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 | U21 : symb | U22 : symb | U23 : symb | U31 : symb | U32 : symb | U41 : symb | U42 : symb | U43 : symb | U51 : symb | U52 : symb | U53 : symb | U61 : symb | U62 : symb | U71 : symb | U72 : symb | __ : symb | a : symb | activate : symb | and : symb | e : symb | i : symb | isList : symb | isNeList : symb | isNePal : symb | isPal : symb | isPalListKind : symb | isQid : symb | n____ : symb | n__a : symb | n__and : symb | n__e : symb | n__i : symb | n__isPalListKind : symb | n__nil : symb | n__o : symb | n__u : symb | nil : symb | o : symb | tt : symb | u : 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 => 1 | M.U21 => 3 | M.U22 => 2 | M.U23 => 1 | M.U31 => 2 | M.U32 => 1 | M.U41 => 3 | M.U42 => 2 | M.U43 => 1 | M.U51 => 3 | M.U52 => 2 | M.U53 => 1 | M.U61 => 2 | M.U62 => 1 | M.U71 => 2 | M.U72 => 1 | M.__ => 2 | M.a => 0 | M.activate => 1 | M.and => 2 | M.e => 0 | M.i => 0 | M.isList => 1 | M.isNeList => 1 | M.isNePal => 1 | M.isPal => 1 | M.isPalListKind => 1 | M.isQid => 1 | M.n____ => 2 | M.n__a => 0 | M.n__and => 2 | M.n__e => 0 | M.n__i => 0 | M.n__isPalListKind => 1 | M.n__nil => 0 | M.n__o => 0 | M.n__u => 0 | M.nil => 0 | M.o => 0 | M.tt => 0 | M.u => 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 x1 := F0 M.U12 (Vcons x1 Vnil). Definition U21 x3 x2 x1 := F0 M.U21 (Vcons x3 (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 x1 := F0 M.U32 (Vcons x1 Vnil). Definition U41 x3 x2 x1 := F0 M.U41 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U42 x2 x1 := F0 M.U42 (Vcons x2 (Vcons x1 Vnil)). Definition U43 x1 := F0 M.U43 (Vcons x1 Vnil). Definition U51 x3 x2 x1 := F0 M.U51 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U52 x2 x1 := F0 M.U52 (Vcons x2 (Vcons x1 Vnil)). Definition U53 x1 := F0 M.U53 (Vcons x1 Vnil). Definition U61 x2 x1 := F0 M.U61 (Vcons x2 (Vcons x1 Vnil)). Definition U62 x1 := F0 M.U62 (Vcons x1 Vnil). Definition U71 x2 x1 := F0 M.U71 (Vcons x2 (Vcons x1 Vnil)). Definition U72 x1 := F0 M.U72 (Vcons x1 Vnil). Definition __ x2 x1 := F0 M.__ (Vcons x2 (Vcons x1 Vnil)). Definition a := F0 M.a Vnil. Definition activate x1 := F0 M.activate (Vcons x1 Vnil). Definition and x2 x1 := F0 M.and (Vcons x2 (Vcons x1 Vnil)). Definition e := F0 M.e Vnil. Definition i := F0 M.i Vnil. Definition isList x1 := F0 M.isList (Vcons x1 Vnil). Definition isNeList x1 := F0 M.isNeList (Vcons x1 Vnil). Definition isNePal x1 := F0 M.isNePal (Vcons x1 Vnil). Definition isPal x1 := F0 M.isPal (Vcons x1 Vnil). Definition isPalListKind x1 := F0 M.isPalListKind (Vcons x1 Vnil). Definition isQid x1 := F0 M.isQid (Vcons x1 Vnil). Definition n____ x2 x1 := F0 M.n____ (Vcons x2 (Vcons x1 Vnil)). Definition n__a := F0 M.n__a Vnil. Definition n__and x2 x1 := F0 M.n__and (Vcons x2 (Vcons x1 Vnil)). Definition n__e := F0 M.n__e Vnil. Definition n__i := F0 M.n__i Vnil. Definition n__isPalListKind x1 := F0 M.n__isPalListKind (Vcons x1 Vnil). Definition n__nil := F0 M.n__nil Vnil. Definition n__o := F0 M.n__o Vnil. Definition n__u := F0 M.n__u Vnil. Definition nil := F0 M.nil Vnil. Definition o := F0 M.o Vnil. Definition tt := F0 M.tt Vnil. Definition u := F0 M.u Vnil. End S0. Definition E := @nil (@ATrs.rule s0). Definition R := R0 (S0.