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_intersect'ii'in *) | id_intersect'ii'in : symb (* id_tautology'i'out *) | id_tautology'i'out : symb (* id_u'6'1 *) | id_u'6'1 : symb (* id_u'3'1 *) | id_u'3'1 : symb (* id_u'12'1 *) | id_u'12'1 : symb (* id_u'2'1 *) | id_u'2'1 : symb (* id_u'9'1 *) | id_u'9'1 : symb (* id_iff *) | id_iff : symb (* id_u'14'1 *) | id_u'14'1 : symb (* id_intersect'ii'out *) | id_intersect'ii'out : symb (* id_u'7'1 *) | id_u'7'1 : symb (* id_x'2d *) | id_x'2d : symb (* id_u'13'1 *) | id_u'13'1 : symb (* id_sequent *) | id_sequent : symb (* id_u'10'1 *) | id_u'10'1 : symb (* id_x'2a *) | id_x'2a : symb (* id_tautology'i'in *) | id_tautology'i'in : symb (* id_cons *) | id_cons : symb (* id_u'6'2 *) | id_u'6'2 : symb (* id_x'2b *) | id_x'2b : symb (* id_u'12'2 *) | id_u'12'2 : symb (* id_reduce'ii'in *) | id_reduce'ii'in : symb (* id_p *) | id_p : symb (* id_u'4'1 *) | id_u'4'1 : symb (* id_u'15'1 *) | id_u'15'1 : symb (* id_u'1'1 *) | id_u'1'1 : symb (* id_u'8'1 *) | id_u'8'1 : symb (* id_reduce'ii'out *) | id_reduce'ii'out : symb (* id_nil *) | id_nil : symb (* id_if *) | id_if : symb (* id_u'11'1 *) | id_u'11'1 : symb (* id_u'5'1 *) | id_u'5'1 : symb (* id_u'16'1 *) | id_u'16'1 : symb . Definition symb_eq_bool (f1 f2:symb) : bool := match f1,f2 with | id_intersect'ii'in,id_intersect'ii'in => true | id_tautology'i'out,id_tautology'i'out => true | id_u'6'1,id_u'6'1 => true | id_u'3'1,id_u'3'1 => true | id_u'12'1,id_u'12'1 => true | id_u'2'1,id_u'2'1 => true | id_u'9'1,id_u'9'1 => true | id_iff,id_iff => true | id_u'14'1,id_u'14'1 => true | id_intersect'ii'out,id_intersect'ii'out => true | id_u'7'1,id_u'7'1 => true | id_x'2d,id_x'2d => true | id_u'13'1,id_u'13'1 => true | id_sequent,id_sequent => true | id_u'10'1,id_u'10'1 => true | id_x'2a,id_x'2a => true | id_tautology'i'in,id_tautology'i'in => true | id_cons,id_cons => true | id_u'6'2,id_u'6'2 => true | id_x'2b,id_x'2b => true | id_u'12'2,id_u'12'2 => true | id_reduce'ii'in,id_reduce'ii'in => true | id_p,id_p => true | id_u'4'1,id_u'4'1 => true | id_u'15'1,id_u'15'1 => true | id_u'1'1,id_u'1'1 => true | id_u'8'1,id_u'8'1 => true | id_reduce'ii'out,id_reduce'ii'out => true | id_nil,id_nil => true | id_if,id_if => true | id_u'11'1,id_u'11'1 => true | id_u'5'1,id_u'5'1 => true | id_u'16'1,id_u'16'1 => 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_intersect'ii'in,id_intersect'ii'in => refl_equal _ | id_tautology'i'out,id_tautology'i'out => refl_equal _ | id_u'6'1,id_u'6'1 => refl_equal _ | id_u'3'1,id_u'3'1 => refl_equal _ | id_u'12'1,id_u'12'1 => refl_equal _ | id_u'2'1,id_u'2'1 => refl_equal _ | id_u'9'1,id_u'9'1 => refl_equal _ | id_iff,id_iff => refl_equal _ | id_u'14'1,id_u'14'1 => refl_equal _ | id_intersect'ii'out,id_intersect'ii'out => refl_equal _ | id_u'7'1,id_u'7'1 => refl_equal _ | id_x'2d,id_x'2d => refl_equal _ | id_u'13'1,id_u'13'1 => refl_equal _ | id_sequent,id_sequent => refl_equal _ | id_u'10'1,id_u'10'1 => refl_equal _ | id_x'2a,id_x'2a => refl_equal _ | id_tautology'i'in,id_tautology'i'in => refl_equal _ | id_cons,id_cons => refl_equal _ | id_u'6'2,id_u'6'2 => refl_equal _ | id_x'2b,id_x'2b => refl_equal _ | id_u'12'2,id_u'12'2 => refl_equal _ | id_reduce'ii'in,id_reduce'ii'in => refl_equal _ | id_p,id_p => refl_equal _ | id_u'4'1,id_u'4'1 => refl_equal _ | id_u'15'1,id_u'15'1 => refl_equal _ | id_u'1'1,id_u'1'1 => refl_equal _ | id_u'8'1,id_u'8'1 => refl_equal _ | id_reduce'ii'out,id_reduce'ii'out => refl_equal _ | id_nil,id_nil => refl_equal _ | id_if,id_if => refl_equal _ | id_u'11'1,id_u'11'1 => refl_equal _ | id_u'5'1,id_u'5'1 => refl_equal _ | id_u'16'1,id_u'16'1 => refl_equal _ | _,_ => _ end;intros abs;discriminate. Defined. Definition arity (f:symb) := match f with | id_intersect'ii'in => term.Free 2 | id_tautology'i'out => term.Free 0 | id_u'6'1 => term.Free 5 | id_u'3'1 => term.Free 1 | id_u'12'1 => term.Free 5 | id_u'2'1 => term.Free 1 | id_u'9'1 => term.Free 1 | id_iff => term.Free 2 | id_u'14'1 => term.Free 1 | id_intersect'ii'out => term.Free 0 | id_u'7'1 => term.Free 1 | id_x'2d => term.Free 1 | id_u'13'1 => term.Free 1 | id_sequent => term.Free 2 | id_u'10'1 => term.Free 1 | id_x'2a => term.Free 2 | id_tautology'i'in => term.Free 1 | id_cons => term.Free 2 | id_u'6'2 => term.Free 1 | id_x'2b => term.Free 2 | id_u'12'2 => term.Free 1 | id_reduce'ii'in => term.Free 2 | id_p => term.Free 1 | id_u'4'1 => term.Free 1 | id_u'15'1 => term.Free 1 | id_u'1'1 => term.Free 1 | id_u'8'1 => term.Free 1 | id_reduce'ii'out => term.Free 0 | id_nil => term.Free 0 | id_if => term.Free 2 | id_u'11'1 => term.Free 1 | id_u'5'1 => term.Free 1 | id_u'16'1 => term.Free 1 end. Definition symb_order (f1 f2:symb) : bool := match f1,f2 with | id_intersect'ii'in,id_intersect'ii'in => true | id_intersect'ii'in,id_tautology'i'out => false | id_intersect'ii'in,id_u'6'1 => false | id_intersect'ii'in,id_u'3'1 => false | id_intersect'ii'in,id_u'12'1 => false | id_intersect'ii'in,id_u'2'1 => false | id_intersect'ii'in,id_u'9'1 => false | id_intersect'ii'in,id_iff => false | id_intersect'ii'in,id_u'14'1 => false | id_intersect'ii'in,id_intersect'ii'out => false | id_intersect'ii'in,id_u'7'1 => false | id_intersect'ii'in,id_x'2d => false | id_intersect'ii'in,id_u'13'1 => false | id_intersect'ii'in,id_sequent => false | id_intersect'ii'in,id_u'10'1 => false | id_intersect'ii'in,id_x'2a => false | id_intersect'ii'in,id_tautology'i'in => false | id_intersect'ii'in,id_cons => false | id_intersect'ii'in,id_u'6'2 => false | id_intersect'ii'in,id_x'2b => false | id_intersect'ii'in,id_u'12'2 => false | id_intersect'ii'in,id_reduce'ii'in => false | id_intersect'ii'in,id_p => false | id_intersect'ii'in,id_u'4'1 => false | id_intersect'ii'in,id_u'15'1 => false | id_intersect'ii'in,id_u'1'1 => false | id_intersect'ii'in,id_u'8'1 => false | id_intersect'ii'in,id_reduce'ii'out => false | id_intersect'ii'in,id_nil => false | id_intersect'ii'in,id_if => false | id_intersect'ii'in,id_u'11'1 => false | id_intersect'ii'in,id_u'5'1 => false | id_intersect'ii'in,id_u'16'1 => false | id_tautology'i'out,id_intersect'ii'in => true | id_tautology'i'out,id_tautology'i'out => true | id_tautology'i'out,id_u'6'1 => false | id_tautology'i'out,id_u'3'1 => false | id_tautology'i'out,id_u'12'1 => false | id_tautology'i'out,id_u'2'1 => false | id_tautology'i'out,id_u'9'1 => false | id_tautology'i'out,id_iff => false | id_tautology'i'out,id_u'14'1 => false | id_tautology'i'out,id_intersect'ii'out => false | id_tautology'i'out,id_u'7'1 => false | id_tautology'i'out,id_x'2d => false | id_tautology'i'out,id_u'13'1 => false | id_tautology'i'out,id_sequent => false | id_tautology'i'out,id_u'10'1 => false | id_tautology'i'out,id_x'2a => false | id_tautology'i'out,id_tautology'i'in => false | id_tautology'i'out,id_cons => false | id_tautology'i'out,id_u'6'2 => false | id_tautology'i'out,id_x'2b => false | id_tautology'i'out,id_u'12'2 => false | id_tautology'i'out,id_reduce'ii'in => false | id_tautology'i'out,id_p => false | id_tautology'i'out,id_u'4'1 => false | id_tautology'i'out,id_u'15'1 => false | id_tautology'i'out,id_u'1'1 => false | id_tautology'i'out,id_u'8'1 => false | id_tautology'i'out,id_reduce'ii'out => false | id_tautology'i'out,id_nil => false | id_tautology'i'out,id_if => false | id_tautology'i'out,id_u'11'1 => false | id_tautology'i'out,id_u'5'1 => false | id_tautology'i'out,id_u'16'1 => false | id_u'6'1,id_intersect'ii'in => true | id_u'6'1,id_tautology'i'out => true | id_u'6'1,id_u'6'1 => true | id_u'6'1,id_u'3'1 => false | id_u'6'1,id_u'12'1 => false | id_u'6'1,id_u'2'1 => false | id_u'6'1,id_u'9'1 => false | id_u'6'1,id_iff => false | id_u'6'1,id_u'14'1 => false | id_u'6'1,id_intersect'ii'out => false | id_u'6'1,id_u'7'1 => false | id_u'6'1,id_x'2d => false | id_u'6'1,id_u'13'1 => false | id_u'6'1,id_sequent => false | id_u'6'1,id_u'10'1 => false | id_u'6'1,id_x'2a => false | id_u'6'1,id_tautology'i'in => false | id_u'6'1,id_cons => false | id_u'6'1,id_u'6'2 => false | id_u'6'1,id_x'2b => false | id_u'6'1,id_u'12'2 => false | id_u'6'1,id_reduce'ii'in => false | id_u'6'1,id_p => false | id_u'6'1,id_u'4'1 => false | id_u'6'1,id_u'15'1 => false | id_u'6'1,id_u'1'1 => false | id_u'6'1,id_u'8'1 => false | id_u'6'1,id_reduce'ii'out => false | id_u'6'1,id_nil => false | id_u'6'1,id_if => false | id_u'6'1,id_u'11'1 => false | id_u'6'1,id_u'5'1 => false | id_u'6'1,id_u'16'1 => false | id_u'3'1,id_intersect'ii'in => true | id_u'3'1,id_tautology'i'out => true | id_u'3'1,id_u'6'1 => true | id_u'3'1,id_u'3'1 => true | id_u'3'1,id_u'12'1 => false | id_u'3'1,id_u'2'1 => false | id_u'3'1,id_u'9'1 => false | id_u'3'1,id_iff => false | id_u'3'1,id_u'14'1 => false | id_u'3'1,id_intersect'ii'out => false | id_u'3'1,id_u'7'1 => false | id_u'3'1,id_x'2d => false | id_u'3'1,id_u'13'1 => false | id_u'3'1,id_sequent => false | id_u'3'1,id_u'10'1 => false | id_u'3'1,id_x'2a => false | id_u'3'1,id_tautology'i'in => false | id_u'3'1,id_cons => false | id_u'3'1,id_u'6'2 => false | id_u'3'1,id_x'2b => false | id_u'3'1,id_u'12'2 => false | id_u'3'1,id_reduce'ii'in => false | id_u'3'1,id_p => false | id_u'3'1,id_u'4'1 => false | id_u'3'1,id_u'15'1 => false | id_u'3'1,id_u'1'1 => false | id_u'3'1,id_u'8'1 => false | id_u'3'1,id_reduce'ii'out => false | id_u'3'1,id_nil => false | id_u'3'1,id_if => false | id_u'3'1,id_u'11'1 => false | id_u'3'1,id_u'5'1 => false | id_u'3'1,id_u'16'1 => false | id_u'12'1,id_intersect'ii'in => true | id_u'12'1,id_tautology'i'out => true | id_u'12'1,id_u'6'1 => true | id_u'12'1,id_u'3'1 => true | id_u'12'1,id_u'12'1 => true | id_u'12'1,id_u'2'1 => false | id_u'12'1,id_u'9'1 => false | id_u'12'1,id_iff => false | id_u'12'1,id_u'14'1 => false | id_u'12'1,id_intersect'ii'out => false | id_u'12'1,id_u'7'1 => false | id_u'12'1,id_x'2d => false | id_u'12'1,id_u'13'1 => false | id_u'12'1,id_sequent => false | id_u'12'1,id_u'10'1 => false | id_u'12'1,id_x'2a => false | id_u'12'1,id_tautology'i'in => false | id_u'12'1,id_cons => false | id_u'12'1,id_u'6'2 => false | id_u'12'1,id_x'2b => false | id_u'12'1,id_u'12'2 => false | id_u'12'1,id_reduce'ii'in => false | id_u'12'1,id_p => false | id_u'12'1,id_u'4'1 => false | id_u'12'1,id_u'15'1 => false | id_u'12'1,id_u'1'1 => false | id_u'12'1,id_u'8'1 => false | id_u'12'1,id_reduce'ii'out => false | id_u'12'1,id_nil => false | id_u'12'1,id_if => false | id_u'12'1,id_u'11'1 => false | id_u'12'1,id_u'5'1 => false | id_u'12'1,id_u'16'1 => false | id_u'2'1,id_intersect'ii'in => true | id_u'2'1,id_tautology'i'out => true | id_u'2'1,id_u'6'1 => true | id_u'2'1,id_u'3'1 => true | id_u'2'1,id_u'12'1 => true | id_u'2'1,id_u'2'1 => true | id_u'2'1,id_u'9'1 => false | id_u'2'1,id_iff => false | id_u'2'1,id_u'14'1 => false | id_u'2'1,id_intersect'ii'out => false | id_u'2'1,id_u'7'1 => false | id_u'2'1,id_x'2d => false | id_u'2'1,id_u'13'1 => false | id_u'2'1,id_sequent => false | id_u'2'1,id_u'10'1 => false | id_u'2'1,id_x'2a => false | id_u'2'1,id_tautology'i'in => false | id_u'2'1,id_cons => false | id_u'2'1,id_u'6'2 => false | id_u'2'1,id_x'2b => false | id_u'2'1,id_u'12'2 => false | id_u'2'1,id_reduce'ii'in => false | id_u'2'1,id_p => false | id_u'2'1,id_u'4'1 => false | id_u'2'1,id_u'15'1 => false | id_u'2'1,id_u'1'1 => false | id_u'2'1,id_u'8'1 => false | id_u'2'1,id_reduce'ii'out => false | id_u'2'1,id_nil => false | id_u'2'1,id_if => false | id_u'2'1,id_u'11'1 => false | id_u'2'1,id_u'5'1 => false | id_u'2'1,id_u'16'1 => false | id_u'9'1,id_intersect'ii'in => true | id_u'9'1,id_tautology'i'out => true | id_u'9'1,id_u'6'1 => true | id_u'9'1,id_u'3'1 => true | id_u'9'1,id_u'12'1 => true | id_u'9'1,id_u'2'1 => true | id_u'9'1,id_u'9'1 => true | id_u'9'1,id_iff => false | id_u'9'1,id_u'14'1 => false | id_u'9'1,id_intersect'ii'out => false | id_u'9'1,id_u'7'1 => false | id_u'9'1,id_x'2d => false | id_u'9'1,id_u'13'1 => false | id_u'9'1,id_sequent => false | id_u'9'1,id_u'10'1 => false | id_u'9'1,id_x'2a => false | id_u'9'1,id_tautology'i'in => false | id_u'9'1,id_cons => false | id_u'9'1,id_u'6'2 => false | id_u'9'1,id_x'2b => false | id_u'9'1,id_u'12'2 => false | id_u'9'1,id_reduce'ii'in => false | id_u'9'1,id_p => false | id_u'9'1,id_u'4'1 => false | id_u'9'1,id_u'15'1 => false | id_u'9'1,id_u'1'1 => false | id_u'9'1,id_u'8'1 => false | id_u'9'1,id_reduce'ii'out => false | id_u'9'1,id_nil => false | id_u'9'1,id_if => false | id_u'9'1,id_u'11'1 => false | id_u'9'1,id_u'5'1 => false | id_u'9'1,id_u'16'1 => false | id_iff,id_intersect'ii'in => true | id_iff,id_tautology'i'out => true | id_iff,id_u'6'1 => true | id_iff,id_u'3'1 => true | id_iff,id_u'12'1 => true | id_iff,id_u'2'1 => true | id_iff,id_u'9'1 => true | id_iff,id_iff => true | id_iff,id_u'14'1 => false | id_iff,id_intersect'ii'out => false | id_iff,id_u'7'1 => false | id_iff,id_x'2d => false | id_iff,id_u'13'1 => false | id_iff,id_sequent => false | id_iff,id_u'10'1 => false | id_iff,id_x'2a => false | id_iff,id_tautology'i'in => false | id_iff,id_cons => false | id_iff,id_u'6'2 => false | id_iff,id_x'2b => false | id_iff,id_u'12'2 => false | id_iff,id_reduce'ii'in => false | id_iff,id_p => false | id_iff,id_u'4'1 => false | id_iff,id_u'15'1 => false | id_iff,id_u'1'1 => false | id_iff,id_u'8'1 => false | id_iff,id_reduce'ii'out => false | id_iff,id_nil => false | id_iff,id_if => false | id_iff,id_u'11'1 => false | id_iff,id_u'5'1 => false | id_iff,id_u'16'1 => false | id_u'14'1,id_intersect'ii'in => true | id_u'14'1,id_tautology'i'out => true | id_u'14'1,id_u'6'1 => true | id_u'14'1,id_u'3'1 => true | id_u'14'1,id_u'12'1 => true | id_u'14'1,id_u'2'1 => true | id_u'14'1,id_u'9'1 => true | id_u'14'1,id_iff => true | id_u'14'1,id_u'14'1 => true | id_u'14'1,id_intersect'ii'out => false | id_u'14'1,id_u'7'1 => false | id_u'14'1,id_x'2d => false | id_u'14'1,id_u'13'1 => false | id_u'14'1,id_sequent => false | id_u'14'1,id_u'10'1 => false | id_u'14'1,id_x'2a => false | id_u'14'1,id_tautology'i'in => false | id_u'14'1,id_cons => false | id_u'14'1,id_u'6'2 => false | id_u'14'1,id_x'2b => false | id_u'14'1,id_u'12'2 => false | id_u'14'1,id_reduce'ii'in => false | id_u'14'1,id_p => false | id_u'14'1,id_u'4'1 => false | id_u'14'1,id_u'15'1 => false | id_u'14'1,id_u'1'1 => false | id_u'14'1,id_u'8'1 => false | id_u'14'1,id_reduce'ii'out => false | id_u'14'1,id_nil => false | id_u'14'1,id_if => false | id_u'14'1,id_u'11'1 => false | id_u'14'1,id_u'5'1 => false | id_u'14'1,id_u'16'1 => false | id_intersect'ii'out,id_intersect'ii'in => true | id_intersect'ii'out,id_tautology'i'out => true | id_intersect'ii'out,id_u'6'1 => true | id_intersect'ii'out,id_u'3'1 => true | id_intersect'ii'out,id_u'12'1 => true | id_intersect'ii'out,id_u'2'1 => true | id_intersect'ii'out,id_u'9'1 => true | id_intersect'ii'out,id_iff => true | id_intersect'ii'out,id_u'14'1 => true | id_intersect'ii'out,id_intersect'ii'out => true | id_intersect'ii'out,id_u'7'1 => false | id_intersect'ii'out,id_x'2d => false | id_intersect'ii'out,id_u'13'1 => false | id_intersect'ii'out,id_sequent => false | id_intersect'ii'out,id_u'10'1 => false | id_intersect'ii'out,id_x'2a => false | id_intersect'ii'out,id_tautology'i'in => false | id_intersect'ii'out,id_cons => false | id_intersect'ii'out,id_u'6'2 => false | id_intersect'ii'out,id_x'2b => false | id_intersect'ii'out,id_u'12'2 => false | id_intersect'ii'out,id_reduce'ii'in => false | id_intersect'ii'out,id_p => false | id_intersect'ii'out,id_u'4'1 => false | id_intersect'ii'out,id_u'15'1 => false | id_intersect'ii'out,id_u'1'1 => false | id_intersect'ii'out,id_u'8'1 => false | id_intersect'ii'out,id_reduce'ii'out => false | id_intersect'ii'out,id_nil => false | id_intersect'ii'out,id_if => false | id_intersect'ii'out,id_u'11'1 => false | id_intersect'ii'out,id_u'5'1 => false | id_intersect'ii'out,id_u'16'1 => false | id_u'7'1,id_intersect'ii'in => true | id_u'7'1,id_tautology'i'out => true | id_u'7'1,id_u'6'1 => true | id_u'7'1,id_u'3'1 => true | id_u'7'1,id_u'12'1 => true | id_u'7'1,id_u'2'1 => true | id_u'7'1,id_u'9'1 => true | id_u'7'1,id_iff => true | id_u'7'1,id_u'14'1 => true | id_u'7'1,id_intersect'ii'out => true | id_u'7'1,id_u'7'1 => true | id_u'7'1,id_x'2d => false | id_u'7'1,id_u'13'1 => false | id_u'7'1,id_sequent => false | id_u'7'1,id_u'10'1 => false | id_u'7'1,id_x'2a => false | id_u'7'1,id_tautology'i'in => false | id_u'7'1,id_cons => false | id_u'7'1,id_u'6'2 => false | id_u'7'1,id_x'2b => false | id_u'7'1,id_u'12'2 => false | id_u'7'1,id_reduce'ii'in => false | id_u'7'1,id_p => false | id_u'7'1,id_u'4'1 => false | id_u'7'1,id_u'15'1 => false | id_u'7'1,id_u'1'1 => false | id_u'7'1,id_u'8'1 => false | id_u'7'1,id_reduce'ii'out => false | id_u'7'1,id_nil => false | id_u'7'1,id_if => false | id_u'7'1,id_u'11'1 => false | id_u'7'1,id_u'5'1 => false | id_u'7'1,id_u'16'1 => false | id_x'2d,id_intersect'ii'in => true | id_x'2d,id_tautology'i'out => true | id_x'2d,id_u'6'1 => true | id_x'2d,id_u'3'1 => true | id_x'2d,id_u'12'1 => true | id_x'2d,id_u'2'1 => true | id_x'2d,id_u'9'1 => true | id_x'2d,id_iff => true | id_x'2d,id_u'14'1 => true | id_x'2d,id_intersect'ii'out => true | id_x'2d,id_u'7'1 => true | id_x'2d,id_x'2d => true | id_x'2d,id_u'13'1 => false | id_x'2d,id_sequent => false | id_x'2d,id_u'10'1 => false | id_x'2d,id_x'2a => false | id_x'2d,id_tautology'i'in => false | id_x'2d,id_cons => false | id_x'2d,id_u'6'2 => false | id_x'2d,id_x'2b => false | id_x'2d,id_u'12'2 => false | id_x'2d,id_reduce'ii'in => false | id_x'2d,id_p => false | id_x'2d,id_u'4'1 => false | id_x'2d,id_u'15'1 => false | id_x'2d,id_u'1'1 => false | id_x'2d,id_u'8'1 => false | id_x'2d,id_reduce'ii'out => false | id_x'2d,id_nil => false | id_x'2d,id_if => false | id_x'2d,id_u'11'1 => false | id_x'2d,id_u'5'1 => false | id_x'2d,id_u'16'1 => false | id_u'13'1,id_intersect'ii'in => true | id_u'13'1,id_tautology'i'out => true | id_u'13'1,id_u'6'1 => true | id_u'13'1,id_u'3'1 => true | id_u'13'1,id_u'12'1 => true | id_u'13'1,id_u'2'1 => true | id_u'13'1,id_u'9'1 => true | id_u'13'1,id_iff => true | id_u'13'1,id_u'14'1 => true | id_u'13'1,id_intersect'ii'out => true | id_u'13'1,id_u'7'1 => true | id_u'13'1,id_x'2d => true | id_u'13'1,id_u'13'1 => true | id_u'13'1,id_sequent => false | id_u'13'1,id_u'10'1 => false | id_u'13'1,id_x'2a => false | id_u'13'1,id_tautology'i'in => false | id_u'13'1,id_cons => false | id_u'13'1,id_u'6'2 => false | id_u'13'1,id_x'2b => false | id_u'13'1,id_u'12'2 => false | id_u'13'1,id_reduce'ii'in => false | id_u'13'1,id_p => false | id_u'13'1,id_u'4'1 => false | id_u'13'1,id_u'15'1 => false | id_u'13'1,id_u'1'1 => false | id_u'13'1,id_u'8'1 => false | id_u'13'1,id_reduce'ii'out => false | id_u'13'1,id_nil => false | id_u'13'1,id_if => false | id_u'13'1,id_u'11'1 => false | id_u'13'1,id_u'5'1 => false | id_u'13'1,id_u'16'1 => false | id_sequent,id_intersect'ii'in => true | id_sequent,id_tautology'i'out => true | id_sequent,id_u'6'1 => true | id_sequent,id_u'3'1 => true | id_sequent,id_u'12'1 => true | id_sequent,id_u'2'1 => true | id_sequent,id_u'9'1 => true | id_sequent,id_iff => true | id_sequent,id_u'14'1 => true | id_sequent,id_intersect'ii'out => true | id_sequent,id_u'7'1 => true | id_sequent,id_x'2d => true | id_sequent,id_u'13'1 => true | id_sequent,id_sequent => true | id_sequent,id_u'10'1 => false | id_sequent,id_x'2a => false | id_sequent,id_tautology'i'in => false | id_sequent,id_cons => false | id_sequent,id_u'6'2 => false | id_sequent,id_x'2b => false | id_sequent,id_u'12'2 => false | id_sequent,id_reduce'ii'in => false | id_sequent,id_p => false | id_sequent,id_u'4'1 => false | id_sequent,id_u'15'1 => false | id_sequent,id_u'1'1 => false | id_sequent,id_u'8'1 => false | id_sequent,id_reduce'ii'out => false | id_sequent,id_nil => false | id_sequent,id_if => false | id_sequent,id_u'11'1 => false | id_sequent,id_u'5'1 => false | id_sequent,id_u'16'1 => false | id_u'10'1,id_intersect'ii'in => true | id_u'10'1,id_tautology'i'out => true | id_u'10'1,id_u'6'1 => true | id_u'10'1,id_u'3'1 => true | id_u'10'1,id_u'12'1 => true | id_u'10'1,id_u'2'1 => true | id_u'10'1,id_u'9'1 => true | id_u'10'1,id_iff => true | id_u'10'1,id_u'14'1 => true | id_u'10'1,id_intersect'ii'out => true | id_u'10'1,id_u'7'1 => true | id_u'10'1,id_x'2d => true | id_u'10'1,id_u'13'1 => true | id_u'10'1,id_sequent => true | id_u'10'1,id_u'10'1 => true | id_u'10'1,id_x'2a => false | id_u'10'1,id_tautology'i'in => false | id_u'10'1,id_cons => false | id_u'10'1,id_u'6'2 => false | id_u'10'1,id_x'2b => false | id_u'10'1,id_u'12'2 => false | id_u'10'1,id_reduce'ii'in => false | id_u'10'1,id_p => false | id_u'10'1,id_u'4'1 => false | id_u'10'1,id_u'15'1 => false | id_u'10'1,id_u'1'1 => false | id_u'10'1,id_u'8'1 => false | id_u'10'1,id_reduce'ii'out => false | id_u'10'1,id_nil => false | id_u'10'1,id_if => false | id_u'10'1,id_u'11'1 => false | id_u'10'1,id_u'5'1 => false | id_u'10'1,id_u'16'1 => false | id_x'2a,id_intersect'ii'in => true | id_x'2a,id_tautology'i'out => true | id_x'2a,id_u'6'1 => true | id_x'2a,id_u'3'1 => true | id_x'2a,id_u'12'1 => true | id_x'2a,id_u'2'1 => true | id_x'2a,id_u'9'1 => true | id_x'2a,id_iff => true | id_x'2a,id_u'14'1 => true | id_x'2a,id_intersect'ii'out => true | id_x'2a,id_u'7'1 => true | id_x'2a,id_x'2d => true | id_x'2a,id_u'13'1 => true | id_x'2a,id_sequent => true | id_x'2a,id_u'10'1 => true | id_x'2a,id_x'2a => true | id_x'2a,id_tautology'i'in => false | id_x'2a,id_cons => false | id_x'2a,id_u'6'2 => false | id_x'2a,id_x'2b => false | id_x'2a,id_u'12'2 => false | id_x'2a,id_reduce'ii'in => false | id_x'2a,id_p => false | id_x'2a,id_u'4'1 => false | id_x'2a,id_u'15'1 => false | id_x'2a,id_u'1'1 => false | id_x'2a,id_u'8'1 => false | id_x'2a,id_reduce'ii'out => false | id_x'2a,id_nil => false | id_x'2a,id_if => false | id_x'2a,id_u'11'1 => false | id_x'2a,id_u'5'1 => false | id_x'2a,id_u'16'1 => false | id_tautology'i'in,id_intersect'ii'in => true | id_tautology'i'in,id_tautology'i'out => true | id_tautology'i'in,id_u'6'1 => true | id_tautology'i'in,id_u'3'1 => true | id_tautology'i'in,id_u'12'1 => true | id_tautology'i'in,id_u'2'1 => true | id_tautology'i'in,id_u'9'1 => true | id_tautology'i'in,id_iff => true | id_tautology'i'in,id_u'14'1 => true | id_tautology'i'in,id_intersect'ii'out => true | id_tautology'i'in,id_u'7'1 => true | id_tautology'i'in,id_x'2d => true | id_tautology'i'in,id_u'13'1 => true | id_tautology'i'in,id_sequent => true | id_tautology'i'in,id_u'10'1 => true | id_tautology'i'in,id_x'2a => true | id_tautology'i'in,id_tautology'i'in => true | id_tautology'i'in,id_cons => false | id_tautology'i'in,id_u'6'2 => false | id_tautology'i'in,id_x'2b => false | id_tautology'i'in,id_u'12'2 => false | id_tautology'i'in,id_reduce'ii'in => false | id_tautology'i'in,id_p => false | id_tautology'i'in,id_u'4'1 => false | id_tautology'i'in,id_u'15'1 => false | id_tautology'i'in,id_u'1'1 => false | id_tautology'i'in,id_u'8'1 => false | id_tautology'i'in,id_reduce'ii'out => false | id_tautology'i'in,id_nil => false | id_tautology'i'in,id_if => false | id_tautology'i'in,id_u'11'1 => false | id_tautology'i'in,id_u'5'1 => false | id_tautology'i'in,id_u'16'1 => false | id_cons,id_intersect'ii'in => true | id_cons,id_tautology'i'out => true | id_cons,id_u'6'1 => true | id_cons,id_u'3'1 => true | id_cons,id_u'12'1 => true | id_cons,id_u'2'1 => true | id_cons,id_u'9'1 => true | id_cons,id_iff => true | id_cons,id_u'14'1 => true | id_cons,id_intersect'ii'out => true | id_cons,id_u'7'1 => true | id_cons,id_x'2d => true | id_cons,id_u'13'1 => true | id_cons,id_sequent => true | id_cons,id_u'10'1 => true | id_cons,id_x'2a => true | id_cons,id_tautology'i'in => true | id_cons,id_cons => true | id_cons,id_u'6'2 => false | id_cons,id_x'2b => false | id_cons,id_u'12'2 => false | id_cons,id_reduce'ii'in => false | id_cons,id_p => false | id_cons,id_u'4'1 => false | id_cons,id_u'15'1 => false | id_cons,id_u'1'1 => false | id_cons,id_u'8'1 => false | id_cons,id_reduce'ii'out => false | id_cons,id_nil => false | id_cons,id_if => false | id_cons,id_u'11'1 => false | id_cons,id_u'5'1 => false | id_cons,id_u'16'1 => false | id_u'6'2,id_intersect'ii'in => true | id_u'6'2,id_tautology'i'out => true | id_u'6'2,id_u'6'1 => true | id_u'6'2,id_u'3'1 => true | id_u'6'2,id_u'12'1 => true | id_u'6'2,id_u'2'1 => true | id_u'6'2,id_u'9'1 => true | id_u'6'2,id_iff => true | id_u'6'2,id_u'14'1 => true | id_u'6'2,id_intersect'ii'out => true | id_u'6'2,id_u'7'1 => true | id_u'6'2,id_x'2d => true | id_u'6'2,id_u'13'1 => true | id_u'6'2,id_sequent => true | id_u'6'2,id_u'10'1 => true | id_u'6'2,id_x'2a => true | id_u'6'2,id_tautology'i'in => true | id_u'6'2,id_cons => true | id_u'6'2,id_u'6'2 => true | id_u'6'2,id_x'2b => false | id_u'6'2,id_u'12'2 => false | id_u'6'2,id_reduce'ii'in => false | id_u'6'2,id_p => false | id_u'6'2,id_u'4'1 => false | id_u'6'2,id_u'15'1 => false | id_u'6'2,id_u'1'1 => false | id_u'6'2,id_u'8'1 => false | id_u'6'2,id_reduce'ii'out => false | id_u'6'2,id_nil => false | id_u'6'2,id_if => false | id_u'6'2,id_u'11'1 => false | id_u'6'2,id_u'5'1 => false | id_u'6'2,id_u'16'1 => false | id_x'2b,id_intersect'ii'in => true | id_x'2b,id_tautology'i'out => true | id_x'2b,id_u'6'1 => true | id_x'2b,id_u'3'1 => true | id_x'2b,id_u'12'1 => true | id_x'2b,id_u'2'1 => true | id_x'2b,id_u'9'1 => true | id_x'2b,id_iff => true | id_x'2b,id_u'14'1 => true | id_x'2b,id_intersect'ii'out => true | id_x'2b,id_u'7'1 => true | id_x'2b,id_x'2d => true | id_x'2b,id_u'13'1 => true | id_x'2b,id_sequent => true | id_x'2b,id_u'10'1 => true | id_x'2b,id_x'2a => true | id_x'2b,id_tautology'i'in => true | id_x'2b,id_cons => true | id_x'2b,id_u'6'2 => true | id_x'2b,id_x'2b => true | id_x'2b,id_u'12'2 => false | id_x'2b,id_reduce'ii'in => false | id_x'2b,id_p => false | id_x'2b,id_u'4'1 => false | id_x'2b,id_u'15'1 => false | id_x'2b,id_u'1'1 => false | id_x'2b,id_u'8'1 => false | id_x'2b,id_reduce'ii'out => false | id_x'2b,id_nil => false | id_x'2b,id_if => false | id_x'2b,id_u'11'1 => false | id_x'2b,id_u'5'1 => false | id_x'2b,id_u'16'1 => false | id_u'12'2,id_intersect'ii'in => true | id_u'12'2,id_tautology'i'out => true | id_u'12'2,id_u'6'1 => true | id_u'12'2,id_u'3'1 => true | id_u'12'2,id_u'12'1 => true | id_u'12'2,id_u'2'1 => true | id_u'12'2,id_u'9'1 => true | id_u'12'2,id_iff => true | id_u'12'2,id_u'14'1 => true | id_u'12'2,id_intersect'ii'out => true | id_u'12'2,id_u'7'1 => true | id_u'12'2,id_x'2d => true | id_u'12'2,id_u'13'1 => true | id_u'12'2,id_sequent => true | id_u'12'2,id_u'10'1 => true | id_u'12'2,id_x'2a => true | id_u'12'2,id_tautology'i'in => true | id_u'12'2,id_cons => true | id_u'12'2,id_u'6'2 => true | id_u'12'2,id_x'2b => true | id_u'12'2,id_u'12'2 => true | id_u'12'2,id_reduce'ii'in => false | id_u'12'2,id_p => false | id_u'12'2,id_u'4'1 => false | id_u'12'2,id_u'15'1 => false | id_u'12'2,id_u'1'1 => false | id_u'12'2,id_u'8'1 => false | id_u'12'2,id_reduce'ii'out => false | id_u'12'2,id_nil => false | id_u'12'2,id_if => false | id_u'12'2,id_u'11'1 => false | id_u'12'2,id_u'5'1 => false | id_u'12'2,id_u'16'1 => false | id_reduce'ii'in,id_intersect'ii'in => true | id_reduce'ii'in,id_tautology'i'out => true | id_reduce'ii'in,id_u'6'1 => true | id_reduce'ii'in,id_u'3'1 => true | id_reduce'ii'in,id_u'12'1 => true | id_reduce'ii'in,id_u'2'1 => true | id_reduce'ii'in,id_u'9'1 => true | id_reduce'ii'in,id_iff => true | id_reduce'ii'in,id_u'14'1 => true | id_reduce'ii'in,id_intersect'ii'out => true | id_reduce'ii'in,id_u'7'1 => true | id_reduce'ii'in,id_x'2d => true | id_reduce'ii'in,id_u'13'1 => true | id_reduce'ii'in,id_sequent => true | id_reduce'ii'in,id_u'10'1 => true | id_reduce'ii'in,id_x'2a => true | id_reduce'ii'in,id_tautology'i'in => true | id_reduce'ii'in,id_cons => true | id_reduce'ii'in,id_u'6'2 => true | id_reduce'ii'in,id_x'2b => true | id_reduce'ii'in,id_u'12'2 => true | id_reduce'ii'in,id_reduce'ii'in => true | id_reduce'ii'in,id_p => false | id_reduce'ii'in,id_u'4'1 => false | id_reduce'ii'in,id_u'15'1 => false | id_reduce'ii'in,id_u'1'1 => false | id_reduce'ii'in,id_u'8'1 => false | id_reduce'ii'in,id_reduce'ii'out => false | id_reduce'ii'in,id_nil => false | id_reduce'ii'in,id_if => false | id_reduce'ii'in,id_u'11'1 => false | id_reduce'ii'in,id_u'5'1 => false | id_reduce'ii'in,id_u'16'1 => false | id_p,id_intersect'ii'in => true | id_p,id_tautology'i'out => true | id_p,id_u'6'1 => true | id_p,id_u'3'1 => true | id_p,id_u'12'1 => true | id_p,id_u'2'1 => true | id_p,id_u'9'1 => true | id_p,id_iff => true | id_p,id_u'14'1 => true | id_p,id_intersect'ii'out => true | id_p,id_u'7'1 => true | id_p,id_x'2d => true | id_p,id_u'13'1 => true | id_p,id_sequent => true | id_p,id_u'10'1 => true | id_p,id_x'2a => true | id_p,id_tautology'i'in => true | id_p,id_cons => true | id_p,id_u'6'2 => true | id_p,id_x'2b => true | id_p,id_u'12'2 => true | id_p,id_reduce'ii'in => true | id_p,id_p => true | id_p,id_u'4'1 => false | id_p,id_u'15'1 => false | id_p,id_u'1'1 => false | id_p,id_u'8'1 => false | id_p,id_reduce'ii'out => false | id_p,id_nil => false | id_p,id_if => false | id_p,id_u'11'1 => false | id_p,id_u'5'1 => false | id_p,id_u'16'1 => false | id_u'4'1,id_intersect'ii'in => true | id_u'4'1,id_tautology'i'out => true | id_u'4'1,id_u'6'1 => true | id_u'4'1,id_u'3'1 => true | id_u'4'1,id_u'12'1 => true | id_u'4'1,id_u'2'1 => true | id_u'4'1,id_u'9'1 => true | id_u'4'1,id_iff => true | id_u'4'1,id_u'14'1 => true | id_u'4'1,id_intersect'ii'out => true | id_u'4'1,id_u'7'1 => true | id_u'4'1,id_x'2d => true | id_u'4'1,id_u'13'1 => true | id_u'4'1,id_sequent => true | id_u'4'1,id_u'10'1 => true | id_u'4'1,id_x'2a => true | id_u'4'1,id_tautology'i'in => true | id_u'4'1,id_cons => true | id_u'4'1,id_u'6'2 => true | id_u'4'1,id_x'2b => true | id_u'4'1,id_u'12'2 => true | id_u'4'1,id_reduce'ii'in => true | id_u'4'1,id_p => true | id_u'4'1,id_u'4'1 => true | id_u'4'1,id_u'15'1 => false | id_u'4'1,id_u'1'1 => false | id_u'4'1,id_u'8'1 => false | id_u'4'1,id_reduce'ii'out => false | id_u'4'1,id_nil => false | id_u'4'1,id_if => false | id_u'4'1,id_u'11'1 => false | id_u'4'1,id_u'5'1 => false | id_u'4'1,id_u'16'1 => false | id_u'15'1,id_intersect'ii'in => true | id_u'15'1,id_tautology'i'out => true | id_u'15'1,id_u'6'1 => true | id_u'15'1,id_u'3'1 => true | id_u'15'1,id_u'12'1 => true | id_u'15'1,id_u'2'1 => true | id_u'15'1,id_u'9'1 => true | id_u'15'1,id_iff => true | id_u'15'1,id_u'14'1 => true | id_u'15'1,id_intersect'ii'out => true | id_u'15'1,id_u'7'1 => true | id_u'15'1,id_x'2d => true | id_u'15'1,id_u'13'1 => true | id_u'15'1,id_sequent => true | id_u'15'1,id_u'10'1 => true | id_u'15'1,id_x'2a => true | id_u'15'1,id_tautology'i'in => true | id_u'15'1,id_cons => true | id_u'15'1,id_u'6'2 => true | id_u'15'1,id_x'2b => true | id_u'15'1,id_u'12'2 => true | id_u'15'1,id_reduce'ii'in => true | id_u'15'1,id_p => true | id_u'15'1,id_u'4'1 => true | id_u'15'1,id_u'15'1 => true | id_u'15'1,id_u'1'1 => false | id_u'15'1,id_u'8'1 => false | id_u'15'1,id_reduce'ii'out => false | id_u'15'1,id_nil => false | id_u'15'1,id_if => false | id_u'15'1,id_u'11'1 => false | id_u'15'1,id_u'5'1 => false | id_u'15'1,id_u'16'1 => false | id_u'1'1,id_intersect'ii'in => true | id_u'1'1,id_tautology'i'out => true | id_u'1'1,id_u'6'1 => true | id_u'1'1,id_u'3'1 => true | id_u'1'1,id_u'12'1 => true | id_u'1'1,id_u'2'1 => true | id_u'1'1,id_u'9'1 => true | id_u'1'1,id_iff => true | id_u'1'1,id_u'14'1 => true | id_u'1'1,id_intersect'ii'out => true | id_u'1'1,id_u'7'1 => true | id_u'1'1,id_x'2d => true | id_u'1'1,id_u'13'1 => true | id_u'1'1,id_sequent => true | id_u'1'1,id_u'10'1 => true | id_u'1'1,id_x'2a => true | id_u'1'1,id_tautology'i'in => true | id_u'1'1,id_cons => true | id_u'1'1,id_u'6'2 => true | id_u'1'1,id_x'2b => true | id_u'1'1,id_u'12'2 => true | id_u'1'1,id_reduce'ii'in => true | id_u'1'1,id_p => true | id_u'1'1,id_u'4'1 => true | id_u'1'1,id_u'15'1 => true | id_u'1'1,id_u'1'1 => true | id_u'1'1,id_u'8'1 => false | id_u'1'1,id_reduce'ii'out => false | id_u'1'1,id_nil => false | id_u'1'1,id_if => false | id_u'1'1,id_u'11'1 => false | id_u'1'1,id_u'5'1 => false | id_u'1'1,id_u'16'1 => false | id_u'8'1,id_intersect'ii'in => true | id_u'8'1,id_tautology'i'out => true | id_u'8'1,id_u'6'1 => true | id_u'8'1,id_u'3'1 => true | id_u'8'1,id_u'12'1 => true | id_u'8'1,id_u'2'1 => true | id_u'8'1,id_u'9'1 => true | id_u'8'1,id_iff => true | id_u'8'1,id_u'14'1 => true | id_u'8'1,id_intersect'ii'out => true | id_u'8'1,id_u'7'1 => true | id_u'8'1,id_x'2d => true | id_u'8'1,id_u'13'1 => true | id_u'8'1,id_sequent => true | id_u'8'1,id_u'10'1 => true | id_u'8'1,id_x'2a => true | id_u'8'1,id_tautology'i'in => true | id_u'8'1,id_cons => true | id_u'8'1,id_u'6'2 => true | id_u'8'1,id_x'2b => true | id_u'8'1,id_u'12'2 => true | id_u'8'1,id_reduce'ii'in => true | id_u'8'1,id_p => true | id_u'8'1,id_u'4'1 => true | id_u'8'1,id_u'15'1 => true | id_u'8'1,id_u'1'1 => true | id_u'8'1,id_u'8'1 => true | id_u'8'1,id_reduce'ii'out => false | id_u'8'1,id_nil => false | id_u'8'1,id_if => false | id_u'8'1,id_u'11'1 => false | id_u'8'1,id_u'5'1 => false | id_u'8'1,id_u'16'1 => false | id_reduce'ii'out,id_intersect'ii'in => true | id_reduce'ii'out,id_tautology'i'out => true | id_reduce'ii'out,id_u'6'1 => true | id_reduce'ii'out,id_u'3'1 => true | id_reduce'ii'out,id_u'12'1 => true | id_reduce'ii'out,id_u'2'1 => true | id_reduce'ii'out,id_u'9'1 => true | id_reduce'ii'out,id_iff => true | id_reduce'ii'out,id_u'14'1 => true | id_reduce'ii'out,id_intersect'ii'out => true | id_reduce'ii'out,id_u'7'1 => true | id_reduce'ii'out,id_x'2d => true | id_reduce'ii'out,id_u'13'1 => true | id_reduce'ii'out,id_sequent => true | id_reduce'ii'out,id_u'10'1 => true | id_reduce'ii'out,id_x'2a => true | id_reduce'ii'out,id_tautology'i'in => true | id_reduce'ii'out,id_cons => true | id_reduce'ii'out,id_u'6'2 => true | id_reduce'ii'out,id_x'2b => true | id_reduce'ii'out,id_u'12'2 => true | id_reduce'ii'out,id_reduce'ii'in => true | id_reduce'ii'out,id_p => true | id_reduce'ii'out,id_u'4'1 => true | id_reduce'ii'out,id_u'15'1 => true | id_reduce'ii'out,id_u'1'1 => true | id_reduce'ii'out,id_u'8'1 => true | id_reduce'ii'out,id_reduce'ii'out => true | id_reduce'ii'out,id_nil => false | id_reduce'ii'out,id_if => false | id_reduce'ii'out,id_u'11'1 => false | id_reduce'ii'out,id_u'5'1 => false | id_reduce'ii'out,id_u'16'1 => false | id_nil,id_intersect'ii'in => true | id_nil,id_tautology'i'out => true | id_nil,id_u'6'1 => true | id_nil,id_u'3'1 => true | id_nil,id_u'12'1 => true | id_nil,id_u'2'1 => true | id_nil,id_u'9'1 => true | id_nil,id_iff => true | id_nil,id_u'14'1 => true | id_nil,id_intersect'ii'out => true | id_nil,id_u'7'1 => true | id_nil,id_x'2d => true | id_nil,id_u'13'1 => true | id_nil,id_sequent => true | id_nil,id_u'10'1 => true | id_nil,id_x'2a => true | id_nil,id_tautology'i'in => true | id_nil,id_cons => true | id_nil,id_u'6'2 => true | id_nil,id_x'2b => true | id_nil,id_u'12'2 => true | id_nil,id_reduce'ii'in => true | id_nil,id_p => true | id_nil,id_u'4'1 => true | id_nil,id_u'15'1 => true | id_nil,id_u'1'1 => true | id_nil,id_u'8'1 => true | id_nil,id_reduce'ii'out => true | id_nil,id_nil => true | id_nil,id_if => false | id_nil,id_u'11'1 => false | id_nil,id_u'5'1 => false | id_nil,id_u'16'1 => false | id_if,id_intersect'ii'in => true | id_if,id_tautology'i'out => true | id_if,id_u'6'1 => true | id_if,id_u'3'1 => true | id_if,id_u'12'1 => true | id_if,id_u'2'1 => true | id_if,id_u'9'1 => true | id_if,id_iff => true | id_if,id_u'14'1 => true | id_if,id_intersect'ii'out => true | id_if,id_u'7'1 => true | id_if,id_x'2d => true | id_if,id_u'13'1 => true | id_if,id_sequent => true | id_if,id_u'10'1 => true | id_if,id_x'2a => true | id_if,id_tautology'i'in => true | id_if,id_cons => true | id_if,id_u'6'2 => true | id_if,id_x'2b => true | id_if,id_u'12'2 => true | id_if,id_reduce'ii'in => true | id_if,id_p => true | id_if,id_u'4'1 => true | id_if,id_u'15'1 => true | id_if,id_u'1'1 => true | id_if,id_u'8'1 => true | id_if,id_reduce'ii'out => true | id_if,id_nil => true | id_if,id_if => true | id_if,id_u'11'1 => false | id_if,id_u'5'1 => false | id_if,id_u'16'1 => false | id_u'11'1,id_intersect'ii'in => true | id_u'11'1,id_tautology'i'out => true | id_u'11'1,id_u'6'1 => true | id_u'11'1,id_u'3'1 => true | id_u'11'1,id_u'12'1 => true | id_u'11'1,id_u'2'1 => true | id_u'11'1,id_u'9'1 => true | id_u'11'1,id_iff => true | id_u'11'1,id_u'14'1 => true | id_u'11'1,id_intersect'ii'out => true | id_u'11'1,id_u'7'1 => true | id_u'11'1,id_x'2d => true | id_u'11'1,id_u'13'1 => true | id_u'11'1,id_sequent => true | id_u'11'1,id_u'10'1 => true | id_u'11'1,id_x'2a => true | id_u'11'1,id_tautology'i'in => true | id_u'11'1,id_cons => true | id_u'11'1,id_u'6'2 => true | id_u'11'1,id_x'2b => true | id_u'11'1,id_u'12'2 => true | id_u'11'1,id_reduce'ii'in => true | id_u'11'1,id_p => true | id_u'11'1,id_u'4'1 => true | id_u'11'1,id_u'15'1 => true | id_u'11'1,id_u'1'1 => true | id_u'11'1,id_u'8'1 => true | id_u'11'1,id_reduce'ii'out => true | id_u'11'1,id_nil => true | id_u'11'1,id_if => true | id_u'11'1,id_u'11'1 => true | id_u'11'1,id_u'5'1 => false | id_u'11'1,id_u'16'1 => false | id_u'5'1,id_intersect'ii'in => true | id_u'5'1,id_tautology'i'out => true | id_u'5'1,id_u'6'1 => true | id_u'5'1,id_u'3'1 => true | id_u'5'1,id_u'12'1 => true | id_u'5'1,id_u'2'1 => true | id_u'5'1,id_u'9'1 => true | id_u'5'1,id_iff => true | id_u'5'1,id_u'14'1 => true | id_u'5'1,id_intersect'ii'out => true | id_u'5'1,id_u'7'1 => true | id_u'5'1,id_x'2d => true | id_u'5'1,id_u'13'1 => true | id_u'5'1,id_sequent => true | id_u'5'1,id_u'10'1 => true | id_u'5'1,id_x'2a => true | id_u'5'1,id_tautology'i'in => true | id_u'5'1,id_cons => true | id_u'5'1,id_u'6'2 => true | id_u'5'1,id_x'2b => true | id_u'5'1,id_u'12'2 => true | id_u'5'1,id_reduce'ii'in => true | id_u'5'1,id_p => true | id_u'5'1,id_u'4'1 => true | id_u'5'1,id_u'15'1 => true | id_u'5'1,id_u'1'1 => true | id_u'5'1,id_u'8'1 => true | id_u'5'1,id_reduce'ii'out => true | id_u'5'1,id_nil => true | id_u'5'1,id_if => true | id_u'5'1,id_u'11'1 => true | id_u'5'1,id_u'5'1 => true | id_u'5'1,id_u'16'1 => false | id_u'16'1,id_intersect'ii'in => true | id_u'16'1,id_tautology'i'out => true | id_u'16'1,id_u'6'1 => true | id_u'16'1,id_u'3'1 => true | id_u'16'1,id_u'12'1 => true | id_u'16'1,id_u'2'1 => true | id_u'16'1,id_u'9'1 => true | id_u'16'1,id_iff => true | id_u'16'1,id_u'14'1 => true | id_u'16'1,id_intersect'ii'out => true | id_u'16'1,id_u'7'1 => true | id_u'16'1,id_x'2d => true | id_u'16'1,id_u'13'1 => true | id_u'16'1,id_sequent => true | id_u'16'1,id_u'10'1 => true | id_u'16'1,id_x'2a => true | id_u'16'1,id_tautology'i'in => true | id_u'16'1,id_cons => true | id_u'16'1,id_u'6'2 => true | id_u'16'1,id_x'2b => true | id_u'16'1,id_u'12'2 => true | id_u'16'1,id_reduce'ii'in => true | id_u'16'1,id_p => true | id_u'16'1,id_u'4'1 => true | id_u'16'1,id_u'15'1 => true | id_u'16'1,id_u'1'1 => true | id_u'16'1,id_u'8'1 => true | id_u'16'1,id_reduce'ii'out => true | id_u'16'1,id_nil => true | id_u'16'1,id_if => true | id_u'16'1,id_u'11'1 => true | id_u'16'1,id_u'5'1 => true | id_u'16'1,id_u'16'1 => 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 := (* intersect'ii'in(cons(X_,X0_),cons(X_,X1_)) -> intersect'ii'out *) | R_xml_0_rule_0 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) (algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil)):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 1):: (algebra.Alg.Var 3)::nil))::nil)) (* intersect'ii'in(Xs_,cons(X0_,Ys_)) -> u'1'1(intersect'ii'in(Xs_,Ys_)) *) | R_xml_0_rule_1 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'1'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Var 4):: (algebra.Alg.Var 5)::nil))::nil)) (algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Var 4):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 2):: (algebra.Alg.Var 5)::nil))::nil)) (* u'1'1(intersect'ii'out) -> intersect'ii'out *) | R_xml_0_rule_2 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) (algebra.Alg.Term algebra.F.id_u'1'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'out nil)::nil)) (* intersect'ii'in(cons(X0_,Xs_),Ys_) -> u'2'1(intersect'ii'in(Xs_,Ys_)) *) | R_xml_0_rule_3 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'2'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Var 4):: (algebra.Alg.Var 5)::nil))::nil)) (algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 2)::(algebra.Alg.Var 4)::nil)):: (algebra.Alg.Var 5)::nil)) (* u'2'1(intersect'ii'out) -> intersect'ii'out *) | R_xml_0_rule_4 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) (algebra.Alg.Term algebra.F.id_u'2'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'out nil)::nil)) (* reduce'ii'in(sequent(cons(if(A_,B_),Fs_),Gs_),NF_) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B_),A_),Fs_),Gs_),NF_)) *) | R_xml_0_rule_5 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'3'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d ((algebra.Alg.Var 7)::nil)):: (algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)) (* u'3'1(reduce'ii'out) -> reduce'ii'out *) | R_xml_0_rule_6 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'3'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) (* reduce'ii'in(sequent(cons(iff(A_,B_),Fs_),Gs_),NF_) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A_,B_),if(B_,A_)),Fs_),Gs_),NF_)) *) | R_xml_0_rule_7 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'4'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil))::(algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 7):: (algebra.Alg.Var 6)::nil))::nil)):: (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff ((algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)) (* u'4'1(reduce'ii'out) -> reduce'ii'out *) | R_xml_0_rule_8 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'4'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) (* reduce'ii'in(sequent(cons(x'2a(F1_,F2_),Fs_),Gs_),NF_) -> u'5'1(reduce'ii'in(sequent(cons(F1_,cons(F2_,Fs_)),Gs_),NF_)) *) | R_xml_0_rule_9 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'5'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 11)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 12):: (algebra.Alg.Var 8)::nil))::nil)):: (algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Var 11):: (algebra.Alg.Var 12)::nil))::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)) (* u'5'1(reduce'ii'out) -> reduce'ii'out *) | R_xml_0_rule_10 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'5'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) (* reduce'ii'in(sequent(cons(x'2b(F1_,F2_),Fs_),Gs_),NF_) -> u'6'1(reduce'ii'in(sequent(cons(F1_,Fs_),Gs_),NF_),F2_,Fs_,Gs_,NF_) *) | R_xml_0_rule_11 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 11)::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)):: (algebra.Alg.Var 12)::(algebra.Alg.Var 8):: (algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Var 11):: (algebra.Alg.Var 12)::nil))::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)) (* u'6'1(reduce'ii'out,F2_,Fs_,Gs_,NF_) -> u'6'2(reduce'ii'in(sequent(cons(F2_,Fs_),Gs_),NF_)) *) | R_xml_0_rule_12 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'6'2 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 12)::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil))::nil)) (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::(algebra.Alg.Var 12):: (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil)) (* u'6'2(reduce'ii'out) -> reduce'ii'out *) | R_xml_0_rule_13 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'6'2 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) (* reduce'ii'in(sequent(cons(x'2d(F1_),Fs_),Gs_),NF_) -> u'7'1(reduce'ii'in(sequent(Fs_,cons(F1_,Gs_)),NF_)) *) | R_xml_0_rule_14 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'7'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 11)::(algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Var 10)::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d ((algebra.Alg.Var 11)::nil)):: (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)) (* u'7'1(reduce'ii'out) -> reduce'ii'out *) | R_xml_0_rule_15 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'7'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) (* reduce'ii'in(sequent(Fs_,cons(if(A_,B_),Gs_)),NF_) -> u'8'1(reduce'ii'in(sequent(Fs_,cons(x'2b(x'2d(B_),A_),Gs_)),NF_)) *) | R_xml_0_rule_16 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'8'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d ((algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 6)::nil)):: (algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Var 10)::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)):: (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil)) (* u'8'1(reduce'ii'out) -> reduce'ii'out *) | R_xml_0_rule_17 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'8'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) (* reduce'ii'in(sequent(Fs_,cons(iff(A_,B_),Gs_)),NF_) -> u'9'1(reduce'ii'in(sequent(Fs_,cons(x'2a(if(A_,B_),if(B_,A_)),Gs_)),NF_)) *) | R_xml_0_rule_18 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'9'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)):: (algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 7):: (algebra.Alg.Var 6)::nil))::nil)):: (algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Var 10)::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)):: (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil)) (* u'9'1(reduce'ii'out) -> reduce'ii'out *) | R_xml_0_rule_19 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'9'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) (* reduce'ii'in(sequent(cons(p(V_),Fs_),Gs_),sequent(Left_,Right_)) -> u'10'1(reduce'ii'in(sequent(Fs_,Gs_),sequent(cons(p(V_),Left_),Right_))) *) | R_xml_0_rule_20 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'10'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8):: (algebra.Alg.Var 9)::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p ((algebra.Alg.Var 13)::nil)):: (algebra.Alg.Var 14)::nil)):: (algebra.Alg.Var 15)::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p ((algebra.Alg.Var 13)::nil)):: (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil)):: (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 14):: (algebra.Alg.Var 15)::nil))::nil)) (* u'10'1(reduce'ii'out) -> reduce'ii'out *) | R_xml_0_rule_21 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'10'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) (* reduce'ii'in(sequent(Fs_,cons(x'2b(G1_,G2_),Gs_)),NF_) -> u'11'1(reduce'ii'in(sequent(Fs_,cons(G1_,cons(G2_,Gs_))),NF_)) *) | R_xml_0_rule_22 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'11'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 16)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 17):: (algebra.Alg.Var 9)::nil))::nil))::nil)):: (algebra.Alg.Var 10)::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Var 16)::(algebra.Alg.Var 17)::nil)):: (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil)) (* u'11'1(reduce'ii'out) -> reduce'ii'out *) | R_xml_0_rule_23 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'11'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) (* reduce'ii'in(sequent(Fs_,cons(x'2a(G1_,G2_),Gs_)),NF_) -> u'12'1(reduce'ii'in(sequent(Fs_,cons(G1_,Gs_)),NF_),Fs_,G2_,Gs_,NF_) *) | R_xml_0_rule_24 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 16)::(algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Var 10)::nil))::(algebra.Alg.Var 8):: (algebra.Alg.Var 17)::(algebra.Alg.Var 9):: (algebra.Alg.Var 10)::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Var 16)::(algebra.Alg.Var 17)::nil)):: (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil)) (* u'12'1(reduce'ii'out,Fs_,G2_,Gs_,NF_) -> u'12'2(reduce'ii'in(sequent(Fs_,cons(G2_,Gs_)),NF_)) *) | R_xml_0_rule_25 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'12'2 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 17)::(algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Var 10)::nil))::nil)) (algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::(algebra.Alg.Var 8):: (algebra.Alg.Var 17)::(algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil)) (* u'12'2(reduce'ii'out) -> reduce'ii'out *) | R_xml_0_rule_26 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'12'2 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) (* reduce'ii'in(sequent(Fs_,cons(x'2d(G1_),Gs_)),NF_) -> u'13'1(reduce'ii'in(sequent(cons(G1_,Fs_),Gs_),NF_)) *) | R_xml_0_rule_27 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'13'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 16)::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d ((algebra.Alg.Var 16)::nil))::(algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Var 10)::nil)) (* u'13'1(reduce'ii'out) -> reduce'ii'out *) | R_xml_0_rule_28 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'13'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) (* reduce'ii'in(sequent(nil,cons(p(V_),Gs_)),sequent(Left_,Right_)) -> u'14'1(reduce'ii'in(sequent(nil,Gs_),sequent(Left_,cons(p(V_),Right_)))) *) | R_xml_0_rule_29 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'14'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Var 9)::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 14):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p ((algebra.Alg.Var 13)::nil)):: (algebra.Alg.Var 15)::nil))::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p ((algebra.Alg.Var 13)::nil))::(algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 14):: (algebra.Alg.Var 15)::nil))::nil)) (* u'14'1(reduce'ii'out) -> reduce'ii'out *) | R_xml_0_rule_30 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'14'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) (* reduce'ii'in(sequent(nil,nil),sequent(F1_,F2_)) -> u'15'1(intersect'ii'in(F1_,F2_)) *) | R_xml_0_rule_31 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'15'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Var 11):: (algebra.Alg.Var 12)::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil):: (algebra.Alg.Term algebra.F.id_nil nil)::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 11):: (algebra.Alg.Var 12)::nil))::nil)) (* u'15'1(intersect'ii'out) -> reduce'ii'out *) | R_xml_0_rule_32 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) (algebra.Alg.Term algebra.F.id_u'15'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'out nil)::nil)) (* tautology'i'in(F_) -> u'16'1(reduce'ii'in(sequent(nil,cons(F_,nil)),sequent(nil,nil))) *) | R_xml_0_rule_33 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_u'16'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 18)::(algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_tautology'i'in ((algebra.Alg.Var 18)::nil)) (* u'16'1(reduce'ii'out) -> tautology'i'out *) | R_xml_0_rule_34 : R_xml_0_rules (algebra.Alg.Term algebra.F.id_tautology'i'out nil) (algebra.Alg.Term algebra.F.id_u'16'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)) . Definition R_xml_0_rule_as_list_0 := ((algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 1)::(algebra.Alg.Var 2)::nil)):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 1):: (algebra.Alg.Var 3)::nil))::nil)), (algebra.Alg.Term algebra.F.id_intersect'ii'out nil))::nil. Definition R_xml_0_rule_as_list_1 := ((algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Var 4):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 2):: (algebra.Alg.Var 5)::nil))::nil)), (algebra.Alg.Term algebra.F.id_u'1'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Var 4):: (algebra.Alg.Var 5)::nil))::nil)))::R_xml_0_rule_as_list_0. Definition R_xml_0_rule_as_list_2 := ((algebra.Alg.Term algebra.F.id_u'1'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_intersect'ii'out nil)):: R_xml_0_rule_as_list_1. Definition R_xml_0_rule_as_list_3 := ((algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 2)::(algebra.Alg.Var 4)::nil)):: (algebra.Alg.Var 5)::nil)), (algebra.Alg.Term algebra.F.id_u'2'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Var 4):: (algebra.Alg.Var 5)::nil))::nil)))::R_xml_0_rule_as_list_2. Definition R_xml_0_rule_as_list_4 := ((algebra.Alg.Term algebra.F.id_u'2'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_intersect'ii'out nil)):: R_xml_0_rule_as_list_3. Definition R_xml_0_rule_as_list_5 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_u'3'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d ((algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 6)::nil)):: (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_4. Definition R_xml_0_rule_as_list_6 := ((algebra.Alg.Term algebra.F.id_u'3'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_5. Definition R_xml_0_rule_as_list_7 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff ((algebra.Alg.Var 6):: (algebra.Alg.Var 7)::nil))::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_u'4'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::(algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 7):: (algebra.Alg.Var 6)::nil))::nil))::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil))::nil))):: R_xml_0_rule_as_list_6. Definition R_xml_0_rule_as_list_8 := ((algebra.Alg.Term algebra.F.id_u'4'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_7. Definition R_xml_0_rule_as_list_9 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Var 11):: (algebra.Alg.Var 12)::nil))::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_u'5'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 11):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 12):: (algebra.Alg.Var 8)::nil))::nil))::(algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_8. Definition R_xml_0_rule_as_list_10 := ((algebra.Alg.Term algebra.F.id_u'5'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_9. Definition R_xml_0_rule_as_list_11 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Var 11):: (algebra.Alg.Var 12)::nil))::(algebra.Alg.Var 8)::nil)):: (algebra.Alg.Var 9)::nil))::(algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 11):: (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil))::(algebra.Alg.Var 12)::(algebra.Alg.Var 8):: (algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))):: R_xml_0_rule_as_list_10. Definition R_xml_0_rule_as_list_12 := ((algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::(algebra.Alg.Var 12):: (algebra.Alg.Var 8)::(algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_u'6'2 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 12):: (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_11. Definition R_xml_0_rule_as_list_13 := ((algebra.Alg.Term algebra.F.id_u'6'2 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_12. Definition R_xml_0_rule_as_list_14 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d ((algebra.Alg.Var 11)::nil)):: (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_u'7'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 11)::(algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_13. Definition R_xml_0_rule_as_list_15 := ((algebra.Alg.Term algebra.F.id_u'7'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_14. Definition R_xml_0_rule_as_list_16 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)):: (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_u'8'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d ((algebra.Alg.Var 7)::nil)):: (algebra.Alg.Var 6)::nil))::(algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_15. Definition R_xml_0_rule_as_list_17 := ((algebra.Alg.Term algebra.F.id_u'8'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_16. Definition R_xml_0_rule_as_list_18 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil)):: (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_u'9'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 6)::(algebra.Alg.Var 7)::nil))::(algebra.Alg.Term algebra.F.id_if ((algebra.Alg.Var 7):: (algebra.Alg.Var 6)::nil))::nil))::(algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_17. Definition R_xml_0_rule_as_list_19 := ((algebra.Alg.Term algebra.F.id_u'9'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_18. Definition R_xml_0_rule_as_list_20 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p ((algebra.Alg.Var 13)::nil)):: (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil)):: (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 14):: (algebra.Alg.Var 15)::nil))::nil)), (algebra.Alg.Term algebra.F.id_u'10'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Var 9)::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p ((algebra.Alg.Var 13)::nil)):: (algebra.Alg.Var 14)::nil))::(algebra.Alg.Var 15)::nil))::nil))::nil))):: R_xml_0_rule_as_list_19. Definition R_xml_0_rule_as_list_21 := ((algebra.Alg.Term algebra.F.id_u'10'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_20. Definition R_xml_0_rule_as_list_22 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Var 16)::(algebra.Alg.Var 17)::nil)):: (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_u'11'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 16)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 17)::(algebra.Alg.Var 9)::nil))::nil))::nil)):: (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_21. Definition R_xml_0_rule_as_list_23 := ((algebra.Alg.Term algebra.F.id_u'11'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_22. Definition R_xml_0_rule_as_list_24 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Var 16)::(algebra.Alg.Var 17)::nil)):: (algebra.Alg.Var 9)::nil))::nil))::(algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 16)::(algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Var 10)::nil))::(algebra.Alg.Var 8)::(algebra.Alg.Var 17):: (algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil))):: R_xml_0_rule_as_list_23. Definition R_xml_0_rule_as_list_25 := ((algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::(algebra.Alg.Var 8):: (algebra.Alg.Var 17)::(algebra.Alg.Var 9)::(algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_u'12'2 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 17)::(algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_24. Definition R_xml_0_rule_as_list_26 := ((algebra.Alg.Term algebra.F.id_u'12'2 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_25. Definition R_xml_0_rule_as_list_27 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 8)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d ((algebra.Alg.Var 16)::nil))::(algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Var 10)::nil)), (algebra.Alg.Term algebra.F.id_u'13'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 16):: (algebra.Alg.Var 8)::nil))::(algebra.Alg.Var 9)::nil)):: (algebra.Alg.Var 10)::nil))::nil)))::R_xml_0_rule_as_list_26. Definition R_xml_0_rule_as_list_28 := ((algebra.Alg.Term algebra.F.id_u'13'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_27. Definition R_xml_0_rule_as_list_29 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p ((algebra.Alg.Var 13)::nil))::(algebra.Alg.Var 9)::nil))::nil)):: (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 14):: (algebra.Alg.Var 15)::nil))::nil)), (algebra.Alg.Term algebra.F.id_u'14'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Var 9)::nil)):: (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 14):: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p ((algebra.Alg.Var 13)::nil)):: (algebra.Alg.Var 15)::nil))::nil))::nil))::nil))):: R_xml_0_rule_as_list_28. Definition R_xml_0_rule_as_list_30 := ((algebra.Alg.Term algebra.F.id_u'14'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_29. Definition R_xml_0_rule_as_list_31 := ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil):: (algebra.Alg.Term algebra.F.id_nil nil)::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Var 11):: (algebra.Alg.Var 12)::nil))::nil)), (algebra.Alg.Term algebra.F.id_u'15'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'in ((algebra.Alg.Var 11):: (algebra.Alg.Var 12)::nil))::nil)))::R_xml_0_rule_as_list_30. Definition R_xml_0_rule_as_list_32 := ((algebra.Alg.Term algebra.F.id_u'15'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)):: R_xml_0_rule_as_list_31. Definition R_xml_0_rule_as_list_33 := ((algebra.Alg.Term algebra.F.id_tautology'i'in ((algebra.Alg.Var 18)::nil)), (algebra.Alg.Term algebra.F.id_u'16'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Var 18)::(algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil):: (algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil))::nil))):: R_xml_0_rule_as_list_32. Definition R_xml_0_rule_as_list_34 := ((algebra.Alg.Term algebra.F.id_u'16'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'out nil)::nil)), (algebra.Alg.Term algebra.F.id_tautology'i'out nil)):: R_xml_0_rule_as_list_33. Definition R_xml_0_rule_as_list := R_xml_0_rule_as_list_34. 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 35. 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. rewrite <- or_ext_generated.or12_equiv in H|-. case H;clear H;intros H. injection H;intros ;subst;constructor 11. 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_12 (x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30 x31:Prop) : Prop := | conj_12 : x20->x21->x22->x23->x24->x25->x26->x27->x28->x29->x30->x31-> and_12 x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30 x31 . 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_12 (t = (algebra.Alg.Term algebra.F.id_tautology'i'out nil) -> t' = (algebra.Alg.Term algebra.F.id_tautology'i'out nil)) (forall x21 x23, t = (algebra.Alg.Term algebra.F.id_iff (x21::x23::nil)) -> exists x20, exists x22, t' = (algebra.Alg.Term algebra.F.id_iff (x20::x22::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x22 x23)) (t = (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) -> t' = (algebra.Alg.Term algebra.F.id_intersect'ii'out nil)) (forall x21, t = (algebra.Alg.Term algebra.F.id_x'2d (x21::nil)) -> exists x20, t' = (algebra.Alg.Term algebra.F.id_x'2d (x20::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)) (forall x21 x23, t = (algebra.Alg.Term algebra.F.id_sequent (x21::x23::nil)) -> exists x20, exists x22, t' = (algebra.Alg.Term algebra.F.id_sequent (x20::x22::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x22 x23)) (forall x21 x23, t = (algebra.Alg.Term algebra.F.id_x'2a (x21::x23::nil)) -> exists x20, exists x22, t' = (algebra.Alg.Term algebra.F.id_x'2a (x20::x22::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x22 x23)) (forall x21 x23, t = (algebra.Alg.Term algebra.F.id_cons (x21::x23::nil)) -> exists x20, exists x22, t' = (algebra.Alg.Term algebra.F.id_cons (x20::x22::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x22 x23)) (forall x21 x23, t = (algebra.Alg.Term algebra.F.id_x'2b (x21::x23::nil)) -> exists x20, exists x22, t' = (algebra.Alg.Term algebra.F.id_x'2b (x20::x22::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x22 x23)) (forall x21, t = (algebra.Alg.Term algebra.F.id_p (x21::nil)) -> exists x20, t' = (algebra.Alg.Term algebra.F.id_p (x20::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)) (t = (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) -> t' = (algebra.Alg.Term algebra.F.id_reduce'ii'out nil)) (t = (algebra.Alg.Term algebra.F.id_nil nil) -> t' = (algebra.Alg.Term algebra.F.id_nil nil)) (forall x21 x23, t = (algebra.Alg.Term algebra.F.id_if (x21::x23::nil)) -> exists x20, exists x22, t' = (algebra.Alg.Term algebra.F.id_if (x20::x22::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x22 x23)). Proof. intros t t' H. induction H as [|y IH z z_to_y] using closure_extension.refl_trans_clos_ind2. constructor 1. intros H;intuition;constructor 1. intros x21 x23 H;exists x21;exists x23;intuition;constructor 1. intros H;intuition;constructor 1. intros x21 H;exists x21;intuition;constructor 1. intros x21 x23 H;exists x21;exists x23;intuition;constructor 1. intros x21 x23 H;exists x21;exists x23;intuition;constructor 1. intros x21 x23 H;exists x21;exists x23;intuition;constructor 1. intros x21 x23 H;exists x21;exists x23;intuition;constructor 1. intros x21 H;exists x21;intuition;constructor 1. intros H;intuition;constructor 1. intros H;intuition;constructor 1. intros x21 x23 H;exists x21;exists x23;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_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if]. constructor. clear H_id_iff H_id_intersect'ii'out H_id_x'2d H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if;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_tautology'i'out H_id_intersect'ii'out H_id_x'2d H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if; intros x21 x23 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 x21 |- _ => destruct (H_id_iff y x23 (refl_equal _)) as [x20 [x22]];intros ; intuition;exists x20;exists x22;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x23 |- _ => destruct (H_id_iff x21 y (refl_equal _)) as [x20 [x22]];intros ; intuition;exists x20;exists x22;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_tautology'i'out H_id_iff H_id_x'2d H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if;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_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if; intros x21 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 x21 |- _ => destruct (H_id_x'2d y (refl_equal _)) as [x20];intros ;intuition; exists x20;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if; intros x21 x23 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 x21 |- _ => destruct (H_id_sequent y x23 (refl_equal _)) as [x20 [x22]];intros ; intuition;exists x20;exists x22;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x23 |- _ => destruct (H_id_sequent x21 y (refl_equal _)) as [x20 [x22]];intros ; intuition;exists x20;exists x22;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d H_id_sequent H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if;intros x21 x23 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 x21 |- _ => destruct (H_id_x'2a y x23 (refl_equal _)) as [x20 [x22]];intros ; intuition;exists x20;exists x22;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x23 |- _ => destruct (H_id_x'2a x21 y (refl_equal _)) as [x20 [x22]];intros ; intuition;exists x20;exists x22;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d H_id_sequent H_id_x'2a H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil H_id_if;intros x21 x23 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 x21 |- _ => destruct (H_id_cons y x23 (refl_equal _)) as [x20 [x22]];intros ; intuition;exists x20;exists x22;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x23 |- _ => destruct (H_id_cons x21 y (refl_equal _)) as [x20 [x22]];intros ; intuition;exists x20;exists x22;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d H_id_sequent H_id_x'2a H_id_cons H_id_p H_id_reduce'ii'out H_id_nil H_id_if;intros x21 x23 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 x21 |- _ => destruct (H_id_x'2b y x23 (refl_equal _)) as [x20 [x22]];intros ; intuition;exists x20;exists x22;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x23 |- _ => destruct (H_id_x'2b x21 y (refl_equal _)) as [x20 [x22]];intros ; intuition;exists x20;exists x22;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . clear H_id_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_reduce'ii'out H_id_nil H_id_if;intros x21 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 x21 |- _ => destruct (H_id_p y (refl_equal _)) as [x20];intros ;intuition; exists x20;intuition;eapply closure_extension.refl_trans_clos_R; eassumption end . clear H_id_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_nil H_id_if; 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_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_if;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_tautology'i'out H_id_iff H_id_intersect'ii'out H_id_x'2d H_id_sequent H_id_x'2a H_id_cons H_id_x'2b H_id_p H_id_reduce'ii'out H_id_nil;intros x21 x23 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 x21 |- _ => destruct (H_id_if y x23 (refl_equal _)) as [x20 [x22]];intros ; intuition;exists x20;exists x22;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . match goal with | H:algebra.EQT.one_step _ ?y x23 |- _ => destruct (H_id_if x21 y (refl_equal _)) as [x20 [x22]];intros ; intuition;exists x20;exists x22;intuition; eapply closure_extension.refl_trans_clos_R;eassumption end . Qed. Lemma id_tautology'i'out_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_tautology'i'out nil) -> t' = (algebra.Alg.Term algebra.F.id_tautology'i'out nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_iff_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x21 x23, t = (algebra.Alg.Term algebra.F.id_iff (x21::x23::nil)) -> exists x20, exists x22, t' = (algebra.Alg.Term algebra.F.id_iff (x20::x22::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x22 x23). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_intersect'ii'out_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_intersect'ii'out nil) -> t' = (algebra.Alg.Term algebra.F.id_intersect'ii'out nil). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_x'2d_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x21, t = (algebra.Alg.Term algebra.F.id_x'2d (x21::nil)) -> exists x20, t' = (algebra.Alg.Term algebra.F.id_x'2d (x20::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_sequent_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x21 x23, t = (algebra.Alg.Term algebra.F.id_sequent (x21::x23::nil)) -> exists x20, exists x22, t' = (algebra.Alg.Term algebra.F.id_sequent (x20::x22::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x22 x23). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_x'2a_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x21 x23, t = (algebra.Alg.Term algebra.F.id_x'2a (x21::x23::nil)) -> exists x20, exists x22, t' = (algebra.Alg.Term algebra.F.id_x'2a (x20::x22::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x22 x23). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_cons_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x21 x23, t = (algebra.Alg.Term algebra.F.id_cons (x21::x23::nil)) -> exists x20, exists x22, t' = (algebra.Alg.Term algebra.F.id_cons (x20::x22::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x22 x23). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_x'2b_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x21 x23, t = (algebra.Alg.Term algebra.F.id_x'2b (x21::x23::nil)) -> exists x20, exists x22, t' = (algebra.Alg.Term algebra.F.id_x'2b (x20::x22::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x22 x23). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_p_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x21, t = (algebra.Alg.Term algebra.F.id_p (x21::nil)) -> exists x20, t' = (algebra.Alg.Term algebra.F.id_p (x20::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21). Proof. intros t t' H. destruct (are_constuctors_of_R_xml_0 H). assumption. Qed. Lemma id_reduce'ii'out_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_reduce'ii'out nil) -> t' = (algebra.Alg.Term algebra.F.id_reduce'ii'out nil). 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_if_is_R_xml_0_constructor : forall t t', (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) t' t) -> forall x21 x23, t = (algebra.Alg.Term algebra.F.id_if (x21::x23::nil)) -> exists x20, exists x22, t' = (algebra.Alg.Term algebra.F.id_if (x20::x22::nil))/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x20 x21)/\ (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_rules) x22 x23). 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_tautology'i'out nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_tautology'i'out_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_iff (?x21::?x20::nil)) |- _ => let x21 := fresh "x" in (let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_iff_is_R_xml_0_constructor H (refl_equal _)) as [x21 [x20 [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_intersect'ii'out nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_intersect'ii'out_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_x'2d (?x20::nil)) |- _ => let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_x'2d_is_R_xml_0_constructor H (refl_equal _)) as [x20 [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_sequent (?x21::?x20::nil)) |- _ => let x21 := fresh "x" in (let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_sequent_is_R_xml_0_constructor H (refl_equal _)) as [x21 [x20 [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_x'2a (?x21::?x20::nil)) |- _ => let x21 := fresh "x" in (let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_x'2a_is_R_xml_0_constructor H (refl_equal _)) as [x21 [x20 [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_cons (?x21::?x20::nil)) |- _ => let x21 := fresh "x" in (let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_cons_is_R_xml_0_constructor H (refl_equal _)) as [x21 [x20 [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_x'2b (?x21::?x20::nil)) |- _ => let x21 := fresh "x" in (let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_x'2b_is_R_xml_0_constructor H (refl_equal _)) as [x21 [x20 [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_p (?x20::nil)) |- _ => let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_p_is_R_xml_0_constructor H (refl_equal _)) as [x20 [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_reduce'ii'out nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_reduce'ii'out_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_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_if (?x21::?x20::nil)) |- _ => let x21 := fresh "x" in (let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_if_is_R_xml_0_constructor H (refl_equal _)) as [x21 [x20 [Heq [Hred2 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_tautology'i'out nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_tautology'i'out_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_iff (?x21::?x20::nil)) |- _ => let x21 := fresh "x" in (let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_iff_is_R_xml_0_constructor H (refl_equal _)) as [x21 [x20 [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_intersect'ii'out nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_intersect'ii'out_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_x'2d (?x20::nil)) |- _ => let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_x'2d_is_R_xml_0_constructor H (refl_equal _)) as [x20 [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_sequent (?x21::?x20::nil)) |- _ => let x21 := fresh "x" in (let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_sequent_is_R_xml_0_constructor H (refl_equal _)) as [x21 [x20 [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_x'2a (?x21::?x20::nil)) |- _ => let x21 := fresh "x" in (let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_x'2a_is_R_xml_0_constructor H (refl_equal _)) as [x21 [x20 [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_cons (?x21::?x20::nil)) |- _ => let x21 := fresh "x" in (let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_cons_is_R_xml_0_constructor H (refl_equal _)) as [x21 [x20 [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_x'2b (?x21::?x20::nil)) |- _ => let x21 := fresh "x" in (let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_x'2b_is_R_xml_0_constructor H (refl_equal _)) as [x21 [x20 [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_p (?x20::nil)) |- _ => let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (destruct (id_p_is_R_xml_0_constructor H (refl_equal _)) as [x20 [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_reduce'ii'out nil) |- _ => let Heq := fresh "Heq" in (set (Heq:=id_reduce'ii'out_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_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_if (?x21::?x20::nil)) |- _ => let x21 := fresh "x" in (let x20 := fresh "x" in (let Heq := fresh "Heq" in (let Hred1 := fresh "Hred" in (let Hred2 := fresh "Hred" in (destruct (id_if_is_R_xml_0_constructor H (refl_equal _)) as [x21 [x20 [Heq [Hred2 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_intersect'ii'in : A ->A ->A. Hypothesis P_id_tautology'i'out : A. Hypothesis P_id_u'6'1 : A ->A ->A ->A ->A ->A. Hypothesis P_id_u'3'1 : A ->A. Hypothesis P_id_u'12'1 : A ->A ->A ->A ->A ->A. Hypothesis P_id_u'2'1 : A ->A. Hypothesis P_id_u'9'1 : A ->A. Hypothesis P_id_iff : A ->A ->A. Hypothesis P_id_u'14'1 : A ->A. Hypothesis P_id_intersect'ii'out : A. Hypothesis P_id_u'7'1 : A ->A. Hypothesis P_id_x'2d : A ->A. Hypothesis P_id_u'13'1 : A ->A. Hypothesis P_id_sequent : A ->A ->A. Hypothesis P_id_u'10'1 : A ->A. Hypothesis P_id_x'2a : A ->A ->A. Hypothesis P_id_tautology'i'in : A ->A. Hypothesis P_id_cons : A ->A ->A. Hypothesis P_id_u'6'2 : A ->A. Hypothesis P_id_x'2b : A ->A ->A. Hypothesis P_id_u'12'2 : A ->A. Hypothesis P_id_reduce'ii'in : A ->A ->A. Hypothesis P_id_p : A ->A. Hypothesis P_id_u'4'1 : A ->A. Hypothesis P_id_u'15'1 : A ->A. Hypothesis P_id_u'1'1 : A ->A. Hypothesis P_id_u'8'1 : A ->A. Hypothesis P_id_reduce'ii'out : A. Hypothesis P_id_nil : A. Hypothesis P_id_if : A ->A ->A. Hypothesis P_id_u'11'1 : A ->A. Hypothesis P_id_u'5'1 : A ->A. Hypothesis P_id_u'16'1 : A ->A. Hypothesis P_id_intersect'ii'in_monotonic : forall x20 x22 x21 x23, (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) -> P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22. Hypothesis P_id_u'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (A0 <= x29)/\ (x29 <= x28) -> (A0 <= x27)/\ (x27 <= x26) -> (A0 <= x25)/\ (x25 <= x24) -> (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) -> P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28. Hypothesis P_id_u'3'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20. Hypothesis P_id_u'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (A0 <= x29)/\ (x29 <= x28) -> (A0 <= x27)/\ (x27 <= x26) -> (A0 <= x25)/\ (x25 <= x24) -> (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) -> P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 x26 x28. Hypothesis P_id_u'2'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20. Hypothesis P_id_u'9'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20. Hypothesis P_id_iff_monotonic : forall x20 x22 x21 x23, (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) ->P_id_iff x21 x23 <= P_id_iff x20 x22. Hypothesis P_id_u'14'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20. Hypothesis P_id_u'7'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20. Hypothesis P_id_x'2d_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20. Hypothesis P_id_u'13'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20. Hypothesis P_id_sequent_monotonic : forall x20 x22 x21 x23, (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) -> P_id_sequent x21 x23 <= P_id_sequent x20 x22. Hypothesis P_id_u'10'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20. Hypothesis P_id_x'2a_monotonic : forall x20 x22 x21 x23, (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) ->P_id_x'2a x21 x23 <= P_id_x'2a x20 x22. Hypothesis P_id_tautology'i'in_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) -> P_id_tautology'i'in x21 <= P_id_tautology'i'in x20. Hypothesis P_id_cons_monotonic : forall x20 x22 x21 x23, (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) ->P_id_cons x21 x23 <= P_id_cons x20 x22. Hypothesis P_id_u'6'2_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20. Hypothesis P_id_x'2b_monotonic : forall x20 x22 x21 x23, (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) ->P_id_x'2b x21 x23 <= P_id_x'2b x20 x22. Hypothesis P_id_u'12'2_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20. Hypothesis P_id_reduce'ii'in_monotonic : forall x20 x22 x21 x23, (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) -> P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22. Hypothesis P_id_p_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20. Hypothesis P_id_u'4'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20. Hypothesis P_id_u'15'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20. Hypothesis P_id_u'1'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20. Hypothesis P_id_u'8'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20. Hypothesis P_id_if_monotonic : forall x20 x22 x21 x23, (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22. Hypothesis P_id_u'11'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20. Hypothesis P_id_u'5'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20. Hypothesis P_id_u'16'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20. Hypothesis P_id_intersect'ii'in_bounded : forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_intersect'ii'in x21 x20. Hypothesis P_id_tautology'i'out_bounded : A0 <= P_id_tautology'i'out . Hypothesis P_id_u'6'1_bounded : forall x24 x20 x22 x21 x23, (A0 <= x20) -> (A0 <= x21) -> (A0 <= x22) -> (A0 <= x23) ->(A0 <= x24) ->A0 <= P_id_u'6'1 x24 x23 x22 x21 x20. Hypothesis P_id_u'3'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'3'1 x20. Hypothesis P_id_u'12'1_bounded : forall x24 x20 x22 x21 x23, (A0 <= x20) -> (A0 <= x21) -> (A0 <= x22) -> (A0 <= x23) ->(A0 <= x24) ->A0 <= P_id_u'12'1 x24 x23 x22 x21 x20. Hypothesis P_id_u'2'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'2'1 x20. Hypothesis P_id_u'9'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'9'1 x20. Hypothesis P_id_iff_bounded : forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_iff x21 x20. Hypothesis P_id_u'14'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'14'1 x20. Hypothesis P_id_intersect'ii'out_bounded : A0 <= P_id_intersect'ii'out . Hypothesis P_id_u'7'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'7'1 x20. Hypothesis P_id_x'2d_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_x'2d x20. Hypothesis P_id_u'13'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'13'1 x20. Hypothesis P_id_sequent_bounded : forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_sequent x21 x20. Hypothesis P_id_u'10'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'10'1 x20. Hypothesis P_id_x'2a_bounded : forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_x'2a x21 x20. Hypothesis P_id_tautology'i'in_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_tautology'i'in x20. Hypothesis P_id_cons_bounded : forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_cons x21 x20. Hypothesis P_id_u'6'2_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'6'2 x20. Hypothesis P_id_x'2b_bounded : forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_x'2b x21 x20. Hypothesis P_id_u'12'2_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'12'2 x20. Hypothesis P_id_reduce'ii'in_bounded : forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_reduce'ii'in x21 x20. Hypothesis P_id_p_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_p x20. Hypothesis P_id_u'4'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'4'1 x20. Hypothesis P_id_u'15'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'15'1 x20. Hypothesis P_id_u'1'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'1'1 x20. Hypothesis P_id_u'8'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'8'1 x20. Hypothesis P_id_reduce'ii'out_bounded : A0 <= P_id_reduce'ii'out . Hypothesis P_id_nil_bounded : A0 <= P_id_nil . Hypothesis P_id_if_bounded : forall x20 x21, (A0 <= x20) ->(A0 <= x21) ->A0 <= P_id_if x21 x20. Hypothesis P_id_u'11'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'11'1 x20. Hypothesis P_id_u'5'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'5'1 x20. Hypothesis P_id_u'16'1_bounded : forall x20, (A0 <= x20) ->A0 <= P_id_u'16'1 x20. Fixpoint measure t { struct t } := match t with | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) => P_id_intersect'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) => P_id_tautology'i'out | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21::x20::nil)) => P_id_u'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_u'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21:: x20::nil)) => P_id_u'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_u'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_u'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) => P_id_iff (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_u'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) => P_id_intersect'ii'out | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_u'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) => P_id_x'2d (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_u'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_sequent (x21::x20::nil)) => P_id_sequent (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_u'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) => P_id_x'2a (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_tautology'i'in (measure x20) | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) => P_id_cons (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_u'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) => P_id_x'2b (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_u'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) => P_id_reduce'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_p (x20::nil)) => P_id_p (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_u'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_u'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_u'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_u'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) => P_id_reduce'ii'out | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) => P_id_if (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_u'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_u'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_u'16'1 (measure x20) | _ => A0 end. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21:: x20::nil)) => P_id_intersect'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) => P_id_tautology'i'out | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22:: x21::x20::nil)) => P_id_u'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_u'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22:: x21::x20::nil)) => P_id_u'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_u'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_u'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) => P_id_iff (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_u'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) => P_id_intersect'ii'out | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_u'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) => P_id_x'2d (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_u'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_sequent (x21::x20::nil)) => P_id_sequent (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_u'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) => P_id_x'2a (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_tautology'i'in (measure x20) | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) => P_id_cons (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_u'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) => P_id_x'2b (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_u'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21:: x20::nil)) => P_id_reduce'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_p (x20::nil)) => P_id_p (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_u'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_u'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_u'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_u'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) => P_id_reduce'ii'out | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) => P_id_if (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_u'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_u'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_u'16'1 (measure x20) | _ => 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_intersect'ii'in => P_id_intersect'ii'in | algebra.F.id_tautology'i'out => P_id_tautology'i'out | algebra.F.id_u'6'1 => P_id_u'6'1 | algebra.F.id_u'3'1 => P_id_u'3'1 | algebra.F.id_u'12'1 => P_id_u'12'1 | algebra.F.id_u'2'1 => P_id_u'2'1 | algebra.F.id_u'9'1 => P_id_u'9'1 | algebra.F.id_iff => P_id_iff | algebra.F.id_u'14'1 => P_id_u'14'1 | algebra.F.id_intersect'ii'out => P_id_intersect'ii'out | algebra.F.id_u'7'1 => P_id_u'7'1 | algebra.F.id_x'2d => P_id_x'2d | algebra.F.id_u'13'1 => P_id_u'13'1 | algebra.F.id_sequent => P_id_sequent | algebra.F.id_u'10'1 => P_id_u'10'1 | algebra.F.id_x'2a => P_id_x'2a | algebra.F.id_tautology'i'in => P_id_tautology'i'in | algebra.F.id_cons => P_id_cons | algebra.F.id_u'6'2 => P_id_u'6'2 | algebra.F.id_x'2b => P_id_x'2b | algebra.F.id_u'12'2 => P_id_u'12'2 | algebra.F.id_reduce'ii'in => P_id_reduce'ii'in | algebra.F.id_p => P_id_p | algebra.F.id_u'4'1 => P_id_u'4'1 | algebra.F.id_u'15'1 => P_id_u'15'1 | algebra.F.id_u'1'1 => P_id_u'1'1 | algebra.F.id_u'8'1 => P_id_u'8'1 | algebra.F.id_reduce'ii'out => P_id_reduce'ii'out | algebra.F.id_nil => P_id_nil | algebra.F.id_if => P_id_if | algebra.F.id_u'11'1 => P_id_u'11'1 | algebra.F.id_u'5'1 => P_id_u'5'1 | algebra.F.id_u'16'1 => P_id_u'16'1 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_intersect'ii'in => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_tautology'i'out => match l with | nil => _ | _::_ => _ end | algebra.F.id_u'6'1 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::nil => _ | _::_::_::_::_::nil => _ | _::_::_::_::_::_::_ => _ end | algebra.F.id_u'3'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'12'1 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::nil => _ | _::_::_::_::_::nil => _ | _::_::_::_::_::_::_ => _ end | algebra.F.id_u'2'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'9'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_iff => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_u'14'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_intersect'ii'out => match l with | nil => _ | _::_ => _ end | algebra.F.id_u'7'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_x'2d => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'13'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_sequent => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_u'10'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_x'2a => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_tautology'i'in => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_cons => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_u'6'2 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_x'2b => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_u'12'2 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_reduce'ii'in => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_p => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'4'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'15'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'1'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'8'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_reduce'ii'out => match l with | nil => _ | _::_ => _ end | algebra.F.id_nil => match l with | nil => _ | _::_ => _ end | algebra.F.id_if => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_u'11'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'5'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'16'1 => 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_intersect'ii'in_bounded;assumption. vm_compute in |-*;intros ;apply P_id_tautology'i'out_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'6'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'3'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'12'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'2'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'9'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_iff_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'14'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_intersect'ii'out_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'7'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_x'2d_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'13'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_sequent_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'10'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_x'2a_bounded;assumption. vm_compute in |-*;intros ;apply P_id_tautology'i'in_bounded;assumption. vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'6'2_bounded;assumption. vm_compute in |-*;intros ;apply P_id_x'2b_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'12'2_bounded;assumption. vm_compute in |-*;intros ;apply P_id_reduce'ii'in_bounded;assumption. vm_compute in |-*;intros ;apply P_id_p_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'4'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'15'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'1'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'8'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_reduce'ii'out_bounded;assumption. vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption. vm_compute in |-*;intros ;apply P_id_if_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'11'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'5'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'16'1_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_intersect'ii'in_monotonic; assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_u'6'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'3'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'12'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'2'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'9'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_iff_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'14'1_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_u'7'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_x'2d_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'13'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_sequent_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'10'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_x'2a_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_tautology'i'in_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_cons_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'6'2_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_x'2b_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'12'2_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_reduce'ii'in_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_p_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'4'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'15'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'1'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'8'1_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_if_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'11'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'5'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'16'1_monotonic;assumption. intros f. case f. vm_compute in |-*;intros ;apply P_id_intersect'ii'in_bounded;assumption. vm_compute in |-*;intros ;apply P_id_tautology'i'out_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'6'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'3'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'12'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'2'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'9'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_iff_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'14'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_intersect'ii'out_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'7'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_x'2d_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'13'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_sequent_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'10'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_x'2a_bounded;assumption. vm_compute in |-*;intros ;apply P_id_tautology'i'in_bounded;assumption. vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'6'2_bounded;assumption. vm_compute in |-*;intros ;apply P_id_x'2b_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'12'2_bounded;assumption. vm_compute in |-*;intros ;apply P_id_reduce'ii'in_bounded;assumption. vm_compute in |-*;intros ;apply P_id_p_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'4'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'15'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'1'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'8'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_reduce'ii'out_bounded;assumption. vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption. vm_compute in |-*;intros ;apply P_id_if_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'11'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'5'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'16'1_bounded;assumption. intros . do 2 (rewrite <- same_measure in |-*). apply rules_monotonic;assumption. assumption. Qed. Hypothesis P_id_U'12'2 : A ->A. Hypothesis P_id_U'6'1 : A ->A ->A ->A ->A ->A. Hypothesis P_id_REDUCE'II'IN : A ->A ->A. Hypothesis P_id_TAUTOLOGY'I'IN : A ->A. Hypothesis P_id_U'9'1 : A ->A. Hypothesis P_id_U'1'1 : A ->A. Hypothesis P_id_U'14'1 : A ->A. Hypothesis P_id_U'7'1 : A ->A. Hypothesis P_id_U'4'1 : A ->A. Hypothesis P_id_U'11'1 : A ->A. Hypothesis P_id_INTERSECT'II'IN : A ->A ->A. Hypothesis P_id_U'13'1 : A ->A. Hypothesis P_id_U'6'2 : A ->A. Hypothesis P_id_U'3'1 : A ->A. Hypothesis P_id_U'16'1 : A ->A. Hypothesis P_id_U'10'1 : A ->A. Hypothesis P_id_U'2'1 : A ->A. Hypothesis P_id_U'15'1 : A ->A. Hypothesis P_id_U'8'1 : A ->A. Hypothesis P_id_U'5'1 : A ->A. Hypothesis P_id_U'12'1 : A ->A ->A ->A ->A ->A. Hypothesis P_id_U'12'2_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20. Hypothesis P_id_U'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (A0 <= x29)/\ (x29 <= x28) -> (A0 <= x27)/\ (x27 <= x26) -> (A0 <= x25)/\ (x25 <= x24) -> (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) -> P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28. Hypothesis P_id_REDUCE'II'IN_monotonic : forall x20 x22 x21 x23, (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) -> P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22. Hypothesis P_id_TAUTOLOGY'I'IN_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) -> P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20. Hypothesis P_id_U'9'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20. Hypothesis P_id_U'1'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20. Hypothesis P_id_U'14'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20. Hypothesis P_id_U'7'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20. Hypothesis P_id_U'4'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20. Hypothesis P_id_U'11'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20. Hypothesis P_id_INTERSECT'II'IN_monotonic : forall x20 x22 x21 x23, (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) -> P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22. Hypothesis P_id_U'13'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20. Hypothesis P_id_U'6'2_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20. Hypothesis P_id_U'3'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20. Hypothesis P_id_U'16'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20. Hypothesis P_id_U'10'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20. Hypothesis P_id_U'2'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20. Hypothesis P_id_U'15'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20. Hypothesis P_id_U'8'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20. Hypothesis P_id_U'5'1_monotonic : forall x20 x21, (A0 <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20. Hypothesis P_id_U'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (A0 <= x29)/\ (x29 <= x28) -> (A0 <= x27)/\ (x27 <= x26) -> (A0 <= x25)/\ (x25 <= x24) -> (A0 <= x23)/\ (x23 <= x22) -> (A0 <= x21)/\ (x21 <= x20) -> P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 x26 x28. Definition marked_measure t := match t with | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_U'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21::x20::nil)) => P_id_U'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) => P_id_REDUCE'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_TAUTOLOGY'I'IN (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_U'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_U'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_U'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_U'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_U'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_U'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) => P_id_INTERSECT'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_U'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_U'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_U'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_U'16'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_U'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_U'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_U'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_U'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_U'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21:: x20::nil)) => P_id_U'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | _ => 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 x20;apply (P_id_U'16'1 x20). 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 x20;apply (P_id_U'5'1 x20). 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 x20;apply (P_id_U'11'1 x20). 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 x20;apply (P_id_U'8'1 x20). 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 x20;apply (P_id_U'1'1 x20). 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 x20;apply (P_id_U'15'1 x20). 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 x20;apply (P_id_U'4'1 x20). 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 x21 x20;apply (P_id_REDUCE'II'IN x21 x20). 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 x20;apply (P_id_U'12'2 x20). 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 x20;apply (P_id_U'6'2 x20). 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 x20;apply (P_id_TAUTOLOGY'I'IN x20). 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 x20;apply (P_id_U'10'1 x20). 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 x20;apply (P_id_U'13'1 x20). 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 x20;apply (P_id_U'7'1 x20). 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 x20;apply (P_id_U'14'1 x20). 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 x20;apply (P_id_U'9'1 x20). 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 x20;apply (P_id_U'2'1 x20). 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 x24 x23 x22 x21 x20; apply (P_id_U'12'1 x24 x23 x22 x21 x20). 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 x20;apply (P_id_U'3'1 x20). 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 x24 x23 x22 x21 x20; apply (P_id_U'6'1 x24 x23 x22 x21 x20). 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 x21 x20;apply (P_id_INTERSECT'II'IN x21 x20). 