Require Import ADPUnif. Require Import ADecomp. Require Import ADuplicateSymb. Require Import AGraph. Require Import APolyInt_MA. Require Import ATrs. Require Import List. Require Import LogicUtil. Require Import MonotonePolynom. Require Import Polynom. Require Import SN. Require Import VecUtil. Open Scope nat_scope. (* termination problem *) Module M. Inductive symb : Type := | U11 : symb | U12 : symb | U13 : symb | U21 : symb | U22 : symb | U23 : symb | U31 : symb | U32 : symb | U33 : symb | U41 : symb | U42 : symb | U43 : symb | U44 : symb | U45 : symb | U46 : symb | U51 : symb | U52 : symb | U61 : symb | U71 : symb | U81 : symb | U82 : symb | U83 : symb | U84 : symb | U85 : symb | U86 : symb | U91 : symb | U92 : symb | U93 : symb | U94 : symb | _0_1 : symb | a__U11 : symb | a__U12 : symb | a__U13 : symb | a__U21 : symb | a__U22 : symb | a__U23 : symb | a__U31 : symb | a__U32 : symb | a__U33 : symb | a__U41 : symb | a__U42 : symb | a__U43 : symb | a__U44 : symb | a__U45 : symb | a__U46 : symb | a__U51 : symb | a__U52 : symb | a__U61 : symb | a__U71 : symb | a__U81 : symb | a__U82 : symb | a__U83 : symb | a__U84 : symb | a__U85 : symb | a__U86 : symb | a__U91 : symb | a__U92 : symb | a__U93 : symb | a__U94 : symb | a__isNat : symb | a__isNatIList : symb | a__isNatIListKind : symb | a__isNatKind : symb | a__isNatList : symb | a__length : symb | a__zeros : symb | cons : symb | isNat : symb | isNatIList : symb | isNatIListKind : symb | isNatKind : symb | isNatList : symb | length : symb | mark : symb | nil : symb | s : symb | tt : symb | zeros : symb. End M. Lemma eq_symb_dec : forall f g : M.symb, {f=g}+{~f=g}. Proof. decide equality. Defined. Open Scope nat_scope. Definition ar (s : M.symb) : nat := match s with | M.U11 => 2 | M.U12 => 2 | M.U13 => 1 | M.U21 => 2 | M.U22 => 2 | M.U23 => 1 | M.U31 => 2 | M.U32 => 2 | M.U33 => 1 | M.U41 => 3 | M.U42 => 3 | M.U43 => 3 | M.U44 => 3 | M.U45 => 2 | M.U46 => 1 | M.U51 => 2 | M.U52 => 1 | M.U61 => 1 | M.U71 => 1 | M.U81 => 3 | M.U82 => 3 | M.U83 => 3 | M.U84 => 3 | M.U85 => 2 | M.U86 => 1 | M.U91 => 3 | M.U92 => 3 | M.U93 => 3 | M.U94 => 2 | M._0_1 => 0 | M.a__U11 => 2 | M.a__U12 => 2 | M.a__U13 => 1 | M.a__U21 => 2 | M.a__U22 => 2 | M.a__U23 => 1 | M.a__U31 => 2 | M.a__U32 => 2 | M.a__U33 => 1 | M.a__U41 => 3 | M.a__U42 => 3 | M.a__U43 => 3 | M.a__U44 => 3 | M.a__U45 => 2 | M.a__U46 => 1 | M.a__U51 => 2 | M.a__U52 => 1 | M.a__U61 => 1 | M.a__U71 => 1 | M.a__U81 => 3 | M.a__U82 => 3 | M.a__U83 => 3 | M.a__U84 => 3 | M.a__U85 => 2 | M.a__U86 => 1 | M.a__U91 => 3 | M.a__U92 => 3 | M.a__U93 => 3 | M.a__U94 => 2 | M.a__isNat => 1 | M.a__isNatIList => 1 | M.a__isNatIListKind => 1 | M.a__isNatKind => 1 | M.a__isNatList => 1 | M.a__length => 1 | M.a__zeros => 0 | M.cons => 2 | M.isNat => 1 | M.isNatIList => 1 | M.isNatIListKind => 1 | M.isNatKind => 1 | M.isNatList => 1 | M.length => 1 | M.mark => 1 | M.nil => 0 | M.s => 1 | M.tt => 0 | M.zeros => 0 end. Definition s0 := ASignature.mkSignature ar eq_symb_dec. Definition s0_p := s0. Definition V0 := @ATerm.Var s0. Definition F0 := @ATerm.Fun s0. Definition R0 := @ATrs.mkRule s0. Module S0. Definition U11 x2 x1 := F0 M.U11 (Vcons x2 (Vcons x1 Vnil)). Definition U12 x2 x1 := F0 M.U12 (Vcons x2 (Vcons x1 Vnil)). Definition U13 x1 := F0 M.U13 (Vcons x1 Vnil). Definition U21 x2 x1 := F0 M.U21 (Vcons x2 (Vcons x1 Vnil)). Definition U22 x2 x1 := F0 M.U22 (Vcons x2 (Vcons x1 Vnil)). Definition U23 x1 := F0 M.U23 (Vcons x1 Vnil). Definition U31 x2 x1 := F0 M.U31 (Vcons x2 (Vcons x1 Vnil)). Definition U32 x2 x1 := F0 M.U32 (Vcons x2 (Vcons x1 Vnil)). Definition U33 x1 := F0 M.U33 (Vcons x1 Vnil). Definition U41 x3 x2 x1 := F0 M.U41 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U42 x3 x2 x1 := F0 M.U42 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U43 x3 x2 x1 := F0 M.U43 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U44 x3 x2 x1 := F0 M.U44 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U45 x2 x1 := F0 M.U45 (Vcons x2 (Vcons x1 Vnil)). Definition U46 x1 := F0 M.U46 (Vcons x1 Vnil). Definition U51 x2 x1 := F0 M.U51 (Vcons x2 (Vcons x1 Vnil)). Definition U52 x1 := F0 M.U52 (Vcons x1 Vnil). Definition U61 x1 := F0 M.U61 (Vcons x1 Vnil). Definition U71 x1 := F0 M.U71 (Vcons x1 Vnil). Definition U81 x3 x2 x1 := F0 M.U81 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U82 x3 x2 x1 := F0 M.U82 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U83 x3 x2 x1 := F0 M.U83 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U84 x3 x2 x1 := F0 M.U84 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U85 x2 x1 := F0 M.U85 (Vcons x2 (Vcons x1 Vnil)). Definition U86 x1 := F0 M.U86 (Vcons x1 Vnil). Definition U91 x3 x2 x1 := F0 M.U91 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U92 x3 x2 x1 := F0 M.U92 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U93 x3 x2 x1 := F0 M.U93 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U94 x2 x1 := F0 M.U94 (Vcons x2 (Vcons x1 Vnil)). Definition _0_1 := F0 M._0_1 Vnil. Definition a__U11 x2 x1 := F0 M.a__U11 (Vcons x2 (Vcons x1 Vnil)). Definition a__U12 x2 x1 := F0 M.a__U12 (Vcons x2 (Vcons x1 Vnil)). Definition a__U13 x1 := F0 M.a__U13 (Vcons x1 Vnil). Definition a__U21 x2 x1 := F0 M.a__U21 (Vcons x2 (Vcons x1 Vnil)). Definition a__U22 x2 x1 := F0 M.a__U22 (Vcons x2 (Vcons x1 Vnil)). Definition a__U23 x1 := F0 M.a__U23 (Vcons x1 Vnil). Definition a__U31 x2 x1 := F0 M.a__U31 (Vcons x2 (Vcons x1 Vnil)). Definition a__U32 x2 x1 := F0 M.a__U32 (Vcons x2 (Vcons x1 Vnil)). Definition a__U33 x1 := F0 M.a__U33 (Vcons x1 Vnil). Definition a__U41 x3 x2 x1 := F0 M.a__U41 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U42 x3 x2 x1 := F0 M.a__U42 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U43 x3 x2 x1 := F0 M.a__U43 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U44 x3 x2 x1 := F0 M.a__U44 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U45 x2 x1 := F0 M.a__U45 (Vcons x2 (Vcons x1 Vnil)). Definition a__U46 x1 := F0 M.a__U46 (Vcons x1 Vnil). Definition a__U51 x2 x1 := F0 M.a__U51 (Vcons x2 (Vcons x1 Vnil)). Definition a__U52 x1 := F0 M.a__U52 (Vcons x1 Vnil). Definition a__U61 x1 := F0 M.a__U61 (Vcons x1 Vnil). Definition a__U71 x1 := F0 M.a__U71 (Vcons x1 Vnil). Definition a__U81 x3 x2 x1 := F0 M.a__U81 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U82 x3 x2 x1 := F0 M.a__U82 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U83 x3 x2 x1 := F0 M.a__U83 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U84 x3 x2 x1 := F0 M.a__U84 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U85 x2 x1 := F0 M.a__U85 (Vcons x2 (Vcons x1 Vnil)). Definition a__U86 x1 := F0 M.a__U86 (Vcons x1 Vnil). Definition a__U91 x3 x2 x1 := F0 M.a__U91 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U92 x3 x2 x1 := F0 M.a__U92 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U93 x3 x2 x1 := F0 M.a__U93 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U94 x2 x1 := F0 M.a__U94 (Vcons x2 (Vcons x1 Vnil)). Definition a__isNat x1 := F0 M.a__isNat (Vcons x1 Vnil). Definition a__isNatIList x1 := F0 M.a__isNatIList (Vcons x1 Vnil). Definition a__isNatIListKind x1 := F0 M.a__isNatIListKind (Vcons x1 Vnil). Definition a__isNatKind x1 := F0 M.a__isNatKind (Vcons x1 Vnil). Definition a__isNatList x1 := F0 M.a__isNatList (Vcons x1 Vnil). Definition a__length x1 := F0 M.a__length (Vcons x1 Vnil). Definition a__zeros := F0 M.a__zeros Vnil. Definition cons x2 x1 := F0 M.cons (Vcons x2 (Vcons x1 Vnil)). Definition isNat x1 := F0 M.isNat (Vcons x1 Vnil). Definition isNatIList x1 := F0 M.isNatIList (Vcons x1 Vnil). Definition isNatIListKind x1 := F0 M.isNatIListKind (Vcons x1 Vnil). Definition isNatKind x1 := F0 M.isNatKind (Vcons x1 Vnil). Definition isNatList x1 := F0 M.isNatList (Vcons x1 Vnil). Definition length x1 := F0 M.length (Vcons x1 Vnil). Definition mark x1 := F0 M.mark (Vcons x1 Vnil). Definition nil := F0 M.nil Vnil. Definition s x1 := F0 M.s (Vcons x1 Vnil). Definition tt := F0 M.tt Vnil. Definition zeros := F0 M.zeros Vnil. End S0. Definition E := @nil (@ATrs.rule s0). Definition R := R0 S0.a__zeros (S0.cons S0._0_1 S0.zeros) :: R0 (S0.a__U11 S0.tt (V0 0)) (S0.a__U12 (S0.a__isNatIListKind (V0 0)) (V0 0)) :: R0 (S0.a__U12 S0.tt (V0 0)) (S0.a__U13 (S0.a__isNatList (V0 0))) :: R0 (S0.a__U13 S0.tt) S0.tt :: R0 (S0.a__U21 S0.tt (V0 0)) (S0.a__U22 (S0.a__isNatKind (V0 0)) (V0 0)) :: R0 (S0.a__U22 S0.tt (V0 0)) (S0.a__U23 (S0.a__isNat (V0 0))) :: R0 (S0.a__U23 S0.tt) S0.tt :: R0 (S0.a__U31 S0.tt (V0 0)) (S0.a__U32 (S0.a__isNatIListKind (V0 0)) (V0 0)) :: R0 (S0.a__U32 S0.tt (V0 0)) (S0.a__U33 (S0.a__isNatList (V0 0))) :: R0 (S0.a__U33 S0.tt) S0.tt :: R0 (S0.a__U41 S0.tt (V0 0) (V0 1)) (S0.a__U42 (S0.a__isNatKind (V0 0)) (V0 0) (V0 1)) :: R0 (S0.a__U42 S0.tt (V0 0) (V0 1)) (S0.a__U43 (S0.a__isNatIListKind (V0 1)) (V0 0) (V0 1)) :: R0 (S0.a__U43 S0.tt (V0 0) (V0 1)) (S0.a__U44 (S0.a__isNatIListKind (V0 1)) (V0 0) (V0 1)) :: R0 (S0.a__U44 S0.tt (V0 0) (V0 1)) (S0.a__U45 (S0.a__isNat (V0 0)) (V0 1)) :: R0 (S0.a__U45 S0.tt (V0 0)) (S0.a__U46 (S0.a__isNatIList (V0 0))) :: R0 (S0.a__U46 S0.tt) S0.tt :: R0 (S0.a__U51 S0.tt (V0 0)) (S0.a__U52 (S0.a__isNatIListKind (V0 0))) :: R0 (S0.a__U52 S0.tt) S0.tt :: R0 (S0.a__U61 S0.tt) S0.tt :: R0 (S0.a__U71 S0.tt) S0.tt :: R0 (S0.a__U81 S0.tt (V0 0) (V0 1)) (S0.a__U82 (S0.a__isNatKind (V0 0)) (V0 0) (V0 1)) :: R0 (S0.a__U82 S0.tt (V0 0) (V0 1)) (S0.a__U83 (S0.a__isNatIListKind (V0 1)) (V0 0) (V0 1)) :: R0 (S0.a__U83 S0.tt (V0 0) (V0 1)) (S0.a__U84 (S0.a__isNatIListKind (V0 1)) (V0 0) (V0 1)) :: R0 (S0.a__U84 S0.tt (V0 0) (V0 1)) (S0.a__U85 (S0.a__isNat (V0 0)) (V0 1)) :: R0 (S0.a__U85 