__ (S0.__ (V0 0) (V0 1)) (V0 2)) (S0.__ (V0 0) (S0.__ (V0 1) (V0 2))) :: R0 (S0.__ (V0 0) S0.nil) (V0 0) :: R0 (S0.__ S0.nil (V0 0)) (V0 0) :: R0 (S0.U11 S0.tt (V0 0)) (S0.U12 (S0.isNeList (S0.activate (V0 0)))) :: R0 (S0.U12 S0.tt) S0.tt :: R0 (S0.U21 S0.tt (V0 0) (V0 1)) (S0.U22 (S0.isList (S0.activate (V0 0))) (S0.activate (V0 1))) :: R0 (S0.U22 S0.tt (V0 0)) (S0.U23 (S0.isList (S0.activate (V0 0)))) :: R0 (S0.U23 S0.tt) S0.tt :: R0 (S0.U31 S0.tt (V0 0)) (S0.U32 (S0.isQid (S0.activate (V0 0)))) :: R0 (S0.U32 S0.tt) S0.tt :: R0 (S0.U41 S0.tt (V0 0) (V0 1)) (S0.U42 (S0.isList (S0.activate (V0 0))) (S0.activate (V0 1))) :: R0 (S0.U42 S0.tt (V0 0)) (S0.U43 (S0.isNeList (S0.activate (V0 0)))) :: R0 (S0.U43 S0.tt) S0.tt :: R0 (S0.U51 S0.tt (V0 0) (V0 1)) (S0.U52 (S0.isNeList (S0.activate (V0 0))) (S0.activate (V0 1))) :: R0 (S0.U52 S0.tt (V0 0)) (S0.U53 (S0.isList (S0.activate (V0 0)))) :: R0 (S0.U53 S0.tt) S0.tt :: R0 (S0.U61 S0.tt (V0 0)) (S0.U62 (S0.isQid (S0.activate (V0 0)))) :: R0 (S0.U62 S0.tt) S0.tt :: R0 (S0.U71 S0.tt (V0 0)) (S0.U72 (S0.isNePal (S0.activate (V0 0)))) :: R0 (S0.U72 S0.tt) S0.tt :: R0 (S0.and S0.tt (V0 0)) (S0.activate (V0 0)) :: R0 (S0.isList (V0 0)) (S0.U11 (S0.isPalListKind (S0.activate (V0 0))) (S0.activate (V0 0))) :: R0 (S0.isList S0.n__nil) S0.tt :: R0 (S0.isList (S0.n____ (V0 0) (V0 1))) (S0.U21 (S0.and (S0.isPalListKind (S0.activate (V0 0))) (S0.n__isPalListKind (S0.activate (V0 1)))) (S0.activate (V0 0)) (S0.activate (V0 1))) :: R0 (S0.isNeList (V0 0)) (S0.U31 (S0.isPalListKind (S0.activate (V0 0))) (S0.activate (V0 0))) :: R0 (S0.isNeList (S0.n____ (V0 0) (V0 1))) (S0.U41 (S0.and (S0.isPalListKind (S0.activate (V0 0))) (S0.n__isPalListKind (S0.activate (V0 1)))) (S0.activate (V0 0)) (S0.activate (V0 1))) :: R0 (S0.isNeList (S0.n____ (V0 0) (V0 1))) (S0.U51 (S0.and (S0.isPalListKind (S0.activate (V0 0))) (S0.n__isPalListKind (S0.activate (V0 1)))) (S0.activate (V0 0)) (S0.activate (V0 1))) :: R0 (S0.isNePal (V0 0)) (S0.U61 (S0.isPalListKind (S0.activate (V0 0))) (S0.activate (V0 0))) :: R0 (S0.isNePal (S0.n____ (V0 0) (S0.__ (V0 1) (V0 0)))) (S0.and (S0.and (S0.isQid (S0.activate (V0 0))) (S0.n__isPalListKind (S0.activate (V0 0)))) (S0.n__and (S0.isPal (S0.activate (V0 1))) (S0.n__isPalListKind (S0.activate (V0 1))))) :: R0 (S0.isPal (V0 0)) (S0.U71 (S0.isPalListKind (S0.activate (V0 0))) (S0.activate (V0 0))) :: R0 (S0.isPal S0.n__nil) S0.tt :: R0 (S0.isPalListKind S0.n__a) S0.tt :: R0 (S0.isPalListKind S0.n__e) S0.tt :: R0 (S0.isPalListKind S0.n__i) S0.tt :: R0 (S0.isPalListKind S0.n__nil) S0.tt :: R0 (S0.isPalListKind S0.n__o) S0.tt :: R0 (S0.isPalListKind S0.n__u) S0.tt :: R0 (S0.isPalListKind (S0.n____ (V0 0) (V0 1))) (S0.and (S0.isPalListKind (S0.activate (V0 0))) (S0.n__isPalListKind (S0.activate (V0 1)))) :: R0 (S0.isQid S0.n__a) S0.tt :: R0 (S0.isQid S0.n__e) S0.tt :: R0 (S0.isQid S0.n__i) S0.tt :: R0 (S0.isQid S0.n__o) S0.tt :: R0 (S0.isQid S0.n__u) S0.tt :: R0 S0.nil S0.n__nil :: R0 (S0.