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_intersect'ii'in => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_tautology'i'out => match l with | nil => _ | _::_ => _ end | algebra.F.id_u'6'1 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::nil => _ | _::_::_::_::_::nil => _ | _::_::_::_::_::_::_ => _ end | algebra.F.id_u'3'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'12'1 => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::nil => _ | _::_::_::_::nil => _ | _::_::_::_::_::nil => _ | _::_::_::_::_::_::_ => _ end | algebra.F.id_u'2'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'9'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_iff => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_u'14'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_intersect'ii'out => match l with | nil => _ | _::_ => _ end | algebra.F.id_u'7'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_x'2d => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'13'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_sequent => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_u'10'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_x'2a => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_tautology'i'in => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_cons => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_u'6'2 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_x'2b => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_u'12'2 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_reduce'ii'in => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_p => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'4'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'15'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'1'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'8'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_reduce'ii'out => match l with | nil => _ | _::_ => _ end | algebra.F.id_nil => match l with | nil => _ | _::_ => _ end | algebra.F.id_if => match l with | nil => _ | _::nil => _ | _::_::nil => _ | _::_::_::_ => _ end | algebra.F.id_u'11'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'5'1 => match l with | nil => _ | _::nil => _ | _::_::_ => _ end | algebra.F.id_u'16'1 => 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 . 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 . 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 . 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 . 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 . 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 . 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 . 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 . 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 . Qed. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_U'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23:: x22::x21::x20::nil)) => P_id_U'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) => P_id_REDUCE'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_TAUTOLOGY'I'IN (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_U'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_U'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_U'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_U'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_U'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_U'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) => P_id_INTERSECT'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_U'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_U'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_U'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_U'16'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_U'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_U'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_U'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_U'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_U'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23:: x22::x21::x20::nil)) => P_id_U'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | _ => 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_intersect'ii'in_monotonic; assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_u'6'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'3'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'12'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'2'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'9'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_iff_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'14'1_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_u'7'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_x'2d_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'13'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_sequent_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'10'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_x'2a_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_tautology'i'in_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_cons_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'6'2_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_x'2b_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'12'2_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_reduce'ii'in_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_p_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'4'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'15'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'1'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'8'1_monotonic;assumption. vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;apply (Aop.(le_refl)). vm_compute in |-*;intros ;apply P_id_if_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'11'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'5'1_monotonic;assumption. vm_compute in |-*;intros ;apply P_id_u'16'1_monotonic;assumption. clear f. intros f. case f. vm_compute in |-*;intros ;apply P_id_intersect'ii'in_bounded;assumption. vm_compute in |-*;intros ;apply P_id_tautology'i'out_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'6'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'3'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'12'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'2'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'9'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_iff_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'14'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_intersect'ii'out_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'7'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_x'2d_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'13'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_sequent_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'10'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_x'2a_bounded;assumption. vm_compute in |-*;intros ;apply P_id_tautology'i'in_bounded;assumption. vm_compute in |-*;intros ;apply P_id_cons_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'6'2_bounded;assumption. vm_compute in |-*;intros ;apply P_id_x'2b_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'12'2_bounded;assumption. vm_compute in |-*;intros ;apply P_id_reduce'ii'in_bounded;assumption. vm_compute in |-*;intros ;apply P_id_p_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'4'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'15'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'1'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'8'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_reduce'ii'out_bounded;assumption. vm_compute in |-*;intros ;apply P_id_nil_bounded;assumption. vm_compute in |-*;intros ;apply P_id_if_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'11'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'5'1_bounded;assumption. vm_compute in |-*;intros ;apply P_id_u'16'1_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_U'16'1_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_U'5'1_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_U'11'1_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_U'8'1_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_U'1'1_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_U'15'1_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_U'4'1_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_REDUCE'II'IN_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_U'12'2_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_U'6'2_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_TAUTOLOGY'I'IN_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_U'10'1_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_U'13'1_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_U'7'1_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_U'14'1_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_U'9'1_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_U'2'1_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_U'12'1_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_U'3'1_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_U'6'1_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_INTERSECT'II'IN_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_intersect'ii'in : Z ->Z ->Z. Hypothesis P_id_tautology'i'out : Z. Hypothesis P_id_u'6'1 : Z ->Z ->Z ->Z ->Z ->Z. Hypothesis P_id_u'3'1 : Z ->Z. Hypothesis P_id_u'12'1 : Z ->Z ->Z ->Z ->Z ->Z. Hypothesis P_id_u'2'1 : Z ->Z. Hypothesis P_id_u'9'1 : Z ->Z. Hypothesis P_id_iff : Z ->Z ->Z. Hypothesis P_id_u'14'1 : Z ->Z. Hypothesis P_id_intersect'ii'out : Z. Hypothesis P_id_u'7'1 : Z ->Z. Hypothesis P_id_x'2d : Z ->Z. Hypothesis P_id_u'13'1 : Z ->Z. Hypothesis P_id_sequent : Z ->Z ->Z. Hypothesis P_id_u'10'1 : Z ->Z. Hypothesis P_id_x'2a : Z ->Z ->Z. Hypothesis P_id_tautology'i'in : Z ->Z. Hypothesis P_id_cons : Z ->Z ->Z. Hypothesis P_id_u'6'2 : Z ->Z. Hypothesis P_id_x'2b : Z ->Z ->Z. Hypothesis P_id_u'12'2 : Z ->Z. Hypothesis P_id_reduce'ii'in : Z ->Z ->Z. Hypothesis P_id_p : Z ->Z. Hypothesis P_id_u'4'1 : Z ->Z. Hypothesis P_id_u'15'1 : Z ->Z. Hypothesis P_id_u'1'1 : Z ->Z. Hypothesis P_id_u'8'1 : Z ->Z. Hypothesis P_id_reduce'ii'out : Z. Hypothesis P_id_nil : Z. Hypothesis P_id_if : Z ->Z ->Z. Hypothesis P_id_u'11'1 : Z ->Z. Hypothesis P_id_u'5'1 : Z ->Z. Hypothesis P_id_u'16'1 : Z ->Z. Hypothesis P_id_intersect'ii'in_monotonic : forall x20 x22 x21 x23, (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22. Hypothesis P_id_u'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (min_value <= x29)/\ (x29 <= x28) -> (min_value <= x27)/\ (x27 <= x26) -> (min_value <= x25)/\ (x25 <= x24) -> (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28. Hypothesis P_id_u'3'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20. Hypothesis P_id_u'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (min_value <= x29)/\ (x29 <= x28) -> (min_value <= x27)/\ (x27 <= x26) -> (min_value <= x25)/\ (x25 <= x24) -> (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 x26 x28. Hypothesis P_id_u'2'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20. Hypothesis P_id_u'9'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20. Hypothesis P_id_iff_monotonic : forall x20 x22 x21 x23, (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_iff x21 x23 <= P_id_iff x20 x22. Hypothesis P_id_u'14'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20. Hypothesis P_id_u'7'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20. Hypothesis P_id_x'2d_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20. Hypothesis P_id_u'13'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20. Hypothesis P_id_sequent_monotonic : forall x20 x22 x21 x23, (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_sequent x21 x23 <= P_id_sequent x20 x22. Hypothesis P_id_u'10'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20. Hypothesis P_id_x'2a_monotonic : forall x20 x22 x21 x23, (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_x'2a x21 x23 <= P_id_x'2a x20 x22. Hypothesis P_id_tautology'i'in_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) -> P_id_tautology'i'in x21 <= P_id_tautology'i'in x20. Hypothesis P_id_cons_monotonic : forall x20 x22 x21 x23, (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_cons x21 x23 <= P_id_cons x20 x22. Hypothesis P_id_u'6'2_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20. Hypothesis P_id_x'2b_monotonic : forall x20 x22 x21 x23, (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_x'2b x21 x23 <= P_id_x'2b x20 x22. Hypothesis P_id_u'12'2_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20. Hypothesis P_id_reduce'ii'in_monotonic : forall x20 x22 x21 x23, (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22. Hypothesis P_id_p_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20. Hypothesis P_id_u'4'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20. Hypothesis P_id_u'15'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20. Hypothesis P_id_u'1'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20. Hypothesis P_id_u'8'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20. Hypothesis P_id_if_monotonic : forall x20 x22 x21 x23, (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22. Hypothesis P_id_u'11'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20. Hypothesis P_id_u'5'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20. Hypothesis P_id_u'16'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20. Hypothesis P_id_intersect'ii'in_bounded : forall x20 x21, (min_value <= x20) -> (min_value <= x21) ->min_value <= P_id_intersect'ii'in x21 x20. Hypothesis P_id_tautology'i'out_bounded : min_value <= P_id_tautology'i'out . Hypothesis P_id_u'6'1_bounded : forall x24 x20 x22 x21 x23, (min_value <= x20) -> (min_value <= x21) -> (min_value <= x22) -> (min_value <= x23) -> (min_value <= x24) ->min_value <= P_id_u'6'1 x24 x23 x22 x21 x20. Hypothesis P_id_u'3'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'3'1 x20. Hypothesis P_id_u'12'1_bounded : forall x24 x20 x22 x21 x23, (min_value <= x20) -> (min_value <= x21) -> (min_value <= x22) -> (min_value <= x23) -> (min_value <= x24) ->min_value <= P_id_u'12'1 x24 x23 x22 x21 x20. Hypothesis P_id_u'2'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'2'1 x20. Hypothesis P_id_u'9'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'9'1 x20. Hypothesis P_id_iff_bounded : forall x20 x21, (min_value <= x20) ->(min_value <= x21) ->min_value <= P_id_iff x21 x20. Hypothesis P_id_u'14'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'14'1 x20. Hypothesis P_id_intersect'ii'out_bounded : min_value <= P_id_intersect'ii'out . Hypothesis P_id_u'7'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'7'1 x20. Hypothesis P_id_x'2d_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_x'2d x20. Hypothesis P_id_u'13'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'13'1 x20. Hypothesis P_id_sequent_bounded : forall x20 x21, (min_value <= x20) -> (min_value <= x21) ->min_value <= P_id_sequent x21 x20. Hypothesis P_id_u'10'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'10'1 x20. Hypothesis P_id_x'2a_bounded : forall x20 x21, (min_value <= x20) ->(min_value <= x21) ->min_value <= P_id_x'2a x21 x20. Hypothesis P_id_tautology'i'in_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_tautology'i'in x20. Hypothesis P_id_cons_bounded : forall x20 x21, (min_value <= x20) ->(min_value <= x21) ->min_value <= P_id_cons x21 x20. Hypothesis P_id_u'6'2_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'6'2 x20. Hypothesis P_id_x'2b_bounded : forall x20 x21, (min_value <= x20) ->(min_value <= x21) ->min_value <= P_id_x'2b x21 x20. Hypothesis P_id_u'12'2_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'12'2 x20. Hypothesis P_id_reduce'ii'in_bounded : forall x20 x21, (min_value <= x20) -> (min_value <= x21) ->min_value <= P_id_reduce'ii'in x21 x20. Hypothesis P_id_p_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_p x20. Hypothesis P_id_u'4'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'4'1 x20. Hypothesis P_id_u'15'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'15'1 x20. Hypothesis P_id_u'1'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'1'1 x20. Hypothesis P_id_u'8'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'8'1 x20. Hypothesis P_id_reduce'ii'out_bounded : min_value <= P_id_reduce'ii'out . Hypothesis P_id_nil_bounded : min_value <= P_id_nil . Hypothesis P_id_if_bounded : forall x20 x21, (min_value <= x20) ->(min_value <= x21) ->min_value <= P_id_if x21 x20. Hypothesis P_id_u'11'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'11'1 x20. Hypothesis P_id_u'5'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'5'1 x20. Hypothesis P_id_u'16'1_bounded : forall x20, (min_value <= x20) ->min_value <= P_id_u'16'1 x20. Definition measure := Interp.measure min_value P_id_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21:: x20::nil)) => P_id_intersect'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) => P_id_tautology'i'out | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22:: x21::x20::nil)) => P_id_u'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_u'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22:: x21::x20::nil)) => P_id_u'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_u'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_u'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) => P_id_iff (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_u'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) => P_id_intersect'ii'out | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_u'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) => P_id_x'2d (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_u'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_sequent (x21::x20::nil)) => P_id_sequent (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_u'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) => P_id_x'2a (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_tautology'i'in (measure x20) | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) => P_id_cons (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_u'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) => P_id_x'2b (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_u'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21:: x20::nil)) => P_id_reduce'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_p (x20::nil)) => P_id_p (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_u'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_u'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_u'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_u'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) => P_id_reduce'ii'out | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) => P_id_if (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_u'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_u'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_u'16'1 (measure x20) | _ => 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_intersect'ii'in_monotonic;assumption. intros ;apply P_id_u'6'1_monotonic;assumption. intros ;apply P_id_u'3'1_monotonic;assumption. intros ;apply P_id_u'12'1_monotonic;assumption. intros ;apply P_id_u'2'1_monotonic;assumption. intros ;apply P_id_u'9'1_monotonic;assumption. intros ;apply P_id_iff_monotonic;assumption. intros ;apply P_id_u'14'1_monotonic;assumption. intros ;apply P_id_u'7'1_monotonic;assumption. intros ;apply P_id_x'2d_monotonic;assumption. intros ;apply P_id_u'13'1_monotonic;assumption. intros ;apply P_id_sequent_monotonic;assumption. intros ;apply P_id_u'10'1_monotonic;assumption. intros ;apply P_id_x'2a_monotonic;assumption. intros ;apply P_id_tautology'i'in_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_u'6'2_monotonic;assumption. intros ;apply P_id_x'2b_monotonic;assumption. intros ;apply P_id_u'12'2_monotonic;assumption. intros ;apply P_id_reduce'ii'in_monotonic;assumption. intros ;apply P_id_p_monotonic;assumption. intros ;apply P_id_u'4'1_monotonic;assumption. intros ;apply P_id_u'15'1_monotonic;assumption. intros ;apply P_id_u'1'1_monotonic;assumption. intros ;apply P_id_u'8'1_monotonic;assumption. intros ;apply P_id_if_monotonic;assumption. intros ;apply P_id_u'11'1_monotonic;assumption. intros ;apply P_id_u'5'1_monotonic;assumption. intros ;apply P_id_u'16'1_monotonic;assumption. intros ;apply P_id_intersect'ii'in_bounded;assumption. intros ;apply P_id_tautology'i'out_bounded;assumption. intros ;apply P_id_u'6'1_bounded;assumption. intros ;apply P_id_u'3'1_bounded;assumption. intros ;apply P_id_u'12'1_bounded;assumption. intros ;apply P_id_u'2'1_bounded;assumption. intros ;apply P_id_u'9'1_bounded;assumption. intros ;apply P_id_iff_bounded;assumption. intros ;apply P_id_u'14'1_bounded;assumption. intros ;apply P_id_intersect'ii'out_bounded;assumption. intros ;apply P_id_u'7'1_bounded;assumption. intros ;apply P_id_x'2d_bounded;assumption. intros ;apply P_id_u'13'1_bounded;assumption. intros ;apply P_id_sequent_bounded;assumption. intros ;apply P_id_u'10'1_bounded;assumption. intros ;apply P_id_x'2a_bounded;assumption. intros ;apply P_id_tautology'i'in_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_u'6'2_bounded;assumption. intros ;apply P_id_x'2b_bounded;assumption. intros ;apply P_id_u'12'2_bounded;assumption. intros ;apply P_id_reduce'ii'in_bounded;assumption. intros ;apply P_id_p_bounded;assumption. intros ;apply P_id_u'4'1_bounded;assumption. intros ;apply P_id_u'15'1_bounded;assumption. intros ;apply P_id_u'1'1_bounded;assumption. intros ;apply P_id_u'8'1_bounded;assumption. intros ;apply P_id_reduce'ii'out_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_if_bounded;assumption. intros ;apply P_id_u'11'1_bounded;assumption. intros ;apply P_id_u'5'1_bounded;assumption. intros ;apply P_id_u'16'1_bounded;assumption. apply rules_monotonic. Qed. Hypothesis P_id_U'12'2 : Z ->Z. Hypothesis P_id_U'6'1 : Z ->Z ->Z ->Z ->Z ->Z. Hypothesis P_id_REDUCE'II'IN : Z ->Z ->Z. Hypothesis P_id_TAUTOLOGY'I'IN : Z ->Z. Hypothesis P_id_U'9'1 : Z ->Z. Hypothesis P_id_U'1'1 : Z ->Z. Hypothesis P_id_U'14'1 : Z ->Z. Hypothesis P_id_U'7'1 : Z ->Z. Hypothesis P_id_U'4'1 : Z ->Z. Hypothesis P_id_U'11'1 : Z ->Z. Hypothesis P_id_INTERSECT'II'IN : Z ->Z ->Z. Hypothesis P_id_U'13'1 : Z ->Z. Hypothesis P_id_U'6'2 : Z ->Z. Hypothesis P_id_U'3'1 : Z ->Z. Hypothesis P_id_U'16'1 : Z ->Z. Hypothesis P_id_U'10'1 : Z ->Z. Hypothesis P_id_U'2'1 : Z ->Z. Hypothesis P_id_U'15'1 : Z ->Z. Hypothesis P_id_U'8'1 : Z ->Z. Hypothesis P_id_U'5'1 : Z ->Z. Hypothesis P_id_U'12'1 : Z ->Z ->Z ->Z ->Z ->Z. Hypothesis P_id_U'12'2_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20. Hypothesis P_id_U'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (min_value <= x29)/\ (x29 <= x28) -> (min_value <= x27)/\ (x27 <= x26) -> (min_value <= x25)/\ (x25 <= x24) -> (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28. Hypothesis P_id_REDUCE'II'IN_monotonic : forall x20 x22 x21 x23, (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22. Hypothesis P_id_TAUTOLOGY'I'IN_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) -> P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20. Hypothesis P_id_U'9'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20. Hypothesis P_id_U'1'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20. Hypothesis P_id_U'14'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20. Hypothesis P_id_U'7'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20. Hypothesis P_id_U'4'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20. Hypothesis P_id_U'11'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20. Hypothesis P_id_INTERSECT'II'IN_monotonic : forall x20 x22 x21 x23, (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22. Hypothesis P_id_U'13'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20. Hypothesis P_id_U'6'2_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20. Hypothesis P_id_U'3'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20. Hypothesis P_id_U'16'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20. Hypothesis P_id_U'10'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20. Hypothesis P_id_U'2'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20. Hypothesis P_id_U'15'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20. Hypothesis P_id_U'8'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20. Hypothesis P_id_U'5'1_monotonic : forall x20 x21, (min_value <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20. Hypothesis P_id_U'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (min_value <= x29)/\ (x29 <= x28) -> (min_value <= x27)/\ (x27 <= x26) -> (min_value <= x25)/\ (x25 <= x24) -> (min_value <= x23)/\ (x23 <= x22) -> (min_value <= x21)/\ (x21 <= x20) -> P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 x26 x28. Definition marked_measure := Interp.marked_measure min_value P_id_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1 P_id_U'12'2 P_id_U'6'1 P_id_REDUCE'II'IN P_id_TAUTOLOGY'I'IN P_id_U'9'1 P_id_U'1'1 P_id_U'14'1 P_id_U'7'1 P_id_U'4'1 P_id_U'11'1 P_id_INTERSECT'II'IN P_id_U'13'1 P_id_U'6'2 P_id_U'3'1 P_id_U'16'1 P_id_U'10'1 P_id_U'2'1 P_id_U'15'1 P_id_U'8'1 P_id_U'5'1 P_id_U'12'1. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_U'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23:: x22::x21::x20::nil)) => P_id_U'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) => P_id_REDUCE'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_TAUTOLOGY'I'IN (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_U'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_U'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_U'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_U'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_U'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_U'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) => P_id_INTERSECT'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_U'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_U'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_U'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_U'16'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_U'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_U'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_U'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_U'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_U'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23:: x22::x21::x20::nil)) => P_id_U'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | _ => 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_intersect'ii'in_monotonic;assumption. intros ;apply P_id_u'6'1_monotonic;assumption. intros ;apply P_id_u'3'1_monotonic;assumption. intros ;apply P_id_u'12'1_monotonic;assumption. intros ;apply P_id_u'2'1_monotonic;assumption. intros ;apply P_id_u'9'1_monotonic;assumption. intros ;apply P_id_iff_monotonic;assumption. intros ;apply P_id_u'14'1_monotonic;assumption. intros ;apply P_id_u'7'1_monotonic;assumption. intros ;apply P_id_x'2d_monotonic;assumption. intros ;apply P_id_u'13'1_monotonic;assumption. intros ;apply P_id_sequent_monotonic;assumption. intros ;apply P_id_u'10'1_monotonic;assumption. intros ;apply P_id_x'2a_monotonic;assumption. intros ;apply P_id_tautology'i'in_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_u'6'2_monotonic;assumption. intros ;apply P_id_x'2b_monotonic;assumption. intros ;apply P_id_u'12'2_monotonic;assumption. intros ;apply P_id_reduce'ii'in_monotonic;assumption. intros ;apply P_id_p_monotonic;assumption. intros ;apply P_id_u'4'1_monotonic;assumption. intros ;apply P_id_u'15'1_monotonic;assumption. intros ;apply P_id_u'1'1_monotonic;assumption. intros ;apply P_id_u'8'1_monotonic;assumption. intros ;apply P_id_if_monotonic;assumption. intros ;apply P_id_u'11'1_monotonic;assumption. intros ;apply P_id_u'5'1_monotonic;assumption. intros ;apply P_id_u'16'1_monotonic;assumption. intros ;apply P_id_intersect'ii'in_bounded;assumption. intros ;apply P_id_tautology'i'out_bounded;assumption. intros ;apply P_id_u'6'1_bounded;assumption. intros ;apply P_id_u'3'1_bounded;assumption. intros ;apply P_id_u'12'1_bounded;assumption. intros ;apply P_id_u'2'1_bounded;assumption. intros ;apply P_id_u'9'1_bounded;assumption. intros ;apply P_id_iff_bounded;assumption. intros ;apply P_id_u'14'1_bounded;assumption. intros ;apply P_id_intersect'ii'out_bounded;assumption. intros ;apply P_id_u'7'1_bounded;assumption. intros ;apply P_id_x'2d_bounded;assumption. intros ;apply P_id_u'13'1_bounded;assumption. intros ;apply P_id_sequent_bounded;assumption. intros ;apply P_id_u'10'1_bounded;assumption. intros ;apply P_id_x'2a_bounded;assumption. intros ;apply P_id_tautology'i'in_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_u'6'2_bounded;assumption. intros ;apply P_id_x'2b_bounded;assumption. intros ;apply P_id_u'12'2_bounded;assumption. intros ;apply P_id_reduce'ii'in_bounded;assumption. intros ;apply P_id_p_bounded;assumption. intros ;apply P_id_u'4'1_bounded;assumption. intros ;apply P_id_u'15'1_bounded;assumption. intros ;apply P_id_u'1'1_bounded;assumption. intros ;apply P_id_u'8'1_bounded;assumption. intros ;apply P_id_reduce'ii'out_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_if_bounded;assumption. intros ;apply P_id_u'11'1_bounded;assumption. intros ;apply P_id_u'5'1_bounded;assumption. intros ;apply P_id_u'16'1_bounded;assumption. apply rules_monotonic. intros ;apply P_id_U'12'2_monotonic;assumption. intros ;apply P_id_U'6'1_monotonic;assumption. intros ;apply P_id_REDUCE'II'IN_monotonic;assumption. intros ;apply P_id_TAUTOLOGY'I'IN_monotonic;assumption. intros ;apply P_id_U'9'1_monotonic;assumption. intros ;apply P_id_U'1'1_monotonic;assumption. intros ;apply P_id_U'14'1_monotonic;assumption. intros ;apply P_id_U'7'1_monotonic;assumption. intros ;apply P_id_U'4'1_monotonic;assumption. intros ;apply P_id_U'11'1_monotonic;assumption. intros ;apply P_id_INTERSECT'II'IN_monotonic;assumption. intros ;apply P_id_U'13'1_monotonic;assumption. intros ;apply P_id_U'6'2_monotonic;assumption. intros ;apply P_id_U'3'1_monotonic;assumption. intros ;apply P_id_U'16'1_monotonic;assumption. intros ;apply P_id_U'10'1_monotonic;assumption. intros ;apply P_id_U'2'1_monotonic;assumption. intros ;apply P_id_U'15'1_monotonic;assumption. intros ;apply P_id_U'8'1_monotonic;assumption. intros ;apply P_id_U'5'1_monotonic;assumption. intros ;apply P_id_U'12'1_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 x4 x20 x2 x5 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x5::nil)) x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'1'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'in (x4::x5::nil))::nil)) (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_1 : forall x4 x20 x2 x5 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x5::nil)) x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_intersect'ii'in (x4:: x5::nil)) (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_2 : forall x4 x20 x2 x5 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x4::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'2'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'in (x4::x5::nil))::nil)) (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_3 : forall x4 x20 x2 x5 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x4::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_intersect'ii'in (x4:: x5::nil)) (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_4 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::x7::nil)):: x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'3'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil))::x6::nil))::x8::nil)):: x9::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_5 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::x7::nil)):: x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil))::x6::nil))::x8::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_6 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6:: x7::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'4'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if (x6::x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7::x6::nil))::nil))::x8::nil)):: x9::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_7 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6:: x7::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7:: x6::nil))::nil))::x8::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_8 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'5'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11::(algebra.Alg.Term algebra.F.id_cons (x12:: x8::nil))::nil))::x9::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_9 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11:: (algebra.Alg.Term algebra.F.id_cons (x12:: x8::nil))::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_10 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11::x8::nil))::x9::nil))::x10::nil))::x12::x8::x9:: x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_11 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11::x8::nil)):: x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_12 : forall x8 x24 x20 x12 x10 x22 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x12 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'6'2 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x12::x8::nil))::x9::nil)):: x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21:: x20::nil)) (* *) | DP_R_xml_0_13 : forall x8 x24 x20 x12 x10 x22 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x12 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x12::x8::nil)):: x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21:: x20::nil)) (* *) | DP_R_xml_0_14 : forall x8 x20 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x11::nil)):: x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'7'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons (x11::x9::nil))::nil)):: x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_15 : forall x8 x20 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x11::nil)):: x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x11:: x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_16 