S0.tt (V0 0)) (S0.a__U86 (S0.a__isNatList (V0 0))) :: R0 (S0.a__U86 S0.tt) S0.tt :: R0 (S0.a__U91 S0.tt (V0 0) (V0 1)) (S0.a__U92 (S0.a__isNatIListKind (V0 0)) (V0 0) (V0 1)) :: R0 (S0.a__U92 S0.tt (V0 0) (V0 1)) (S0.a__U93 (S0.a__isNat (V0 1)) (V0 0) (V0 1)) :: R0 (S0.a__U93 S0.tt (V0 0) (V0 1)) (S0.a__U94 (S0.a__isNatKind (V0 1)) (V0 0)) :: R0 (S0.a__U94 S0.tt (V0 0)) (S0.s (S0.a__length (S0.mark (V0 0)))) :: R0 (S0.a__isNat S0._0_1) S0.tt :: R0 (S0.a__isNat (S0.length (V0 0))) (S0.a__U11 (S0.a__isNatIListKind (V0 0)) (V0 0)) :: R0 (S0.a__isNat (S0.s (V0 0))) (S0.a__U21 (S0.a__isNatKind (V0 0)) (V0 0)) :: R0 (S0.a__isNatIList (V0 0)) (S0.a__U31 (S0.a__isNatIListKind (V0 0)) (V0 0)) :: R0 (S0.a__isNatIList S0.zeros) S0.tt :: R0 (S0.a__isNatIList (S0.cons (V0 0) (V0 1))) (S0.a__U41 (S0.a__isNatKind (V0 0)) (V0 0) (V0 1)) :: R0 (S0.a__isNatIListKind S0.nil) S0.tt :: R0 (S0.a__isNatIListKind S0.zeros) S0.tt :: R0 (S0.a__isNatIListKind (S0.cons (V0 0) (V0 1))) (S0.a__U51 (S0.a__isNatKind (V0 0)) (V0 1)) :: R0 (S0.a__isNatKind S0._0_1) S0.tt :: R0 (S0.a__isNatKind (S0.length (V0 0))) (S0.a__U61 (S0.a__isNatIListKind (V0 0))) :: R0 (S0.a__isNatKind (S0.s (V0 0))) (S0.a__U71 (S0.a__isNatKind (V0 0))) :: R0 (S0.a__isNatList S0.nil) S0.tt :: R0 (S0.a__isNatList (S0.cons (V0 0) (V0 1))) (S0.a__U81 (S0.a__isNatKind (V0 0)) (V0 0) (V0 1)) :: R0 (S0.a__length S0.nil) S0._0_1 :: R0 (S0.a__length (S0.cons (V0 0) (V0 1))) (S0.a__U91 (S0.a__isNatList (V0 1)) (V0 1) (V0 0)) :: R0 (S0.mark S0.zeros) S0.a__zeros :: R0 (S0.mark (S0.U11 (V0 0) (V0 1))) (S0.a__U11 (S0.mark (V0 0)) (V0 1)) :: R0 (S0.mark (S0.U12 (V0 0) (V0 1))) (S0.a__U12 (S0.mark (V0 0)) (V0 1)) :: R0 (S0.mark (S0.isNatIListKind (V0 0))) (S0.a__isNatIListKind (V0 0)) :: R0 (S0.mark (S0.U13 (V0 0))) (S0.a__U13 (S0.mark (V0 0))) :: R0 (S0.mark (S0.isNatList (V0 0))) (S0.a__isNatList (V0 0)) :: R0 (S0.mark (S0.U21 (V0 0) (V0 1))) (S0.a__U21 (S0.mark (V0 0)) (V0 1)) :: R0 (S0.mark (S0.U22 (V0 0) (V0 1))) (S0.a__U22 (S0.mark (V0 0)) (V0 1)) :: R0 (S0.mark (S0.isNatKind (V0 0))) (S0.a__isNatKind (V0 0)) :: R0 (S0.mark (S0.U23 (V0 0))) (S0.a__U23 (S0.mark (V0 0))) :: R0 (S0.mark (S0.isNat (V0 0))) (S0.a__isNat (V0 0)) :: R0 (S0.mark (S0.U31 (V0 0) (V0 1))) (S0.a__U31 (S0.mark (V0 0)) (V0 1)) :: R0 (S0.mark (S0.U32 (V0 0) (V0 1))) (S0.a__U32 (S0.mark (V0 0)) (V0 1)) :: R0 (S0.mark (S0.U33 (V0 0))) (S0.a__U33 (S0.mark (V0 0))) :: R0 (S0.mark (S0.U41 (V0 0) (V0 1) (V0 2))) (S0.a__U41 (S0.mark (V0 0)) (V0 1) (V0 2)) :: R0 (S0.mark (S0.U42 (V0 0) (V0 1) (V0 2))) (S0.a__U42 (S0.mark (V0 0)) (V0 1) (V0 2)) :: R0 (S0.mark (S0.U43 (V0 0) (V0 1) (V0 2))) (S0.a__U43 (S0.mark (V0 0)) (V0 1) (V0 2)) :: R0 (S0.mark (S0.U44 (V0 0) (V0 1) (V0 2))) (S0.a__U44 (S0.mark (V0 0)) (V0 1) (V0 2)) :: R0 (S0.mark (S0.U45 (V0 0) (V0 1))) (S0.a__U45 (S0.mark (V0 0)) (V0 1)) :: R0 (S0.mark (S0.U46 (V0 0))) (S0.a__U46 (S0.mark (V0 0))) :: R0 (S0.mark (S0.isNatIList (V0 0))) (S0.a__isNatIList (V0 0)) :: R0 (S0.mark (S0.U51 (V0 0) (V0 1))) (S0.a__U51 (S0.mark (V0 0)) (V0 1)) :: R0 (S0.mark (S0.U52 (V0 0))) (S0.a__U52 (S0.mark (V0 0))) :: R0 (S0.mark (S0.U61 (V0 0))) (S0.a__U61 (S0.mark (V0 0))) :: R0 (S0.mark (S0.U71 (V0 0))) (S0.a__U71 (S0.mark (V0 0))) :: R0 (S0.mark (S0.U81 (V0 0) (V0 1) (V0 2))) (S0.a__U81 (S0.mark (V0 0)) (V0 1) (V0 2)) :: R0 (S0.mark (S0.U82 (V0 0) (V0 1) (V0 2))) (S0.a__U82 (S0.mark (V0 0)) (V0 1) (V0 2)) :: R0 (S0.mark (S0.U83 (V0 0) (V0 1) (V0 2))) (S0.a__U83 (S0.mark (V0 0)) (V0 1) (V0 2)) :: R0 (S0.mark (S0.U84 (V0 0) (V0 1) (V0 2))) (S0.a__U84 (S0.mark (V0 0)) (V0 1) (V0 2)) :: R0 (S0.mark (S0.U85 (V0 0) (V0 1))) (S0.a__U85 (S0.mark (V0 0)) (V0 1)) :: R0 (S0.mark (S0.U86 (V0 0))) (S0.a__U86 (S0.mark (V0 0))) :: R0 (S0.mark (S0.U91 (V0 0) (V0 1) (V0 2))) (S0.a__U91 (S0.mark (V0 0)) (V0 1) (V0 2)) :: R0 (S0.mark (S0.U92 (V0 0) (V0 1) (V0 2))) (S0.a__U92 (S0.mark (V0 0)) (V0 1) (V0 2)) :: R0 (S0.mark (S0.U93 (V0 0) (V0 1) (V0 2))) (S0.a__U93 (S0.mark (V0 0)) (V0 1) (V0 2)) :: R0 (S0.mark (S0.U94 (V0 0) (V0 1))) (S0.a__U94 (S0.mark (V0 0)) (V0 1)) :: R0 (S0.mark (S0.length (V0 0))) (S0.a__length (S0.mark (V0 0))) :: R0 (S0.mark (S0.cons (V0 0) (V0 1))) (S0.cons (S0.mark (V0 0)) (V0 1)) :: R0 (S0.mark S0._0_1) S0._0_1 :: R0 (S0.mark S0.tt) S0.tt :: R0 (S0.mark (S0.s (V0 0))) (S0.s (S0.mark (V0 0))) :: R0 (S0.mark S0.nil) S0.nil :: R0 S0.a__zeros S0.zeros :: R0 (S0.a__U11 (V0 0) (V0 1)) (S0.U11 (V0 0) (V0 1)) :: R0 (S0.a__U12 (V0 0) (V0 1)) (S0.U12 (V0 0) (V0 1)) :: R0 (S0.a__isNatIListKind (V0 0)) (S0.isNatIListKind (V0 0)) :: R0 (S0.a__U13 (V0 0)) (S0.U13 (V0 0)) :: R0 (S0.a__isNatList (V0 0)) (S0.isNatList (V0 0)) :: R0 (S0.a__U21 (V0 0) (V0 1)) (S0.U21 (V0 0) (V0 1)) :: R0 (S0.a__U22 (V0 0) (V0 1)) (S0.U22 (V0 0) (V0 1)) :: R0 (S0.a__isNatKind (V0 0)) (S0.isNatKind (V0 0)) :: R0 (S0.a__U23 (V0 0)) (S0.U23 (V0 0)) :: R0 (S0.a__isNat (V0 0)) (S0.isNat (V0 0)) :: R0 (S0.a__U31 (V0 0) (V0 1)) (S0.U31 (V0 0) (V0 1)) :: R0 (S0.a__U32 (V0 0) (V0 1)) (S0.U32 (V0 0) (V0 1)) :: R0 (S0.a__U33 (V0 0)) (S0.U33 (V0 0)) :: R0 (S0.a__U41 (V0 0) (V0 1) (V0 2)) (S0.U41 (V0 0) (V0 1) (V0 2)) :: R0 (S0.a__U42 (V0 0) (V0 1) (V0 2)) (S0.U42 (V0 0) (V0 1) (V0 2)) :: R0 (S0.a__U43 (V0 0) (V0 1) (V0 2)) (S0.U43 (V0 0) (V0 1) (V0 2)) :: R0 (S0.a__U44 (V0 0) (V0 1) (V0 2)) (S0.U44 (V0 0) (V0 1) (V0 2)) :: R0 (S0.a__U45 (V0 0) (V0 1)) (S0.U45 (V0 0) (V0 1)) :: R0 (S0.a__U46 (V0 0)) (S0.U46 (V0 0)) :: R0 (S0.a__isNatIList (V0 0)) (S0.isNatIList (V0 0)) :: R0 (S0.a__U51 (V0 0) (V0 1)) (S0.U51 (V0 0) (V0 1)) :: R0 (S0.a__U52 (V0 0)) (S0.U52 (V0 0)) :: R0 (S0.a__U61 (V0 0)) (S0.U61 (V0 0)) :: R0 (S0.a__U71 (V0 0)) (S0.U71 (V0 0)) :: R0 (S0.a__U81 (V0 0) (V0 1) (V0 2)) (S0.U81 (V0 0) (V0 1) (V0 2)) :: R0 (S0.a__U82 (V0 0) (V0 1) (V0 2)) (S0.U82 (V0 0) (V0 1) (V0 2)) :: R0 (S0.a__U83 (V0 0) (V0 1) (V0 2)) (S0.U83 (V0 0) (V0 1) (V0 2)) :: R0 (S0.a__U84 (V0 0) (V0 1) (V0 2)) (S0.U84 (V0 0) (V0 1) (V0 2)) :: R0 (S0.a__U85 (V0 0) (V0 1)) (S0.U85 (V0 0) (V0 1)) :: R0 (S0.a__U86 (V0 0)) (S0.U86 (V0 0)) :: R0 (S0.a__U91 (V0 0) (V0 1) (V0 2)) (S0.U91 (V0 0) (V0 1) (V0 2)) :: R0 (S0.a__U92 (V0 0) (V0 1) (V0 2)) (S0.U92 (V0 0) (V0 1) (V0 2)) :: R0 (S0.a__U93 (V0 0) (V0 1) (V0 2)) (S0.U93 (V0 0) (V0 1) (V0 2)) :: R0 (S0.a__U94 (V0 0) (V0 1)) (S0.U94 (V0 0) (V0 1)) :: R0 (S0.a__length (V0 0)) (S0.length (V0 0)) :: @nil (@ATrs.rule s0). Definition rel := ATrs.red_mod E R. (* symbol marking *) Definition s1 := dup_sig s0. Definition s1_p := s0. Definition V1 := @ATerm.Var s1. Definition F1 := @ATerm.Fun s1. Definition R1 := @ATrs.mkRule s1. Module S1. Definition hU11 x2 x1 := F1 (hd_symb s1_p M.U11) (Vcons x2 (Vcons x1 Vnil)). Definition U11 x2 x1 := F1 (int_symb s1_p M.U11) (Vcons x2 (Vcons x1 Vnil)). Definition hU12 x2 x1 := F1 (hd_symb s1_p M.U12) (Vcons x2 (Vcons x1 Vnil)). Definition U12 x2 x1 := F1 (int_symb s1_p M.U12) (Vcons x2 (Vcons x1 Vnil)). Definition hU13 x1 := F1 (hd_symb s1_p M.U13) (Vcons x1 Vnil). Definition U13 x1 := F1 (int_symb s1_p M.U13) (Vcons x1 Vnil). Definition hU21 x2 x1 := F1 (hd_symb s1_p M.U21) (Vcons x2 (Vcons x1 Vnil)). Definition U21 x2 x1 := F1 (int_symb s1_p M.U21) (Vcons x2 (Vcons x1 Vnil)). Definition hU22 x2 x1 := F1 (hd_symb s1_p M.U22) (Vcons x2 (Vcons x1 Vnil)). Definition U22 x2 x1 := F1 (int_symb s1_p M.U22) (Vcons x2 (Vcons x1 Vnil)). Definition hU23 x1 := F1 (hd_symb s1_p M.U23) (Vcons x1 Vnil). Definition U23 x1 := F1 (int_symb s1_p M.U23) (Vcons x1 Vnil). Definition hU31 x2 x1 := F1 (hd_symb s1_p M.U31) (Vcons x2 (Vcons x1 Vnil)). Definition U31 x2 x1 := F1 (int_symb s1_p M.U31) (Vcons x2 (Vcons x1 Vnil)). Definition hU32 x2 x1 := F1 (hd_symb s1_p M.U32) (Vcons x2 (Vcons x1 Vnil)). Definition U32 x2 x1 := F1 (int_symb s1_p M.U32) (Vcons x2 (Vcons x1 Vnil)). Definition hU33 x1 := F1 (hd_symb s1_p M.U33) (Vcons x1 Vnil). Definition U33 x1 := F1 (int_symb s1_p M.U33) (Vcons x1 Vnil). Definition hU41 x3 x2 x1 := F1 (hd_symb s1_p M.U41) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U41 x3 x2 x1 := F1 (int_symb s1_p M.U41) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition hU42 x3 x2 x1 := F1 (hd_symb s1_p M.U42) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U42 x3 x2 x1 := F1 (int_symb s1_p M.U42) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition hU43 x3 x2 x1 := F1 (hd_symb s1_p M.U43) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U43 x3 x2 x1 := F1 (int_symb s1_p M.U43) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition hU44 x3 x2 x1 := F1 (hd_symb s1_p M.U44) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U44 x3 x2 x1 := F1 (int_symb s1_p M.U44) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition hU45 x2 x1 := F1 (hd_symb s1_p M.U45) (Vcons x2 (Vcons x1 Vnil)). Definition U45 x2 x1 := F1 (int_symb s1_p M.U45) (Vcons x2 (Vcons x1 Vnil)). Definition hU46 x1 := F1 (hd_symb s1_p M.U46) (Vcons x1 Vnil). Definition U46 x1 := F1 (int_symb s1_p M.U46) (Vcons x1 Vnil). Definition hU51 x2 x1 := F1 (hd_symb s1_p M.U51) (Vcons x2 (Vcons x1 Vnil)). Definition U51 x2 x1 := F1 (int_symb s1_p M.U51) (Vcons x2 (Vcons x1 Vnil)). Definition hU52 x1 := F1 (hd_symb s1_p M.U52) (Vcons x1 Vnil). Definition U52 x1 := F1 (int_symb s1_p M.U52) (Vcons x1 Vnil). Definition hU61 x1 := F1 (hd_symb s1_p M.U61) (Vcons x1 Vnil). Definition U61 x1 := F1 (int_symb s1_p M.U61) (Vcons x1 Vnil). Definition hU71 x1 := F1 (hd_symb s1_p M.U71) (Vcons x1 Vnil). Definition U71 x1 := F1 (int_symb s1_p M.U71) (Vcons x1 Vnil). Definition hU81 x3 x2 x1 := F1 (hd_symb s1_p M.U81) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U81 x3 x2 x1 := F1 (int_symb s1_p M.U81) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition hU82 x3 x2 x1 := F1 (hd_symb s1_p M.U82) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U82 x3 x2 x1 := F1 (int_symb s1_p M.U82) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition hU83 x3 x2 x1 := F1 (hd_symb s1_p M.U83) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U83 x3 x2 x1 := F1 (int_symb s1_p M.U83) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition hU84 x3 x2 x1 := F1 (hd_symb s1_p M.U84) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U84 x3 x2 x1 := F1 (int_symb s1_p M.U84) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition hU85 x2 x1 := F1 (hd_symb s1_p M.U85) (Vcons x2 (Vcons x1 Vnil)). Definition U85 x2 x1 := F1 (int_symb s1_p M.U85) (Vcons x2 (Vcons x1 Vnil)). Definition hU86 x1 := F1 (hd_symb s1_p M.U86) (Vcons x1 Vnil). Definition U86 x1 := F1 (int_symb s1_p M.U86) (Vcons x1 Vnil). Definition hU91 x3 x2 x1 := F1 (hd_symb s1_p M.U91) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U91 x3 x2 x1 := F1 (int_symb s1_p M.U91) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition hU92 x3 x2 x1 := F1 (hd_symb s1_p M.U92) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U92 x3 x2 x1 := F1 (int_symb s1_p M.U92) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition hU93 x3 x2 x1 := F1 (hd_symb s1_p M.U93) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U93 x3 x2 x1 := F1 (int_symb s1_p M.U93) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition hU94 x2 x1 := F1 (hd_symb s1_p M.U94) (Vcons x2 (Vcons x1 Vnil)). Definition U94 x2 x1 := F1 (int_symb s1_p M.U94) (Vcons x2 (Vcons x1 Vnil)). Definition h_0_1 := F1 (hd_symb s1_p M._0_1) Vnil. Definition _0_1 := F1 (int_symb s1_p M._0_1) Vnil. Definition ha__U11 x2 x1 := F1 (hd_symb s1_p M.a__U11) (Vcons x2 (Vcons x1 Vnil)). Definition a__U11 x2 x1 := F1 (int_symb s1_p M.a__U11) (Vcons x2 (Vcons x1 Vnil)). Definition ha__U12 x2 x1 := F1 (hd_symb s1_p M.a__U12) (Vcons x2 (Vcons x1 Vnil)). Definition a__U12 x2 x1 := F1 (int_symb s1_p M.a__U12) (Vcons x2 (Vcons x1 Vnil)). Definition ha__U13 x1 := F1 (hd_symb s1_p M.a__U13) (Vcons x1 Vnil). Definition a__U13 x1 := F1 (int_symb s1_p M.a__U13) (Vcons x1 Vnil). Definition ha__U21 x2 x1 := F1 (hd_symb s1_p M.a__U21) (Vcons x2 (Vcons x1 Vnil)). Definition a__U21 x2 x1 := F1 (int_symb s1_p M.a__U21) (Vcons x2 (Vcons x1 Vnil)). Definition ha__U22 x2 x1 := F1 (hd_symb s1_p M.a__U22) (Vcons x2 (Vcons x1 Vnil)). Definition a__U22 x2 x1 := F1 (int_symb s1_p M.a__U22) (Vcons x2 (Vcons x1 Vnil)). Definition ha__U23 x1 := F1 (hd_symb s1_p M.a__U23) (Vcons x1 Vnil). Definition a__U23 x1 := F1 (int_symb s1_p M.a__U23) (Vcons x1 Vnil). Definition ha__U31 x2 x1 := F1 (hd_symb s1_p M.a__U31) (Vcons x2 (Vcons x1 Vnil)). Definition a__U31 x2 x1 := F1 (int_symb s1_p M.a__U31) (Vcons x2 (Vcons x1 Vnil)). Definition ha__U32 x2 x1 := F1 (hd_symb s1_p M.a__U32) (Vcons x2 (Vcons x1 Vnil)). Definition a__U32 x2 x1 := F1 (int_symb s1_p M.a__U32) (Vcons x2 (Vcons x1 Vnil)). Definition ha__U33 x1 := F1 (hd_symb s1_p M.a__U33) (Vcons x1 Vnil). Definition a__U33 x1 := F1 (int_symb s1_p M.a__U33) (Vcons x1 Vnil). Definition ha__U41 x3 x2 x1 := F1 (hd_symb s1_p M.a__U41) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U41 x3 x2 x1 := F1 (int_symb s1_p M.a__U41) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition ha__U42 x3 x2 x1 := F1 (hd_symb s1_p M.a__U42) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U42 x3 x2 x1 := F1 (int_symb s1_p M.a__U42) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition ha__U43 x3 x2 x1 := F1 (hd_symb s1_p M.a__U43) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U43 x3 x2 x1 := F1 (int_symb s1_p M.a__U43) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition ha__U44 x3 x2 x1 := F1 (hd_symb s1_p M.a__U44) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U44 x3 x2 x1 := F1 (int_symb s1_p M.a__U44) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition ha__U45 x2 x1 := F1 (hd_symb s1_p M.a__U45) (Vcons x2 (Vcons x1 Vnil)). Definition a__U45 x2 x1 := F1 (int_symb s1_p M.a__U45) (Vcons x2 (Vcons x1 Vnil)). Definition ha__U46 x1 := F1 (hd_symb s1_p M.a__U46) (Vcons x1 Vnil). Definition a__U46 x1 := F1 (int_symb s1_p M.a__U46) (Vcons x1 Vnil). Definition ha__U51 x2 x1 := F1 (hd_symb s1_p M.a__U51) (Vcons x2 (Vcons x1 Vnil)). Definition a__U51 x2 x1 := F1 (int_symb s1_p M.a__U51) (Vcons x2 (Vcons x1 Vnil)). Definition ha__U52 x1 := F1 (hd_symb s1_p M.a__U52) (Vcons x1 Vnil). Definition a__U52 x1 := F1 (int_symb s1_p M.a__U52) (Vcons x1 Vnil). Definition ha__U61 x1 := F1 (hd_symb s1_p M.a__U61) (Vcons x1 Vnil). Definition a__U61 x1 := F1 (int_symb s1_p M.a__U61) (Vcons x1 Vnil). Definition ha__U71 x1 := F1 (hd_symb s1_p M.a__U71) (Vcons x1 Vnil). Definition a__U71 x1 := F1 (int_symb s1_p M.a__U71) (Vcons x1 Vnil). Definition ha__U81 x3 x2 x1 := F1 (hd_symb s1_p M.a__U81) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U81 x3 x2 x1 := F1 (int_symb s1_p M.a__U81) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition ha__U82 x3 x2 x1 := F1 (hd_symb s1_p M.a__U82) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U82 x3 x2 x1 := F1 (int_symb s1_p M.a__U82) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition ha__U83 x3 x2 x1 := F1 (hd_symb s1_p M.a__U83) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U83 x3 x2 x1 := F1 (int_symb s1_p M.a__U83) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition ha__U84 x3 x2 x1 := F1 (hd_symb s1_p M.a__U84) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U84 x3 x2 x1 := F1 (int_symb s1_p M.a__U84) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition ha__U85 x2 x1 := F1 (hd_symb s1_p M.a__U85) (Vcons x2 (Vcons x1 Vnil)). Definition a__U85 x2 x1 := F1 (int_symb s1_p M.a__U85) (Vcons x2 (Vcons x1 Vnil)). Definition ha__U86 x1 := F1 (hd_symb s1_p M.a__U86) (Vcons x1 Vnil). Definition a__U86 x1 := F1 (int_symb s1_p M.a__U86) (Vcons x1 Vnil). Definition ha__U91 x3 x2 x1 := F1 (hd_symb s1_p M.a__U91) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U91 x3 x2 x1 := F1 (int_symb s1_p M.a__U91) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition ha__U92 x3 x2 x1 := F1 (hd_symb s1_p M.a__U92) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U92 x3 x2 x1 := F1 (int_symb s1_p M.a__U92) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition ha__U93 x3 x2 x1 := F1 (hd_symb s1_p M.a__U93) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition a__U93 x3 x2 x1 := F1 (int_symb s1_p M.a__U93) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition ha__U94 x2 x1 := F1 (hd_symb s1_p M.a__U94) (Vcons x2 (Vcons x1 Vnil)). Definition a__U94 x2 x1 := F1 (int_symb s1_p M.a__U94) (Vcons x2 (Vcons x1 Vnil)). Definition ha__isNat x1 := F1 (hd_symb s1_p M.a__isNat) (Vcons x1 Vnil). Definition a__isNat x1 := F1 (int_symb s1_p M.a__isNat) (Vcons x1 Vnil). Definition ha__isNatIList x1 := F1 (hd_symb s1_p M.a__isNatIList) (Vcons x1 Vnil). Definition a__isNatIList x1 := F1 (int_symb s1_p M.a__isNatIList) (Vcons x1 Vnil). Definition ha__isNatIListKind x1 := F1 (hd_symb s1_p M.a__isNatIListKind) (Vcons x1 Vnil). Definition a__isNatIListKind x1 := F1 (int_symb s1_p M.a__isNatIListKind) (Vcons x1 Vnil). Definition ha__isNatKind x1 := F1 (hd_symb s1_p M.a__isNatKind) (Vcons x1 Vnil). Definition a__isNatKind x1 := F1 (int_symb s1_p M.a__isNatKind) (Vcons x1 Vnil). Definition ha__isNatList x1 := F1 (hd_symb s1_p M.a__isNatList) (Vcons x1 Vnil). Definition a__isNatList x1 := F1 (int_symb s1_p M.a__isNatList) (Vcons x1 Vnil). Definition ha__length x1 := F1 (hd_symb s1_p M.a__length) (Vcons x1 Vnil). Definition a__length x1 := F1 (int_symb s1_p M.a__length) (Vcons x1 Vnil). Definition ha__zeros := F1 (hd_symb s1_p M.a__zeros) Vnil. Definition a__zeros := F1 (int_symb s1_p M.a__zeros) Vnil. Definition hcons x2 x1 := F1 (hd_symb s1_p M.cons) (Vcons x2 (Vcons x1 Vnil)). Definition cons x2 x1 := F1 (int_symb s1_p M.cons) (Vcons x2 (Vcons x1 Vnil)). Definition hisNat x1 := F1 (hd_symb s1_p M.isNat) (Vcons x1 Vnil). Definition isNat x1 := F1 (int_symb s1_p M.isNat) (Vcons x1 Vnil). Definition hisNatIList x1 := F1 (hd_symb s1_p M.isNatIList) (Vcons x1 Vnil). Definition isNatIList x1 := F1 (int_symb s1_p M.isNatIList) (Vcons x1 Vnil). Definition hisNatIListKind x1 := F1 (hd_symb s1_p M.isNatIListKind) (Vcons x1 Vnil). Definition isNatIListKind x1 := F1 (int_symb s1_p M.isNatIListKind) (Vcons x1 Vnil). Definition hisNatKind x1 := F1 (hd_symb s1_p M.isNatKind) (Vcons x1 Vnil). Definition isNatKind x1 := F1 (int_symb s1_p M.isNatKind) (Vcons x1 Vnil). Definition hisNatList x1 := F1 (hd_symb s1_p M.isNatList) (Vcons x1 Vnil). Definition isNatList x1 := F1 (int_symb s1_p M.isNatList) (Vcons x1 Vnil). Definition hlength x1 := F1 (hd_symb s1_p M.length) (Vcons x1 Vnil). Definition length x1 := F1 (int_symb s1_p M.length) (Vcons x1 Vnil). Definition hmark x1 := F1 (hd_symb s1_p M.mark) (Vcons x1 Vnil). Definition mark x1 := F1 (int_symb s1_p M.mark) (Vcons x1 Vnil). Definition hnil := F1 (hd_symb s1_p M.nil) Vnil. Definition nil := F1 (int_symb s1_p M.nil) Vnil. Definition hs x1 := F1 (hd_symb s1_p M.s) (Vcons x1 Vnil). Definition s x1 := F1 (int_symb s1_p M.s) (Vcons x1 Vnil). Definition htt := F1 (hd_symb s1_p M.tt) Vnil. Definition tt := F1 (int_symb s1_p M.tt) Vnil. Definition hzeros := F1 (hd_symb s1_p M.zeros) Vnil. Definition zeros := F1 (int_symb s1_p M.zeros) Vnil. End S1. (* graph decomposition 1 *) Definition cs1 : list (list (@ATrs.rule s1)) := ( R1 (S1.hmark (S1.U86 (V1 0))) (S1.ha__U86 (S1.mark (V1 0))) :: nil) :: ( R1 (S1.hmark (S1.U71 (V1 0))) (S1.ha__U71 (S1.mark (V1 0))) :: nil) :: ( R1 (S1.hmark (S1.U61 (V1 0))) (S1.ha__U61 (S1.mark (V1 0))) :: nil) :: ( R1 (S1.hmark (S1.U52 (V1 0))) (S1.ha__U52 (S1.mark (V1 0))) :: nil) :: ( R1 (S1.hmark (S1.U46 (V1 0))) (S1.ha__U46 (S1.mark (V1 0))) :: nil) :: ( R1 (S1.hmark (S1.U33 (V1 0))) (S1.ha__U33 (S1.mark (V1 0))) :: nil) :: ( R1 (S1.hmark (S1.U23 (V1 0))) (S1.ha__U23 (S1.mark (V1 0))) :: nil) :: ( R1 (S1.hmark (S1.U13 (V1 0))) (S1.ha__U13 (S1.mark (V1 0))) :: nil) :: ( R1 (S1.hmark (S1.zeros)) (S1.ha__zeros) :: nil) :: ( R1 (S1.ha__isNatKind (S1.s (V1 0))) (S1.ha__U71 (S1.a__isNatKind (V1 0))) :: nil) :: ( R1 (S1.ha__isNatKind (S1.length (V1 0))) (S1.ha__U61 (S1.a__isNatIListKind (V1 0))) :: nil) :: ( R1 (S1.ha__U85 (S1.tt) (V1 0)) (S1.ha__U86 (S1.a__isNatList (V1 0))) :: nil) :: ( R1 (S1.ha__U51 (S1.tt) (V1 0)) (S1.ha__U52 (S1.a__isNatIListKind (V1 0))) :: nil) :: ( R1 (S1.ha__U51 (S1.tt) (V1 0)) (S1.ha__isNatIListKind (V1 0)) :: R1 (S1.ha__isNatIListKind (S1.cons (V1 0) (V1 1))) (S1.ha__U51 (S1.a__isNatKind (V1 0)) (V1 1)) :: R1 (S1.ha__isNatIListKind (S1.cons (V1 0) (V1 1))) (S1.ha__isNatKind (V1 0)) :: R1 (S1.ha__isNatKind (S1.length (V1 0))) (S1.ha__isNatIListKind (V1 0)) :: R1 (S1.ha__isNatKind (S1.s (V1 0))) (S1.ha__isNatKind (V1 0)) :: nil) :: ( R1 (S1.hmark (S1.isNatKind (V1 0))) (S1.ha__isNatKind (V1 0)) :: nil) :: ( R1 (S1.ha__isNatList (S1.cons (V1 0) (V1 1))) (S1.ha__isNatKind (V1 0)) :: nil) :: ( R1 (S1.hmark (S1.isNatIListKind (V1 0))) (S1.ha__isNatIListKind (V1 0)) :: nil) :: ( R1 (S1.ha__isNatIList (S1.cons (V1 0) (V1 1))) (S1.ha__isNatKind (V1 0)) :: nil) :: ( R1 (S1.ha__isNat (S1.s (V1 0))) (S1.ha__isNatKind (V1 0)) :: nil) :: ( R1 (S1.ha__U93 (S1.tt) (V1 0) (V1 1)) (S1.ha__isNatKind (V1 1)) :: nil) :: ( R1 (S1.ha__U81 (S1.tt) (V1 0) (V1 1)) (S1.ha__isNatKind (V1 0)) :: nil) :: ( R1 (S1.ha__isNatIList (V1 0)) (S1.ha__isNatIListKind (V1 0)) :: nil) :: ( R1 (S1.ha__isNat (S1.length (V1 0))) (S1.ha__isNatIListKind (V1 0)) :: nil) :: ( R1 (S1.ha__U91 (S1.tt) (V1 0) (V1 1)) (S1.ha__isNatIListKind (V1 0)) :: nil) :: ( R1 (S1.ha__U83 (S1.tt) (V1 0) (V1 1)) (S1.ha__isNatIListKind (V1 1)) :: nil) :: ( R1 (S1.ha__U82 (S1.tt) (V1 0) (V1 1)) (S1.ha__isNatIListKind (V1 1)) :: nil) :: ( R1 (S1.hmark (S1.U51 (V1 0) (V1 1))) (S1.ha__U51 (S1.mark (V1 0)) (V1 1)) :: nil) :: ( R1 (S1.ha__U45 (S1.tt) (V1 0)) (S1.ha__U46 (S1.a__isNatIList (V1 0))) :: nil) :: ( R1 (S1.ha__U43 (S1.tt) (V1 0) (V1 1)) (S1.ha__isNatIListKind (V1 1)) :: nil) :: ( R1 (S1.ha__U42 (S1.tt) (V1 0) (V1 1)) (S1.ha__isNatIListKind (V1 1)) :: nil) :: ( R1 (S1.ha__U41 (S1.tt) (V1 0) (V1 1)) (S1.ha__isNatKind (V1 0)) :: nil) :: ( R1 (S1.ha__U32 (S1.tt) (V1 0)) (S1.ha__U33 (S1.a__isNatList (V1 0))) :: nil) :: ( R1 (S1.ha__U31 (S1.tt) (V1 0)) (S1.ha__isNatIListKind (V1 0)) :: nil) :: ( R1 (S1.ha__U22 (S1.tt) (V1 0)) (S1.ha__U23 (S1.a__isNat (V1 0))) :: nil) :: ( R1 (S1.ha__U21 (S1.tt) (V1 0)) (S1.ha__isNatKind (V1 0)) :: nil) :: ( R1 (S1.ha__U12 (S1.tt) (V1 0)) (S1.ha__U13 (S1.a__isNatList (V1 0))) :: nil) :: ( R1 (S1.ha__U11 (S1.tt) (V1 0)) (S1.ha__isNatIListKind (V1 0)) :: nil) :: ( R1 (S1.ha__U12 (S1.tt) (V1 0)) (S1.ha__isNatList (V1 0)) :: R1 (S1.ha__isNatList (S1.cons (V1 0) (V1 1))) (S1.ha__U81 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1)) :: R1 (S1.ha__U81 (S1.tt) (V1 0) (V1 1)) (S1.ha__U82 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1)) :: R1 (S1.ha__U82 (S1.tt) (V1 0) (V1 1)) (S1.ha__U83 (S1.a__isNatIListKind (V1 1)) (V1 0) (V1 1)) :: R1 (S1.ha__U83 (S1.tt) (V1 0) (V1 1)) (S1.ha__U84 (S1.a__isNatIListKind (V1 1)) (V1 0) (V1 1)) :: R1 (S1.ha__U84 (S1.tt) (V1 0) (V1 1)) (S1.ha__U85 (S1.a__isNat (V1 0)) (V1 1)) :: R1 (S1.ha__U85 (S1.tt) (V1 0)) (S1.ha__isNatList (V1 0)) :: R1 (S1.ha__U84 (S1.tt) (V1 0) (V1 1)) (S1.ha__isNat (V1 0)) :: R1 (S1.ha__isNat (S1.length (V1 0))) (S1.ha__U11 (S1.a__isNatIListKind (V1 0)) (V1 0)) :: R1 (S1.ha__U11 (S1.tt) (V1 0)) (S1.ha__U12 (S1.a__isNatIListKind (V1 0)) (V1 0)) :: R1 (S1.ha__isNat (S1.s (V1 0))) (S1.ha__U21 (S1.a__isNatKind (V1 0)) (V1 0)) :: R1 (S1.ha__U21 (S1.tt) (V1 0)) (S1.ha__U22 (S1.a__isNatKind (V1 0)) (V1 0)) :: R1 (S1.ha__U22 (S1.tt) (V1 0)) (S1.ha__isNat (V1 0)) :: nil) :: ( R1 (S1.hmark (S1.U11 (V1 0) (V1 1))) (S1.ha__U11 (S1.mark (V1 0)) (V1 1)) :: nil) :: ( R1 (S1.hmark (S1.U22 (V1 0) (V1 1))) (S1.ha__U22 (S1.mark (V1 0)) (V1 1)) :: nil) :: ( R1 (S1.hmark (S1.U21 (V1 0) (V1 1))) (S1.ha__U21 (S1.mark (V1 0)) (V1 1)) :: nil) :: ( R1 (S1.hmark (S1.isNat (V1 0))) (S1.ha__isNat (V1 0)) :: nil) :: ( R1 (S1.ha__U92 (S1.tt) (V1 0) (V1 1)) (S1.ha__isNat (V1 1)) :: nil) :: ( R1 (S1.hmark (S1.U83 (V1 0) (V1 1) (V1 2))) (S1.ha__U83 (S1.mark (V1 0)) (V1 1) (V1 2)) :: nil) :: ( R1 (S1.hmark (S1.U82 (V1 0) (V1 1) (V1 2))) (S1.ha__U82 (S1.mark (V1 0)) (V1 1) (V1 2)) :: nil) :: ( R1 (S1.hmark (S1.U81 (V1 0) (V1 1) (V1 2))) (S1.ha__U81 (S1.mark (V1 0)) (V1 1) (V1 2)) :: nil) :: ( R1 (S1.hmark (S1.isNatList (V1 0))) (S1.ha__isNatList (V1 0)) :: nil) :: ( R1 (S1.ha__length (S1.cons (V1 0) (V1 1))) (S1.ha__isNatList (V1 1)) :: nil) :: ( R1 (S1.hmark (S1.U85 (V1 0) (V1 1))) (S1.ha__U85 (S1.mark (V1 0)) (V1 1)) :: nil) :: ( R1 (S1.hmark (S1.U84 (V1 0) (V1 1) (V1 2))) (S1.ha__U84 (S1.mark (V1 0)) (V1 1) (V1 2)) :: nil) :: ( R1 (S1.ha__U32 (S1.tt) (V1 0)) (S1.ha__isNatList (V1 0)) :: nil) :: ( R1 (S1.hmark (S1.U32 (V1 0) (V1 1))) (S1.ha__U32 (S1.mark (V1 0)) (V1 1)) :: nil) :: ( R1 (S1.ha__U31 (S1.tt) (V1 0)) (S1.ha__U32 (S1.a__isNatIListKind (V1 0)) (V1 0)) :: nil) :: ( R1 (S1.hmark (S1.U31 (V1 0) (V1 1))) (S1.ha__U31 (S1.mark (V1 0)) (V1 1)) :: nil) :: ( R1 (S1.ha__isNatIList (V1 0)) (S1.ha__U31 (S1.a__isNatIListKind (V1 0)) (V1 0)) :: nil) :: ( R1 (S1.hmark (S1.U12 (V1 0) (V1 1))) (S1.ha__U12 (S1.mark (V1 0)) (V1 1)) :: nil) :: ( R1 (S1.ha__U44 (S1.tt) (V1 0) (V1 1)) (S1.ha__isNat (V1 0)) :: nil) :: ( R1 (S1.ha__U44 (S1.tt) (V1 0) (V1 1)) (S1.ha__U45 (S1.a__isNat (V1 0)) (V1 1)) :: R1 (S1.ha__U45 (S1.tt) (V1 0)) (S1.ha__isNatIList (V1 0)) :: R1 (S1.ha__isNatIList (S1.cons (V1 0) (V1 1))) (S1.ha__U41 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1)) :: R1 (S1.ha__U41 (S1.tt) (V1 0) (V1 1)) (S1.ha__U42 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1)) :: R1 (S1.ha__U42 (S1.tt) (V1 0) (V1 1)) (S1.ha__U43 (S1.a__isNatIListKind (V1 1)) (V1 0) (V1 1)) :: R1 (S1.ha__U43 (S1.tt) (V1 0) (V1 1)) (S1.ha__U44 (S1.a__isNatIListKind (V1 1)) (V1 0) (V1 1)) :: nil) :: ( R1 (S1.hmark (S1.U43 (V1 0) (V1 1) (V1 2))) (S1.ha__U43 (S1.mark (V1 0)) (V1 1) (V1 2)) :: nil) :: ( R1 (S1.hmark (S1.U42 (V1 0) (V1 1) (V1 2))) (S1.ha__U42 (S1.mark (V1 0)) (V1 1) (V1 2)) :: nil) :: ( R1 (S1.hmark (S1.U41 (V1 0) (V1 1) (V1 2))) (S1.ha__U41 (S1.mark (V1 0)) (V1 1) (V1 2)) :: nil) :: ( R1 (S1.hmark (S1.isNatIList (V1 0))) (S1.ha__isNatIList (V1 0)) :: nil) :: ( R1 (S1.hmark (S1.U45 (V1 0) (V1 1))) (S1.ha__U45 (S1.mark (V1 0)) (V1 1)) :: nil) :: ( R1 (S1.hmark (S1.U44 (V1 0) (V1 1) (V1 2))) (S1.ha__U44 (S1.mark (V1 0)) (V1 1) (V1 2)) :: nil) :: ( R1 (S1.hmark (S1.U11 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U12 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U13 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U21 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U22 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U23 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U31 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U32 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U33 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U41 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U42 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U43 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U44 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U45 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U46 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U51 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U52 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U61 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U71 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U81 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U82 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U83 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U84 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U85 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U86 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U91 (V1 0) (V1 1) (V1 2))) (S1.ha__U91 (S1.mark (V1 0)) (V1 1) (V1 2)) :: R1 (S1.ha__U91 (S1.tt) (V1 0) (V1 1)) (S1.ha__U92 (S1.a__isNatIListKind (V1 0)) (V1 0) (V1 1)) :: R1 (S1.ha__U92 (S1.tt) (V1 0) (V1 1)) (S1.ha__U93 (S1.a__isNat (V1 1)) (V1 0) (V1 1)) :: R1 (S1.ha__U93 (S1.tt) (V1 0) (V1 1)) (S1.ha__U94 (S1.a__isNatKind (V1 1)) (V1 0)) :: R1 (S1.ha__U94 (S1.tt) (V1 0)) (S1.ha__length (S1.mark (V1 0))) :: R1 (S1.ha__length (S1.cons (V1 0) (V1 1))) (S1.ha__U91 (S1.a__isNatList (V1 1)) (V1 1) (V1 0)) :: R1 (S1.ha__U94 (S1.tt) (V1 0)) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U91 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U92 (V1 0) (V1 1) (V1 2))) (S1.ha__U92 (S1.mark (V1 0)) (V1 1) (V1 2)) :: R1 (S1.hmark (S1.U92 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U93 (V1 0) (V1 1) (V1 2))) (S1.ha__U93 (S1.mark (V1 0)) (V1 1) (V1 2)) :: R1 (S1.hmark (S1.U93 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U94 (V1 0) (V1 1))) (S1.ha__U94 (S1.mark (V1 0)) (V1 1)) :: R1 (S1.hmark (S1.U94 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.length (V1 0))) (S1.ha__length (S1.mark (V1 0))) :: R1 (S1.hmark (S1.length (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.cons (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.s (V1 0))) (S1.hmark (V1 0)) :: nil) :: nil. (* polynomial