__ (V0 0) (V0 1)) (S0.n____ (V0 0) (V0 1)) :: R0 (S0.isPalListKind (V0 0)) (S0.n__isPalListKind (V0 0)) :: R0 (S0.and (V0 0) (V0 1)) (S0.n__and (V0 0) (V0 1)) :: R0 S0.a S0.n__a :: R0 S0.e S0.n__e :: R0 S0.i S0.n__i :: R0 S0.o S0.n__o :: R0 S0.u S0.n__u :: R0 (S0.activate S0.n__nil) S0.nil :: R0 (S0.activate (S0.n____ (V0 0) (V0 1))) (S0.__ (V0 0) (V0 1)) :: R0 (S0.activate (S0.n__isPalListKind (V0 0))) (S0.isPalListKind (V0 0)) :: R0 (S0.activate (S0.n__and (V0 0) (V0 1))) (S0.and (V0 0) (V0 1)) :: R0 (S0.activate S0.n__a) S0.a :: R0 (S0.activate S0.n__e) S0.e :: R0 (S0.activate S0.n__i) S0.i :: R0 (S0.activate S0.n__o) S0.o :: R0 (S0.activate S0.n__u) S0.u :: R0 (S0.activate (V0 0)) (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 x1 := F1 (hd_symb s1_p M.U12) (Vcons x1 Vnil). Definition U12 x1 := F1 (int_symb s1_p M.U12) (Vcons x1 Vnil). Definition hU21 x3 x2 x1 := F1 (hd_symb s1_p M.U21) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U21 x3 x2 x1 := F1 (int_symb s1_p M.U21) (Vcons x3 (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 x1 := F1 (hd_symb s1_p M.U32) (Vcons x1 Vnil). Definition U32 x1 := F1 (int_symb s1_p M.U32) (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 x2 x1 := F1 (hd_symb s1_p M.U42) (Vcons x2 (Vcons x1 Vnil)). Definition U42 x2 x1 := F1 (int_symb s1_p M.U42) (Vcons x2 (Vcons x1 Vnil)). Definition hU43 x1 := F1 (hd_symb s1_p M.U43) (Vcons x1 Vnil). Definition U43 x1 := F1 (int_symb s1_p M.U43) (Vcons x1 Vnil). Definition hU51 x3 x2 x1 := F1 (hd_symb s1_p M.U51) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition U51 x3 x2 x1 := F1 (int_symb s1_p M.U51) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))). Definition hU52 x2 x1 := F1 (hd_symb s1_p M.U52) (Vcons x2 (Vcons x1 Vnil)). Definition U52 x2 x1 := F1 (int_symb s1_p M.U52) (Vcons x2 (Vcons x1 Vnil)). Definition hU53 x1 := F1 (hd_symb s1_p M.U53) (Vcons x1 Vnil). Definition U53 x1 := F1 (int_symb s1_p M.U53) (Vcons x1 Vnil). Definition hU61 x2 x1 := F1 (hd_symb s1_p M.U61) (Vcons x2 (Vcons x1 Vnil)). Definition U61 x2 x1 := F1 (int_symb s1_p M.U61) (Vcons x2 (Vcons x1 Vnil)). Definition hU62 x1 := F1 (hd_symb s1_p M.U62) (Vcons x1 Vnil). Definition U62 x1 := F1 (int_symb s1_p M.U62) (Vcons x1 Vnil). Definition hU71 x2 x1 := F1 (hd_symb s1_p M.U71) (Vcons x2 (Vcons x1 Vnil)). Definition U71 x2 x1 := F1 (int_symb s1_p M.U71) (Vcons x2 (Vcons x1 Vnil)). Definition hU72 x1 := F1 (hd_symb s1_p M.U72) (Vcons x1 Vnil). Definition U72 x1 := F1 (int_symb s1_p M.U72) (Vcons x1 Vnil). Definition h__ x2 x1 := F1 (hd_symb s1_p M.__) (Vcons x2 (Vcons x1 Vnil)). Definition __ x2 x1 := F1 (int_symb s1_p M.