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::x7::nil)):: x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'8'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil)):: x6::nil))::x9::nil))::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_17 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6::x7::nil)):: x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil))::x6::nil))::x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_18 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6:: x7::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'9'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if (x6::x7::nil)):: (algebra.Alg.Term algebra.F.id_if (x7::x6::nil))::nil)):: x9::nil))::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_19 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6:: x7::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7:: x6::nil))::nil))::x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_20 : forall x8 x20 x14 x9 x21 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_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil)):: x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'10'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::x9::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil)):: x14::nil))::x15::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_21 : forall x8 x20 x14 x9 x21 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_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil)):: x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::x9::nil)):: (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x14::nil))::x15::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_22 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'11'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons (x16::(algebra.Alg.Term algebra.F.id_cons (x17::x9::nil))::nil))::nil)):: x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_23 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x16:: (algebra.Alg.Term algebra.F.id_cons (x17:: x9::nil))::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_24 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons (x16::x9::nil))::nil))::x10::nil))::x8:: x17::x9::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_25 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x16:: x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_26 : forall x8 x24 x20 x10 x22 x17 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x17 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'12'2 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x17:: x9::nil))::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21:: x20::nil)) (* *) | DP_R_xml_0_27 : forall x8 x24 x20 x10 x22 x17 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x17 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x17:: x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21:: x20::nil)) (* *) | DP_R_xml_0_28 : forall x16 x8 x20 x10 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x16::nil)):: x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'13'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x16::x8::nil))::x9::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_29 : forall x16 x8 x20 x10 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x16::nil)):: x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x16::x8::nil)):: x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_30 : forall x20 x14 x9 x21 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_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'14'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::x9::nil))::(algebra.Alg.Term algebra.F.id_sequent (x14::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil)):: x15::nil))::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_31 : forall x20 x14 x9 x21 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_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::x9::nil)):: (algebra.Alg.Term algebra.F.id_sequent (x14:: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x15::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_32 : forall x20 x12 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil nil)::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x11::x12::nil)) x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'15'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'in (x11::x12::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_33 : forall x20 x12 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil nil)::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x11::x12::nil)) x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_intersect'ii'in (x11:: x12::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_34 : forall x20 x18, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x18 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_u'16'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons (x18:: (algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil)):: (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) (* *) | DP_R_xml_0_35 : forall x20 x18, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x18 x20) -> DP_R_xml_0 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons (x18::(algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil):: (algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil)) (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::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 x20 x18, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x18 x20) -> DP_R_xml_0_non_scc_1 (algebra.Alg.Term algebra.F.id_u'16'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil):: (algebra.Alg.Term algebra.F.id_cons (x18:: (algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::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 x20 x12 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil nil)::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x11::x12::nil)) x20) -> DP_R_xml_0_non_scc_2 (algebra.Alg.Term algebra.F.id_u'15'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'in (x11:: x12::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 x20 x14 x9 x21 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_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) -> DP_R_xml_0_non_scc_3 (algebra.Alg.Term algebra.F.id_u'14'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil):: x9::nil))::(algebra.Alg.Term algebra.F.id_sequent (x14::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil)):: x15::nil))::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 x16 x8 x20 x10 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x16::nil)):: x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_non_scc_4 (algebra.Alg.Term algebra.F.id_u'13'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x16:: x8::nil))::x9::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 x8 x24 x20 x10 x22 x17 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x17 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_non_scc_5 (algebra.Alg.Term algebra.F.id_u'12'2 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons (x17::x9::nil))::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21:: x20::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 x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_non_scc_6 (algebra.Alg.Term algebra.F.id_u'11'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x16:: (algebra.Alg.Term algebra.F.id_cons (x17:: x9::nil))::nil))::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 x8 x20 x14 x9 x21 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_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil)):: x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) -> DP_R_xml_0_non_scc_7 (algebra.Alg.Term algebra.F.id_u'10'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: x9::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x14::nil)):: x15::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6:: x7::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_non_scc_8 (algebra.Alg.Term algebra.F.id_u'9'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7::x6::nil))::nil))::x9::nil))::nil)):: x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_non_scc_9 (algebra.Alg.Term algebra.F.id_u'8'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil))::x6::nil))::x9::nil))::nil)):: x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 x8 x20 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x11::nil)):: x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_non_scc_10 (algebra.Alg.Term algebra.F.id_u'7'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x11:: x9::nil))::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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 x24 x20 x12 x10 x22 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x12 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_non_scc_11 (algebra.Alg.Term algebra.F.id_u'6'2 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x12:: x8::nil))::x9::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21:: x20::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 x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_non_scc_12 (algebra.Alg.Term algebra.F.id_u'5'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11:: (algebra.Alg.Term algebra.F.id_cons (x12:: x8::nil))::nil))::x9::nil))::x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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_non_scc_13 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_13_0 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6:: x7::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_non_scc_13 (algebra.Alg.Term algebra.F.id_u'4'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7::x6::nil))::nil))::x8::nil))::x9::nil)):: x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) . Lemma acc_DP_R_xml_0_non_scc_13 : forall x y, (DP_R_xml_0_non_scc_13 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_14 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_non_scc_14_0 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_non_scc_14 (algebra.Alg.Term algebra.F.id_u'3'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil))::x6::nil))::x8::nil))::x9::nil)):: x10::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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; (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 x20 x2 x5 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x4::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x20) -> DP_R_xml_0_non_scc_15 (algebra.Alg.Term algebra.F.id_u'2'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'in (x4:: x5::nil))::nil)) (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::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; (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 x20 x2 x5 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x5::nil)) x20) -> DP_R_xml_0_non_scc_16 (algebra.Alg.Term algebra.F.id_u'1'1 ((algebra.Alg.Term algebra.F.id_intersect'ii'in (x4:: x5::nil))::nil)) (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::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; (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_17 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_17_0 : forall x4 x20 x2 x5 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x4::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x5 x20) -> DP_R_xml_0_scc_17 (algebra.Alg.Term algebra.F.id_intersect'ii'in (x4::x5::nil)) (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_17_1 : forall x4 x20 x2 x5 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x4 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_cons (x2::x5::nil)) x20) -> DP_R_xml_0_scc_17 (algebra.Alg.Term algebra.F.id_intersect'ii'in (x4::x5::nil)) (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) . Module WF_DP_R_xml_0_scc_17. Open Scope Z_scope. Import ring_extention. Notation Local "a <= b" := (Zle a b). Notation Local "a < b" := (Zlt a b). Definition P_id_intersect'ii'in (x20:Z) (x21:Z) := 1* x21. Definition P_id_tautology'i'out := 0. Definition P_id_u'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 1 + 2* x22 + 1* x23. Definition P_id_u'3'1 (x20:Z) := 0. Definition P_id_u'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 1* x22 + 1* x23. Definition P_id_u'2'1 (x20:Z) := 1* x20. Definition P_id_u'9'1 (x20:Z) := 0. Definition P_id_iff (x20:Z) (x21:Z) := 0. Definition P_id_u'14'1 (x20:Z) := 3. Definition P_id_intersect'ii'out := 0. Definition P_id_u'7'1 (x20:Z) := 1. Definition P_id_x'2d (x20:Z) := 0. Definition P_id_u'13'1 (x20:Z) := 1. Definition P_id_sequent (x20:Z) (x21:Z) := 1* x20 + 2* x21. Definition P_id_u'10'1 (x20:Z) := 0. Definition P_id_x'2a (x20:Z) (x21:Z) := 1* x21. Definition P_id_tautology'i'in (x20:Z) := 2. Definition P_id_cons (x20:Z) (x21:Z) := 1 + 1* x20 + 2* x21. Definition P_id_u'6'2 (x20:Z) := 1. Definition P_id_x'2b (x20:Z) (x21:Z) := 0. Definition P_id_u'12'2 (x20:Z) := 0. Definition P_id_reduce'ii'in (x20:Z) (x21:Z) := 1* x20. Definition P_id_p (x20:Z) := 3 + 1* x20. Definition P_id_u'4'1 (x20:Z) := 1. Definition P_id_u'15'1 (x20:Z) := 0. Definition P_id_u'1'1 (x20:Z) := 1. Definition P_id_u'8'1 (x20:Z) := 2. Definition P_id_reduce'ii'out := 0. Definition P_id_nil := 0. Definition P_id_if (x20:Z) (x21:Z) := 0. Definition P_id_u'11'1 (x20:Z) := 2. Definition P_id_u'5'1 (x20:Z) := 0. Definition P_id_u'16'1 (x20:Z) := 1. Lemma P_id_intersect'ii'in_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'3'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'2'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'9'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_iff_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_iff x21 x23 <= P_id_iff x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'14'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'7'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2d_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'13'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_sequent_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_sequent x21 x23 <= P_id_sequent x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'10'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2a_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_x'2a x21 x23 <= P_id_x'2a x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'in_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) -> P_id_tautology'i'in x21 <= P_id_tautology'i'in x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_cons x21 x23 <= P_id_cons x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'6'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2b_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_x'2b x21 x23 <= P_id_x'2b x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'12'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'in_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_p_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'4'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'15'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'1'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'8'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_if_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'11'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'5'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'16'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_intersect'ii'in_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_intersect'ii'in x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'out_bounded : 0 <= P_id_tautology'i'out . 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'6'1_bounded : forall x24 x20 x22 x21 x23, (0 <= x20) -> (0 <= x21) -> (0 <= x22) -> (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'6'1 x24 x23 x22 x21 x20. 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'3'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'3'1 x20. 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'12'1_bounded : forall x24 x20 x22 x21 x23, (0 <= x20) -> (0 <= x21) -> (0 <= x22) -> (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'12'1 x24 x23 x22 x21 x20. 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'2'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'2'1 x20. 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'9'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'9'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_iff_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_iff x21 x20. 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'14'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'14'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_intersect'ii'out_bounded : 0 <= P_id_intersect'ii'out . 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'7'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'7'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2d_bounded : forall x20, (0 <= x20) ->0 <= P_id_x'2d x20. 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'13'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'13'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_sequent_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_sequent x21 x20. 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'10'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'10'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2a_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2a x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'in_bounded : forall x20, (0 <= x20) ->0 <= P_id_tautology'i'in x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_cons x21 x20. 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'6'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'6'2 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2b_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2b x21 x20. 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'12'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'12'2 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'in_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_reduce'ii'in x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_p_bounded : forall x20, (0 <= x20) ->0 <= P_id_p x20. 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'4'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'4'1 x20. 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'15'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'15'1 x20. 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'1'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'1'1 x20. 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'8'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'8'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'out_bounded : 0 <= P_id_reduce'ii'out . 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_if_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_if x21 x20. 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'11'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'11'1 x20. 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'5'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'5'1 x20. 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'16'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'16'1 x20. 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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21:: x20::nil)) => P_id_intersect'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) => P_id_tautology'i'out | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22:: x21::x20::nil)) => P_id_u'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_u'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22:: x21::x20::nil)) => P_id_u'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_u'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_u'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) => P_id_iff (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_u'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) => P_id_intersect'ii'out | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_u'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) => P_id_x'2d (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_u'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_sequent (x21::x20::nil)) => P_id_sequent (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_u'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) => P_id_x'2a (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_tautology'i'in (measure x20) | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) => P_id_cons (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_u'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) => P_id_x'2b (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_u'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21:: x20::nil)) => P_id_reduce'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_p (x20::nil)) => P_id_p (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_u'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_u'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_u'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_u'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) => P_id_reduce'ii'out | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) => P_id_if (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_u'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_u'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_u'16'1 (measure x20) | _ => 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_intersect'ii'in_monotonic;assumption. intros ;apply P_id_u'6'1_monotonic;assumption. intros ;apply P_id_u'3'1_monotonic;assumption. intros ;apply P_id_u'12'1_monotonic;assumption. intros ;apply P_id_u'2'1_monotonic;assumption. intros ;apply P_id_u'9'1_monotonic;assumption. intros ;apply P_id_iff_monotonic;assumption. intros ;apply P_id_u'14'1_monotonic;assumption. intros ;apply P_id_u'7'1_monotonic;assumption. intros ;apply P_id_x'2d_monotonic;assumption. intros ;apply P_id_u'13'1_monotonic;assumption. intros ;apply P_id_sequent_monotonic;assumption. intros ;apply P_id_u'10'1_monotonic;assumption. intros ;apply P_id_x'2a_monotonic;assumption. intros ;apply P_id_tautology'i'in_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_u'6'2_monotonic;assumption. intros ;apply P_id_x'2b_monotonic;assumption. intros ;apply P_id_u'12'2_monotonic;assumption. intros ;apply P_id_reduce'ii'in_monotonic;assumption. intros ;apply P_id_p_monotonic;assumption. intros ;apply P_id_u'4'1_monotonic;assumption. intros ;apply P_id_u'15'1_monotonic;assumption. intros ;apply P_id_u'1'1_monotonic;assumption. intros ;apply P_id_u'8'1_monotonic;assumption. intros ;apply P_id_if_monotonic;assumption. intros ;apply P_id_u'11'1_monotonic;assumption. intros ;apply P_id_u'5'1_monotonic;assumption. intros ;apply P_id_u'16'1_monotonic;assumption. intros ;apply P_id_intersect'ii'in_bounded;assumption. intros ;apply P_id_tautology'i'out_bounded;assumption. intros ;apply P_id_u'6'1_bounded;assumption. intros ;apply P_id_u'3'1_bounded;assumption. intros ;apply P_id_u'12'1_bounded;assumption. intros ;apply P_id_u'2'1_bounded;assumption. intros ;apply P_id_u'9'1_bounded;assumption. intros ;apply P_id_iff_bounded;assumption. intros ;apply P_id_u'14'1_bounded;assumption. intros ;apply P_id_intersect'ii'out_bounded;assumption. intros ;apply P_id_u'7'1_bounded;assumption. intros ;apply P_id_x'2d_bounded;assumption. intros ;apply P_id_u'13'1_bounded;assumption. intros ;apply P_id_sequent_bounded;assumption. intros ;apply P_id_u'10'1_bounded;assumption. intros ;apply P_id_x'2a_bounded;assumption. intros ;apply P_id_tautology'i'in_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_u'6'2_bounded;assumption. intros ;apply P_id_x'2b_bounded;assumption. intros ;apply P_id_u'12'2_bounded;assumption. intros ;apply P_id_reduce'ii'in_bounded;assumption. intros ;apply P_id_p_bounded;assumption. intros ;apply P_id_u'4'1_bounded;assumption. intros ;apply P_id_u'15'1_bounded;assumption. intros ;apply P_id_u'1'1_bounded;assumption. intros ;apply P_id_u'8'1_bounded;assumption. intros ;apply P_id_reduce'ii'out_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_if_bounded;assumption. intros ;apply P_id_u'11'1_bounded;assumption. intros ;apply P_id_u'5'1_bounded;assumption. intros ;apply P_id_u'16'1_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_U'12'2 (x20:Z) := 0. Definition P_id_U'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0. Definition P_id_REDUCE'II'IN (x20:Z) (x21:Z) := 0. Definition P_id_TAUTOLOGY'I'IN (x20:Z) := 0. Definition P_id_U'9'1 (x20:Z) := 0. Definition P_id_U'1'1 (x20:Z) := 0. Definition P_id_U'14'1 (x20:Z) := 0. Definition P_id_U'7'1 (x20:Z) := 0. Definition P_id_U'4'1 (x20:Z) := 0. Definition P_id_U'11'1 (x20:Z) := 0. Definition P_id_INTERSECT'II'IN (x20:Z) (x21:Z) := 1* x20 + 1* x21. Definition P_id_U'13'1 (x20:Z) := 0. Definition P_id_U'6'2 (x20:Z) := 0. Definition P_id_U'3'1 (x20:Z) := 0. Definition P_id_U'16'1 (x20:Z) := 0. Definition P_id_U'10'1 (x20:Z) := 0. Definition P_id_U'2'1 (x20:Z) := 0. Definition P_id_U'15'1 (x20:Z) := 0. Definition P_id_U'8'1 (x20:Z) := 0. Definition P_id_U'5'1 (x20:Z) := 0. Definition P_id_U'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0. Lemma P_id_U'12'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_REDUCE'II'IN_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_TAUTOLOGY'I'IN_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) -> P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'9'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'1'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'14'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'7'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'4'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'11'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_INTERSECT'II'IN_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'13'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'6'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'3'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'16'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'10'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'2'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'15'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'8'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'5'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. 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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1 P_id_U'12'2 P_id_U'6'1 P_id_REDUCE'II'IN P_id_TAUTOLOGY'I'IN P_id_U'9'1 P_id_U'1'1 P_id_U'14'1 P_id_U'7'1 P_id_U'4'1 P_id_U'11'1 P_id_INTERSECT'II'IN P_id_U'13'1 P_id_U'6'2 P_id_U'3'1 P_id_U'16'1 P_id_U'10'1 P_id_U'2'1 P_id_U'15'1 P_id_U'8'1 P_id_U'5'1 P_id_U'12'1. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_U'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23:: x22::x21::x20::nil)) => P_id_U'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) => P_id_REDUCE'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_TAUTOLOGY'I'IN (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_U'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_U'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_U'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_U'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_U'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_U'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) => P_id_INTERSECT'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_U'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_U'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_U'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_U'16'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_U'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_U'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_U'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_U'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_U'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23:: x22::x21::x20::nil)) => P_id_U'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | _ => 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_intersect'ii'in_monotonic;assumption. intros ;apply P_id_u'6'1_monotonic;assumption. intros ;apply P_id_u'3'1_monotonic;assumption. intros ;apply P_id_u'12'1_monotonic;assumption. intros ;apply P_id_u'2'1_monotonic;assumption. intros ;apply P_id_u'9'1_monotonic;assumption. intros ;apply P_id_iff_monotonic;assumption. intros ;apply P_id_u'14'1_monotonic;assumption. intros ;apply P_id_u'7'1_monotonic;assumption. intros ;apply P_id_x'2d_monotonic;assumption. intros ;apply P_id_u'13'1_monotonic;assumption. intros ;apply P_id_sequent_monotonic;assumption. intros ;apply P_id_u'10'1_monotonic;assumption. intros ;apply P_id_x'2a_monotonic;assumption. intros ;apply P_id_tautology'i'in_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_u'6'2_monotonic;assumption. intros ;apply P_id_x'2b_monotonic;assumption. intros ;apply P_id_u'12'2_monotonic;assumption. intros ;apply P_id_reduce'ii'in_monotonic;assumption. intros ;apply P_id_p_monotonic;assumption. intros ;apply P_id_u'4'1_monotonic;assumption. intros ;apply P_id_u'15'1_monotonic;assumption. intros ;apply P_id_u'1'1_monotonic;assumption. intros ;apply P_id_u'8'1_monotonic;assumption. intros ;apply P_id_if_monotonic;assumption. intros ;apply P_id_u'11'1_monotonic;assumption. intros ;apply P_id_u'5'1_monotonic;assumption. intros ;apply P_id_u'16'1_monotonic;assumption. intros ;apply P_id_intersect'ii'in_bounded;assumption. intros ;apply P_id_tautology'i'out_bounded;assumption. intros ;apply P_id_u'6'1_bounded;assumption. intros ;apply P_id_u'3'1_bounded;assumption. intros ;apply P_id_u'12'1_bounded;assumption. intros ;apply P_id_u'2'1_bounded;assumption. intros ;apply P_id_u'9'1_bounded;assumption. intros ;apply P_id_iff_bounded;assumption. intros ;apply P_id_u'14'1_bounded;assumption. intros ;apply P_id_intersect'ii'out_bounded;assumption. intros ;apply P_id_u'7'1_bounded;assumption. intros ;apply P_id_x'2d_bounded;assumption. intros ;apply P_id_u'13'1_bounded;assumption. intros ;apply P_id_sequent_bounded;assumption. intros ;apply P_id_u'10'1_bounded;assumption. intros ;apply P_id_x'2a_bounded;assumption. intros ;apply P_id_tautology'i'in_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_u'6'2_bounded;assumption. intros ;apply P_id_x'2b_bounded;assumption. intros ;apply P_id_u'12'2_bounded;assumption. intros ;apply P_id_reduce'ii'in_bounded;assumption. intros ;apply P_id_p_bounded;assumption. intros ;apply P_id_u'4'1_bounded;assumption. intros ;apply P_id_u'15'1_bounded;assumption. intros ;apply P_id_u'1'1_bounded;assumption. intros ;apply P_id_u'8'1_bounded;assumption. intros ;apply P_id_reduce'ii'out_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_if_bounded;assumption. intros ;apply P_id_u'11'1_bounded;assumption. intros ;apply P_id_u'5'1_bounded;assumption. intros ;apply P_id_u'16'1_bounded;assumption. apply rules_monotonic. intros ;apply P_id_U'12'2_monotonic;assumption. intros ;apply P_id_U'6'1_monotonic;assumption. intros ;apply P_id_REDUCE'II'IN_monotonic;assumption. intros ;apply P_id_TAUTOLOGY'I'IN_monotonic;assumption. intros ;apply P_id_U'9'1_monotonic;assumption. intros ;apply P_id_U'1'1_monotonic;assumption. intros ;apply P_id_U'14'1_monotonic;assumption. intros ;apply P_id_U'7'1_monotonic;assumption. intros ;apply P_id_U'4'1_monotonic;assumption. intros ;apply P_id_U'11'1_monotonic;assumption. intros ;apply P_id_INTERSECT'II'IN_monotonic;assumption. intros ;apply P_id_U'13'1_monotonic;assumption. intros ;apply P_id_U'6'2_monotonic;assumption. intros ;apply P_id_U'3'1_monotonic;assumption. intros ;apply P_id_U'16'1_monotonic;assumption. intros ;apply P_id_U'10'1_monotonic;assumption. intros ;apply P_id_U'2'1_monotonic;assumption. intros ;apply P_id_U'15'1_monotonic;assumption. intros ;apply P_id_U'8'1_monotonic;assumption. intros ;apply P_id_U'5'1_monotonic;assumption. intros ;apply P_id_U'12'1_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.DP_R_xml_0_scc_17. 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_17. Definition wf_DP_R_xml_0_scc_17 := WF_DP_R_xml_0_scc_17.wf. Lemma acc_DP_R_xml_0_scc_17 : forall x y, (DP_R_xml_0_scc_17 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_17). 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_16; 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' ))))). apply wf_DP_R_xml_0_scc_17. 