interpretation 1 *) Module PIS1 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => nil | (hd_symb M.a__isNatIListKind) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => nil | (hd_symb M.a__isNatKind) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => nil | (hd_symb M.a__U51) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U51) => nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => nil | (hd_symb M.U11) => nil | (int_symb M.U11) => nil | (hd_symb M.U12) => nil | (int_symb M.U12) => nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => nil | (hd_symb M.U21) => nil | (int_symb M.U21) => nil | (hd_symb M.U22) => nil | (int_symb M.U22) => nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => nil | (hd_symb M.U32) => nil | (int_symb M.U32) => nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => nil | (hd_symb M.U42) => nil | (int_symb M.U42) => nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U44) => nil | (int_symb M.U44) => nil | (hd_symb M.U45) => nil | (int_symb M.U45) => nil | (hd_symb M.U46) => nil | (int_symb M.U46) => nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => nil | (hd_symb M.U51) => nil | (int_symb M.U51) => nil | (hd_symb M.U52) => nil | (int_symb M.U52) => nil | (hd_symb M.U61) => nil | (int_symb M.U61) => nil | (hd_symb M.U71) => nil | (int_symb M.U71) => nil | (hd_symb M.U81) => nil | (int_symb M.U81) => nil | (hd_symb M.U82) => nil | (int_symb M.U82) => nil | (hd_symb M.U83) => nil | (int_symb M.U83) => nil | (hd_symb M.U84) => nil | (int_symb M.U84) => nil | (hd_symb M.U85) => nil | (int_symb M.U85) => nil | (hd_symb M.U86) => nil | (int_symb M.U86) => nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS1. Module PI1 := PolyInt PIS1. (* graph decomposition 2 *) Definition cs2 : list (list (@ATrs.rule s1)) := ( R1 (S1.ha__isNatKind (S1.s (V1 0))) (S1.ha__isNatKind (V1 0)) :: nil) :: ( R1 (S1.ha__isNatIListKind (S1.cons (V1 0) (V1 1))) (S1.ha__U51 (S1.a__isNatKind (V1 0)) (V1 1)) :: R1 (S1.ha__U51 (S1.tt) (V1 0)) (S1.ha__isNatIListKind (V1 0)) :: nil) :: nil. (* polynomial interpretation 2 *) Module PIS2 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => (1%Z, Vnil) :: nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatKind) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.a__isNatKind) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (1%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS2. Module PI2 := PolyInt PIS2. (* polynomial interpretation 3 *) Module PIS3 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => (3%Z, Vnil) :: nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => (2%Z, Vnil) :: nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => nil | (hd_symb M.a__isNatIListKind) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U51) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U51) => nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => nil | (hd_symb M.s) => nil | (int_symb M.s) => nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => nil | (hd_symb M.mark) => nil | (int_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (1%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => nil | (hd_symb M.U21) => nil | (int_symb M.U21) => nil | (hd_symb M.U22) => nil | (int_symb M.U22) => nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => nil | (hd_symb M.U32) => nil | (int_symb M.U32) => nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => nil | (hd_symb M.U42) => nil | (int_symb M.U42) => nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U44) => nil | (int_symb M.U44) => nil | (hd_symb M.U45) => nil | (int_symb M.U45) => nil | (hd_symb M.U46) => nil | (int_symb M.U46) => nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => nil | (hd_symb M.U52) => nil | (int_symb M.U52) => nil | (hd_symb M.U61) => nil | (int_symb M.U61) => nil | (hd_symb M.U71) => nil | (int_symb M.U71) => nil | (hd_symb M.U81) => nil | (int_symb M.U81) => nil | (hd_symb M.U82) => nil | (int_symb M.U82) => nil | (hd_symb M.U83) => nil | (int_symb M.U83) => nil | (hd_symb M.U84) => nil | (int_symb M.U84) => nil | (hd_symb M.U85) => nil | (int_symb M.U85) => nil | (hd_symb M.U86) => nil | (int_symb M.U86) => nil | (hd_symb M.U91) => nil | (int_symb M.U91) => nil | (hd_symb M.U92) => nil | (int_symb M.U92) => nil | (hd_symb M.U93) => nil | (int_symb M.U93) => nil | (hd_symb M.U94) => nil | (int_symb M.U94) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS3. Module PI3 := PolyInt PIS3. (* graph decomposition 3 *) Definition cs3 : list (list (@ATrs.rule s1)) := ( R1 (S1.ha__U51 (S1.tt) (V1 0)) (S1.ha__isNatIListKind (V1 0)) :: nil) :: nil. (* polynomial interpretation 4 *) Module PIS4 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U11) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U12) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => nil | (hd_symb M.a__isNatList) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.a__isNatList) => nil | (hd_symb M.a__U21) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U21) => nil | (hd_symb M.a__U22) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U22) => nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => nil | (hd_symb M.a__isNat) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.a__isNat) => nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.a__U81) => nil | (hd_symb M.a__U82) => (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.a__U82) => nil | (hd_symb M.a__U83) => (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.a__U83) => nil | (hd_symb M.a__U84) => (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.a__U84) => nil | (hd_symb M.a__U85) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U85) => nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => nil | (hd_symb M.U21) => nil | (int_symb M.U21) => nil | (hd_symb M.U22) => nil | (int_symb M.U22) => nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => nil | (hd_symb M.U32) => nil | (int_symb M.U32) => nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => nil | (hd_symb M.U42) => nil | (int_symb M.U42) => nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U44) => nil | (int_symb M.U44) => nil | (hd_symb M.U45) => nil | (int_symb M.U45) => nil | (hd_symb M.U46) => nil | (int_symb M.U46) => nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => nil | (hd_symb M.U51) => nil | (int_symb M.U51) => nil | (hd_symb M.U52) => nil | (int_symb M.U52) => nil | (hd_symb M.U61) => nil | (int_symb M.U61) => nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => nil | (hd_symb M.U82) => nil | (int_symb M.U82) => nil | (hd_symb M.U83) => nil | (int_symb M.U83) => nil | (hd_symb M.U84) => nil | (int_symb M.U84) => nil | (hd_symb M.U85) => nil | (int_symb M.U85) => nil | (hd_symb M.U86) => nil | (int_symb M.U86) => nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS4. Module PI4 := PolyInt PIS4. (* graph decomposition 4 *) Definition cs4 : list (list (@ATrs.rule s1)) := ( R1 (S1.ha__U11 (S1.tt) (V1 0)) (S1.ha__U12 (S1.a__isNatIListKind (V1 0)) (V1 0)) :: nil) :: ( R1 (S1.ha__isNat (S1.length (V1 0))) (S1.ha__U11 (S1.a__isNatIListKind (V1 0)) (V1 0)) :: nil) :: ( R1 (S1.ha__isNat (S1.s (V1 0))) (S1.ha__U21 (S1.a__isNatKind (V1 0)) (V1 0)) :: R1 (S1.ha__U21 (S1.tt) (V1 0)) (S1.ha__U22 (S1.a__isNatKind (V1 0)) (V1 0)) :: R1 (S1.ha__U22 (S1.tt) (V1 0)) (S1.ha__isNat (V1 0)) :: nil) :: ( R1 (S1.ha__U84 (S1.tt) (V1 0) (V1 1)) (S1.ha__isNat (V1 0)) :: nil) :: ( R1 (S1.ha__U81 (S1.tt) (V1 0) (V1 1)) (S1.ha__U82 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1)) :: R1 (S1.ha__U82 (S1.tt) (V1 0) (V1 1)) (S1.ha__U83 (S1.a__isNatIListKind (V1 1)) (V1 0) (V1 1)) :: R1 (S1.ha__U83 (S1.tt) (V1 0) (V1 1)) (S1.ha__U84 (S1.a__isNatIListKind (V1 1)) (V1 0) (V1 1)) :: R1 (S1.ha__U84 (S1.tt) (V1 0) (V1 1)) (S1.ha__U85 (S1.a__isNat (V1 0)) (V1 1)) :: R1 (S1.ha__U85 (S1.tt) (V1 0)) (S1.ha__isNatList (V1 0)) :: R1 (S1.ha__isNatList (S1.cons (V1 0) (V1 1))) (S1.ha__U81 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1)) :: nil) :: nil. (* polynomial interpretation 5 *) Module PIS5 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => (1%Z, Vnil) :: nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => (1%Z, Vnil) :: nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U21) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U21) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U22) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U22) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNat) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.a__isNat) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (1%Z, (Vcons 0 Vnil)) :: (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (2%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (1%Z, (Vcons 0 Vnil)) :: (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS5. Module PI5 := PolyInt PIS5. (* graph decomposition 5 *) Definition cs5 : list (list (@ATrs.rule s1)) := ( R1 (S1.ha__U22 (S1.tt) (V1 0)) (S1.ha__isNat (V1 0)) :: nil) :: nil. (* polynomial interpretation 6 *) Module PIS6 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => (2%Z, Vnil) :: nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => (1%Z, Vnil) :: nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => nil | (hd_symb M.a__isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.a__isNatList) => nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (1%Z, (Vcons 0 Vnil)) :: (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U81) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.a__U81) => nil | (hd_symb M.a__U82) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.a__U82) => nil | (hd_symb