__) (Vcons x2 (Vcons x1 Vnil)). Definition ha := F1 (hd_symb s1_p M.a) Vnil. Definition a := F1 (int_symb s1_p M.a) Vnil. Definition hactivate x1 := F1 (hd_symb s1_p M.activate) (Vcons x1 Vnil). Definition activate x1 := F1 (int_symb s1_p M.activate) (Vcons x1 Vnil). Definition hand x2 x1 := F1 (hd_symb s1_p M.and) (Vcons x2 (Vcons x1 Vnil)). Definition and x2 x1 := F1 (int_symb s1_p M.and) (Vcons x2 (Vcons x1 Vnil)). Definition he := F1 (hd_symb s1_p M.e) Vnil. Definition e := F1 (int_symb s1_p M.e) Vnil. Definition hi := F1 (hd_symb s1_p M.i) Vnil. Definition i := F1 (int_symb s1_p M.i) Vnil. Definition hisList x1 := F1 (hd_symb s1_p M.isList) (Vcons x1 Vnil). Definition isList x1 := F1 (int_symb s1_p M.isList) (Vcons x1 Vnil). Definition hisNeList x1 := F1 (hd_symb s1_p M.isNeList) (Vcons x1 Vnil). Definition isNeList x1 := F1 (int_symb s1_p M.isNeList) (Vcons x1 Vnil). Definition hisNePal x1 := F1 (hd_symb s1_p M.isNePal) (Vcons x1 Vnil). Definition isNePal x1 := F1 (int_symb s1_p M.isNePal) (Vcons x1 Vnil). Definition hisPal x1 := F1 (hd_symb s1_p M.isPal) (Vcons x1 Vnil). Definition isPal x1 := F1 (int_symb s1_p M.isPal) (Vcons x1 Vnil). Definition hisPalListKind x1 := F1 (hd_symb s1_p M.isPalListKind) (Vcons x1 Vnil). Definition isPalListKind x1 := F1 (int_symb s1_p M.isPalListKind) (Vcons x1 Vnil). Definition hisQid x1 := F1 (hd_symb s1_p M.isQid) (Vcons x1 Vnil). Definition isQid x1 := F1 (int_symb s1_p M.isQid) (Vcons x1 Vnil). Definition hn____ x2 x1 := F1 (hd_symb s1_p M.n____) (Vcons x2 (Vcons x1 Vnil)). Definition n____ x2 x1 := F1 (int_symb s1_p M.n____) (Vcons x2 (Vcons x1 Vnil)). Definition hn__a := F1 (hd_symb s1_p M.n__a) Vnil. Definition n__a := F1 (int_symb s1_p M.n__a) Vnil. Definition hn__and x2 x1 := F1 (hd_symb s1_p M.n__and) (Vcons x2 (Vcons x1 Vnil)). Definition n__and x2 x1 := F1 (int_symb s1_p M.n__and) (Vcons x2 (Vcons x1 Vnil)). Definition hn__e := F1 (hd_symb s1_p M.n__e) Vnil. Definition n__e := F1 (int_symb s1_p M.n__e) Vnil. Definition hn__i := F1 (hd_symb s1_p M.n__i) Vnil. Definition n__i := F1 (int_symb s1_p M.n__i) Vnil. Definition hn__isPalListKind x1 := F1 (hd_symb s1_p M.n__isPalListKind) (Vcons x1 Vnil). Definition n__isPalListKind x1 := F1 (int_symb s1_p M.n__isPalListKind) (Vcons x1 Vnil). Definition hn__nil := F1 (hd_symb s1_p M.n__nil) Vnil. Definition n__nil := F1 (int_symb s1_p M.n__nil) Vnil. Definition hn__o := F1 (hd_symb s1_p M.n__o) Vnil. Definition n__o := F1 (int_symb s1_p M.n__o) Vnil. Definition hn__u := F1 (hd_symb s1_p M.n__u) Vnil. Definition n__u := F1 (int_symb s1_p M.n__u) Vnil. Definition hnil := F1 (hd_symb s1_p M.nil) Vnil. Definition nil := F1 (int_symb s1_p M.nil) Vnil. Definition ho := F1 (hd_symb s1_p M.o) Vnil. Definition o := F1 (int_symb s1_p M.o) Vnil. Definition htt := F1 (hd_symb s1_p M.tt) Vnil. Definition tt := F1 (int_symb s1_p M.tt) Vnil. Definition hu := F1 (hd_symb s1_p M.u) Vnil. Definition u := F1 (int_symb s1_p M.u) Vnil. End S1. (* graph decomposition 1 *) Definition cs1 : list (list (@ATrs.rule s1)) := ( R1 (S1.hactivate (S1.n__u)) (S1.hu) :: nil) :: ( R1 (S1.hactivate (S1.n__o)) (S1.ho) :: nil) :: ( R1 (S1.hactivate (S1.n__i)) (S1.hi) :: nil) :: ( R1 (S1.hactivate (S1.n__e)) (S1.he) :: nil) :: ( R1 (S1.hactivate (S1.n__a)) (S1.ha) :: nil) :: ( R1 (S1.hactivate (S1.n__nil)) (S1.hnil) :: nil) :: ( R1 (S1.hisNePal (S1.n____ (V1 0) (S1.