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 x20 x12 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil nil)::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x11::x12::nil)) x20) -> DP_R_xml_0_non_scc_18 (algebra.Alg.Term algebra.F.id_intersect'ii'in (x11::x12::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::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_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' ))|| ((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_19 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_19_0 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11:: x8::nil))::x9::nil))::x10::nil))::x12::x8::x9:: x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_1 : forall x8 x24 x20 x12 x10 x22 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x12 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x12:: x8::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21:: x20::nil)) (* *) | DP_R_xml_0_scc_19_2 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil)):: x6::nil))::x8::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_3 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11:: x8::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_4 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6:: x7::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7:: x6::nil))::nil))::x8::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_5 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11:: (algebra.Alg.Term algebra.F.id_cons (x12:: x8::nil))::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_6 : forall x8 x20 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x11::nil)):: x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x11:: x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_7 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil)):: x6::nil))::x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_8 : forall x8 x20 x14 x9 x21 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_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil)):: x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: x9::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil)):: x14::nil))::x15::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_9 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6:: x7::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7:: x6::nil))::nil))::x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_10 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x16:: x9::nil))::nil))::x10::nil))::x8::x17::x9:: x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_11 : forall x8 x24 x20 x10 x22 x17 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x17 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x17:: x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21:: x20::nil)) (* *) | DP_R_xml_0_scc_19_12 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x16:: (algebra.Alg.Term algebra.F.id_cons (x17:: x9::nil))::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_13 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x16:: x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_14 : forall x16 x8 x20 x10 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x16::nil)):: x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x16:: x8::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_15 : forall x20 x14 x9 x21 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_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) -> DP_R_xml_0_scc_19 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil):: x9::nil))::(algebra.Alg.Term algebra.F.id_sequent (x14::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil)):: x15::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) . Module WF_DP_R_xml_0_scc_19. Inductive DP_R_xml_0_scc_19_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_19_large_0 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11:: x8::nil))::x9::nil))::x10::nil))::x12::x8:: x9::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_1 : forall x8 x24 x20 x12 x10 x22 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x12 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x12::x8::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21:: x20::nil)) (* *) | DP_R_xml_0_scc_19_large_2 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil))::x6::nil))::x8::nil)):: x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_3 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11:: x8::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_4 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11:: (algebra.Alg.Term algebra.F.id_cons (x12:: x8::nil))::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_5 : forall x8 x20 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x11::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons (x11::x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_6 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil))::x6::nil))::x9::nil))::nil)):: x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_7 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons (x16::x9::nil))::nil))::x10::nil))::x8:: x17::x9::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_8 : forall x8 x24 x20 x10 x22 x17 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x17 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons (x17:: x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21:: x20::nil)) (* *) | DP_R_xml_0_scc_19_large_9 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons (x16::(algebra.Alg.Term algebra.F.id_cons (x17::x9::nil))::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_10 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons (x16::x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_11 : forall x16 x8 x20 x10 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x16::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x16:: x8::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) . Inductive DP_R_xml_0_scc_19_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_19_strict_0 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6:: x7::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_strict (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7::x6::nil))::nil)):: x8::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_strict_1 : forall x8 x20 x14 x9 x21 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_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil)):: x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) -> DP_R_xml_0_scc_19_strict (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::x9::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x14::nil)):: x15::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_strict_2 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_iff (x6:: x7::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_strict (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::(algebra.Alg.Term algebra.F.id_if (x7::x6::nil))::nil)):: x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_strict_3 : forall x20 x14 x9 x21 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_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x14::x15::nil)) x20) -> DP_R_xml_0_scc_19_strict (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil):: x9::nil))::(algebra.Alg.Term algebra.F.id_sequent (x14:: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_p (x13::nil))::x15::nil))::nil))::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) . Module WF_DP_R_xml_0_scc_19_large. Inductive DP_R_xml_0_scc_19_large_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_19_large_large_0 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11::x8::nil)):: x9::nil))::x10::nil))::x12::x8::x9:: x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_large_1 : forall x8 x24 x20 x12 x10 x22 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x12 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x12:: x8::nil))::x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21:: x20::nil)) (* *) | DP_R_xml_0_scc_19_large_large_2 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11::x8::nil)):: x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_large_3 : forall x8 x24 x20 x10 x22 x17 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x17 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x17:: x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21:: x20::nil)) (* *) | DP_R_xml_0_scc_19_large_large_4 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x16::(algebra.Alg.Term algebra.F.id_cons (x17:: x9::nil))::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) . Inductive DP_R_xml_0_scc_19_large_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_19_large_strict_0 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil)):: x6::nil))::x8::nil))::x9::nil)):: x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_strict_1 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11:: (algebra.Alg.Term algebra.F.id_cons (x12:: x8::nil))::nil))::x9::nil)):: x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_strict_2 : forall x8 x20 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x11::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x11:: x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_strict_3 : forall x8 x20 x10 x6 x9 x21 x7, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_if (x6:: x7::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b ((algebra.Alg.Term algebra.F.id_x'2d (x7::nil)):: x6::nil))::x9::nil))::nil)):: x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_strict_4 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term algebra.F.id_u'12'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x16:: x9::nil))::nil))::x10::nil))::x8:: x17::x9::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_strict_5 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2a (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x16:: x9::nil))::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_strict_6 : forall x16 x8 x20 x10 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2d (x16::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_strict (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x16::x8::nil)):: x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) . Module WF_DP_R_xml_0_scc_19_large_large. Inductive DP_R_xml_0_scc_19_large_large_scc_1 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_19_large_large_scc_1_0 : forall x8 x24 x20 x12 x10 x22 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x12 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large_scc_1 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x12:: x8::nil))::x9::nil)):: x10::nil)) (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21:: x20::nil)) (* *) | DP_R_xml_0_scc_19_large_large_scc_1_1 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large_scc_1 (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11:: x8::nil))::x9::nil)):: x10::nil))::x12::x8::x9:: x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_large_scc_1_2 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large_scc_1 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11:: x8::nil))::x9::nil)):: x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_large_scc_1_3 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large_scc_1 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x16:: (algebra.Alg.Term algebra.F.id_cons (x17:: x9::nil))::nil))::nil)):: x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) . Module WF_DP_R_xml_0_scc_19_large_large_scc_1. Inductive DP_R_xml_0_scc_19_large_large_scc_1_large : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_19_large_large_scc_1_large_0 : forall x16 x8 x20 x10 x17 x9 x21, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent (x8::(algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x16:: x17::nil))::x9::nil))::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large_scc_1_large (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x16:: (algebra.Alg.Term algebra.F.id_cons (x17:: x9::nil))::nil))::nil)):: x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) . Inductive DP_R_xml_0_scc_19_large_large_scc_1_strict : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_19_large_large_scc_1_strict_0 : forall x8 x24 x20 x12 x10 x22 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x12 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large_scc_1_strict (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x12::x8::nil)):: x9::nil)):: x10::nil)) (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22::x21:: x20::nil)) (* *) | DP_R_xml_0_scc_19_large_large_scc_1_strict_1 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large_scc_1_strict (algebra.Alg.Term algebra.F.id_u'6'1 ((algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11::x8::nil)):: x9::nil)):: x10::nil))::x12::x8:: x9::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) (* *) | DP_R_xml_0_scc_19_large_large_scc_1_strict_2 : forall x8 x20 x12 x10 x9 x21 x11, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons ((algebra.Alg.Term algebra.F.id_x'2b (x11:: x12::nil))::x8::nil))::x9::nil)) x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large_scc_1_strict (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_cons (x11::x8::nil)):: x9::nil))::x10::nil)) (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) . Module WF_DP_R_xml_0_scc_19_large_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_intersect'ii'in (x20:Z) (x21:Z) := 1. Definition P_id_tautology'i'out := 1. Definition P_id_u'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 1* x24. Definition P_id_u'3'1 (x20:Z) := 0. Definition P_id_u'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 1* x24. Definition P_id_u'2'1 (x20:Z) := 0. Definition P_id_u'9'1 (x20:Z) := 1* x20. Definition P_id_iff (x20:Z) (x21:Z) := 0. Definition P_id_u'14'1 (x20:Z) := 0. Definition P_id_intersect'ii'out := 0. Definition P_id_u'7'1 (x20:Z) := 1* x20. Definition P_id_x'2d (x20:Z) := 0. Definition P_id_u'13'1 (x20:Z) := 0. Definition P_id_sequent (x20:Z) (x21:Z) := 2* x21. Definition P_id_u'10'1 (x20:Z) := 1* x20. Definition P_id_x'2a (x20:Z) (x21:Z) := 0. Definition P_id_tautology'i'in (x20:Z) := 2 + 3* x20. Definition P_id_cons (x20:Z) (x21:Z) := 2* x20. Definition P_id_u'6'2 (x20:Z) := 0. Definition P_id_x'2b (x20:Z) (x21:Z) := 2 + 2* x20. Definition P_id_u'12'2 (x20:Z) := 0. Definition P_id_reduce'ii'in (x20:Z) (x21:Z) := 1* x21. Definition P_id_p (x20:Z) := 0. Definition P_id_u'4'1 (x20:Z) := 1* x20. Definition P_id_u'15'1 (x20:Z) := 0. Definition P_id_u'1'1 (x20:Z) := 1* x20. Definition P_id_u'8'1 (x20:Z) := 1* x20. Definition P_id_reduce'ii'out := 0. Definition P_id_nil := 0. Definition P_id_if (x20:Z) (x21:Z) := 0. Definition P_id_u'11'1 (x20:Z) := 1* x20. Definition P_id_u'5'1 (x20:Z) := 0. Definition P_id_u'16'1 (x20:Z) := 1. Lemma P_id_intersect'ii'in_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'3'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'2'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'9'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_iff_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_iff x21 x23 <= P_id_iff x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'14'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'7'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2d_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'13'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_sequent_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_sequent x21 x23 <= P_id_sequent x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'10'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2a_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_x'2a x21 x23 <= P_id_x'2a x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'in_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) -> P_id_tautology'i'in x21 <= P_id_tautology'i'in x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_cons x21 x23 <= P_id_cons x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'6'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2b_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_x'2b x21 x23 <= P_id_x'2b x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'12'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'in_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_p_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'4'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'15'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'1'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'8'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_if_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'11'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'5'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'16'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_intersect'ii'in_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_intersect'ii'in x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'out_bounded : 0 <= P_id_tautology'i'out . 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'6'1_bounded : forall x24 x20 x22 x21 x23, (0 <= x20) -> (0 <= x21) -> (0 <= x22) -> (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'6'1 x24 x23 x22 x21 x20. 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'3'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'3'1 x20. 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'12'1_bounded : forall x24 x20 x22 x21 x23, (0 <= x20) -> (0 <= x21) -> (0 <= x22) -> (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'12'1 x24 x23 x22 x21 x20. 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'2'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'2'1 x20. 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'9'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'9'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_iff_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_iff x21 x20. 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'14'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'14'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_intersect'ii'out_bounded : 0 <= P_id_intersect'ii'out . 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'7'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'7'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2d_bounded : forall x20, (0 <= x20) ->0 <= P_id_x'2d x20. 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'13'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'13'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_sequent_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_sequent x21 x20. 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'10'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'10'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2a_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2a x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'in_bounded : forall x20, (0 <= x20) ->0 <= P_id_tautology'i'in x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_cons x21 x20. 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'6'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'6'2 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2b_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2b x21 x20. 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'12'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'12'2 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'in_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_reduce'ii'in x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_p_bounded : forall x20, (0 <= x20) ->0 <= P_id_p x20. 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'4'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'4'1 x20. 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'15'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'15'1 x20. 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'1'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'1'1 x20. 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'8'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'8'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'out_bounded : 0 <= P_id_reduce'ii'out . 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_if_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_if x21 x20. 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'11'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'11'1 x20. 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'5'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'5'1 x20. 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'16'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'16'1 x20. 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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) => P_id_intersect'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) => P_id_tautology'i'out | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23:: x22::x21::x20::nil)) => P_id_u'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_u'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23:: x22::x21::x20::nil)) => P_id_u'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_u'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_u'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) => P_id_iff (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_u'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) => P_id_intersect'ii'out | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_u'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) => P_id_x'2d (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_u'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_sequent (x21:: x20::nil)) => P_id_sequent (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_u'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) => P_id_x'2a (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_tautology'i'in (measure x20) | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) => P_id_cons (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_u'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) => P_id_x'2b (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_u'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21:: x20::nil)) => P_id_reduce'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_p (x20::nil)) => P_id_p (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_u'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_u'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_u'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_u'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) => P_id_reduce'ii'out | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) => P_id_if (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_u'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_u'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_u'16'1 (measure x20) | _ => 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_intersect'ii'in_monotonic;assumption. intros ;apply P_id_u'6'1_monotonic;assumption. intros ;apply P_id_u'3'1_monotonic;assumption. intros ;apply P_id_u'12'1_monotonic;assumption. intros ;apply P_id_u'2'1_monotonic;assumption. intros ;apply P_id_u'9'1_monotonic;assumption. intros ;apply P_id_iff_monotonic;assumption. intros ;apply P_id_u'14'1_monotonic;assumption. intros ;apply P_id_u'7'1_monotonic;assumption. intros ;apply P_id_x'2d_monotonic;assumption. intros ;apply P_id_u'13'1_monotonic;assumption. intros ;apply P_id_sequent_monotonic;assumption. intros ;apply P_id_u'10'1_monotonic;assumption. intros ;apply P_id_x'2a_monotonic;assumption. intros ;apply P_id_tautology'i'in_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_u'6'2_monotonic;assumption. intros ;apply P_id_x'2b_monotonic;assumption. intros ;apply P_id_u'12'2_monotonic;assumption. intros ;apply P_id_reduce'ii'in_monotonic;assumption. intros ;apply P_id_p_monotonic;assumption. intros ;apply P_id_u'4'1_monotonic;assumption. intros ;apply P_id_u'15'1_monotonic;assumption. intros ;apply P_id_u'1'1_monotonic;assumption. intros ;apply P_id_u'8'1_monotonic;assumption. intros ;apply P_id_if_monotonic;assumption. intros ;apply P_id_u'11'1_monotonic;assumption. intros ;apply P_id_u'5'1_monotonic;assumption. intros ;apply P_id_u'16'1_monotonic;assumption. intros ;apply P_id_intersect'ii'in_bounded;assumption. intros ;apply P_id_tautology'i'out_bounded;assumption. intros ;apply P_id_u'6'1_bounded;assumption. intros ;apply P_id_u'3'1_bounded;assumption. intros ;apply P_id_u'12'1_bounded;assumption. intros ;apply P_id_u'2'1_bounded;assumption. intros ;apply P_id_u'9'1_bounded;assumption. intros ;apply P_id_iff_bounded;assumption. intros ;apply P_id_u'14'1_bounded;assumption. intros ;apply P_id_intersect'ii'out_bounded;assumption. intros ;apply P_id_u'7'1_bounded;assumption. intros ;apply P_id_x'2d_bounded;assumption. intros ;apply P_id_u'13'1_bounded;assumption. intros ;apply P_id_sequent_bounded;assumption. intros ;apply P_id_u'10'1_bounded;assumption. intros ;apply P_id_x'2a_bounded;assumption. intros ;apply P_id_tautology'i'in_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_u'6'2_bounded;assumption. intros ;apply P_id_x'2b_bounded;assumption. intros ;apply P_id_u'12'2_bounded;assumption. intros ;apply P_id_reduce'ii'in_bounded;assumption. intros ;apply P_id_p_bounded;assumption. intros ;apply P_id_u'4'1_bounded;assumption. intros ;apply P_id_u'15'1_bounded;assumption. intros ;apply P_id_u'1'1_bounded;assumption. intros ;apply P_id_u'8'1_bounded;assumption. intros ;apply P_id_reduce'ii'out_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_if_bounded;assumption. intros ;apply P_id_u'11'1_bounded;assumption. intros ;apply P_id_u'5'1_bounded;assumption. intros ;apply P_id_u'16'1_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_U'12'2 (x20:Z) := 0. Definition P_id_U'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0. Definition P_id_REDUCE'II'IN (x20:Z) (x21:Z) := 2* x20. Definition P_id_TAUTOLOGY'I'IN (x20:Z) := 0. Definition P_id_U'9'1 (x20:Z) := 0. Definition P_id_U'1'1 (x20:Z) := 0. Definition P_id_U'14'1 (x20:Z) := 0. Definition P_id_U'7'1 (x20:Z) := 0. Definition P_id_U'4'1 (x20:Z) := 0. Definition P_id_U'11'1 (x20:Z) := 0. Definition P_id_INTERSECT'II'IN (x20:Z) (x21:Z) := 0. Definition P_id_U'13'1 (x20:Z) := 0. Definition P_id_U'6'2 (x20:Z) := 0. Definition P_id_U'3'1 (x20:Z) := 0. Definition P_id_U'16'1 (x20:Z) := 0. Definition P_id_U'10'1 (x20:Z) := 0. Definition P_id_U'2'1 (x20:Z) := 0. Definition P_id_U'15'1 (x20:Z) := 0. Definition P_id_U'8'1 (x20:Z) := 0. Definition P_id_U'5'1 (x20:Z) := 0. Definition P_id_U'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0. Lemma P_id_U'12'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_REDUCE'II'IN_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_TAUTOLOGY'I'IN_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) -> P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'9'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'1'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'14'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'7'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'4'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'11'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_INTERSECT'II'IN_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'13'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'6'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'3'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'16'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'10'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'2'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'15'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'8'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'5'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. 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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1 P_id_U'12'2 P_id_U'6'1 P_id_REDUCE'II'IN P_id_TAUTOLOGY'I'IN P_id_U'9'1 P_id_U'1'1 P_id_U'14'1 P_id_U'7'1 P_id_U'4'1 P_id_U'11'1 P_id_INTERSECT'II'IN P_id_U'13'1 P_id_U'6'2 P_id_U'3'1 P_id_U'16'1 P_id_U'10'1 P_id_U'2'1 P_id_U'15'1 P_id_U'8'1 P_id_U'5'1 P_id_U'12'1. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_U'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'1 (x24:: x23::x22::x21::x20::nil)) => P_id_U'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) => P_id_REDUCE'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_TAUTOLOGY'I'IN (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_U'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_U'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_U'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_U'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_U'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_U'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21:: x20::nil)) => P_id_INTERSECT'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_U'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_U'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_U'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_U'16'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_U'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_U'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_U'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_U'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_U'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24:: x23::x22::x21::x20::nil)) => P_id_U'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | _ => 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_intersect'ii'in_monotonic;assumption. intros ;apply P_id_u'6'1_monotonic;assumption. intros ;apply P_id_u'3'1_monotonic;assumption. intros ;apply P_id_u'12'1_monotonic;assumption. intros ;apply P_id_u'2'1_monotonic;assumption. intros ;apply P_id_u'9'1_monotonic;assumption. intros ;apply P_id_iff_monotonic;assumption. intros ;apply P_id_u'14'1_monotonic;assumption. intros ;apply P_id_u'7'1_monotonic;assumption. intros ;apply P_id_x'2d_monotonic;assumption. intros ;apply P_id_u'13'1_monotonic;assumption. intros ;apply P_id_sequent_monotonic;assumption. intros ;apply P_id_u'10'1_monotonic;assumption. intros ;apply P_id_x'2a_monotonic;assumption. intros ;apply P_id_tautology'i'in_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_u'6'2_monotonic;assumption. intros ;apply P_id_x'2b_monotonic;assumption. intros ;apply P_id_u'12'2_monotonic;assumption. intros ;apply P_id_reduce'ii'in_monotonic;assumption. intros ;apply P_id_p_monotonic;assumption. intros ;apply P_id_u'4'1_monotonic;assumption. intros ;apply P_id_u'15'1_monotonic;assumption. intros ;apply P_id_u'1'1_monotonic;assumption. intros ;apply P_id_u'8'1_monotonic;assumption. intros ;apply P_id_if_monotonic;assumption. intros ;apply P_id_u'11'1_monotonic;assumption. intros ;apply P_id_u'5'1_monotonic;assumption. intros ;apply P_id_u'16'1_monotonic;assumption. intros ;apply P_id_intersect'ii'in_bounded;assumption. intros ;apply P_id_tautology'i'out_bounded;assumption. intros ;apply P_id_u'6'1_bounded;assumption. intros ;apply P_id_u'3'1_bounded;assumption. intros ;apply P_id_u'12'1_bounded;assumption. intros ;apply P_id_u'2'1_bounded;assumption. intros ;apply P_id_u'9'1_bounded;assumption. intros ;apply P_id_iff_bounded;assumption. intros ;apply P_id_u'14'1_bounded;assumption. intros ;apply P_id_intersect'ii'out_bounded;assumption. intros ;apply P_id_u'7'1_bounded;assumption. intros ;apply P_id_x'2d_bounded;assumption. intros ;apply P_id_u'13'1_bounded;assumption. intros ;apply P_id_sequent_bounded;assumption. intros ;apply P_id_u'10'1_bounded;assumption. intros ;apply P_id_x'2a_bounded;assumption. intros ;apply P_id_tautology'i'in_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_u'6'2_bounded;assumption. intros ;apply P_id_x'2b_bounded;assumption. intros ;apply P_id_u'12'2_bounded;assumption. intros ;apply P_id_reduce'ii'in_bounded;assumption. intros ;apply P_id_p_bounded;assumption. intros ;apply P_id_u'4'1_bounded;assumption. intros ;apply P_id_u'15'1_bounded;assumption. intros ;apply P_id_u'1'1_bounded;assumption. intros ;apply P_id_u'8'1_bounded;assumption. intros ;apply P_id_reduce'ii'out_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_if_bounded;assumption. intros ;apply P_id_u'11'1_bounded;assumption. intros ;apply P_id_u'5'1_bounded;assumption. intros ;apply P_id_u'16'1_bounded;assumption. apply rules_monotonic. intros ;apply P_id_U'12'2_monotonic;assumption. intros ;apply P_id_U'6'1_monotonic;assumption. intros ;apply P_id_REDUCE'II'IN_monotonic;assumption. intros ;apply P_id_TAUTOLOGY'I'IN_monotonic;assumption. intros ;apply P_id_U'9'1_monotonic;assumption. intros ;apply P_id_U'1'1_monotonic;assumption. intros ;apply P_id_U'14'1_monotonic;assumption. intros ;apply P_id_U'7'1_monotonic;assumption. intros ;apply P_id_U'4'1_monotonic;assumption. intros ;apply P_id_U'11'1_monotonic;assumption. intros ;apply P_id_INTERSECT'II'IN_monotonic;assumption. intros ;apply P_id_U'13'1_monotonic;assumption. intros ;apply P_id_U'6'2_monotonic;assumption. intros ;apply P_id_U'3'1_monotonic;assumption. intros ;apply P_id_U'16'1_monotonic;assumption. intros ;apply P_id_U'10'1_monotonic;assumption. intros ;apply P_id_U'2'1_monotonic;assumption. intros ;apply P_id_U'15'1_monotonic;assumption. intros ;apply P_id_U'8'1_monotonic;assumption. intros ;apply P_id_U'5'1_monotonic;assumption. intros ;apply P_id_U'12'1_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_19_large_large_scc_1.DP_R_xml_0_scc_19_large_large_scc_1_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_19_large_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_intersect'ii'in (x20:Z) (x21:Z) := 0. Definition P_id_tautology'i'out := 0. Definition P_id_u'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0. Definition P_id_u'3'1 (x20:Z) := 0. Definition P_id_u'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0. Definition P_id_u'2'1 (x20:Z) := 0. Definition P_id_u'9'1 (x20:Z) := 0. Definition P_id_iff (x20:Z) (x21:Z) := 0. Definition P_id_u'14'1 (x20:Z) := 0. Definition P_id_intersect'ii'out := 0. Definition P_id_u'7'1 (x20:Z) := 0. Definition P_id_x'2d (x20:Z) := 0. Definition P_id_u'13'1 (x20:Z) := 0. Definition P_id_sequent (x20:Z) (x21:Z) := 1* x20. Definition P_id_u'10'1 (x20:Z) := 0. Definition P_id_x'2a (x20:Z) (x21:Z) := 0. Definition P_id_tautology'i'in (x20:Z) := 3 + 3* x20. Definition P_id_cons (x20:Z) (x21:Z) := 1* x20. Definition P_id_u'6'2 (x20:Z) := 0. Definition P_id_x'2b (x20:Z) (x21:Z) := 2 + 1* x20 + 2* x21. Definition P_id_u'12'2 (x20:Z) := 0. Definition P_id_reduce'ii'in (x20:Z) (x21:Z) := 0. Definition P_id_p (x20:Z) := 0. Definition P_id_u'4'1 (x20:Z) := 0. Definition P_id_u'15'1 (x20:Z) := 0. Definition P_id_u'1'1 (x20:Z) := 0. Definition P_id_u'8'1 (x20:Z) := 0. Definition P_id_reduce'ii'out := 0. Definition P_id_nil := 0. Definition P_id_if (x20:Z) (x21:Z) := 0. Definition P_id_u'11'1 (x20:Z) := 0. Definition P_id_u'5'1 (x20:Z) := 0. Definition P_id_u'16'1 (x20:Z) := 0. Lemma P_id_intersect'ii'in_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'3'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'2'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'9'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_iff_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_iff x21 x23 <= P_id_iff x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'14'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'7'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2d_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'13'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_sequent_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_sequent x21 x23 <= P_id_sequent x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'10'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2a_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_x'2a x21 x23 <= P_id_x'2a x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'in_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) -> P_id_tautology'i'in x21 <= P_id_tautology'i'in x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_cons x21 x23 <= P_id_cons x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'6'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2b_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_x'2b x21 x23 <= P_id_x'2b x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'12'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'in_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_p_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'4'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'15'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'1'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'8'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_if_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'11'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'5'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'16'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_intersect'ii'in_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_intersect'ii'in x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'out_bounded : 0 <= P_id_tautology'i'out . 