U26 (x13:Z) := 0. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14:: x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14:: x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14:: x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15:: x14::x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14:: x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14:: x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15:: x14::x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14:: x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14:: x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15:: x14::x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14:: x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15:: x14::x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14:: x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14:: x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15:: x14::x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14:: x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14:: x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14:: x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15:: x14::x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14:: x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15:: x14::x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14:: x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14:: x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14:: x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14:: x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14:: x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14:: x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15:: x14::x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14:: x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14:: x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15:: x14::x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14:: x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15:: x14::x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14:: x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14:: x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15:: x14::x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14:: x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15:: x14::x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14:: x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14:: x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15:: x14::x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 2* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large.wf . Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large.DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large) . clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx'; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 2 + 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_a__U56 (x13:Z) := 1* x13. Definition P_id_a__U26 (x13:Z) := 1 + 1* x13. Definition P_id_U61 (x13:Z) (x14:Z) := 1* x14. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 1* x13. Definition P_id_U45 (x13:Z) (x14:Z) := 1 + 1* x13 + 1* x14. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 1* x14 + 1* x15. Definition P_id_U81 (x13:Z) (x14:Z) := 0. Definition P_id_isNeList (x13:Z) := 1* x13. Definition P_id_a__U11 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x14 + 1* x15. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 2* x15. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 1* x14 + 1* x15. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_a__isNePal (x13:Z) := 2* x13. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 1 + 2* x13 + 1* x14 + 1* x15. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 2* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 1* x14. Definition P_id_a__U32 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 1 + 2* x14 + 1* x15. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 1 + 1* x13. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 1* x14 + 1* x15. Definition P_id_a__U12 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 1* x14 + 1* x15. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 1* x14 + 1* x15. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 1 + 1* x13 + 1* x14. Definition P_id_U55 (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_a__U92 (x13:Z) := 1* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_isList (x13:Z) := 1* x13. Definition P_id___ (x13:Z) (x14:Z) := 2 + 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 1* x14. Definition P_id_a__U31 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U62 (x13:Z) (x14:Z) := 1* x14. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 1* x13. Definition P_id_U46 (x13:Z) := 1 + 1* x13. Definition P_id_a__U81 (x13:Z) (x14:Z) := 0. Definition P_id_a__U45 (x13:Z) (x14:Z) := 1 + 1* x13 + 1* x14. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 1 + 2* x14 + 1* x15. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 1* x14 + 1* x15. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x14 + 1* x15. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 1 + 2* x13 + 1* x14 + 1* x15. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_U92 (x13:Z) := 1* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 1 + 1* x13 + 1* x14. Definition P_id_nil := 0. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 2* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 2* x15. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 1* x14 + 1* x15. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_isNePal (x13:Z) := 2* x13. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 1* x14 + 1* x15. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 1* x14 + 1* x15. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 1* x14 + 1* x15. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 1* x13. Definition P_id_a__isList (x13:Z) := 1* x13. Definition P_id_U56 (x13:Z) := 1* x13. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U26 (x13:Z) := 1 + 1* x13. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14:: x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14:: x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14:: x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14:: x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14:: x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14:: x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14:: x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14:: x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14:: x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14:: x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14:: x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14:: x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14:: x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14:: x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14:: x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14:: x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14:: x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14:: x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14:: x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14:: x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14:: x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14:: x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14:: x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14:: x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14:: x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14:: x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14:: x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14:: x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 1* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large.wf . Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large.DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large) . clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx'; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 1 + 1* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U56 (x13:Z) := 1* x13. Definition P_id_a__U26 (x13:Z) := 1* x13. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 1* x13. Definition P_id_U45 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_U81 (x13:Z) (x14:Z) := 0. Definition P_id_isNeList (x13:Z) := 1* x13. Definition P_id_a__U11 (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 1* x14 + 1* x15. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_a__isNePal (x13:Z) := 0. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 1* x14 + 1* x15. Definition P_id_U91 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 2* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 1. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 1 + 2* x13 + 1* x14 + 1* x15. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 1* x14 + 1* x15. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 1* x13. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 1* x14 + 1* x15. Definition P_id_a__U12 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 1* x14 + 1* x15. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U55 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_a__U92 (x13:Z) := 2* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_isList (x13:Z) := 1* x13. Definition P_id___ (x13:Z) (x14:Z) := 1 + 1* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 1* x13. Definition P_id_U46 (x13:Z) := 1* x13. Definition P_id_a__U81 (x13:Z) (x14:Z) := 0. Definition P_id_a__U45 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 1 + 2* x13 + 1* x14 + 1* x15. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 1* x14 + 1* x15. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_a__U91 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_U92 (x13:Z) := 2* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_nil := 0. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 2* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 2. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 1* x14 + 1* x15. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 1* x14 + 1* x15. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 1* x14 + 1* x15. Definition P_id_isNePal (x13:Z) := 0. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 1* x14 + 1* x15. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 1* x14 + 1* x15. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 1* x13. Definition P_id_a__isList (x13:Z) := 1* x13. Definition P_id_U56 (x13:Z) := 1* x13. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U26 (x13:Z) := 1* x13. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) -> (0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14:: x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14:: x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14:: x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14:: x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14:: x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14:: x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14:: x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14:: x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14:: x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14:: x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14:: x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14:: x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14:: x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14:: x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14::x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14:: x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14:: x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14:: x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14:: x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14:: x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14:: x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14:: x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14:: x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14:: x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14:: x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14:: x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14:: x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14:: x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 2* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14::x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14::x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14::x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14::x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14::x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14::x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14::x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14::x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14::x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14::x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14::x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14::x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14::x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat ( apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large.wf. Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large.DP_R_xml_0_scc_84_large_scc_1_large_large_large_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large) . clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx'; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 1 + 3* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U56 (x13:Z) := 1* x13. Definition P_id_a__U26 (x13:Z) := 1* x13. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 1* x13. Definition P_id_U45 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_U81 (x13:Z) (x14:Z) := 0. Definition P_id_isNeList (x13:Z) := 1* x13. Definition P_id_a__U11 (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 2* x14 + 1* x15. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 2. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 0. Definition P_id_U11 (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 1* x15. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_a__isNePal (x13:Z) := 1* x13. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 1* x15. Definition P_id_U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 2* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 1* x15. Definition P_id_a__U82 (x13:Z) (x14:Z) := 0. Definition P_id_a__U46 (x13:Z) := 1* x13. Definition P_id_U83 (x13:Z) := 0. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 1* x15. Definition P_id_a__U12 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 1* x15. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 1* x15. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U55 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_a__U92 (x13:Z) := 1* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_isList (x13:Z) := 1* x13. Definition P_id___ (x13:Z) (x14:Z) := 1 + 3* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 1* x13. Definition P_id_U46 (x13:Z) := 1* x13. Definition P_id_a__U81 (x13:Z) (x14:Z) := 0. Definition P_id_a__U45 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U82 (x13:Z) (x14:Z) := 0. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 1* x15. Definition P_id_a__U72 (x13:Z) (x14:Z) := 0. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 1 + 1* x13 + 2* x14 + 1* x15. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 1* x14 + 1* x15. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_U92 (x13:Z) := 1* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_nil := 0. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 2* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 2. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 1* x15. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 1* x15. Definition P_id_a__U83 (x13:Z) := 0. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 1* x15. Definition P_id_isNePal (x13:Z) := 1* x13. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 1* x15. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 1* x13 + 2* x14 + 1* x15. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 2* x13 + 2* x14 + 1* x15. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 1* x13. Definition P_id_a__isList (x13:Z) := 1* x13. Definition P_id_U56 (x13:Z) := 1* x13. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 1* x13 + 1* x14. Definition P_id_U26 (x13:Z) := 1* x13. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14::x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14::x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14::x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14::x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14:: x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14::x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14::x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14:: x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14::x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14:: x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14:: x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14:: x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14:: x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14::x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14::x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14::x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14::x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14::x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14:: x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14:: x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14::x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14::x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14:: x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14::x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14:: x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14:: x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14:: x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14:: x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 2* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15:: x14::x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15:: x14::x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15:: x14::x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15:: x14::x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15:: x14::x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15:: x14::x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15:: x14::x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15:: x14::x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15:: x14::x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15:: x14::x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15:: x14::x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15:: x14::x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15:: x14::x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos );(assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos );(assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large.wf. Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large_large.DP_R_xml_0_scc_84_large_scc_1_large_large_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large_large_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx'; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U56 (x13:Z) := 2* x13. Definition P_id_a__U26 (x13:Z) := 1* x13. Definition P_id_U61 (x13:Z) (x14:Z) := 0. Definition P_id_e := 1. Definition P_id_a__U13 (x13:Z) := 2* x13. Definition P_id_U45 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U74 (x13:Z) := 0. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U81 (x13:Z) (x14:Z) := 3. Definition P_id_isNeList (x13:Z) := 0. Definition P_id_a__U11 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U11 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__isNePal (x13:Z) := 0. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 1* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__U82 (x13:Z) (x14:Z) := 3. Definition P_id_a__U46 (x13:Z) := 2* x13. Definition P_id_U83 (x13:Z) := 3 + 2* x13. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U12 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U73 (x13:Z) (x14:Z) := 0. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_isPal (x13:Z) := 3. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U55 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U92 (x13:Z) := 2* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_isList (x13:Z) := 0. Definition P_id___ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 0. Definition P_id_a__U31 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 0. Definition P_id_U46 (x13:Z) := 2* x13. Definition P_id_a__U81 (x13:Z) (x14:Z) := 3. Definition P_id_a__U45 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U82 (x13:Z) (x14:Z) := 3. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__U72 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U73 (x13:Z) (x14:Z) := 0. Definition P_id_U12 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U92 (x13:Z) := 2* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 1* x13. Definition P_id_nil := 0. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 1* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U83 (x13:Z) := 3 + 2* x13. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_isNePal (x13:Z) := 0. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__isPal (x13:Z) := 3. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U74 (x13:Z) := 0. Definition P_id_U13 (x13:Z) := 2* x13. Definition P_id_a__isList (x13:Z) := 0. Definition P_id_U56 (x13:Z) := 2* x13. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U26 (x13:Z) := 1* x13. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14::x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14::x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14::x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14::x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14:: x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14::x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14::x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14:: x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14::x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14:: x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14:: x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14:: x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14:: x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14::x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14::x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14::x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14::x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14::x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14:: x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14:: x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14::x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14::x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14:: x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14::x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14:: x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14:: x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14:: x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14:: x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 2* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15:: x14::x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15:: x14::x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15:: x14::x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15:: x14::x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15:: x14::x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15:: x14::x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15:: x14::x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15:: x14::x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15:: x14::x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15:: x14::x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15:: x14::x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15:: x14::x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15:: x14::x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b) . Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large_large.wf. Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1_large.DP_R_xml_0_scc_84_large_scc_1_large_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx'; apply DP_R_xml_0_scc_84_large_scc_1_large_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 1 + 3* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U56 (x13:Z) := 2* x13. Definition P_id_a__U26 (x13:Z) := 2* x13. Definition P_id_U61 (x13:Z) (x14:Z) := 3* x13. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 2* x13. Definition P_id_U45 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U74 (x13:Z) := 1 + 2* x13. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U81 (x13:Z) (x14:Z) := 1* x14. Definition P_id_isNeList (x13:Z) := 0. Definition P_id_a__U11 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 1 + 3* x15. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 1 + 3* x14. Definition P_id_U11 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__isNePal (x13:Z) := 1* x13. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 1* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 0. Definition P_id_a__U32 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U63 (x13:Z) := 0. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U82 (x13:Z) (x14:Z) := 1* x14. Definition P_id_a__U46 (x13:Z) := 1* x13. Definition P_id_U83 (x13:Z) := 1* x13. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U12 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U73 (x13:Z) (x14:Z) := 1 + 2* x13. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_isPal (x13:Z) := 1* x13. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U55 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U92 (x13:Z) := 2* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_isList (x13:Z) := 0. Definition P_id___ (x13:Z) (x14:Z) := 1 + 3* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 3* x13. Definition P_id_a__U31 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U62 (x13:Z) (x14:Z) := 0. Definition P_id_i := 1. Definition P_id_a__isNeList (x13:Z) := 0. Definition P_id_U46 (x13:Z) := 1* x13. Definition P_id_a__U81 (x13:Z) (x14:Z) := 1* x14. Definition P_id_a__U45 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U82 (x13:Z) (x14:Z) := 1* x14. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U72 (x13:Z) (x14:Z) := 1 + 3* x14. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U73 (x13:Z) (x14:Z) := 1 + 2* x13. Definition P_id_U12 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U92 (x13:Z) := 2* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 2* x13. Definition P_id_nil := 0. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 0. Definition P_id_a__U33 (x13:Z) := 1* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 1 + 3* x15. Definition P_id_u := 1. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U83 (x13:Z) := 1* x13. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_isNePal (x13:Z) := 1* x13. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__isPal (x13:Z) := 1* x13. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U74 (x13:Z) := 1 + 2* x13. Definition P_id_U13 (x13:Z) := 2* x13. Definition P_id_a__isList (x13:Z) := 0. Definition P_id_U56 (x13:Z) := 2* x13. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U26 (x13:Z) := 2* x13. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14::x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14::x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14::x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14::x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14:: x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14::x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14::x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14:: x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14::x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14:: x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14:: x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14:: x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14:: x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14::x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14::x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14::x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14::x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14::x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14:: x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14:: x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14::x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14::x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14:: x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14::x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14:: x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14:: x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14:: x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14:: x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 2* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15:: x14::x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15:: x14::x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15:: x14::x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15:: x14::x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15:: x14::x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15:: x14::x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15:: x14::x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15:: x14::x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15:: x14::x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15:: x14::x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15:: x14::x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15:: x14::x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15:: x14::x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large_large := WF_DP_R_xml_0_scc_84_large_scc_1_large_large.wf. Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large_scc_1.DP_R_xml_0_scc_84_large_scc_1_large . Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_84_large_scc_1_large_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U56 (x13:Z) := 2* x13. Definition P_id_a__U26 (x13:Z) := 1* x13. Definition P_id_U61 (x13:Z) (x14:Z) := 1 + 1* x13. Definition P_id_e := 1. Definition P_id_a__U13 (x13:Z) := 2* x13. Definition P_id_U45 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U74 (x13:Z) := 1* x13. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U81 (x13:Z) (x14:Z) := 2 + 1* x13. Definition P_id_isNeList (x13:Z) := 0. Definition P_id_a__U11 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 2 + 2* x13. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 2 + 2* x13. Definition P_id_U11 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__isNePal (x13:Z) := 2. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 1* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 1 + 2* x13. Definition P_id_a__U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U63 (x13:Z) := 1 + 1* x13. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U82 (x13:Z) (x14:Z) := 2 + 2* x13. Definition P_id_a__U46 (x13:Z) := 2* x13. Definition P_id_U83 (x13:Z) := 1* x13. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U12 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U73 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_isPal (x13:Z) := 2. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U55 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U92 (x13:Z) := 1* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_isList (x13:Z) := 0. Definition P_id___ (x13:Z) (x14:Z) := 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 1 + 1* x13. Definition P_id_a__U31 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U62 (x13:Z) (x14:Z) := 1 + 2* x13. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 0. Definition P_id_U46 (x13:Z) := 2* x13. Definition P_id_a__U81 (x13:Z) (x14:Z) := 2 + 1* x13. Definition P_id_a__U45 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U82 (x13:Z) (x14:Z) := 2 + 2* x13. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__U72 (x13:Z) (x14:Z) := 2 + 2* x13. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U73 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U12 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U92 (x13:Z) := 1* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 2* x13. Definition P_id_nil := 2. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 1 + 1* x13. Definition P_id_a__U33 (x13:Z) := 1* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 2 + 2* x13. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__U83 (x13:Z) := 1* x13. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_isNePal (x13:Z) := 2. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__isPal (x13:Z) := 2. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U74 (x13:Z) := 1* x13. Definition P_id_U13 (x13:Z) := 2* x13. Definition P_id_a__isList (x13:Z) := 0. Definition P_id_U56 (x13:Z) := 2* x13. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U26 (x13:Z) := 1* x13. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14::x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14::x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14::x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14::x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14:: x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14::x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14::x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14:: x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14::x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14:: x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14:: x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14:: x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14:: x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14::x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14::x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14::x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14::x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14::x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14:: x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14:: x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14::x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14::x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14:: x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14::x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14:: x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14:: x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14:: x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14:: x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 0. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 1* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15:: x14::x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15:: x14::x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15:: x14::x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15:: x14::x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15:: x14::x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15:: x14::x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15:: x14::x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15:: x14::x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15:: x14::x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15:: x14::x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15:: x14::x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15:: x14::x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15:: x14::x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_scc_1_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large_scc_1_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large_scc_1_large := WF_DP_R_xml_0_scc_84_large_scc_1_large.wf. Lemma wf : well_founded WF_DP_R_xml_0_scc_84_large.DP_R_xml_0_scc_84_large_scc_1. Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_84_large_scc_1_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_scc_1_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large_scc_1_large. Qed. End WF_DP_R_xml_0_scc_84_large_scc_1. Definition wf_DP_R_xml_0_scc_84_large_scc_1 := WF_DP_R_xml_0_scc_84_large_scc_1.wf. Lemma acc_DP_R_xml_0_scc_84_large_scc_1 : forall x y, (DP_R_xml_0_scc_84_large_scc_1 x y) -> Acc WF_DP_R_xml_0_scc_84.DP_R_xml_0_scc_84_large x. Proof. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large_scc_1). intros x' _ Hrec y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply Hrec;econstructor eassumption)|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). apply wf_DP_R_xml_0_scc_84_large_scc_1. Qed. Inductive DP_R_xml_0_scc_84_large_non_scc_2 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_non_scc_2_0 : forall x14 x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_nil nil) x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x13) -> DP_R_xml_0_scc_84_large_non_scc_2 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_84_large_non_scc_2 : forall x y, (DP_R_xml_0_scc_84_large_non_scc_2 x y) -> Acc WF_DP_R_xml_0_scc_84.DP_R_xml_0_scc_84_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_84_large_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Inductive DP_R_xml_0_scc_84_large_non_scc_3 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_84_large_non_scc_3_0 : forall x14 x1 x13, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x1 x14) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_nil nil) x13) -> DP_R_xml_0_scc_84_large_non_scc_3 (algebra.Alg.Term algebra.F.id_mark (x1::nil)) (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) . Lemma acc_DP_R_xml_0_scc_84_large_non_scc_3 : forall x y, (DP_R_xml_0_scc_84_large_non_scc_3 x y) -> Acc WF_DP_R_xml_0_scc_84.DP_R_xml_0_scc_84_large x. Proof. intros x y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_84_large_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))). Qed. Lemma wf : well_founded WF_DP_R_xml_0_scc_84.DP_R_xml_0_scc_84_large. Proof. constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_84_large_non_scc_3; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_84_large_non_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_84_large_non_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_84_large_non_scc_0; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_84_large_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_84_large_scc_0; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )||(fail))))))). Qed. End WF_DP_R_xml_0_scc_84_large. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_a____ (x13:Z) (x14:Z) := 1 + 2* x13 + 1* x14. Definition P_id_U31 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U56 (x13:Z) := 2* x13. Definition P_id_a__U26 (x13:Z) := 1* x13. Definition P_id_U61 (x13:Z) (x14:Z) := 1* x13. Definition P_id_e := 0. Definition P_id_a__U13 (x13:Z) := 1* x13. Definition P_id_U45 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U74 (x13:Z) := 2* x13. Definition P_id_a__U44 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U81 (x13:Z) (x14:Z) := 1* x13. Definition P_id_isNeList (x13:Z) := 0. Definition P_id_a__U11 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U41 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__U71 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__isQid (x13:Z) := 0. Definition P_id_U72 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U11 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U23 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U53 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__isNePal (x13:Z) := 0. Definition P_id_a__U52 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U24 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_mark (x13:Z) := 1* x13. Definition P_id_U33 (x13:Z) := 1* x13. Definition P_id_a__U62 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U63 (x13:Z) := 1* x13. Definition P_id_o := 0. Definition P_id_a__U21 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U51 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U82 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U46 (x13:Z) := 1* x13. Definition P_id_U83 (x13:Z) := 2* x13. Definition P_id_U22 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U12 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U43 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__U73 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U42 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_isPal (x13:Z) := 0. Definition P_id_isPalListKind (x13:Z) := 0. Definition P_id_a__U25 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U55 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U92 (x13:Z) := 1* x13. Definition P_id_a__U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_isList (x13:Z) := 0. Definition P_id___ (x13:Z) (x14:Z) := 1 + 2* x13 + 1* x14. Definition P_id_U32 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U61 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U31 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U62 (x13:Z) (x14:Z) := 1* x13. Definition P_id_i := 0. Definition P_id_a__isNeList (x13:Z) := 0. Definition P_id_U46 (x13:Z) := 1* x13. Definition P_id_a__U81 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U45 (x13:Z) (x14:Z) := 1* x13. Definition P_id_U82 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U21 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_tt := 0. Definition P_id_U42 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_a__U72 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U41 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U73 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U12 (x13:Z) (x14:Z) := 2* x13. Definition P_id_a__U24 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U54 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__U91 (x13:Z) (x14:Z) := 1* x13. Definition P_id_a__U53 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U92 (x13:Z) := 1* x13. Definition P_id_U25 (x13:Z) (x14:Z) := 2* x13. Definition P_id_nil := 0. Definition P_id_isQid (x13:Z) := 0. Definition P_id_a__U63 (x13:Z) := 1* x13. Definition P_id_a__U33 (x13:Z) := 1* x13. Definition P_id_U71 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_u := 0. Definition P_id_a__U22 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_U52 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__U83 (x13:Z) := 2* x13. Definition P_id_a__U51 (x13:Z) (x14:Z) (x15:Z) := 2* x13. Definition P_id_isNePal (x13:Z) := 0. Definition P_id_U23 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__isPalListKind (x13:Z) := 0. Definition P_id_U44 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_a__isPal (x13:Z) := 0. Definition P_id_a__U43 (x13:Z) (x14:Z) (x15:Z) := 1* x13. Definition P_id_U74 (x13:Z) := 2* x13. Definition P_id_U13 (x13:Z) := 1* x13. Definition P_id_a__isList (x13:Z) := 0. Definition P_id_U56 (x13:Z) := 2* x13. Definition P_id_a := 0. Definition P_id_a__U55 (x13:Z) (x14:Z) := 2* x13. Definition P_id_U26 (x13:Z) := 1* x13. Lemma P_id_a_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a____ x14 x16 <= P_id_a____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U31 x14 x16 <= P_id_U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U56 x14 <= P_id_a__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U26 x14 <= P_id_a__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U61 x14 x16 <= P_id_U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U13 x14 <= P_id_a__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U45 x14 x16 <= P_id_U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U74 x14 <= P_id_a__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U44 x14 x16 x18 <= P_id_a__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U81 x14 x16 <= P_id_U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNeList x14 <= P_id_isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U11 x14 x16 <= P_id_a__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U41 x14 x16 x18 <= P_id_U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U71 x14 x16 x18 <= P_id_a__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isQid x14 <= P_id_a__isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U72 x14 x16 <= P_id_U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U11 x14 x16 <= P_id_U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U23 x14 x16 x18 <= P_id_a__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U53 x14 x16 x18 <= P_id_U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isNePal x14 <= P_id_a__isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U52 x14 x16 x18 <= P_id_a__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U91 x14 x16 <= P_id_U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U24 x14 x16 x18 <= P_id_U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_mark x14 <= P_id_mark x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U33 x14 <= P_id_U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U62 x14 x16 <= P_id_a__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U32 x14 x16 <= P_id_a__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U63 x14 <= P_id_U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U21 x14 x16 x18 <= P_id_a__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U51 x14 x16 x18 <= P_id_U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U82 x14 x16 <= P_id_a__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U46 x14 <= P_id_a__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U83 x14 <= P_id_U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U22 x14 x16 x18 <= P_id_U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U12 x14 x16 <= P_id_a__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U43 x14 x16 x18 <= P_id_U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U73 x14 x16 <= P_id_a__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U42 x14 x16 x18 <= P_id_a__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isPal x14 <= P_id_isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_isPalListKind x14 <= P_id_isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U25 x14 x16 <= P_id_a__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U55 x14 x16 <= P_id_U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U92 x14 <= P_id_a__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U54 x14 x16 x18 <= P_id_a__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isList x14 <= P_id_isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id___ x14 x16 <= P_id___ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U32 x14 x16 <= P_id_U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U61 x14 x16 <= P_id_a__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U31 x14 x16 <= P_id_a__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U62 x14 x16 <= P_id_U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isNeList x14 <= P_id_a__isNeList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U46 x14 <= P_id_U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U81 x14 x16 <= P_id_a__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U45 x14 x16 <= P_id_a__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U82 x14 x16 <= P_id_U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U21 x14 x16 x18 <= P_id_U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U42 x14 x16 x18 <= P_id_U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U72 x14 x16 <= P_id_a__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U41 x14 x16 x18 <= P_id_a__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U73 x14 x16 <= P_id_U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U12 x14 x16 <= P_id_U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U24 x14 x16 x18 <= P_id_a__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U54 x14 x16 x18 <= P_id_U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U91 x14 x16 <= P_id_a__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U53 x14 x16 x18 <= P_id_a__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U92 x14 <= P_id_U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_U25 x14 x16 <= P_id_U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isQid x14 <= P_id_isQid x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U63 x14 <= P_id_a__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U33 x14 <= P_id_a__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U71 x14 x16 x18 <= P_id_U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U22 x14 x16 x18 <= P_id_a__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U52 x14 x16 x18 <= P_id_U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__U83 x14 <= P_id_a__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U51 x14 x16 x18 <= P_id_a__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_isNePal x14 <= P_id_isNePal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U23 x14 x16 x18 <= P_id_U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_a__isPalListKind x14 <= P_id_a__isPalListKind x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_U44 x14 x16 x18 <= P_id_U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isPal x14 <= P_id_a__isPal x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_a__U43 x14 x16 x18 <= P_id_a__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U74 x14 <= P_id_U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U13 x14 <= P_id_U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_a__isList x14 <= P_id_a__isList x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U56 x14 <= P_id_U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_a__U55 x14 x16 <= P_id_a__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_U26 x14 <= P_id_U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a____ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_e_bounded : 0 <= P_id_e . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U11_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U11 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_mark_bounded : forall x13, (0 <= x13) ->0 <= P_id_mark x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_o_bounded : 0 <= P_id_o . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id____bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id___ x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U32_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U32 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U61_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U61 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U31_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U31 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U62_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U62 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_i_bounded : 0 <= P_id_i . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isNeList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isNeList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U46_bounded : forall x13, (0 <= x13) ->0 <= P_id_U46 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U81_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U81 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U45_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U45 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U82_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U82 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U21_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U21 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tt_bounded : 0 <= P_id_tt . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U42_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U42 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U72_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U72 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U41_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U41 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U73_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U73 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U12_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U12 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U24_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U24 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U54_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U54 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U91_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U91 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U53_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U53 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U92_bounded : forall x13, (0 <= x13) ->0 <= P_id_U92 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U25_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_U25 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_nil_bounded : 0 <= P_id_nil . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isQid_bounded : forall x13, (0 <= x13) ->0 <= P_id_isQid x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U63_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U63 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U33_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U33 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U71_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U71 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u_bounded : 0 <= P_id_u . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U22_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U22 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U52_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U52 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U83_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__U83 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U51_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U51 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_isNePal_bounded : forall x13, (0 <= x13) ->0 <= P_id_isNePal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U23_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U23 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPalListKind_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPalListKind x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U44_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_U44 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isPal_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isPal x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U43_bounded : forall x14 x13 x15, (0 <= x13) ->(0 <= x14) ->(0 <= x15) ->0 <= P_id_a__U43 x15 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U74_bounded : forall x13, (0 <= x13) ->0 <= P_id_U74 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U13_bounded : forall x13, (0 <= x13) ->0 <= P_id_U13 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__isList_bounded : forall x13, (0 <= x13) ->0 <= P_id_a__isList x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U56_bounded : forall x13, (0 <= x13) ->0 <= P_id_U56 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a_bounded : 0 <= P_id_a . Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_a__U55_bounded : forall x14 x13, (0 <= x13) ->(0 <= x14) ->0 <= P_id_a__U55 x14 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U26_bounded : forall x13, (0 <= x13) ->0 <= P_id_U26 x13. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition measure := InterpZ.measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_a____ (x14::x13::nil)) => P_id_a____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U31 (x14::x13::nil)) => P_id_U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_a__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_a__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_U61 (x14::x13::nil)) => P_id_U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_e nil) => P_id_e | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_a__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_U45 (x14::x13::nil)) => P_id_U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_a__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_a__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U81 (x14::x13::nil)) => P_id_U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNeList (x13::nil)) => P_id_isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14::x13::nil)) => P_id_a__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U41 (x15::x14::x13::nil)) => P_id_U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_a__U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_a__isQid (measure x13) | (algebra.Alg.Term algebra.F.id_U72 (x14::x13::nil)) => P_id_U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U11 (x14::x13::nil)) => P_id_U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_a__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U53 (x15::x14::x13::nil)) => P_id_U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_a__isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_a__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U91 (x14::x13::nil)) => P_id_U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U24 (x15::x14::x13::nil)) => P_id_U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_mark (measure x13) | (algebra.Alg.Term algebra.F.id_U33 (x13::nil)) => P_id_U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14::x13::nil)) => P_id_a__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14::x13::nil)) => P_id_a__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U63 (x13::nil)) => P_id_U63 (measure x13) | (algebra.Alg.Term algebra.F.id_o nil) => P_id_o | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_a__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U51 (x15::x14::x13::nil)) => P_id_U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14::x13::nil)) => P_id_a__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_a__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_U83 (x13::nil)) => P_id_U83 (measure x13) | (algebra.Alg.Term algebra.F.id_U22 (x15::x14::x13::nil)) => P_id_U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14::x13::nil)) => P_id_a__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U43 (x15::x14::x13::nil)) => P_id_U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14::x13::nil)) => P_id_a__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_a__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isPal (x13::nil)) => P_id_isPal (measure x13) | (algebra.Alg.Term algebra.F.id_isPalListKind (x13::nil)) => P_id_isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14::x13::nil)) => P_id_a__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U55 (x14::x13::nil)) => P_id_U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_a__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_a__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isList (x13::nil)) => P_id_isList (measure x13) | (algebra.Alg.Term algebra.F.id___ (x14::x13::nil)) => P_id___ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U32 (x14::x13::nil)) => P_id_U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14::x13::nil)) => P_id_a__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14::x13::nil)) => P_id_a__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U62 (x14::x13::nil)) => P_id_U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_i nil) => P_id_i | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_a__isNeList (measure x13) | (algebra.Alg.Term algebra.F.id_U46 (x13::nil)) => P_id_U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14::x13::nil)) => P_id_a__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14::x13::nil)) => P_id_a__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U82 (x14::x13::nil)) => P_id_U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U21 (x15::x14::x13::nil)) => P_id_U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_tt nil) => P_id_tt | (algebra.Alg.Term algebra.F.id_U42 (x15::x14::x13::nil)) => P_id_U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U72 (x14::x13::nil)) => P_id_a__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_a__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U73 (x14::x13::nil)) => P_id_U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U12 (x14::x13::nil)) => P_id_U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_a__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U54 (x15::x14::x13::nil)) => P_id_U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14::x13::nil)) => P_id_a__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_a__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U92 (x13::nil)) => P_id_U92 (measure x13) | (algebra.Alg.Term algebra.F.id_U25 (x14::x13::nil)) => P_id_U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_isQid (x13::nil)) => P_id_isQid (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_a__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_a__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_U71 (x15::x14::x13::nil)) => P_id_U71 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_u nil) => P_id_u | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_a__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U52 (x15::x14::x13::nil)) => P_id_U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_a__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_a__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_isNePal (x13::nil)) => P_id_isNePal (measure x13) | (algebra.Alg.Term algebra.F.id_U23 (x15::x14::x13::nil)) => P_id_U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_a__isPalListKind (measure x13) | (algebra.Alg.Term algebra.F.id_U44 (x15::x14::x13::nil)) => P_id_U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_a__isPal (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_a__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U74 (x13::nil)) => P_id_U74 (measure x13) | (algebra.Alg.Term algebra.F.id_U13 (x13::nil)) => P_id_U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_a__isList (measure x13) | (algebra.Alg.Term algebra.F.id_U56 (x13::nil)) => P_id_U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a nil) => P_id_a | (algebra.Alg.Term algebra.F.id_a__U55 (x14::x13::nil)) => P_id_a__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_U26 (x13::nil)) => P_id_U26 (measure x13) | _ => 0 end. Proof. intros t;case t;intros ;apply refl_equal. Qed. Lemma measure_bounded : forall t, 0 <= measure t. Proof. unfold measure in |-*. apply InterpZ.measure_bounded; cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Ltac generate_pos_hyp := match goal with | H:context [measure ?x] |- _ => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) | |- context [measure ?x] => let v := fresh "v" in (let H := fresh "h" in (set (H:=measure_bounded x) in *;set (v:=measure x) in *; clearbody H;clearbody v)) end . Lemma rules_monotonic : forall l r, (algebra.EQT.axiom R_xml_0_deep_rew.R_xml_0_rules r l) -> measure r <= measure l. Proof. intros l r H. fold measure in |-*. inversion H;clear H;subst;inversion H0;clear H0;subst; simpl algebra.EQT.T.apply_subst in |-*; repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) end );repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma measure_star_monotonic : forall l r, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) r l) ->measure r <= measure l. Proof. unfold measure in *. apply InterpZ.measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_A__U72 (x13:Z) (x14:Z) := 0. Definition P_id_A__U11 (x13:Z) (x14:Z) := 0. Definition P_id_A__U41 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U91 (x13:Z) (x14:Z) := 0. Definition P_id_A__U24 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U53 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U81 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISNELIST (x13:Z) := 0. Definition P_id_A__U45 (x13:Z) (x14:Z) := 0. Definition P_id_A__U31 (x13:Z) (x14:Z) := 0. Definition P_id_A__U61 (x13:Z) (x14:Z) := 0. Definition P_id_A__ISPAL (x13:Z) := 0. Definition P_id_A__ISPALLISTKIND (x13:Z) := 0. Definition P_id_A__U43 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISLIST (x13:Z) := 0. Definition P_id_A__U55 (x13:Z) (x14:Z) := 0. Definition P_id_A__U83 (x13:Z) := 0. Definition P_id_A__U22 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U51 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U33 (x13:Z) := 0. Definition P_id_A____ (x13:Z) (x14:Z) := 2 + 2* x13 + 1* x14. Definition P_id_A__U63 (x13:Z) := 0. Definition P_id_A__U73 (x13:Z) (x14:Z) := 0. Definition P_id_A__U12 (x13:Z) (x14:Z) := 0. Definition P_id_A__U42 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U92 (x13:Z) := 0. Definition P_id_A__U25 (x13:Z) (x14:Z) := 0. Definition P_id_A__U54 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U82 (x13:Z) (x14:Z) := 0. Definition P_id_A__U21 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U46 (x13:Z) := 0. Definition P_id_A__U32 (x13:Z) (x14:Z) := 0. Definition P_id_A__U62 (x13:Z) (x14:Z) := 0. Definition P_id_A__U74 (x13:Z) := 0. Definition P_id_A__U13 (x13:Z) := 0. Definition P_id_A__U44 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U26 (x13:Z) := 0. Definition P_id_A__U56 (x13:Z) := 0. Definition