__ (V1 1) (V1 0)))) (S1.hisQid (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hU71 (S1.tt) (V1 0)) (S1.hU72 (S1.isNePal (S1.activate (V1 0)))) :: nil) :: ( R1 (S1.hU61 (S1.tt) (V1 0)) (S1.hisQid (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hU61 (S1.tt) (V1 0)) (S1.hU62 (S1.isQid (S1.activate (V1 0)))) :: nil) :: ( R1 (S1.hU52 (S1.tt) (V1 0)) (S1.hU53 (S1.isList (S1.activate (V1 0)))) :: nil) :: ( R1 (S1.hU42 (S1.tt) (V1 0)) (S1.hU43 (S1.isNeList (S1.activate (V1 0)))) :: nil) :: ( R1 (S1.hU31 (S1.tt) (V1 0)) (S1.hisQid (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hU31 (S1.tt) (V1 0)) (S1.hU32 (S1.isQid (S1.activate (V1 0)))) :: nil) :: ( R1 (S1.hU22 (S1.tt) (V1 0)) (S1.hU23 (S1.isList (S1.activate (V1 0)))) :: nil) :: ( R1 (S1.hU11 (S1.tt) (V1 0)) (S1.hU12 (S1.isNeList (S1.activate (V1 0)))) :: nil) :: ( R1 (S1.h__ (S1.__ (V1 0) (V1 1)) (V1 2)) (S1.h__ (V1 1) (V1 2)) :: R1 (S1.h__ (S1.__ (V1 0) (V1 1)) (V1 2)) (S1.h__ (V1 0) (S1.__ (V1 1) (V1 2))) :: nil) :: ( R1 (S1.hactivate (S1.n____ (V1 0) (V1 1))) (S1.h__ (V1 0) (V1 1)) :: nil) :: ( R1 (S1.hactivate (S1.n__isPalListKind (V1 0))) (S1.hisPalListKind (V1 0)) :: R1 (S1.hisPalListKind (S1.n____ (V1 0) (V1 1))) (S1.hand (S1.isPalListKind (S1.activate (V1 0))) (S1.n__isPalListKind (S1.activate (V1 1)))) :: R1 (S1.hand (S1.tt) (V1 0)) (S1.hactivate (V1 0)) :: R1 (S1.hactivate (S1.n__and (V1 0) (V1 1))) (S1.hand (V1 0) (V1 1)) :: R1 (S1.hisPalListKind (S1.n____ (V1 0) (V1 1))) (S1.hisPalListKind (S1.activate (V1 0))) :: R1 (S1.hisPalListKind (S1.n____ (V1 0) (V1 1))) (S1.hactivate (V1 0)) :: R1 (S1.hisPalListKind (S1.n____ (V1 0) (V1 1))) (S1.hactivate (V1 1)) :: nil) :: ( R1 (S1.hisPal (V1 0)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hisNePal (S1.n____ (V1 0) (S1.__ (V1 1) (V1 0)))) (S1.hactivate (V1 1)) :: nil) :: ( R1 (S1.hisNePal (S1.n____ (V1 0) (S1.__ (V1 1) (V1 0)))) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hisNePal (V1 0)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1))) (S1.hactivate (V1 1)) :: nil) :: ( R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1))) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hisNeList (V1 0)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hisList (S1.n____ (V1 0) (V1 1))) (S1.hactivate (V1 1)) :: nil) :: ( R1 (S1.hisList (S1.n____ (V1 0) (V1 1))) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hisList (V1 0)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hisPal (V1 0)) (S1.hisPalListKind (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hisNePal (V1 0)) (S1.hisPalListKind (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1))) (S1.hisPalListKind (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hisNeList (V1 0)) (S1.hisPalListKind (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hisList (S1.n____ (V1 0) (V1 1))) (S1.hisPalListKind (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hisList (V1 0)) (S1.hisPalListKind (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hisNePal (S1.n____ (V1 0) (S1.