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'6'1_bounded : forall x24 x20 x22 x21 x23, (0 <= x20) -> (0 <= x21) -> (0 <= x22) -> (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'6'1 x24 x23 x22 x21 x20. 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'3'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'3'1 x20. 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'12'1_bounded : forall x24 x20 x22 x21 x23, (0 <= x20) -> (0 <= x21) -> (0 <= x22) -> (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'12'1 x24 x23 x22 x21 x20. 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'2'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'2'1 x20. 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'9'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'9'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_iff_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_iff x21 x20. 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'14'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'14'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_intersect'ii'out_bounded : 0 <= P_id_intersect'ii'out . 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'7'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'7'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2d_bounded : forall x20, (0 <= x20) ->0 <= P_id_x'2d x20. 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'13'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'13'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_sequent_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_sequent x21 x20. 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'10'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'10'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2a_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2a x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'in_bounded : forall x20, (0 <= x20) ->0 <= P_id_tautology'i'in x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_cons x21 x20. 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'6'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'6'2 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2b_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2b x21 x20. 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'12'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'12'2 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'in_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_reduce'ii'in x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_p_bounded : forall x20, (0 <= x20) ->0 <= P_id_p x20. 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'4'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'4'1 x20. 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'15'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'15'1 x20. 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'1'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'1'1 x20. 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'8'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'8'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'out_bounded : 0 <= P_id_reduce'ii'out . 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_if_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_if x21 x20. 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'11'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'11'1 x20. 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'5'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'5'1 x20. 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'16'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'16'1 x20. 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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21:: x20::nil)) => P_id_intersect'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) => P_id_tautology'i'out | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22:: x21::x20::nil)) => P_id_u'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_u'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23:: x22::x21::x20::nil)) => P_id_u'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_u'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_u'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) => P_id_iff (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_u'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) => P_id_intersect'ii'out | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_u'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) => P_id_x'2d (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_u'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_sequent (x21:: x20::nil)) => P_id_sequent (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_u'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) => P_id_x'2a (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_tautology'i'in (measure x20) | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) => P_id_cons (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_u'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) => P_id_x'2b (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_u'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21:: x20::nil)) => P_id_reduce'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_p (x20::nil)) => P_id_p (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_u'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_u'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_u'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_u'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) => P_id_reduce'ii'out | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) => P_id_if (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_u'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_u'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_u'16'1 (measure x20) | _ => 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_intersect'ii'in_monotonic;assumption. intros ;apply P_id_u'6'1_monotonic;assumption. intros ;apply P_id_u'3'1_monotonic;assumption. intros ;apply P_id_u'12'1_monotonic;assumption. intros ;apply P_id_u'2'1_monotonic;assumption. intros ;apply P_id_u'9'1_monotonic;assumption. intros ;apply P_id_iff_monotonic;assumption. intros ;apply P_id_u'14'1_monotonic;assumption. intros ;apply P_id_u'7'1_monotonic;assumption. intros ;apply P_id_x'2d_monotonic;assumption. intros ;apply P_id_u'13'1_monotonic;assumption. intros ;apply P_id_sequent_monotonic;assumption. intros ;apply P_id_u'10'1_monotonic;assumption. intros ;apply P_id_x'2a_monotonic;assumption. intros ;apply P_id_tautology'i'in_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_u'6'2_monotonic;assumption. intros ;apply P_id_x'2b_monotonic;assumption. intros ;apply P_id_u'12'2_monotonic;assumption. intros ;apply P_id_reduce'ii'in_monotonic;assumption. intros ;apply P_id_p_monotonic;assumption. intros ;apply P_id_u'4'1_monotonic;assumption. intros ;apply P_id_u'15'1_monotonic;assumption. intros ;apply P_id_u'1'1_monotonic;assumption. intros ;apply P_id_u'8'1_monotonic;assumption. intros ;apply P_id_if_monotonic;assumption. intros ;apply P_id_u'11'1_monotonic;assumption. intros ;apply P_id_u'5'1_monotonic;assumption. intros ;apply P_id_u'16'1_monotonic;assumption. intros ;apply P_id_intersect'ii'in_bounded;assumption. intros ;apply P_id_tautology'i'out_bounded;assumption. intros ;apply P_id_u'6'1_bounded;assumption. intros ;apply P_id_u'3'1_bounded;assumption. intros ;apply P_id_u'12'1_bounded;assumption. intros ;apply P_id_u'2'1_bounded;assumption. intros ;apply P_id_u'9'1_bounded;assumption. intros ;apply P_id_iff_bounded;assumption. intros ;apply P_id_u'14'1_bounded;assumption. intros ;apply P_id_intersect'ii'out_bounded;assumption. intros ;apply P_id_u'7'1_bounded;assumption. intros ;apply P_id_x'2d_bounded;assumption. intros ;apply P_id_u'13'1_bounded;assumption. intros ;apply P_id_sequent_bounded;assumption. intros ;apply P_id_u'10'1_bounded;assumption. intros ;apply P_id_x'2a_bounded;assumption. intros ;apply P_id_tautology'i'in_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_u'6'2_bounded;assumption. intros ;apply P_id_x'2b_bounded;assumption. intros ;apply P_id_u'12'2_bounded;assumption. intros ;apply P_id_reduce'ii'in_bounded;assumption. intros ;apply P_id_p_bounded;assumption. intros ;apply P_id_u'4'1_bounded;assumption. intros ;apply P_id_u'15'1_bounded;assumption. intros ;apply P_id_u'1'1_bounded;assumption. intros ;apply P_id_u'8'1_bounded;assumption. intros ;apply P_id_reduce'ii'out_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_if_bounded;assumption. intros ;apply P_id_u'11'1_bounded;assumption. intros ;apply P_id_u'5'1_bounded;assumption. intros ;apply P_id_u'16'1_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_U'12'2 (x20:Z) := 0. Definition P_id_U'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 2 + 3* x21 + 1* x24. Definition P_id_REDUCE'II'IN (x20:Z) (x21:Z) := 2* x20 + 1* x21. Definition P_id_TAUTOLOGY'I'IN (x20:Z) := 0. Definition P_id_U'9'1 (x20:Z) := 0. Definition P_id_U'1'1 (x20:Z) := 0. Definition P_id_U'14'1 (x20:Z) := 0. Definition P_id_U'7'1 (x20:Z) := 0. Definition P_id_U'4'1 (x20:Z) := 0. Definition P_id_U'11'1 (x20:Z) := 0. Definition P_id_INTERSECT'II'IN (x20:Z) (x21:Z) := 0. Definition P_id_U'13'1 (x20:Z) := 0. Definition P_id_U'6'2 (x20:Z) := 0. Definition P_id_U'3'1 (x20:Z) := 0. Definition P_id_U'16'1 (x20:Z) := 0. Definition P_id_U'10'1 (x20:Z) := 0. Definition P_id_U'2'1 (x20:Z) := 0. Definition P_id_U'15'1 (x20:Z) := 0. Definition P_id_U'8'1 (x20:Z) := 0. Definition P_id_U'5'1 (x20:Z) := 0. Definition P_id_U'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0. Lemma P_id_U'12'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_REDUCE'II'IN_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_TAUTOLOGY'I'IN_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) -> P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'9'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'1'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'14'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'7'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'4'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'11'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_INTERSECT'II'IN_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'13'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'6'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'3'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'16'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'10'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'2'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'15'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'8'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'5'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. 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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1 P_id_U'12'2 P_id_U'6'1 P_id_REDUCE'II'IN P_id_TAUTOLOGY'I'IN P_id_U'9'1 P_id_U'1'1 P_id_U'14'1 P_id_U'7'1 P_id_U'4'1 P_id_U'11'1 P_id_INTERSECT'II'IN P_id_U'13'1 P_id_U'6'2 P_id_U'3'1 P_id_U'16'1 P_id_U'10'1 P_id_U'2'1 P_id_U'15'1 P_id_U'8'1 P_id_U'5'1 P_id_U'12'1 . Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_U'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'1 (x24:: x23::x22::x21::x20::nil)) => P_id_U'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) => P_id_REDUCE'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_TAUTOLOGY'I'IN (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_U'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_U'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_U'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_U'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_U'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_U'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) => P_id_INTERSECT'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_U'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_U'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_U'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_U'16'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_U'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_U'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_U'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_U'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_U'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24:: x23::x22::x21::x20::nil)) => P_id_U'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | _ => 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_intersect'ii'in_monotonic;assumption. intros ;apply P_id_u'6'1_monotonic;assumption. intros ;apply P_id_u'3'1_monotonic;assumption. intros ;apply P_id_u'12'1_monotonic;assumption. intros ;apply P_id_u'2'1_monotonic;assumption. intros ;apply P_id_u'9'1_monotonic;assumption. intros ;apply P_id_iff_monotonic;assumption. intros ;apply P_id_u'14'1_monotonic;assumption. intros ;apply P_id_u'7'1_monotonic;assumption. intros ;apply P_id_x'2d_monotonic;assumption. intros ;apply P_id_u'13'1_monotonic;assumption. intros ;apply P_id_sequent_monotonic;assumption. intros ;apply P_id_u'10'1_monotonic;assumption. intros ;apply P_id_x'2a_monotonic;assumption. intros ;apply P_id_tautology'i'in_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_u'6'2_monotonic;assumption. intros ;apply P_id_x'2b_monotonic;assumption. intros ;apply P_id_u'12'2_monotonic;assumption. intros ;apply P_id_reduce'ii'in_monotonic;assumption. intros ;apply P_id_p_monotonic;assumption. intros ;apply P_id_u'4'1_monotonic;assumption. intros ;apply P_id_u'15'1_monotonic;assumption. intros ;apply P_id_u'1'1_monotonic;assumption. intros ;apply P_id_u'8'1_monotonic;assumption. intros ;apply P_id_if_monotonic;assumption. intros ;apply P_id_u'11'1_monotonic;assumption. intros ;apply P_id_u'5'1_monotonic;assumption. intros ;apply P_id_u'16'1_monotonic;assumption. intros ;apply P_id_intersect'ii'in_bounded;assumption. intros ;apply P_id_tautology'i'out_bounded;assumption. intros ;apply P_id_u'6'1_bounded;assumption. intros ;apply P_id_u'3'1_bounded;assumption. intros ;apply P_id_u'12'1_bounded;assumption. intros ;apply P_id_u'2'1_bounded;assumption. intros ;apply P_id_u'9'1_bounded;assumption. intros ;apply P_id_iff_bounded;assumption. intros ;apply P_id_u'14'1_bounded;assumption. intros ;apply P_id_intersect'ii'out_bounded;assumption. intros ;apply P_id_u'7'1_bounded;assumption. intros ;apply P_id_x'2d_bounded;assumption. intros ;apply P_id_u'13'1_bounded;assumption. intros ;apply P_id_sequent_bounded;assumption. intros ;apply P_id_u'10'1_bounded;assumption. intros ;apply P_id_x'2a_bounded;assumption. intros ;apply P_id_tautology'i'in_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_u'6'2_bounded;assumption. intros ;apply P_id_x'2b_bounded;assumption. intros ;apply P_id_u'12'2_bounded;assumption. intros ;apply P_id_reduce'ii'in_bounded;assumption. intros ;apply P_id_p_bounded;assumption. intros ;apply P_id_u'4'1_bounded;assumption. intros ;apply P_id_u'15'1_bounded;assumption. intros ;apply P_id_u'1'1_bounded;assumption. intros ;apply P_id_u'8'1_bounded;assumption. intros ;apply P_id_reduce'ii'out_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_if_bounded;assumption. intros ;apply P_id_u'11'1_bounded;assumption. intros ;apply P_id_u'5'1_bounded;assumption. intros ;apply P_id_u'16'1_bounded;assumption. apply rules_monotonic. intros ;apply P_id_U'12'2_monotonic;assumption. intros ;apply P_id_U'6'1_monotonic;assumption. intros ;apply P_id_REDUCE'II'IN_monotonic;assumption. intros ;apply P_id_TAUTOLOGY'I'IN_monotonic;assumption. intros ;apply P_id_U'9'1_monotonic;assumption. intros ;apply P_id_U'1'1_monotonic;assumption. intros ;apply P_id_U'14'1_monotonic;assumption. intros ;apply P_id_U'7'1_monotonic;assumption. intros ;apply P_id_U'4'1_monotonic;assumption. intros ;apply P_id_U'11'1_monotonic;assumption. intros ;apply P_id_INTERSECT'II'IN_monotonic;assumption. intros ;apply P_id_U'13'1_monotonic;assumption. intros ;apply P_id_U'6'2_monotonic;assumption. intros ;apply P_id_U'3'1_monotonic;assumption. intros ;apply P_id_U'16'1_monotonic;assumption. intros ;apply P_id_U'10'1_monotonic;assumption. intros ;apply P_id_U'2'1_monotonic;assumption. intros ;apply P_id_U'15'1_monotonic;assumption. intros ;apply P_id_U'8'1_monotonic;assumption. intros ;apply P_id_U'5'1_monotonic;assumption. intros ;apply P_id_U'12'1_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_19_large_large_scc_1_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_19_large_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_19_large_large_scc_1_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_19_large_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_19_large_large_scc_1_large := WF_DP_R_xml_0_scc_19_large_large_scc_1_large.wf. Lemma wf : well_founded WF_DP_R_xml_0_scc_19_large_large.DP_R_xml_0_scc_19_large_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_19_large_large_scc_1_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_19_large_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_19_large_large_scc_1_large_in_le; econstructor eassumption])). apply wf_DP_R_xml_0_scc_19_large_large_scc_1_large. Qed. End WF_DP_R_xml_0_scc_19_large_large_scc_1. Definition wf_DP_R_xml_0_scc_19_large_large_scc_1 := WF_DP_R_xml_0_scc_19_large_large_scc_1.wf. Lemma acc_DP_R_xml_0_scc_19_large_large_scc_1 : forall x y, (DP_R_xml_0_scc_19_large_large_scc_1 x y) -> Acc WF_DP_R_xml_0_scc_19_large.DP_R_xml_0_scc_19_large_large x. Proof. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_19_large_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_19_large_large_scc_1. Qed. Inductive DP_R_xml_0_scc_19_large_large_non_scc_2 : algebra.Alg.term ->algebra.Alg.term ->Prop := (* *) | DP_R_xml_0_scc_19_large_large_non_scc_2_0 : forall x8 x24 x20 x10 x22 x17 x9 x21 x23, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) x24) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x8 x23) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x17 x22) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x9 x21) -> (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x10 x20) -> DP_R_xml_0_scc_19_large_large_non_scc_2 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent (x8:: (algebra.Alg.Term algebra.F.id_cons (x17:: x9::nil))::nil)):: x10::nil)) (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22::x21:: x20::nil)) . Lemma acc_DP_R_xml_0_scc_19_large_large_non_scc_2 : forall x y, (DP_R_xml_0_scc_19_large_large_non_scc_2 x y) -> Acc WF_DP_R_xml_0_scc_19_large.DP_R_xml_0_scc_19_large_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_19_large_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_19_large.DP_R_xml_0_scc_19_large_large. Proof. constructor;intros _y _h;inversion _h;clear _h;subst; (eapply acc_DP_R_xml_0_scc_19_large_large_non_scc_2; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_19_large_large_non_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_19_large_large_non_scc_0; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_19_large_large_scc_1; econstructor (eassumption)||(algebra.Alg_ext.star_refl' ))|| ((eapply acc_DP_R_xml_0_scc_19_large_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_19_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_intersect'ii'in (x20:Z) (x21:Z) := 0. Definition P_id_tautology'i'out := 0. Definition P_id_u'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0. Definition P_id_u'3'1 (x20:Z) := 0. Definition P_id_u'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0. Definition P_id_u'2'1 (x20:Z) := 0. Definition P_id_u'9'1 (x20:Z) := 0. Definition P_id_iff (x20:Z) (x21:Z) := 0. Definition P_id_u'14'1 (x20:Z) := 0. Definition P_id_intersect'ii'out := 0. Definition P_id_u'7'1 (x20:Z) := 0. Definition P_id_x'2d (x20:Z) := 1 + 1* x20. Definition P_id_u'13'1 (x20:Z) := 0. Definition P_id_sequent (x20:Z) (x21:Z) := 1* x20 + 1* x21. Definition P_id_u'10'1 (x20:Z) := 0. Definition P_id_x'2a (x20:Z) (x21:Z) := 3 + 1* x20 + 1* x21. Definition P_id_tautology'i'in (x20:Z) := 2* x20. Definition P_id_cons (x20:Z) (x21:Z) := 1* x20 + 1* x21. Definition P_id_u'6'2 (x20:Z) := 0. Definition P_id_x'2b (x20:Z) (x21:Z) := 1* x20 + 2* x21. Definition P_id_u'12'2 (x20:Z) := 0. Definition P_id_reduce'ii'in (x20:Z) (x21:Z) := 0. Definition P_id_p (x20:Z) := 0. Definition P_id_u'4'1 (x20:Z) := 0. Definition P_id_u'15'1 (x20:Z) := 0. Definition P_id_u'1'1 (x20:Z) := 0. Definition P_id_u'8'1 (x20:Z) := 0. Definition P_id_reduce'ii'out := 0. Definition P_id_nil := 0. Definition P_id_if (x20:Z) (x21:Z) := 2 + 2* x20 + 2* x21. Definition P_id_u'11'1 (x20:Z) := 0. Definition P_id_u'5'1 (x20:Z) := 0. Definition P_id_u'16'1 (x20:Z) := 0. Lemma P_id_intersect'ii'in_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'3'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'2'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'9'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_iff_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_iff x21 x23 <= P_id_iff x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'14'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'7'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2d_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'13'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_sequent_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_sequent x21 x23 <= P_id_sequent x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'10'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2a_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_x'2a x21 x23 <= P_id_x'2a x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'in_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) -> P_id_tautology'i'in x21 <= P_id_tautology'i'in x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_cons x21 x23 <= P_id_cons x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'6'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2b_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_x'2b x21 x23 <= P_id_x'2b x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'12'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'in_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_p_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'4'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'15'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'1'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'8'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_if_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'11'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'5'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'16'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_intersect'ii'in_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_intersect'ii'in x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'out_bounded : 0 <= P_id_tautology'i'out . 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'6'1_bounded : forall x24 x20 x22 x21 x23, (0 <= x20) -> (0 <= x21) -> (0 <= x22) -> (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'6'1 x24 x23 x22 x21 x20. 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'3'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'3'1 x20. 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'12'1_bounded : forall x24 x20 x22 x21 x23, (0 <= x20) -> (0 <= x21) -> (0 <= x22) -> (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'12'1 x24 x23 x22 x21 x20. 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'2'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'2'1 x20. 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'9'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'9'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_iff_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_iff x21 x20. 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'14'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'14'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_intersect'ii'out_bounded : 0 <= P_id_intersect'ii'out . 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'7'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'7'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2d_bounded : forall x20, (0 <= x20) ->0 <= P_id_x'2d x20. 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'13'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'13'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_sequent_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_sequent x21 x20. 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'10'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'10'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2a_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2a x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'in_bounded : forall x20, (0 <= x20) ->0 <= P_id_tautology'i'in x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_cons x21 x20. 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'6'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'6'2 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2b_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2b x21 x20. 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'12'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'12'2 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'in_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_reduce'ii'in x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_p_bounded : forall x20, (0 <= x20) ->0 <= P_id_p x20. 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'4'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'4'1 x20. 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'15'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'15'1 x20. 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'1'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'1'1 x20. 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'8'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'8'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'out_bounded : 0 <= P_id_reduce'ii'out . 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_if_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_if x21 x20. 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'11'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'11'1 x20. 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'5'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'5'1 x20. 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'16'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'16'1 x20. 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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21:: x20::nil)) => P_id_intersect'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) => P_id_tautology'i'out | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22:: x21::x20::nil)) => P_id_u'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_u'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22:: x21::x20::nil)) => P_id_u'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_u'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_u'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) => P_id_iff (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_u'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) => P_id_intersect'ii'out | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_u'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) => P_id_x'2d (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_u'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_sequent (x21::x20::nil)) => P_id_sequent (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_u'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) => P_id_x'2a (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_tautology'i'in (measure x20) | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) => P_id_cons (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_u'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) => P_id_x'2b (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_u'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21:: x20::nil)) => P_id_reduce'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_p (x20::nil)) => P_id_p (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_u'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_u'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_u'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_u'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) => P_id_reduce'ii'out | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) => P_id_if (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_u'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_u'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_u'16'1 (measure x20) | _ => 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_intersect'ii'in_monotonic;assumption. intros ;apply P_id_u'6'1_monotonic;assumption. intros ;apply P_id_u'3'1_monotonic;assumption. intros ;apply P_id_u'12'1_monotonic;assumption. intros ;apply P_id_u'2'1_monotonic;assumption. intros ;apply P_id_u'9'1_monotonic;assumption. intros ;apply P_id_iff_monotonic;assumption. intros ;apply P_id_u'14'1_monotonic;assumption. intros ;apply P_id_u'7'1_monotonic;assumption. intros ;apply P_id_x'2d_monotonic;assumption. intros ;apply P_id_u'13'1_monotonic;assumption. intros ;apply P_id_sequent_monotonic;assumption. intros ;apply P_id_u'10'1_monotonic;assumption. intros ;apply P_id_x'2a_monotonic;assumption. intros ;apply P_id_tautology'i'in_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_u'6'2_monotonic;assumption. intros ;apply P_id_x'2b_monotonic;assumption. intros ;apply P_id_u'12'2_monotonic;assumption. intros ;apply P_id_reduce'ii'in_monotonic;assumption. intros ;apply P_id_p_monotonic;assumption. intros ;apply P_id_u'4'1_monotonic;assumption. intros ;apply P_id_u'15'1_monotonic;assumption. intros ;apply P_id_u'1'1_monotonic;assumption. intros ;apply P_id_u'8'1_monotonic;assumption. intros ;apply P_id_if_monotonic;assumption. intros ;apply P_id_u'11'1_monotonic;assumption. intros ;apply P_id_u'5'1_monotonic;assumption. intros ;apply P_id_u'16'1_monotonic;assumption. intros ;apply P_id_intersect'ii'in_bounded;assumption. intros ;apply P_id_tautology'i'out_bounded;assumption. intros ;apply P_id_u'6'1_bounded;assumption. intros ;apply P_id_u'3'1_bounded;assumption. intros ;apply P_id_u'12'1_bounded;assumption. intros ;apply P_id_u'2'1_bounded;assumption. intros ;apply P_id_u'9'1_bounded;assumption. intros ;apply P_id_iff_bounded;assumption. intros ;apply P_id_u'14'1_bounded;assumption. intros ;apply P_id_intersect'ii'out_bounded;assumption. intros ;apply P_id_u'7'1_bounded;assumption. intros ;apply P_id_x'2d_bounded;assumption. intros ;apply P_id_u'13'1_bounded;assumption. intros ;apply P_id_sequent_bounded;assumption. intros ;apply P_id_u'10'1_bounded;assumption. intros ;apply P_id_x'2a_bounded;assumption. intros ;apply P_id_tautology'i'in_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_u'6'2_bounded;assumption. intros ;apply P_id_x'2b_bounded;assumption. intros ;apply P_id_u'12'2_bounded;assumption. intros ;apply P_id_reduce'ii'in_bounded;assumption. intros ;apply P_id_p_bounded;assumption. intros ;apply P_id_u'4'1_bounded;assumption. intros ;apply P_id_u'15'1_bounded;assumption. intros ;apply P_id_u'1'1_bounded;assumption. intros ;apply P_id_u'8'1_bounded;assumption. intros ;apply P_id_reduce'ii'out_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_if_bounded;assumption. intros ;apply P_id_u'11'1_bounded;assumption. intros ;apply P_id_u'5'1_bounded;assumption. intros ;apply P_id_u'16'1_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_U'12'2 (x20:Z) := 0. Definition P_id_U'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 3* x21 + 2* x22 + 2* x23. Definition P_id_REDUCE'II'IN (x20:Z) (x21:Z) := 2* x20. Definition P_id_TAUTOLOGY'I'IN (x20:Z) := 0. Definition P_id_U'9'1 (x20:Z) := 0. Definition P_id_U'1'1 (x20:Z) := 0. Definition P_id_U'14'1 (x20:Z) := 0. Definition P_id_U'7'1 (x20:Z) := 0. Definition P_id_U'4'1 (x20:Z) := 0. Definition P_id_U'11'1 (x20:Z) := 0. Definition P_id_INTERSECT'II'IN (x20:Z) (x21:Z) := 0. Definition P_id_U'13'1 (x20:Z) := 0. Definition P_id_U'6'2 (x20:Z) := 0. Definition P_id_U'3'1 (x20:Z) := 0. Definition P_id_U'16'1 (x20:Z) := 0. Definition P_id_U'10'1 (x20:Z) := 0. Definition P_id_U'2'1 (x20:Z) := 0. Definition P_id_U'15'1 (x20:Z) := 0. Definition P_id_U'8'1 (x20:Z) := 0. Definition P_id_U'5'1 (x20:Z) := 0. Definition P_id_U'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 2* x21 + 2* x22 + 2* x23. Lemma P_id_U'12'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_REDUCE'II'IN_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_TAUTOLOGY'I'IN_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) -> P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'9'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'1'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'14'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'7'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'4'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'11'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_INTERSECT'II'IN_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'13'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'6'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'3'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'16'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'10'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'2'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'15'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'8'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'5'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. 