P_id_A__ISNEPAL (x13:Z) := 0. Definition P_id_A__U23 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__U52 (x13:Z) (x14:Z) (x15:Z) := 0. Definition P_id_A__ISQID (x13:Z) := 0. Definition P_id_MARK (x13:Z) := 2 + 1* x13. Definition P_id_A__U71 (x13:Z) (x14:Z) (x15:Z) := 0. Lemma P_id_A__U72_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U72 x14 x16 <= P_id_A__U72 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U11_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U11 x14 x16 <= P_id_A__U11 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U41_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U41 x14 x16 x18 <= P_id_A__U41 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U91_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U91 x14 x16 <= P_id_A__U91 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U24_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U24 x14 x16 x18 <= P_id_A__U24 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U53_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U53 x14 x16 x18 <= P_id_A__U53 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U81_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U81 x14 x16 <= P_id_A__U81 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNELIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISNELIST x14 <= P_id_A__ISNELIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U45_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U45 x14 x16 <= P_id_A__U45 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U31_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U31 x14 x16 <= P_id_A__U31 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U61_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U61 x14 x16 <= P_id_A__U61 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISPAL x14 <= P_id_A__ISPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISPALLISTKIND_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) -> P_id_A__ISPALLISTKIND x14 <= P_id_A__ISPALLISTKIND x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U43_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U43 x14 x16 x18 <= P_id_A__U43 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISLIST_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISLIST x14 <= P_id_A__ISLIST x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U55_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U55 x14 x16 <= P_id_A__U55 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U83_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U83 x14 <= P_id_A__U83 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U22_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U22 x14 x16 x18 <= P_id_A__U22 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U51_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U51 x14 x16 x18 <= P_id_A__U51 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U33_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U33 x14 <= P_id_A__U33 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A_____monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A____ x14 x16 <= P_id_A____ x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U63_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U63 x14 <= P_id_A__U63 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U73_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U73 x14 x16 <= P_id_A__U73 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U12_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U12 x14 x16 <= P_id_A__U12 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U42_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U42 x14 x16 x18 <= P_id_A__U42 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U92_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U92 x14 <= P_id_A__U92 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U25_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U25 x14 x16 <= P_id_A__U25 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U54_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U54 x14 x16 x18 <= P_id_A__U54 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U82_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U82 x14 x16 <= P_id_A__U82 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U21_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U21 x14 x16 x18 <= P_id_A__U21 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U46_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U46 x14 <= P_id_A__U46 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U32_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U32 x14 x16 <= P_id_A__U32 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U62_monotonic : forall x16 x14 x13 x15, (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) ->P_id_A__U62 x14 x16 <= P_id_A__U62 x13 x15. Proof. intros x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U74_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U74 x14 <= P_id_A__U74 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U13_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U13 x14 <= P_id_A__U13 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U44_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U44 x14 x16 x18 <= P_id_A__U44 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U26_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U26 x14 <= P_id_A__U26 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U56_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__U56 x14 <= P_id_A__U56 x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISNEPAL_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISNEPAL x14 <= P_id_A__ISNEPAL x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U23_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U23 x14 x16 x18 <= P_id_A__U23 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U52_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U52 x14 x16 x18 <= P_id_A__U52 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__ISQID_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_A__ISQID x14 <= P_id_A__ISQID x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_MARK_monotonic : forall x14 x13, (0 <= x14)/\ (x14 <= x13) ->P_id_MARK x14 <= P_id_MARK x13. Proof. intros x14 x13. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_A__U71_monotonic : forall x16 x18 x14 x17 x13 x15, (0 <= x18)/\ (x18 <= x17) -> (0 <= x16)/\ (x16 <= x15) -> (0 <= x14)/\ (x14 <= x13) -> P_id_A__U71 x14 x16 x18 <= P_id_A__U71 x13 x15 x17. Proof. intros x18 x17 x16 x15 x14 x13. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition marked_measure := InterpZ.marked_measure 0 P_id_a____ P_id_U31 P_id_a__U56 P_id_a__U26 P_id_U61 P_id_e P_id_a__U13 P_id_U45 P_id_a__U74 P_id_a__U44 P_id_U81 P_id_isNeList P_id_a__U11 P_id_U41 P_id_a__U71 P_id_a__isQid P_id_U72 P_id_U11 P_id_a__U23 P_id_U53 P_id_a__isNePal P_id_a__U52 P_id_U91 P_id_U24 P_id_mark P_id_U33 P_id_a__U62 P_id_a__U32 P_id_U63 P_id_o P_id_a__U21 P_id_U51 P_id_a__U82 P_id_a__U46 P_id_U83 P_id_U22 P_id_a__U12 P_id_U43 P_id_a__U73 P_id_a__U42 P_id_isPal P_id_isPalListKind P_id_a__U25 P_id_U55 P_id_a__U92 P_id_a__U54 P_id_isList P_id___ P_id_U32 P_id_a__U61 P_id_a__U31 P_id_U62 P_id_i P_id_a__isNeList P_id_U46 P_id_a__U81 P_id_a__U45 P_id_U82 P_id_U21 P_id_tt P_id_U42 P_id_a__U72 P_id_a__U41 P_id_U73 P_id_U12 P_id_a__U24 P_id_U54 P_id_a__U91 P_id_a__U53 P_id_U92 P_id_U25 P_id_nil P_id_isQid P_id_a__U63 P_id_a__U33 P_id_U71 P_id_u P_id_a__U22 P_id_U52 P_id_a__U83 P_id_a__U51 P_id_isNePal P_id_U23 P_id_a__isPalListKind P_id_U44 P_id_a__isPal P_id_a__U43 P_id_U74 P_id_U13 P_id_a__isList P_id_U56 P_id_a P_id_a__U55 P_id_U26 P_id_A__U72 P_id_A__U11 P_id_A__U41 P_id_A__U91 P_id_A__U24 P_id_A__U53 P_id_A__U81 P_id_A__ISNELIST P_id_A__U45 P_id_A__U31 P_id_A__U61 P_id_A__ISPAL P_id_A__ISPALLISTKIND P_id_A__U43 P_id_A__ISLIST P_id_A__U55 P_id_A__U83 P_id_A__U22 P_id_A__U51 P_id_A__U33 P_id_A____ P_id_A__U63 P_id_A__U73 P_id_A__U12 P_id_A__U42 P_id_A__U92 P_id_A__U25 P_id_A__U54 P_id_A__U82 P_id_A__U21 P_id_A__U46 P_id_A__U32 P_id_A__U62 P_id_A__U74 P_id_A__U13 P_id_A__U44 P_id_A__U26 P_id_A__U56 P_id_A__ISNEPAL P_id_A__U23 P_id_A__U52 P_id_A__ISQID P_id_MARK P_id_A__U71. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_a__U72 (x14:: x13::nil)) => P_id_A__U72 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U11 (x14:: x13::nil)) => P_id_A__U11 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U41 (x15::x14:: x13::nil)) => P_id_A__U41 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U91 (x14:: x13::nil)) => P_id_A__U91 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U24 (x15::x14:: x13::nil)) => P_id_A__U24 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U53 (x15::x14:: x13::nil)) => P_id_A__U53 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U81 (x14:: x13::nil)) => P_id_A__U81 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNeList (x13::nil)) => P_id_A__ISNELIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U45 (x14:: x13::nil)) => P_id_A__U45 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U31 (x14:: x13::nil)) => P_id_A__U31 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U61 (x14:: x13::nil)) => P_id_A__U61 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPal (x13::nil)) => P_id_A__ISPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__isPalListKind (x13::nil)) => P_id_A__ISPALLISTKIND (measure x13) | (algebra.Alg.Term algebra.F.id_a__U43 (x15::x14:: x13::nil)) => P_id_A__U43 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isList (x13::nil)) => P_id_A__ISLIST (measure x13) | (algebra.Alg.Term algebra.F.id_a__U55 (x14:: x13::nil)) => P_id_A__U55 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U83 (x13::nil)) => P_id_A__U83 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U22 (x15::x14:: x13::nil)) => P_id_A__U22 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U51 (x15::x14:: x13::nil)) => P_id_A__U51 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U33 (x13::nil)) => P_id_A__U33 (measure x13) | (algebra.Alg.Term algebra.F.id_a____ (x14:: x13::nil)) => P_id_A____ (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U63 (x13::nil)) => P_id_A__U63 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U73 (x14:: x13::nil)) => P_id_A__U73 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U12 (x14:: x13::nil)) => P_id_A__U12 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U42 (x15::x14:: x13::nil)) => P_id_A__U42 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U92 (x13::nil)) => P_id_A__U92 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U25 (x14:: x13::nil)) => P_id_A__U25 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U54 (x15::x14:: x13::nil)) => P_id_A__U54 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U82 (x14:: x13::nil)) => P_id_A__U82 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U21 (x15::x14:: x13::nil)) => P_id_A__U21 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U46 (x13::nil)) => P_id_A__U46 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U32 (x14:: x13::nil)) => P_id_A__U32 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U62 (x14:: x13::nil)) => P_id_A__U62 (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U74 (x13::nil)) => P_id_A__U74 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U13 (x13::nil)) => P_id_A__U13 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U44 (x15::x14:: x13::nil)) => P_id_A__U44 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U26 (x13::nil)) => P_id_A__U26 (measure x13) | (algebra.Alg.Term algebra.F.id_a__U56 (x13::nil)) => P_id_A__U56 (measure x13) | (algebra.Alg.Term algebra.F.id_a__isNePal (x13::nil)) => P_id_A__ISNEPAL (measure x13) | (algebra.Alg.Term algebra.F.id_a__U23 (x15::x14:: x13::nil)) => P_id_A__U23 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__U52 (x15::x14:: x13::nil)) => P_id_A__U52 (measure x15) (measure x14) (measure x13) | (algebra.Alg.Term algebra.F.id_a__isQid (x13::nil)) => P_id_A__ISQID (measure x13) | (algebra.Alg.Term algebra.F.id_mark (x13::nil)) => P_id_MARK (measure x13) | (algebra.Alg.Term algebra.F.id_a__U71 (x15::x14:: x13::nil)) => P_id_A__U71 (measure x15) (measure x14) (measure x13) | _ => measure t end. Proof. reflexivity . Qed. Lemma marked_measure_star_monotonic : forall f l1 l2, (closure.refl_trans_clos (closure.one_step_list (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) ) l1 l2) -> marked_measure (algebra.Alg.Term f l1) <= marked_measure (algebra.Alg.Term f l2). Proof. unfold marked_measure in *. apply InterpZ.marked_measure_star_monotonic. intros ;apply P_id_a_____monotonic;assumption. intros ;apply P_id_U31_monotonic;assumption. intros ;apply P_id_a__U56_monotonic;assumption. intros ;apply P_id_a__U26_monotonic;assumption. intros ;apply P_id_U61_monotonic;assumption. intros ;apply P_id_a__U13_monotonic;assumption. intros ;apply P_id_U45_monotonic;assumption. intros ;apply P_id_a__U74_monotonic;assumption. intros ;apply P_id_a__U44_monotonic;assumption. intros ;apply P_id_U81_monotonic;assumption. intros ;apply P_id_isNeList_monotonic;assumption. intros ;apply P_id_a__U11_monotonic;assumption. intros ;apply P_id_U41_monotonic;assumption. intros ;apply P_id_a__U71_monotonic;assumption. intros ;apply P_id_a__isQid_monotonic;assumption. intros ;apply P_id_U72_monotonic;assumption. intros ;apply P_id_U11_monotonic;assumption. intros ;apply P_id_a__U23_monotonic;assumption. intros ;apply P_id_U53_monotonic;assumption. intros ;apply P_id_a__isNePal_monotonic;assumption. intros ;apply P_id_a__U52_monotonic;assumption. intros ;apply P_id_U91_monotonic;assumption. intros ;apply P_id_U24_monotonic;assumption. intros ;apply P_id_mark_monotonic;assumption. intros ;apply P_id_U33_monotonic;assumption. intros ;apply P_id_a__U62_monotonic;assumption. intros ;apply P_id_a__U32_monotonic;assumption. intros ;apply P_id_U63_monotonic;assumption. intros ;apply P_id_a__U21_monotonic;assumption. intros ;apply P_id_U51_monotonic;assumption. intros ;apply P_id_a__U82_monotonic;assumption. intros ;apply P_id_a__U46_monotonic;assumption. intros ;apply P_id_U83_monotonic;assumption. intros ;apply P_id_U22_monotonic;assumption. intros ;apply P_id_a__U12_monotonic;assumption. intros ;apply P_id_U43_monotonic;assumption. intros ;apply P_id_a__U73_monotonic;assumption. intros ;apply P_id_a__U42_monotonic;assumption. intros ;apply P_id_isPal_monotonic;assumption. intros ;apply P_id_isPalListKind_monotonic;assumption. intros ;apply P_id_a__U25_monotonic;assumption. intros ;apply P_id_U55_monotonic;assumption. intros ;apply P_id_a__U92_monotonic;assumption. intros ;apply P_id_a__U54_monotonic;assumption. intros ;apply P_id_isList_monotonic;assumption. intros ;apply P_id____monotonic;assumption. intros ;apply P_id_U32_monotonic;assumption. intros ;apply P_id_a__U61_monotonic;assumption. intros ;apply P_id_a__U31_monotonic;assumption. intros ;apply P_id_U62_monotonic;assumption. intros ;apply P_id_a__isNeList_monotonic;assumption. intros ;apply P_id_U46_monotonic;assumption. intros ;apply P_id_a__U81_monotonic;assumption. intros ;apply P_id_a__U45_monotonic;assumption. intros ;apply P_id_U82_monotonic;assumption. intros ;apply P_id_U21_monotonic;assumption. intros ;apply P_id_U42_monotonic;assumption. intros ;apply P_id_a__U72_monotonic;assumption. intros ;apply P_id_a__U41_monotonic;assumption. intros ;apply P_id_U73_monotonic;assumption. intros ;apply P_id_U12_monotonic;assumption. intros ;apply P_id_a__U24_monotonic;assumption. intros ;apply P_id_U54_monotonic;assumption. intros ;apply P_id_a__U91_monotonic;assumption. intros ;apply P_id_a__U53_monotonic;assumption. intros ;apply P_id_U92_monotonic;assumption. intros ;apply P_id_U25_monotonic;assumption. intros ;apply P_id_isQid_monotonic;assumption. intros ;apply P_id_a__U63_monotonic;assumption. intros ;apply P_id_a__U33_monotonic;assumption. intros ;apply P_id_U71_monotonic;assumption. intros ;apply P_id_a__U22_monotonic;assumption. intros ;apply P_id_U52_monotonic;assumption. intros ;apply P_id_a__U83_monotonic;assumption. intros ;apply P_id_a__U51_monotonic;assumption. intros ;apply P_id_isNePal_monotonic;assumption. intros ;apply P_id_U23_monotonic;assumption. intros ;apply P_id_a__isPalListKind_monotonic;assumption. intros ;apply P_id_U44_monotonic;assumption. intros ;apply P_id_a__isPal_monotonic;assumption. intros ;apply P_id_a__U43_monotonic;assumption. intros ;apply P_id_U74_monotonic;assumption. intros ;apply P_id_U13_monotonic;assumption. intros ;apply P_id_a__isList_monotonic;assumption. intros ;apply P_id_U56_monotonic;assumption. intros ;apply P_id_a__U55_monotonic;assumption. intros ;apply P_id_U26_monotonic;assumption. intros ;apply P_id_a_____bounded;assumption. intros ;apply P_id_U31_bounded;assumption. intros ;apply P_id_a__U56_bounded;assumption. intros ;apply P_id_a__U26_bounded;assumption. intros ;apply P_id_U61_bounded;assumption. intros ;apply P_id_e_bounded;assumption. intros ;apply P_id_a__U13_bounded;assumption. intros ;apply P_id_U45_bounded;assumption. intros ;apply P_id_a__U74_bounded;assumption. intros ;apply P_id_a__U44_bounded;assumption. intros ;apply P_id_U81_bounded;assumption. intros ;apply P_id_isNeList_bounded;assumption. intros ;apply P_id_a__U11_bounded;assumption. intros ;apply P_id_U41_bounded;assumption. intros ;apply P_id_a__U71_bounded;assumption. intros ;apply P_id_a__isQid_bounded;assumption. intros ;apply P_id_U72_bounded;assumption. intros ;apply P_id_U11_bounded;assumption. intros ;apply P_id_a__U23_bounded;assumption. intros ;apply P_id_U53_bounded;assumption. intros ;apply P_id_a__isNePal_bounded;assumption. intros ;apply P_id_a__U52_bounded;assumption. intros ;apply P_id_U91_bounded;assumption. intros ;apply P_id_U24_bounded;assumption. intros ;apply P_id_mark_bounded;assumption. intros ;apply P_id_U33_bounded;assumption. intros ;apply P_id_a__U62_bounded;assumption. intros ;apply P_id_a__U32_bounded;assumption. intros ;apply P_id_U63_bounded;assumption. intros ;apply P_id_o_bounded;assumption. intros ;apply P_id_a__U21_bounded;assumption. intros ;apply P_id_U51_bounded;assumption. intros ;apply P_id_a__U82_bounded;assumption. intros ;apply P_id_a__U46_bounded;assumption. intros ;apply P_id_U83_bounded;assumption. intros ;apply P_id_U22_bounded;assumption. intros ;apply P_id_a__U12_bounded;assumption. intros ;apply P_id_U43_bounded;assumption. intros ;apply P_id_a__U73_bounded;assumption. intros ;apply P_id_a__U42_bounded;assumption. intros ;apply P_id_isPal_bounded;assumption. intros ;apply P_id_isPalListKind_bounded;assumption. intros ;apply P_id_a__U25_bounded;assumption. intros ;apply P_id_U55_bounded;assumption. intros ;apply P_id_a__U92_bounded;assumption. intros ;apply P_id_a__U54_bounded;assumption. intros ;apply P_id_isList_bounded;assumption. intros ;apply P_id____bounded;assumption. intros ;apply P_id_U32_bounded;assumption. intros ;apply P_id_a__U61_bounded;assumption. intros ;apply P_id_a__U31_bounded;assumption. intros ;apply P_id_U62_bounded;assumption. intros ;apply P_id_i_bounded;assumption. intros ;apply P_id_a__isNeList_bounded;assumption. intros ;apply P_id_U46_bounded;assumption. intros ;apply P_id_a__U81_bounded;assumption. intros ;apply P_id_a__U45_bounded;assumption. intros ;apply P_id_U82_bounded;assumption. intros ;apply P_id_U21_bounded;assumption. intros ;apply P_id_tt_bounded;assumption. intros ;apply P_id_U42_bounded;assumption. intros ;apply P_id_a__U72_bounded;assumption. intros ;apply P_id_a__U41_bounded;assumption. intros ;apply P_id_U73_bounded;assumption. intros ;apply P_id_U12_bounded;assumption. intros ;apply P_id_a__U24_bounded;assumption. intros ;apply P_id_U54_bounded;assumption. intros ;apply P_id_a__U91_bounded;assumption. intros ;apply P_id_a__U53_bounded;assumption. intros ;apply P_id_U92_bounded;assumption. intros ;apply P_id_U25_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_isQid_bounded;assumption. intros ;apply P_id_a__U63_bounded;assumption. intros ;apply P_id_a__U33_bounded;assumption. intros ;apply P_id_U71_bounded;assumption. intros ;apply P_id_u_bounded;assumption. intros ;apply P_id_a__U22_bounded;assumption. intros ;apply P_id_U52_bounded;assumption. intros ;apply P_id_a__U83_bounded;assumption. intros ;apply P_id_a__U51_bounded;assumption. intros ;apply P_id_isNePal_bounded;assumption. intros ;apply P_id_U23_bounded;assumption. intros ;apply P_id_a__isPalListKind_bounded;assumption. intros ;apply P_id_U44_bounded;assumption. intros ;apply P_id_a__isPal_bounded;assumption. intros ;apply P_id_a__U43_bounded;assumption. intros ;apply P_id_U74_bounded;assumption. intros ;apply P_id_U13_bounded;assumption. intros ;apply P_id_a__isList_bounded;assumption. intros ;apply P_id_U56_bounded;assumption. intros ;apply P_id_a_bounded;assumption. intros ;apply P_id_a__U55_bounded;assumption. intros ;apply P_id_U26_bounded;assumption. apply rules_monotonic. intros ;apply P_id_A__U72_monotonic;assumption. intros ;apply P_id_A__U11_monotonic;assumption. intros ;apply P_id_A__U41_monotonic;assumption. intros ;apply P_id_A__U91_monotonic;assumption. intros ;apply P_id_A__U24_monotonic;assumption. intros ;apply P_id_A__U53_monotonic;assumption. intros ;apply P_id_A__U81_monotonic;assumption. intros ;apply P_id_A__ISNELIST_monotonic;assumption. intros ;apply P_id_A__U45_monotonic;assumption. intros ;apply P_id_A__U31_monotonic;assumption. intros ;apply P_id_A__U61_monotonic;assumption. intros ;apply P_id_A__ISPAL_monotonic;assumption. intros ;apply P_id_A__ISPALLISTKIND_monotonic;assumption. intros ;apply P_id_A__U43_monotonic;assumption. intros ;apply P_id_A__ISLIST_monotonic;assumption. intros ;apply P_id_A__U55_monotonic;assumption. intros ;apply P_id_A__U83_monotonic;assumption. intros ;apply P_id_A__U22_monotonic;assumption. intros ;apply P_id_A__U51_monotonic;assumption. intros ;apply P_id_A__U33_monotonic;assumption. intros ;apply P_id_A_____monotonic;assumption. intros ;apply P_id_A__U63_monotonic;assumption. intros ;apply P_id_A__U73_monotonic;assumption. intros ;apply P_id_A__U12_monotonic;assumption. intros ;apply P_id_A__U42_monotonic;assumption. intros ;apply P_id_A__U92_monotonic;assumption. intros ;apply P_id_A__U25_monotonic;assumption. intros ;apply P_id_A__U54_monotonic;assumption. intros ;apply P_id_A__U82_monotonic;assumption. intros ;apply P_id_A__U21_monotonic;assumption. intros ;apply P_id_A__U46_monotonic;assumption. intros ;apply P_id_A__U32_monotonic;assumption. intros ;apply P_id_A__U62_monotonic;assumption. intros ;apply P_id_A__U74_monotonic;assumption. intros ;apply P_id_A__U13_monotonic;assumption. intros ;apply P_id_A__U44_monotonic;assumption. intros ;apply P_id_A__U26_monotonic;assumption. intros ;apply P_id_A__U56_monotonic;assumption. intros ;apply P_id_A__ISNEPAL_monotonic;assumption. intros ;apply P_id_A__U23_monotonic;assumption. intros ;apply P_id_A__U52_monotonic;assumption. intros ;apply P_id_A__ISQID_monotonic;assumption. intros ;apply P_id_MARK_monotonic;assumption. intros ;apply P_id_A__U71_monotonic;assumption. Qed. Ltac rewrite_and_unfold := do 2 (rewrite marked_measure_equation); repeat ( match goal with | |- context [measure (algebra.Alg.Term ?f ?t)] => rewrite (measure_equation (algebra.Alg.Term f t)) | H:context [measure (algebra.Alg.Term ?f ?t)] |- _ => rewrite (measure_equation (algebra.Alg.Term f t)) in H|- end ). Definition lt a b := (Zwf.Zwf 0) (marked_measure a) (marked_measure b). Definition le a b := marked_measure a <= marked_measure b. Lemma lt_le_compat : forall a b c, (lt a b) ->(le b c) ->lt a c. Proof. unfold lt, le in *. intros a b c. apply (interp.le_lt_compat_right (interp.o_Z 0)). Qed. Lemma wf_lt : well_founded lt. Proof. unfold lt in *. apply Inverse_Image.wf_inverse_image with (B:=Z). apply Zwf.Zwf_well_founded. Qed. Lemma DP_R_xml_0_scc_84_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_strict lt. Proof. unfold Relation_Definitions.inclusion, lt in *. intros a b H;destruct H; match goal with | |- (Zwf.Zwf 0) _ (marked_measure (algebra.Alg.Term ?f ?l)) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_lt_compat_right (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma DP_R_xml_0_scc_84_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_84_large le. Proof. unfold Relation_Definitions.inclusion, le, Zwf.Zwf in *. intros a b H;destruct H; match goal with | |- _ <= marked_measure (algebra.Alg.Term ?f ?l) => let l'' := algebra.Alg_ext.find_replacement l in ((apply (interp.le_trans (interp.o_Z 0)) with (marked_measure (algebra.Alg.Term f l''));[idtac| apply marked_measure_star_monotonic; repeat (apply algebra.EQT_ext.one_step_list_refl_trans_clos); (assumption)||(constructor 1)])) end ;clear ;rewrite_and_unfold ;repeat (generate_pos_hyp ); cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Definition wf_DP_R_xml_0_scc_84_large := WF_DP_R_xml_0_scc_84_large.wf. Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_84. Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_84_strict_in_lt; econstructor eassumption)|| ((apply IHx;[econstructor eassumption| intros y' Hlt;apply IHx';apply lt_le_compat with (1:=Hlt) ; apply DP_R_xml_0_scc_84_large_in_le;econstructor eassumption])). apply wf_DP_R_xml_0_scc_84_large. Qed. End WF_DP_R_xml_0_scc_84. Definition wf_DP_R_xml_0_scc_84 := WF_DP_R_xml_0_scc_84.wf. Lemma acc_DP_R_xml_0_scc_84 : forall x y, (DP_R_xml_0_scc_84 x y) ->Acc WF_R_xml_0_deep_rew.DP_R_xml_0 x. Proof. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_84). intros x' _ Hrec y h. inversion h;clear h;subst; constructor;intros _y _h;inversion _h;clear _h;subst; (eapply Hrec;econstructor eassumption)|| ((eapply acc_DP_R_xml_0_non_scc_83; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_82; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_81; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_80; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_79; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_78; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_77; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_76; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_75; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_74; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_73; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_72; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_71; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_70; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_69; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_68; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_67; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_66; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_65; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_56; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_53; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_42; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_41; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_40; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_39; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_38; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_37; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_34; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_31; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_26; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_21; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_14; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_10; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_9; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_8; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_7; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_6; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_5; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_4; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_3; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_2; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_1; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| (eapply Hrec; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))))))))))))))))))))))))))))))))))))))))))))). apply wf_DP_R_xml_0_scc_84. Qed. Lemma wf : well_founded WF_R_xml_0_deep_rew.DP_R_xml_0. Proof. constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_non_scc_83; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_82; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_81; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_80; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_79; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_78; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_77; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_76; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_75; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_74; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_73; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_72; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_71; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_70; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_69; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_68; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_67; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_66; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_65; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_64; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_63; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_62; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_61; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_60; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_59; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_58; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_57; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_56; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_55; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_54; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_53; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_52; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_51; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_50; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_49; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_48; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_47; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_46; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_45; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_44; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_43; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_42; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_41; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_40; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_39; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_38; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_37; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_36; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_35; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_34; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_33; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_32; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_31; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_30; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_29; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_28; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_27; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_26; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_25; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_24; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_23; econstructor (eassumption)|| (algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_22; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_non_scc_21; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| (( eapply acc_DP_R_xml_0_non_scc_20; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_19; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_18; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_17; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_16; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_15; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_14; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_13; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_12; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_11; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_10; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_9; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_8; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_7; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_6; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_5; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_4; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_3; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_2; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_1; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_non_scc_0; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_84; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_83; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_82; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_81; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_80; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_79; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_78; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_77; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_76; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_75; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_74; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_73; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_72; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_71; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_70; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_69; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_68; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_67; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_66; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_65; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_64; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_63; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_62; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_61; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_60; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_59; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_58; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_57; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_56; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_55; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_54; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_53; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_52; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_51; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_50; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_49; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_48; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_47; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_46; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_45; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_44; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_43; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_42; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_41; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_40; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_39; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_38; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_37; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_36; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_35; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_34; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_33; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_32; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_31; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_30; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_29; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_28; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_27; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_26; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_25; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_24; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_23; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_22; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_21; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_20; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_19; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_18; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_17; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_16; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_15; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_14; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_13; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_12; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_11; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_10; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_9; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_8; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_7; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_6; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_5; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_4; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_3; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_2; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_1; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( eapply acc_DP_R_xml_0_scc_0; econstructor ( eassumption)|| ( algebra.Alg_ext.star_refl' ))|| ( ( R_xml_0_deep_rew.impossible_star_reduction_R_xml_0 )|| ( fail)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))). Qed. End WF_DP_R_xml_0. Definition wf_H := WF_DP_R_xml_0.wf. Lemma wf : well_founded (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules). Proof. apply ddp.dp_criterion. apply R_xml_0_deep_rew.R_xml_0_non_var. apply R_xml_0_deep_rew.R_xml_0_reg. intros ; apply (ddp.constructor_defined_dec _ _ R_xml_0_deep_rew.R_xml_0_rules_included). refine (Inclusion.wf_incl _ _ _ _ wf_H). intros x y H. destruct (R_xml_0_dp_step_spec H) as [f [l1 [l2 [H1 [H2 H3]]]]]. destruct (ddp.dp_list_complete _ _ R_xml_0_deep_rew.R_xml_0_rules_included _ _ H3) as [x' [y' [sigma [h1 [h2 h3]]]]]. clear H3. subst. vm_compute in h3|-. let e := type of h3 in (dp_concl_tac h2 h3 ltac:(fun _ => idtac) e). Qed. End WF_R_xml_0_deep_rew. (* *** Local Variables: *** *** coq-prog-name: "coqtop" *** *** coq-prog-args: ("-emacs-U" "-I" "$COCCINELLE/examples" "-I" "$COCCINELLE/term_algebra" "-I" "$COCCINELLE/term_orderings" "-I" "$COCCINELLE/basis" "-I" "$COCCINELLE/list_extensions" "-I" "$COCCINELLE/examples/cime_trace/") *** *** compile-command: "coqc -I $COCCINELLE/term_algebra -I $COCCINELLE/term_orderings -I $COCCINELLE/basis -I $COCCINELLE/list_extensions -I $COCCINELLE/examples/cime_trace/ -I $COCCINELLE/examples/ c_output/strat/tpdb-5.0___TRS___TRCSR___PALINDROME_complete_noand_GM.trs/a3pat.v" *** *** End: *** *)