__ (V1 1) (V1 0)))) (S1.hand (S1.isQid (S1.activate (V1 0))) (S1.n__isPalListKind (S1.activate (V1 0)))) :: nil) :: ( R1 (S1.hisNePal (S1.n____ (V1 0) (S1.__ (V1 1) (V1 0)))) (S1.hand (S1.and (S1.isQid (S1.activate (V1 0))) (S1.n__isPalListKind (S1.activate (V1 0)))) (S1.n__and (S1.isPal (S1.activate (V1 1))) (S1.n__isPalListKind (S1.activate (V1 1))))) :: nil) :: ( R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1))) (S1.hand (S1.isPalListKind (S1.activate (V1 0))) (S1.n__isPalListKind (S1.activate (V1 1)))) :: nil) :: ( R1 (S1.hisList (S1.n____ (V1 0) (V1 1))) (S1.hand (S1.isPalListKind (S1.activate (V1 0))) (S1.n__isPalListKind (S1.activate (V1 1)))) :: nil) :: ( R1 (S1.hU71 (S1.tt) (V1 0)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hU61 (S1.tt) (V1 0)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hisNePal (V1 0)) (S1.hU61 (S1.isPalListKind (S1.activate (V1 0))) (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hisNePal (S1.n____ (V1 0) (S1.__ (V1 1) (V1 0)))) (S1.hisPal (S1.activate (V1 1))) :: R1 (S1.hisPal (V1 0)) (S1.hU71 (S1.isPalListKind (S1.activate (V1 0))) (S1.activate (V1 0))) :: R1 (S1.hU71 (S1.tt) (V1 0)) (S1.hisNePal (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hU52 (S1.tt) (V1 0)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hU51 (S1.tt) (V1 0) (V1 1)) (S1.hactivate (V1 1)) :: nil) :: ( R1 (S1.hU51 (S1.tt) (V1 0) (V1 1)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hU42 (S1.tt) (V1 0)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hU41 (S1.tt) (V1 0) (V1 1)) (S1.hactivate (V1 1)) :: nil) :: ( R1 (S1.hU41 (S1.tt) (V1 0) (V1 1)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hU31 (S1.tt) (V1 0)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hisNeList (V1 0)) (S1.hU31 (S1.isPalListKind (S1.activate (V1 0))) (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hU22 (S1.tt) (V1 0)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hU21 (S1.tt) (V1 0) (V1 1)) (S1.hactivate (V1 1)) :: nil) :: ( R1 (S1.hU21 (S1.tt) (V1 0) (V1 1)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hU11 (S1.tt) (V1 0)) (S1.hactivate (V1 0)) :: nil) :: ( R1 (S1.hU11 (S1.tt) (V1 0)) (S1.hisNeList (S1.activate (V1 0))) :: R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1))) (S1.hU41 (S1.and (S1.isPalListKind (S1.activate (V1 0))) (S1.n__isPalListKind (S1.activate (V1 1)))) (S1.activate (V1 0)) (S1.activate (V1 1))) :: R1 (S1.hU41 (S1.tt) (V1 0) (V1 1)) (S1.hU42 (S1.isList (S1.activate (V1 0))) (S1.activate (V1 1))) :: R1 (S1.hU42 (S1.tt) (V1 0)) (S1.hisNeList (S1.activate (V1 0))) :: R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1))) (S1.hU51 (S1.and (S1.isPalListKind (S1.activate (V1 0))) (S1.n__isPalListKind (S1.activate (V1 1)))) (S1.activate (V1 0)) (S1.activate (V1 1))) :: R1 (S1.hU51 (S1.tt) (V1 0) (V1 1)) (S1.hU52 (S1.isNeList (S1.activate (V1 0))) (S1.activate (V1 1))) :: R1 (S1.hU52 (S1.tt) (V1 0)) (S1.hisList (S1.activate (V1 0))) :: R1 (S1.hisList (V1 0)) (S1.hU11 (S1.isPalListKind (S1.activate (V1 0))) (S1.activate (V1 0))) :: R1 (S1.hisList (S1.n____ (V1 0) (V1 1))) (S1.hU21 (S1.and (S1.isPalListKind (S1.activate (V1 0))) (S1.n__isPalListKind (S1.activate (V1 1)))) (S1.activate (V1 0)) (S1.activate (V1 1))) :: R1 (S1.hU21 (S1.tt) (V1 0) (V1 1)) (S1.hU22 (S1.isList (S1.activate (V1 0))) (S1.activate (V1 1))) :: R1 (S1.hU22 (S1.tt) (V1 0)) (S1.hisList (S1.activate (V1 0))) :: R1 (S1.hU21 (S1.tt) (V1 0) (V1 1)) (S1.hisList (S1.activate (V1 0))) :: R1 (S1.hU51 (S1.tt) (V1 0) (V1 1)) (S1.hisNeList (S1.activate (V1 0))) :: R1 (S1.hU41 (S1.tt) (V1 0) (V1 1)) (S1.hisList (S1.activate (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.__) => (3%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (int_symb M.__) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (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 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.U12) => nil | (int_symb M.U12) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNeList) => nil | (int_symb M.isNeList) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.activate) => nil | (int_symb M.activate) => (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (1%Z, (Vcons 0 (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.isList) => nil | (int_symb M.isList) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (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 Vnil)) :: nil | (hd_symb M.isQid) => nil | (int_symb M.isQid) => (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 Vnil))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U51) => nil | (int_symb M.U51) => nil | (hd_symb M.U52) => nil | (int_symb M.U52) => nil | (hd_symb M.U53) => nil | (int_symb M.U53) => nil | (hd_symb M.U61) => nil | (int_symb M.U61) => nil | (hd_symb M.U62) => nil | (int_symb M.U62) => nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (3%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U72) => nil | (int_symb M.U72) => (3%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.isNePal) => nil | (int_symb M.isNePal) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.and) => nil | (int_symb M.and) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isPalListKind) => nil | (int_symb M.isPalListKind) => nil | (hd_symb M.n__nil) => nil | (int_symb M.n__nil) => (2%Z, Vnil) :: nil | (hd_symb M.n____) => nil | (int_symb M.n____) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.n__isPalListKind) => nil | (int_symb M.n__isPalListKind) => nil | (hd_symb M.n__and) => nil | (int_symb M.n__and) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isPal) => nil | (int_symb M.isPal) => (3%Z, (Vcons 0 Vnil)) :: (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.n__a) => nil | (int_symb M.n__a) => nil | (hd_symb M.n__e) => nil | (int_symb M.n__e) => (2%Z, Vnil) :: nil | (hd_symb M.n__i) => nil | (int_symb M.n__i) => (2%Z, Vnil) :: nil | (hd_symb M.n__o) => nil | (int_symb M.n__o) => (2%Z, Vnil) :: nil | (hd_symb M.n__u) => nil | (int_symb M.n__u) => nil | (hd_symb M.a) => nil | (int_symb M.a) => nil | (hd_symb M.e) => nil | (int_symb M.e) => (2%Z, Vnil) :: nil | (hd_symb M.i) => nil | (int_symb M.i) => (2%Z, Vnil) :: nil | (hd_symb M.o) => nil | (int_symb M.o) => (3%Z, Vnil) :: nil | (hd_symb M.u) => nil | (int_symb M.u) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS1. Module PI1 := PolyInt PIS1. (* 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.