M.a__U83) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.a__U83) => nil | (hd_symb M.a__U84) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.a__U84) => nil | (hd_symb M.a__U85) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U85) => nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => nil | (hd_symb M.s) => nil | (int_symb M.s) => nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => nil | (hd_symb M.mark) => nil | (int_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (2%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => nil | (hd_symb M.U12) => nil | (int_symb M.U12) => nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => nil | (hd_symb M.U21) => nil | (int_symb M.U21) => nil | (hd_symb M.U22) => nil | (int_symb M.U22) => nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => nil | (hd_symb M.U46) => nil | (int_symb M.U46) => nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => nil | (hd_symb M.U52) => nil | (int_symb M.U52) => nil | (hd_symb M.U61) => nil | (int_symb M.U61) => nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => nil | (hd_symb M.U82) => nil | (int_symb M.U82) => nil | (hd_symb M.U83) => nil | (int_symb M.U83) => nil | (hd_symb M.U84) => nil | (int_symb M.U84) => nil | (hd_symb M.U85) => nil | (int_symb M.U85) => nil | (hd_symb M.U86) => nil | (int_symb M.U86) => nil | (hd_symb M.U91) => nil | (int_symb M.U91) => nil | (hd_symb M.U92) => nil | (int_symb M.U92) => nil | (hd_symb M.U93) => nil | (int_symb M.U93) => nil | (hd_symb M.U94) => nil | (int_symb M.U94) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS6. Module PI6 := PolyInt PIS6. (* graph decomposition 6 *) Definition cs6 : list (list (@ATrs.rule s1)) := ( R1 (S1.ha__U85 (S1.tt) (V1 0)) (S1.ha__isNatList (V1 0)) :: nil) :: ( R1 (S1.ha__U84 (S1.tt) (V1 0) (V1 1)) (S1.ha__U85 (S1.a__isNat (V1 0)) (V1 1)) :: nil) :: ( R1 (S1.ha__U83 (S1.tt) (V1 0) (V1 1)) (S1.ha__U84 (S1.a__isNatIListKind (V1 1)) (V1 0) (V1 1)) :: nil) :: ( R1 (S1.ha__U82 (S1.tt) (V1 0) (V1 1)) (S1.ha__U83 (S1.a__isNatIListKind (V1 1)) (V1 0) (V1 1)) :: nil) :: ( R1 (S1.ha__U81 (S1.tt) (V1 0) (V1 1)) (S1.ha__U82 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1)) :: nil) :: nil. (* polynomial interpretation 7 *) Module PIS7 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => (3%Z, Vnil) :: nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => (2%Z, Vnil) :: nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.a__U41) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.a__U42) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.a__U43) => nil | (hd_symb M.a__U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.a__U44) => nil | (hd_symb M.a__U45) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U45) => nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => nil | (hd_symb M.a__isNatIList) => (1%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.a__isNatIList) => (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => nil | (hd_symb M.s) => nil | (int_symb M.s) => nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => nil | (hd_symb M.mark) => nil | (int_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => nil | (hd_symb M.nil) => nil | (int_symb M.nil) => nil | (hd_symb M.U11) => nil | (int_symb M.U11) => nil | (hd_symb M.U12) => nil | (int_symb M.U12) => nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => nil | (hd_symb M.U21) => nil | (int_symb M.U21) => nil | (hd_symb M.U22) => nil | (int_symb M.U22) => nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U44) => nil | (int_symb M.U44) => nil | (hd_symb M.U45) => nil | (int_symb M.U45) => nil | (hd_symb M.U46) => nil | (int_symb M.U46) => nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => nil | (hd_symb M.U52) => nil | (int_symb M.U52) => nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => nil | (hd_symb M.U81) => nil | (int_symb M.U81) => nil | (hd_symb M.U82) => nil | (int_symb M.U82) => nil | (hd_symb M.U83) => nil | (int_symb M.U83) => nil | (hd_symb M.U84) => nil | (int_symb M.U84) => nil | (hd_symb M.U85) => nil | (int_symb M.U85) => nil | (hd_symb M.U86) => nil | (int_symb M.U86) => nil | (hd_symb M.U91) => nil | (int_symb M.U91) => nil | (hd_symb M.U92) => nil | (int_symb M.U92) => nil | (hd_symb M.U93) => nil | (int_symb M.U93) => nil | (hd_symb M.U94) => nil | (int_symb M.U94) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS7. Module PI7 := PolyInt PIS7. (* graph decomposition 7 *) Definition cs7 : list (list (@ATrs.rule s1)) := ( R1 (S1.ha__isNatIList (S1.cons (V1 0) (V1 1))) (S1.ha__U41 (S1.a__isNatKind (V1 0)) (V1 0) (V1 1)) :: nil) :: ( R1 (S1.ha__U45 (S1.tt) (V1 0)) (S1.ha__isNatIList (V1 0)) :: nil) :: ( R1 (S1.ha__U44 (S1.tt) (V1 0) (V1 1)) (S1.ha__U45 (S1.a__isNat (V1 0)) (V1 1)) :: nil) :: ( R1 (S1.ha__U43 (S1.tt) (V1 0) (V1 1)) (S1.ha__U44 (S1.a__isNatIListKind (V1 1)) (V1 0) (V1 1)) :: nil) :: ( R1 (S1.ha__U42 (S1.tt) (V1 0) (V1 1)) (S1.ha__U43 (S1.a__isNatIListKind (V1 1)) (V1 0) (V1 1)) :: nil) :: nil. (* polynomial interpretation 8 *) Module PIS8 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (int_symb M.a__U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U92) => (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (int_symb M.a__U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (int_symb M.a__U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => (1%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.a__length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (2%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS8. Module PI8 := PolyInt PIS8. (* graph decomposition 8 *) Definition cs8 : list (list (@ATrs.rule s1)) := ( R1 (S1.hmark (S1.U12 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U11 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U13 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U21 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U22 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U23 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U31 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U32 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U33 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U41 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U42 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U43 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U44 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U45 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U46 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U51 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U52 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U61 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U71 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U81 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U82 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U83 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U84 (V1 0) (V1 1) (V1 2))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U85 (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.U86 (V1 0))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.cons (V1 0) (V1 1))) (S1.hmark (V1 0)) :: R1 (S1.hmark (S1.s (V1 0))) (S1.hmark (V1 0)) :: nil) :: ( R1 (S1.ha__U94 (S1.tt) (V1 0)) (S1.hmark (V1 0)) :: nil) :: ( R1 (S1.ha__U94 (S1.tt) (V1 0)) (S1.ha__length (S1.mark (V1 0))) :: R1 (S1.ha__length (S1.cons (V1 0) (V1 1))) (S1.ha__U91 (S1.a__isNatList (V1 1)) (V1 1) (V1 0)) :: R1 (S1.ha__U91 (S1.tt) (V1 0) (V1 1)) (S1.ha__U92 (S1.a__isNatIListKind (V1 0)) (V1 0) (V1 1)) :: R1 (S1.ha__U92 (S1.tt) (V1 0) (V1 1)) (S1.ha__U93 (S1.a__isNat (V1 1)) (V1 0) (V1 1)) :: R1 (S1.ha__U93 (S1.tt) (V1 0) (V1 1)) (S1.ha__U94 (S1.a__isNatKind (V1 1)) (V1 0)) :: nil) :: nil. (* polynomial interpretation 9 *) Module PIS9 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => (1%Z, Vnil) :: nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => (1%Z, Vnil) :: nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => nil | (hd_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (2%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => nil | (hd_symb M.U92) => nil | (int_symb M.U92) => nil | (hd_symb M.U93) => nil | (int_symb M.U93) => nil | (hd_symb M.U94) => nil | (int_symb M.U94) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS9. Module PI9 := PolyInt PIS9. (* polynomial interpretation 10 *) Module PIS10 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => nil | (hd_symb M.s) => nil | (int_symb M.s) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => nil | (hd_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (2%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => nil | (hd_symb M.U92) => nil | (int_symb M.U92) => nil | (hd_symb M.U93) => nil | (int_symb M.U93) => nil | (hd_symb M.U94) => nil | (int_symb M.U94) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS10. Module PI10 := PolyInt PIS10. (* polynomial interpretation 11 *) Module PIS11 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => (1%Z, Vnil) :: nil | (hd_symb M.cons) => nil | (int_symb M.cons) => nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => (1%Z, Vnil) :: nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => nil | (hd_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => nil | (hd_symb M.nil) => nil | (int_symb M.nil) => nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => nil | (hd_symb M.U92) => nil | (int_symb M.U92) => nil | (hd_symb M.U93) => nil | (int_symb M.U93) => nil | (hd_symb M.U94) => nil | (int_symb M.U94) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS11. Module PI11 := PolyInt PIS11. (* polynomial interpretation 