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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1 P_id_U'12'2 P_id_U'6'1 P_id_REDUCE'II'IN P_id_TAUTOLOGY'I'IN P_id_U'9'1 P_id_U'1'1 P_id_U'14'1 P_id_U'7'1 P_id_U'4'1 P_id_U'11'1 P_id_INTERSECT'II'IN P_id_U'13'1 P_id_U'6'2 P_id_U'3'1 P_id_U'16'1 P_id_U'10'1 P_id_U'2'1 P_id_U'15'1 P_id_U'8'1 P_id_U'5'1 P_id_U'12'1. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_U'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23:: x22::x21::x20::nil)) => P_id_U'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) => P_id_REDUCE'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_TAUTOLOGY'I'IN (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_U'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_U'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_U'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_U'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_U'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_U'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) => P_id_INTERSECT'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_U'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_U'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_U'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_U'16'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_U'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_U'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_U'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_U'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_U'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24:: x23::x22::x21::x20::nil)) => P_id_U'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | _ => 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_intersect'ii'in_monotonic;assumption. intros ;apply P_id_u'6'1_monotonic;assumption. intros ;apply P_id_u'3'1_monotonic;assumption. intros ;apply P_id_u'12'1_monotonic;assumption. intros ;apply P_id_u'2'1_monotonic;assumption. intros ;apply P_id_u'9'1_monotonic;assumption. intros ;apply P_id_iff_monotonic;assumption. intros ;apply P_id_u'14'1_monotonic;assumption. intros ;apply P_id_u'7'1_monotonic;assumption. intros ;apply P_id_x'2d_monotonic;assumption. intros ;apply P_id_u'13'1_monotonic;assumption. intros ;apply P_id_sequent_monotonic;assumption. intros ;apply P_id_u'10'1_monotonic;assumption. intros ;apply P_id_x'2a_monotonic;assumption. intros ;apply P_id_tautology'i'in_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_u'6'2_monotonic;assumption. intros ;apply P_id_x'2b_monotonic;assumption. intros ;apply P_id_u'12'2_monotonic;assumption. intros ;apply P_id_reduce'ii'in_monotonic;assumption. intros ;apply P_id_p_monotonic;assumption. intros ;apply P_id_u'4'1_monotonic;assumption. intros ;apply P_id_u'15'1_monotonic;assumption. intros ;apply P_id_u'1'1_monotonic;assumption. intros ;apply P_id_u'8'1_monotonic;assumption. intros ;apply P_id_if_monotonic;assumption. intros ;apply P_id_u'11'1_monotonic;assumption. intros ;apply P_id_u'5'1_monotonic;assumption. intros ;apply P_id_u'16'1_monotonic;assumption. intros ;apply P_id_intersect'ii'in_bounded;assumption. intros ;apply P_id_tautology'i'out_bounded;assumption. intros ;apply P_id_u'6'1_bounded;assumption. intros ;apply P_id_u'3'1_bounded;assumption. intros ;apply P_id_u'12'1_bounded;assumption. intros ;apply P_id_u'2'1_bounded;assumption. intros ;apply P_id_u'9'1_bounded;assumption. intros ;apply P_id_iff_bounded;assumption. intros ;apply P_id_u'14'1_bounded;assumption. intros ;apply P_id_intersect'ii'out_bounded;assumption. intros ;apply P_id_u'7'1_bounded;assumption. intros ;apply P_id_x'2d_bounded;assumption. intros ;apply P_id_u'13'1_bounded;assumption. intros ;apply P_id_sequent_bounded;assumption. intros ;apply P_id_u'10'1_bounded;assumption. intros ;apply P_id_x'2a_bounded;assumption. intros ;apply P_id_tautology'i'in_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_u'6'2_bounded;assumption. intros ;apply P_id_x'2b_bounded;assumption. intros ;apply P_id_u'12'2_bounded;assumption. intros ;apply P_id_reduce'ii'in_bounded;assumption. intros ;apply P_id_p_bounded;assumption. intros ;apply P_id_u'4'1_bounded;assumption. intros ;apply P_id_u'15'1_bounded;assumption. intros ;apply P_id_u'1'1_bounded;assumption. intros ;apply P_id_u'8'1_bounded;assumption. intros ;apply P_id_reduce'ii'out_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_if_bounded;assumption. intros ;apply P_id_u'11'1_bounded;assumption. intros ;apply P_id_u'5'1_bounded;assumption. intros ;apply P_id_u'16'1_bounded;assumption. apply rules_monotonic. intros ;apply P_id_U'12'2_monotonic;assumption. intros ;apply P_id_U'6'1_monotonic;assumption. intros ;apply P_id_REDUCE'II'IN_monotonic;assumption. intros ;apply P_id_TAUTOLOGY'I'IN_monotonic;assumption. intros ;apply P_id_U'9'1_monotonic;assumption. intros ;apply P_id_U'1'1_monotonic;assumption. intros ;apply P_id_U'14'1_monotonic;assumption. intros ;apply P_id_U'7'1_monotonic;assumption. intros ;apply P_id_U'4'1_monotonic;assumption. intros ;apply P_id_U'11'1_monotonic;assumption. intros ;apply P_id_INTERSECT'II'IN_monotonic;assumption. intros ;apply P_id_U'13'1_monotonic;assumption. intros ;apply P_id_U'6'2_monotonic;assumption. intros ;apply P_id_U'3'1_monotonic;assumption. intros ;apply P_id_U'16'1_monotonic;assumption. intros ;apply P_id_U'10'1_monotonic;assumption. intros ;apply P_id_U'2'1_monotonic;assumption. intros ;apply P_id_U'15'1_monotonic;assumption. intros ;apply P_id_U'8'1_monotonic;assumption. intros ;apply P_id_U'5'1_monotonic;assumption. intros ;apply P_id_U'12'1_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_19_large_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_19_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_19_large_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_19_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_19_large_large := WF_DP_R_xml_0_scc_19_large_large.wf. Lemma wf : well_founded WF_DP_R_xml_0_scc_19.DP_R_xml_0_scc_19_large. Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_19_large_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_19_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_19_large_large_in_le;econstructor eassumption])). apply wf_DP_R_xml_0_scc_19_large_large. Qed. End WF_DP_R_xml_0_scc_19_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_intersect'ii'in (x20:Z) (x21:Z) := 0. Definition P_id_tautology'i'out := 0. Definition P_id_u'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0. Definition P_id_u'3'1 (x20:Z) := 0. Definition P_id_u'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 0. Definition P_id_u'2'1 (x20:Z) := 0. Definition P_id_u'9'1 (x20:Z) := 0. Definition P_id_iff (x20:Z) (x21:Z) := 3 + 3* x20 + 3* x21. Definition P_id_u'14'1 (x20:Z) := 0. Definition P_id_intersect'ii'out := 0. Definition P_id_u'7'1 (x20:Z) := 0. Definition P_id_x'2d (x20:Z) := 2* x20. Definition P_id_u'13'1 (x20:Z) := 0. Definition P_id_sequent (x20:Z) (x21:Z) := 1* x20 + 1* x21. Definition P_id_u'10'1 (x20:Z) := 0. Definition P_id_x'2a (x20:Z) (x21:Z) := 1* x20 + 1* x21. Definition P_id_tautology'i'in (x20:Z) := 2 + 1* x20. Definition P_id_cons (x20:Z) (x21:Z) := 1* x20 + 1* x21. Definition P_id_u'6'2 (x20:Z) := 0. Definition P_id_x'2b (x20:Z) (x21:Z) := 1* x20 + 1* x21. Definition P_id_u'12'2 (x20:Z) := 0. Definition P_id_reduce'ii'in (x20:Z) (x21:Z) := 0. Definition P_id_p (x20:Z) := 3. Definition P_id_u'4'1 (x20:Z) := 0. Definition P_id_u'15'1 (x20:Z) := 0. Definition P_id_u'1'1 (x20:Z) := 0. Definition P_id_u'8'1 (x20:Z) := 0. Definition P_id_reduce'ii'out := 0. Definition P_id_nil := 0. Definition P_id_if (x20:Z) (x21:Z) := 1* x20 + 2* x21. Definition P_id_u'11'1 (x20:Z) := 0. Definition P_id_u'5'1 (x20:Z) := 0. Definition P_id_u'16'1 (x20:Z) := 1. Lemma P_id_intersect'ii'in_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_intersect'ii'in x21 x23 <= P_id_intersect'ii'in x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_u'6'1 x21 x23 x25 x27 x29 <= P_id_u'6'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'3'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'3'1 x21 <= P_id_u'3'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_u'12'1 x21 x23 x25 x27 x29 <= P_id_u'12'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'2'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'2'1 x21 <= P_id_u'2'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'9'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'9'1 x21 <= P_id_u'9'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_iff_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_iff x21 x23 <= P_id_iff x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'14'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'14'1 x21 <= P_id_u'14'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'7'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'7'1 x21 <= P_id_u'7'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2d_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_x'2d x21 <= P_id_x'2d x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'13'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'13'1 x21 <= P_id_u'13'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_sequent_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_sequent x21 x23 <= P_id_sequent x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'10'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'10'1 x21 <= P_id_u'10'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2a_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_x'2a x21 x23 <= P_id_x'2a x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'in_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) -> P_id_tautology'i'in x21 <= P_id_tautology'i'in x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_cons x21 x23 <= P_id_cons x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'6'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'6'2 x21 <= P_id_u'6'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2b_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_x'2b x21 x23 <= P_id_x'2b x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'12'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'12'2 x21 <= P_id_u'12'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'in_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_reduce'ii'in x21 x23 <= P_id_reduce'ii'in x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_p_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_p x21 <= P_id_p x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'4'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'4'1 x21 <= P_id_u'4'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'15'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'15'1 x21 <= P_id_u'15'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'1'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'1'1 x21 <= P_id_u'1'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'8'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'8'1 x21 <= P_id_u'8'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_if_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) ->P_id_if x21 x23 <= P_id_if x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'11'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'11'1 x21 <= P_id_u'11'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'5'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'5'1 x21 <= P_id_u'5'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_u'16'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_u'16'1 x21 <= P_id_u'16'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_intersect'ii'in_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_intersect'ii'in x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'out_bounded : 0 <= P_id_tautology'i'out . 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'6'1_bounded : forall x24 x20 x22 x21 x23, (0 <= x20) -> (0 <= x21) -> (0 <= x22) -> (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'6'1 x24 x23 x22 x21 x20. 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'3'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'3'1 x20. 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'12'1_bounded : forall x24 x20 x22 x21 x23, (0 <= x20) -> (0 <= x21) -> (0 <= x22) -> (0 <= x23) ->(0 <= x24) ->0 <= P_id_u'12'1 x24 x23 x22 x21 x20. 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'2'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'2'1 x20. 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'9'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'9'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_iff_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_iff x21 x20. 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'14'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'14'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_intersect'ii'out_bounded : 0 <= P_id_intersect'ii'out . 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'7'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'7'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2d_bounded : forall x20, (0 <= x20) ->0 <= P_id_x'2d x20. 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'13'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'13'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_sequent_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_sequent x21 x20. 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'10'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'10'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2a_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2a x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_tautology'i'in_bounded : forall x20, (0 <= x20) ->0 <= P_id_tautology'i'in x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_cons_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_cons x21 x20. 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'6'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'6'2 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_x'2b_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_x'2b x21 x20. 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'12'2_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'12'2 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'in_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_reduce'ii'in x21 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_p_bounded : forall x20, (0 <= x20) ->0 <= P_id_p x20. 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'4'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'4'1 x20. 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'15'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'15'1 x20. 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'1'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'1'1 x20. 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'8'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'8'1 x20. Proof. intros . cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_reduce'ii'out_bounded : 0 <= P_id_reduce'ii'out . 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_if_bounded : forall x20 x21, (0 <= x20) ->(0 <= x21) ->0 <= P_id_if x21 x20. 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'11'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'11'1 x20. 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'5'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'5'1 x20. 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'16'1_bounded : forall x20, (0 <= x20) ->0 <= P_id_u'16'1 x20. 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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1. Lemma measure_equation : forall t, measure t = match t with | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21:: x20::nil)) => P_id_intersect'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'out nil) => P_id_tautology'i'out | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23::x22:: x21::x20::nil)) => P_id_u'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_u'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23::x22:: x21::x20::nil)) => P_id_u'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_u'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_u'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_iff (x21::x20::nil)) => P_id_iff (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_u'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'out nil) => P_id_intersect'ii'out | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_u'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2d (x20::nil)) => P_id_x'2d (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_u'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_sequent (x21::x20::nil)) => P_id_sequent (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_u'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2a (x21::x20::nil)) => P_id_x'2a (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_tautology'i'in (measure x20) | (algebra.Alg.Term algebra.F.id_cons (x21::x20::nil)) => P_id_cons (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_u'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_x'2b (x21::x20::nil)) => P_id_x'2b (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_u'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21:: x20::nil)) => P_id_reduce'ii'in (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_p (x20::nil)) => P_id_p (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_u'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_u'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_u'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_u'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'out nil) => P_id_reduce'ii'out | (algebra.Alg.Term algebra.F.id_nil nil) => P_id_nil | (algebra.Alg.Term algebra.F.id_if (x21::x20::nil)) => P_id_if (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_u'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_u'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_u'16'1 (measure x20) | _ => 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_intersect'ii'in_monotonic;assumption. intros ;apply P_id_u'6'1_monotonic;assumption. intros ;apply P_id_u'3'1_monotonic;assumption. intros ;apply P_id_u'12'1_monotonic;assumption. intros ;apply P_id_u'2'1_monotonic;assumption. intros ;apply P_id_u'9'1_monotonic;assumption. intros ;apply P_id_iff_monotonic;assumption. intros ;apply P_id_u'14'1_monotonic;assumption. intros ;apply P_id_u'7'1_monotonic;assumption. intros ;apply P_id_x'2d_monotonic;assumption. intros ;apply P_id_u'13'1_monotonic;assumption. intros ;apply P_id_sequent_monotonic;assumption. intros ;apply P_id_u'10'1_monotonic;assumption. intros ;apply P_id_x'2a_monotonic;assumption. intros ;apply P_id_tautology'i'in_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_u'6'2_monotonic;assumption. intros ;apply P_id_x'2b_monotonic;assumption. intros ;apply P_id_u'12'2_monotonic;assumption. intros ;apply P_id_reduce'ii'in_monotonic;assumption. intros ;apply P_id_p_monotonic;assumption. intros ;apply P_id_u'4'1_monotonic;assumption. intros ;apply P_id_u'15'1_monotonic;assumption. intros ;apply P_id_u'1'1_monotonic;assumption. intros ;apply P_id_u'8'1_monotonic;assumption. intros ;apply P_id_if_monotonic;assumption. intros ;apply P_id_u'11'1_monotonic;assumption. intros ;apply P_id_u'5'1_monotonic;assumption. intros ;apply P_id_u'16'1_monotonic;assumption. intros ;apply P_id_intersect'ii'in_bounded;assumption. intros ;apply P_id_tautology'i'out_bounded;assumption. intros ;apply P_id_u'6'1_bounded;assumption. intros ;apply P_id_u'3'1_bounded;assumption. intros ;apply P_id_u'12'1_bounded;assumption. intros ;apply P_id_u'2'1_bounded;assumption. intros ;apply P_id_u'9'1_bounded;assumption. intros ;apply P_id_iff_bounded;assumption. intros ;apply P_id_u'14'1_bounded;assumption. intros ;apply P_id_intersect'ii'out_bounded;assumption. intros ;apply P_id_u'7'1_bounded;assumption. intros ;apply P_id_x'2d_bounded;assumption. intros ;apply P_id_u'13'1_bounded;assumption. intros ;apply P_id_sequent_bounded;assumption. intros ;apply P_id_u'10'1_bounded;assumption. intros ;apply P_id_x'2a_bounded;assumption. intros ;apply P_id_tautology'i'in_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_u'6'2_bounded;assumption. intros ;apply P_id_x'2b_bounded;assumption. intros ;apply P_id_u'12'2_bounded;assumption. intros ;apply P_id_reduce'ii'in_bounded;assumption. intros ;apply P_id_p_bounded;assumption. intros ;apply P_id_u'4'1_bounded;assumption. intros ;apply P_id_u'15'1_bounded;assumption. intros ;apply P_id_u'1'1_bounded;assumption. intros ;apply P_id_u'8'1_bounded;assumption. intros ;apply P_id_reduce'ii'out_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_if_bounded;assumption. intros ;apply P_id_u'11'1_bounded;assumption. intros ;apply P_id_u'5'1_bounded;assumption. intros ;apply P_id_u'16'1_bounded;assumption. apply rules_monotonic. Qed. Definition P_id_U'12'2 (x20:Z) := 0. Definition P_id_U'6'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 2* x21 + 2* x22 + 2* x23 + 1* x24. Definition P_id_REDUCE'II'IN (x20:Z) (x21:Z) := 2* x20 + 1* x21. Definition P_id_TAUTOLOGY'I'IN (x20:Z) := 0. Definition P_id_U'9'1 (x20:Z) := 0. Definition P_id_U'1'1 (x20:Z) := 0. Definition P_id_U'14'1 (x20:Z) := 0. Definition P_id_U'7'1 (x20:Z) := 0. Definition P_id_U'4'1 (x20:Z) := 0. Definition P_id_U'11'1 (x20:Z) := 0. Definition P_id_INTERSECT'II'IN (x20:Z) (x21:Z) := 0. Definition P_id_U'13'1 (x20:Z) := 0. Definition P_id_U'6'2 (x20:Z) := 0. Definition P_id_U'3'1 (x20:Z) := 0. Definition P_id_U'16'1 (x20:Z) := 0. Definition P_id_U'10'1 (x20:Z) := 0. Definition P_id_U'2'1 (x20:Z) := 0. Definition P_id_U'15'1 (x20:Z) := 0. Definition P_id_U'8'1 (x20:Z) := 0. Definition P_id_U'5'1 (x20:Z) := 0. Definition P_id_U'12'1 (x20:Z) (x21:Z) (x22:Z) (x23:Z) (x24:Z) := 2* x21 + 2* x22 + 2* x23 + 1* x24. Lemma P_id_U'12'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'12'2 x21 <= P_id_U'12'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'6'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_U'6'1 x21 x23 x25 x27 x29 <= P_id_U'6'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_REDUCE'II'IN_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_REDUCE'II'IN x21 x23 <= P_id_REDUCE'II'IN x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_TAUTOLOGY'I'IN_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) -> P_id_TAUTOLOGY'I'IN x21 <= P_id_TAUTOLOGY'I'IN x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'9'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'9'1 x21 <= P_id_U'9'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'1'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'1'1 x21 <= P_id_U'1'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'14'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'14'1 x21 <= P_id_U'14'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'7'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'7'1 x21 <= P_id_U'7'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'4'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'4'1 x21 <= P_id_U'4'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'11'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'11'1 x21 <= P_id_U'11'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_INTERSECT'II'IN_monotonic : forall x20 x22 x21 x23, (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_INTERSECT'II'IN x21 x23 <= P_id_INTERSECT'II'IN x20 x22. Proof. intros x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'13'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'13'1 x21 <= P_id_U'13'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'6'2_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'6'2 x21 <= P_id_U'6'2 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'3'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'3'1 x21 <= P_id_U'3'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'16'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'16'1 x21 <= P_id_U'16'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'10'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'10'1 x21 <= P_id_U'10'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'2'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'2'1 x21 <= P_id_U'2'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'15'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'15'1 x21 <= P_id_U'15'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'8'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'8'1 x21 <= P_id_U'8'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'5'1_monotonic : forall x20 x21, (0 <= x21)/\ (x21 <= x20) ->P_id_U'5'1 x21 <= P_id_U'5'1 x20. Proof. intros x21 x20. intros [H_1 H_0]. cbv beta iota zeta delta - [Zle Zlt Zplus Zmult] ;intuition; (auto with zarith)||(repeat (translate_vars );prove_ineq ). Qed. Lemma P_id_U'12'1_monotonic : forall x24 x20 x28 x26 x22 x25 x21 x29 x27 x23, (0 <= x29)/\ (x29 <= x28) -> (0 <= x27)/\ (x27 <= x26) -> (0 <= x25)/\ (x25 <= x24) -> (0 <= x23)/\ (x23 <= x22) -> (0 <= x21)/\ (x21 <= x20) -> P_id_U'12'1 x21 x23 x25 x27 x29 <= P_id_U'12'1 x20 x22 x24 x26 x28. Proof. intros x29 x28 x27 x26 x25 x24 x23 x22 x21 x20. intros [H_1 H_0]. intros [H_3 H_2]. intros [H_5 H_4]. intros [H_7 H_6]. intros [H_9 H_8]. 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_intersect'ii'in P_id_tautology'i'out P_id_u'6'1 P_id_u'3'1 P_id_u'12'1 P_id_u'2'1 P_id_u'9'1 P_id_iff P_id_u'14'1 P_id_intersect'ii'out P_id_u'7'1 P_id_x'2d P_id_u'13'1 P_id_sequent P_id_u'10'1 P_id_x'2a P_id_tautology'i'in P_id_cons P_id_u'6'2 P_id_x'2b P_id_u'12'2 P_id_reduce'ii'in P_id_p P_id_u'4'1 P_id_u'15'1 P_id_u'1'1 P_id_u'8'1 P_id_reduce'ii'out P_id_nil P_id_if P_id_u'11'1 P_id_u'5'1 P_id_u'16'1 P_id_U'12'2 P_id_U'6'1 P_id_REDUCE'II'IN P_id_TAUTOLOGY'I'IN P_id_U'9'1 P_id_U'1'1 P_id_U'14'1 P_id_U'7'1 P_id_U'4'1 P_id_U'11'1 P_id_INTERSECT'II'IN P_id_U'13'1 P_id_U'6'2 P_id_U'3'1 P_id_U'16'1 P_id_U'10'1 P_id_U'2'1 P_id_U'15'1 P_id_U'8'1 P_id_U'5'1 P_id_U'12'1. Lemma marked_measure_equation : forall t, marked_measure t = match t with | (algebra.Alg.Term algebra.F.id_u'12'2 (x20::nil)) => P_id_U'12'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'1 (x24::x23:: x22::x21::x20::nil)) => P_id_U'6'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_reduce'ii'in (x21::x20::nil)) => P_id_REDUCE'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::nil)) => P_id_TAUTOLOGY'I'IN (measure x20) | (algebra.Alg.Term algebra.F.id_u'9'1 (x20::nil)) => P_id_U'9'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'1'1 (x20::nil)) => P_id_U'1'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'14'1 (x20::nil)) => P_id_U'14'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'7'1 (x20::nil)) => P_id_U'7'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'4'1 (x20::nil)) => P_id_U'4'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'11'1 (x20::nil)) => P_id_U'11'1 (measure x20) | (algebra.Alg.Term algebra.F.id_intersect'ii'in (x21::x20::nil)) => P_id_INTERSECT'II'IN (measure x21) (measure x20) | (algebra.Alg.Term algebra.F.id_u'13'1 (x20::nil)) => P_id_U'13'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'6'2 (x20::nil)) => P_id_U'6'2 (measure x20) | (algebra.Alg.Term algebra.F.id_u'3'1 (x20::nil)) => P_id_U'3'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'16'1 (x20::nil)) => P_id_U'16'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'10'1 (x20::nil)) => P_id_U'10'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'2'1 (x20::nil)) => P_id_U'2'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'15'1 (x20::nil)) => P_id_U'15'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'8'1 (x20::nil)) => P_id_U'8'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'5'1 (x20::nil)) => P_id_U'5'1 (measure x20) | (algebra.Alg.Term algebra.F.id_u'12'1 (x24::x23:: x22::x21::x20::nil)) => P_id_U'12'1 (measure x24) (measure x23) (measure x22) (measure x21) (measure x20) | _ => 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_intersect'ii'in_monotonic;assumption. intros ;apply P_id_u'6'1_monotonic;assumption. intros ;apply P_id_u'3'1_monotonic;assumption. intros ;apply P_id_u'12'1_monotonic;assumption. intros ;apply P_id_u'2'1_monotonic;assumption. intros ;apply P_id_u'9'1_monotonic;assumption. intros ;apply P_id_iff_monotonic;assumption. intros ;apply P_id_u'14'1_monotonic;assumption. intros ;apply P_id_u'7'1_monotonic;assumption. intros ;apply P_id_x'2d_monotonic;assumption. intros ;apply P_id_u'13'1_monotonic;assumption. intros ;apply P_id_sequent_monotonic;assumption. intros ;apply P_id_u'10'1_monotonic;assumption. intros ;apply P_id_x'2a_monotonic;assumption. intros ;apply P_id_tautology'i'in_monotonic;assumption. intros ;apply P_id_cons_monotonic;assumption. intros ;apply P_id_u'6'2_monotonic;assumption. intros ;apply P_id_x'2b_monotonic;assumption. intros ;apply P_id_u'12'2_monotonic;assumption. intros ;apply P_id_reduce'ii'in_monotonic;assumption. intros ;apply P_id_p_monotonic;assumption. intros ;apply P_id_u'4'1_monotonic;assumption. intros ;apply P_id_u'15'1_monotonic;assumption. intros ;apply P_id_u'1'1_monotonic;assumption. intros ;apply P_id_u'8'1_monotonic;assumption. intros ;apply P_id_if_monotonic;assumption. intros ;apply P_id_u'11'1_monotonic;assumption. intros ;apply P_id_u'5'1_monotonic;assumption. intros ;apply P_id_u'16'1_monotonic;assumption. intros ;apply P_id_intersect'ii'in_bounded;assumption. intros ;apply P_id_tautology'i'out_bounded;assumption. intros ;apply P_id_u'6'1_bounded;assumption. intros ;apply P_id_u'3'1_bounded;assumption. intros ;apply P_id_u'12'1_bounded;assumption. intros ;apply P_id_u'2'1_bounded;assumption. intros ;apply P_id_u'9'1_bounded;assumption. intros ;apply P_id_iff_bounded;assumption. intros ;apply P_id_u'14'1_bounded;assumption. intros ;apply P_id_intersect'ii'out_bounded;assumption. intros ;apply P_id_u'7'1_bounded;assumption. intros ;apply P_id_x'2d_bounded;assumption. intros ;apply P_id_u'13'1_bounded;assumption. intros ;apply P_id_sequent_bounded;assumption. intros ;apply P_id_u'10'1_bounded;assumption. intros ;apply P_id_x'2a_bounded;assumption. intros ;apply P_id_tautology'i'in_bounded;assumption. intros ;apply P_id_cons_bounded;assumption. intros ;apply P_id_u'6'2_bounded;assumption. intros ;apply P_id_x'2b_bounded;assumption. intros ;apply P_id_u'12'2_bounded;assumption. intros ;apply P_id_reduce'ii'in_bounded;assumption. intros ;apply P_id_p_bounded;assumption. intros ;apply P_id_u'4'1_bounded;assumption. intros ;apply P_id_u'15'1_bounded;assumption. intros ;apply P_id_u'1'1_bounded;assumption. intros ;apply P_id_u'8'1_bounded;assumption. intros ;apply P_id_reduce'ii'out_bounded;assumption. intros ;apply P_id_nil_bounded;assumption. intros ;apply P_id_if_bounded;assumption. intros ;apply P_id_u'11'1_bounded;assumption. intros ;apply P_id_u'5'1_bounded;assumption. intros ;apply P_id_u'16'1_bounded;assumption. apply rules_monotonic. intros ;apply P_id_U'12'2_monotonic;assumption. intros ;apply P_id_U'6'1_monotonic;assumption. intros ;apply P_id_REDUCE'II'IN_monotonic;assumption. intros ;apply P_id_TAUTOLOGY'I'IN_monotonic;assumption. intros ;apply P_id_U'9'1_monotonic;assumption. intros ;apply P_id_U'1'1_monotonic;assumption. intros ;apply P_id_U'14'1_monotonic;assumption. intros ;apply P_id_U'7'1_monotonic;assumption. intros ;apply P_id_U'4'1_monotonic;assumption. intros ;apply P_id_U'11'1_monotonic;assumption. intros ;apply P_id_INTERSECT'II'IN_monotonic;assumption. intros ;apply P_id_U'13'1_monotonic;assumption. intros ;apply P_id_U'6'2_monotonic;assumption. intros ;apply P_id_U'3'1_monotonic;assumption. intros ;apply P_id_U'16'1_monotonic;assumption. intros ;apply P_id_U'10'1_monotonic;assumption. intros ;apply P_id_U'2'1_monotonic;assumption. intros ;apply P_id_U'15'1_monotonic;assumption. intros ;apply P_id_U'8'1_monotonic;assumption. intros ;apply P_id_U'5'1_monotonic;assumption. intros ;apply P_id_U'12'1_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_19_strict_in_lt : Relation_Definitions.inclusion _ DP_R_xml_0_scc_19_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_19_large_in_le : Relation_Definitions.inclusion _ DP_R_xml_0_scc_19_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_19_large := WF_DP_R_xml_0_scc_19_large.wf. Lemma wf : well_founded WF_DP_R_xml_0.DP_R_xml_0_scc_19. Proof. intros x. apply (well_founded_ind wf_lt). clear x. intros x. pattern x. apply (@Acc_ind _ DP_R_xml_0_scc_19_large). clear x. intros x _ IHx IHx'. constructor. intros y H. destruct H; (apply IHx';apply DP_R_xml_0_scc_19_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_19_large_in_le;econstructor eassumption])). apply wf_DP_R_xml_0_scc_19_large. Qed. End WF_DP_R_xml_0_scc_19. Definition wf_DP_R_xml_0_scc_19 := WF_DP_R_xml_0_scc_19.wf. Lemma acc_DP_R_xml_0_scc_19 : forall x y, (DP_R_xml_0_scc_19 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_19). 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_18; 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' ))|| ((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_19. 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 x20 x18, (closure.refl_trans_clos (algebra.EQT.one_step R_xml_0_deep_rew.R_xml_0_rules) x18 x20) -> DP_R_xml_0_non_scc_20 (algebra.Alg.Term algebra.F.id_reduce'ii'in ((algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil):: (algebra.Alg.Term algebra.F.id_cons (x18:: (algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil))::(algebra.Alg.Term algebra.F.id_sequent ((algebra.Alg.Term algebra.F.id_nil nil)::(algebra.Alg.Term algebra.F.id_nil nil)::nil))::nil)) (algebra.Alg.Term algebra.F.id_tautology'i'in (x20::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_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_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_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_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' ))|| ((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_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_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_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___secret05___cime5.trs/a3pat.v" *** *** End: *** *)