12 *) Module PIS12 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => nil | (hd_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => nil | (hd_symb M.nil) => nil | (int_symb M.nil) => nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => nil | (hd_symb M.U92) => nil | (int_symb M.U92) => nil | (hd_symb M.U93) => nil | (int_symb M.U93) => nil | (hd_symb M.U94) => nil | (int_symb M.U94) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS12. Module PI12 := PolyInt PIS12. (* polynomial interpretation 13 *) Module PIS13 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => (1%Z, Vnil) :: nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => nil | (hd_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => nil | (hd_symb M.nil) => nil | (int_symb M.nil) => nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => nil | (hd_symb M.U94) => nil | (int_symb M.U94) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS13. Module PI13 := PolyInt PIS13. (* polynomial interpretation 14 *) Module PIS14 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => nil | (hd_symb M.U32) => nil | (int_symb M.U32) => nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U44) => nil | (int_symb M.U44) => nil | (hd_symb M.U45) => nil | (int_symb M.U45) => nil | (hd_symb M.U46) => nil | (int_symb M.U46) => nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS14. Module PI14 := PolyInt PIS14. (* polynomial interpretation 15 *) Module PIS15 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => nil | (hd_symb M.U32) => nil | (int_symb M.U32) => nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => nil | (hd_symb M.U42) => nil | (int_symb M.U42) => nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U44) => nil | (int_symb M.U44) => nil | (hd_symb M.U45) => nil | (int_symb M.U45) => nil | (hd_symb M.U46) => nil | (int_symb M.U46) => nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS15. Module PI15 := PolyInt PIS15. (* polynomial interpretation 16 *) Module PIS16 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => nil | (hd_symb M.U32) => nil | (int_symb M.U32) => nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => nil | (hd_symb M.U45) => nil | (int_symb M.U45) => nil | (hd_symb M.U46) => nil | (int_symb M.U46) => nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS16. Module PI16 := PolyInt PIS16. (* polynomial interpretation 17 *) Module PIS17 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => (2%Z, Vnil) :: nil | (hd_symb M.cons) => nil | (int_symb M.cons) => nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => (1%Z, Vnil) :: nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (3%Z, (Vcons 0 (Vcons 0 Vnil))) :: (3%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (3%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (3%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => nil | (hd_symb M.s) => nil | (int_symb M.s) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => nil | (hd_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (2%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (3%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => nil | (hd_symb M.U92) => nil | (int_symb M.U92) => nil | (hd_symb M.U93) => nil | (int_symb M.U93) => nil | (hd_symb M.U94) => nil | (int_symb M.U94) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS17. Module PI17 := PolyInt PIS17. (* polynomial interpretation 18 *) Module PIS18 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => (2%Z, Vnil) :: nil | (hd_symb M.cons) => nil | (int_symb M.cons) => nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => (2%Z, Vnil) :: nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (3%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => nil | (hd_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (1%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => nil | (hd_symb M.U32) => nil | (int_symb M.U32) => nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => nil | (hd_symb M.U42) => nil | (int_symb M.U42) => nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U44) => nil | (int_symb M.U44) => nil | (hd_symb M.U45) => nil | (int_symb M.U45) => nil | (hd_symb M.U46) => nil | (int_symb M.U46) => nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (3%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => nil | (hd_symb M.U82) => nil | (int_symb M.U82) => nil | (hd_symb M.U83) => nil | (int_symb M.U83) => nil | (hd_symb M.U84) => nil | (int_symb M.U84) => nil | (hd_symb M.U85) => nil | (int_symb M.U85) => nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => nil | (hd_symb M.U92) => nil | (int_symb M.U92) => nil | (hd_symb M.U93) => nil | (int_symb M.U93) => nil | (hd_symb M.U94) => nil | (int_symb M.U94) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS18. Module PI18 := PolyInt PIS18. (* polynomial interpretation 19 *) Module PIS19 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => (2%Z, Vnil) :: nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => (1%Z, Vnil) :: nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => nil | (hd_symb M.s) => nil | (int_symb M.s) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => nil | (hd_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => nil | (hd_symb M.nil) => nil | (int_symb M.nil) => nil | (hd_symb M.U11) => nil | (int_symb M.U11) => nil | (hd_symb M.U12) => nil | (int_symb M.U12) => nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => nil | (hd_symb M.U13) => nil | (int_symb M.U13) => nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => nil | (hd_symb M.U22) => nil | (int_symb M.U22) => nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => nil | (hd_symb M.U32) => nil | (int_symb M.U32) => nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => nil | (hd_symb M.U42) => nil | (int_symb M.U42) => nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U44) => nil | (int_symb M.U44) => nil | (hd_symb M.U45) => nil | (int_symb M.U45) => nil | (hd_symb M.U46) => nil | (int_symb M.U46) => nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => nil | (hd_symb M.U92) => nil | (int_symb M.U92) => nil | (hd_symb M.U93) => nil | (int_symb M.U93) => nil | (hd_symb M.U94) => nil | (int_symb M.U94) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS19. Module PI19 := PolyInt PIS19. (* polynomial interpretation 20 *) Module PIS20 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (2%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => nil | (hd_symb M.U11) => nil | (int_symb M.U11) => nil | (hd_symb M.U12) => nil | (int_symb M.U12) => nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U13) => nil | (int_symb M.U13) => nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => nil | (hd_symb M.U22) => nil | (int_symb M.U22) => nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U44) => nil | (int_symb M.U44) => nil | (hd_symb M.U45) => nil | (int_symb M.U45) => nil | (hd_symb M.U46) => nil | (int_symb M.U46) => nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => nil | (hd_symb M.U85) => nil | (int_symb M.U85) => nil | (hd_symb M.U86) => nil | (int_symb M.U86) => nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS20. Module PI20 := PolyInt PIS20. (* polynomial interpretation 21 *) Module PIS21 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => (2%Z, Vnil) :: nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => (1%Z, Vnil) :: nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => nil | (hd_symb M.s) => nil | (int_symb M.s) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => nil | (hd_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => nil | (hd_symb M.nil) => nil | (int_symb M.nil) => nil | (hd_symb M.U11) => nil | (int_symb M.U11) => nil | (hd_symb M.U12) => nil | (int_symb M.U12) => nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U13) => nil | (int_symb M.U13) => nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => nil | (hd_symb M.U22) => nil | (int_symb M.U22) => nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U44) => nil | (int_symb M.U44) => nil | (hd_symb M.U45) => nil | (int_symb M.U45) => nil | (hd_symb M.U46) => nil | (int_symb M.U46) => nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => nil | (hd_symb M.U84) => nil | (int_symb M.U84) => nil | (hd_symb M.U85) => nil | (int_symb M.U85) => nil | (hd_symb M.U86) => nil | (int_symb M.U86) => nil | (hd_symb M.U91) => nil | (int_symb M.U91) => nil | (hd_symb M.U92) => nil | (int_symb M.U92) => nil | (hd_symb M.U93) => nil | (int_symb M.U93) => nil | (hd_symb M.U94) => nil | (int_symb M.U94) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS21. Module PI21 := PolyInt PIS21. (* polynomial interpretation 22 *) Module PIS22 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => (1%Z, Vnil) :: nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => (2%Z, Vnil) :: nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (2%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => (3%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (2%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => (3%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (1%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (2%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (3%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS22. Module PI22 := PolyInt PIS22. (* polynomial interpretation 23 *) Module PIS23 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => (1%Z, Vnil) :: nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => (1%Z, Vnil) :: nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (1%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS23. Module PI23 := PolyInt PIS23. (* polynomial interpretation 24 *) Module PIS24 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => (2%Z, Vnil) :: nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => (1%Z, Vnil) :: nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => nil | (int_symb M.a__U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U92) => nil | (int_symb M.a__U92) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => nil | (int_symb M.a__U93) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => nil | (int_symb M.a__U94) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__length) => nil | (int_symb M.a__length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (1%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS24. Module PI24 := PolyInt PIS24. (* polynomial interpretation 25 *) Module PIS25 (*<: TPolyInt*). Definition sig := s1. Definition trsInt f := match f as f return poly (@ASignature.arity s1 f) with | (hd_symb M.a__zeros) => nil | (int_symb M.a__zeros) => nil | (hd_symb M.cons) => nil | (int_symb M.cons) => (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M._0_1) => nil | (int_symb M._0_1) => (2%Z, Vnil) :: nil | (hd_symb M.zeros) => nil | (int_symb M.zeros) => nil | (hd_symb M.a__U11) => nil | (int_symb M.a__U11) => (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => (2%Z, Vnil) :: nil | (hd_symb M.a__U12) => nil | (int_symb M.a__U12) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__isNatIListKind) => nil | (int_symb M.a__isNatIListKind) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U13) => nil | (int_symb M.a__U13) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__isNatList) => nil | (int_symb M.a__isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U21) => nil | (int_symb M.a__U21) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U22) => nil | (int_symb M.a__U22) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__isNatKind) => nil | (int_symb M.a__isNatKind) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U23) => nil | (int_symb M.a__U23) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNat) => nil | (int_symb M.a__isNat) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U31) => nil | (int_symb M.a__U31) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U32) => nil | (int_symb M.a__U32) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U33) => nil | (int_symb M.a__U33) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U41) => nil | (int_symb M.a__U41) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U42) => nil | (int_symb M.a__U42) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U43) => nil | (int_symb M.a__U43) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U44) => nil | (int_symb M.a__U44) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U45) => nil | (int_symb M.a__U45) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U46) => nil | (int_symb M.a__U46) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__isNatIList) => nil | (int_symb M.a__isNatIList) => (2%Z, (Vcons 0 Vnil)) :: (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U51) => nil | (int_symb M.a__U51) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.a__U52) => nil | (int_symb M.a__U52) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U61) => nil | (int_symb M.a__U61) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U71) => nil | (int_symb M.a__U71) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__U81) => nil | (int_symb M.a__U81) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U82) => nil | (int_symb M.a__U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U83) => nil | (int_symb M.a__U83) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U84) => nil | (int_symb M.a__U84) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.a__U85) => nil | (int_symb M.a__U85) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.a__U86) => nil | (int_symb M.a__U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.a__U91) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (int_symb M.a__U91) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U92) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (int_symb M.a__U92) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U93) => (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (int_symb M.a__U93) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.a__U94) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.a__U94) => (3%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.s) => nil | (int_symb M.s) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.a__length) => (1%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.a__length) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.mark) => nil | (int_symb M.mark) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.length) => nil | (int_symb M.length) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (2%Z, Vnil) :: nil | (hd_symb M.U11) => nil | (int_symb M.U11) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isNatIListKind) => nil | (int_symb M.isNatIListKind) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U13) => nil | (int_symb M.U13) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.isNatList) => nil | (int_symb M.isNatList) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.isNatKind) => nil | (int_symb M.isNatKind) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNat) => nil | (int_symb M.isNat) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U31) => nil | (int_symb M.U31) => (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U32) => nil | (int_symb M.U32) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U33) => nil | (int_symb M.U33) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U44) => nil | (int_symb M.U44) => (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U45) => nil | (int_symb M.U45) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U46) => nil | (int_symb M.U46) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNatIList) => nil | (int_symb M.isNatIList) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U81) => nil | (int_symb M.U81) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U82) => nil | (int_symb M.U82) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U83) => nil | (int_symb M.U83) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U84) => nil | (int_symb M.U84) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U85) => nil | (int_symb M.U85) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U86) => nil | (int_symb M.U86) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U91) => nil | (int_symb M.U91) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U92) => nil | (int_symb M.U92) => (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U93) => nil | (int_symb M.U93) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U94) => nil | (int_symb M.U94) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS25. Module PI25 := PolyInt PIS25. (* graph decomposition 9 *) Definition cs9 : list (list (@ATrs.rule s1)) := ( R1 (S1.ha__length (S1.cons (V1 0) (V1 1))) (S1.ha__U91 (S1.a__isNatList (V1 1)) (V1 1) (V1 0)) :: nil) :: ( R1 (S1.ha__U94 (S1.tt) (V1 0)) (S1.ha__length (S1.mark (V1 0))) :: nil) :: ( R1 (S1.ha__U93 (S1.tt) (V1 0) (V1 1)) (S1.ha__U94 (S1.a__isNatKind (V1 1)) (V1 0)) :: nil) :: nil. (* termination proof *) Lemma termination : WF rel. Proof. unfold rel. dp_trans. mark. let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R; graph_decomp (dpg_unif_N 100 R D) cs1; subst D; subst R. dpg_unif_N_correct. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. right. PI1.prove_termination. let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R; graph_decomp (dpg_unif_N 100 R D) cs2; subst D; subst R. dpg_unif_N_correct. right. PI2.prove_termination. termination_trivial. right. PI3.prove_termination. let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R; graph_decomp (dpg_unif_N 100 R D) cs3; subst D; subst R. dpg_unif_N_correct. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. right. PI4.prove_termination. let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R; graph_decomp (dpg_unif_N 100 R D) cs4; subst D; subst R. dpg_unif_N_correct. left. co_scc. left. co_scc. right. PI5.prove_termination. let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R; graph_decomp (dpg_unif_N 100 R D) cs5; subst D; subst R. dpg_unif_N_correct. left. co_scc. left. co_scc. right. PI6.prove_termination. let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R; graph_decomp (dpg_unif_N 100 R D) cs6; subst D; subst R. dpg_unif_N_correct. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. right. PI7.prove_termination. let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R; graph_decomp (dpg_unif_N 100 R D) cs7; subst D; subst R. dpg_unif_N_correct. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. left. co_scc. right. PI8.prove_termination. let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R; graph_decomp (dpg_unif_N 100 R D) cs8; subst D; subst R. dpg_unif_N_correct. right. PI9.prove_termination. PI10.prove_termination. PI11.prove_termination. PI12.prove_termination. PI13.prove_termination. PI14.prove_termination. PI15.prove_termination. PI16.prove_termination. PI17.prove_termination. PI18.prove_termination. PI19.prove_termination. PI20.prove_termination. PI21.prove_termination. PI22.prove_termination. PI23.prove_termination. PI24.prove_termination. termination_trivial. left. co_scc. right. PI25.prove_termination. let D := fresh "D" in let R := fresh "R" in set_rules_to D; set_mod_rules_to R; graph_decomp (dpg_unif_N 100 R D) cs9; subst D; subst R. dpg_unif_N_correct. left. co_scc. left. co_scc. left. co_scc. Qed.