__) => nil | (int_symb M.__) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (3%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (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.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.U12) => nil | (int_symb M.U12) => nil | (hd_symb M.isNeList) => nil | (int_symb M.isNeList) => (1%Z, (Vcons 0 Vnil)) :: (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.activate) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.activate) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.isList) => nil | (int_symb M.isList) => (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => (2%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.isQid) => nil | (int_symb M.isQid) => 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.U51) => nil | (int_symb M.U51) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: nil | (hd_symb M.U53) => nil | (int_symb M.U53) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U62) => nil | (int_symb M.U62) => nil | (hd_symb M.U71) => nil | (int_symb M.U71) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U72) => nil | (int_symb M.U72) => nil | (hd_symb M.isNePal) => nil | (int_symb M.isNePal) => (2%Z, (Vcons 0 Vnil)) :: (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.and) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.and) => (3%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isPalListKind) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.isPalListKind) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.n__nil) => nil | (int_symb M.n__nil) => nil | (hd_symb M.n____) => nil | (int_symb M.n____) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (3%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.n__isPalListKind) => nil | (int_symb M.n__isPalListKind) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.n__and) => nil | (int_symb M.n__and) => (3%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isPal) => nil | (int_symb M.isPal) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.n__a) => nil | (int_symb M.n__a) => nil | (hd_symb M.n__e) => nil | (int_symb M.n__e) => (2%Z, Vnil) :: nil | (hd_symb M.n__i) => nil | (int_symb M.n__i) => nil | (hd_symb M.n__o) => nil | (int_symb M.n__o) => nil | (hd_symb M.n__u) => nil | (int_symb M.n__u) => nil | (hd_symb M.a) => nil | (int_symb M.a) => nil | (hd_symb M.e) => nil | (int_symb M.e) => (2%Z, Vnil) :: nil | (hd_symb M.i) => nil | (int_symb M.i) => nil | (hd_symb M.o) => nil | (int_symb M.o) => nil | (hd_symb M.u) => nil | (int_symb M.u) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS2. Module PI2 := PolyInt PIS2. (* graph decomposition 2 *) Definition cs2 : list (list (@ATrs.rule s1)) := ( R1 (S1.hactivate (S1.n__isPalListKind (V1 0))) (S1.hisPalListKind (V1 0)) :: nil) :: ( R1 (S1.hactivate (S1.n__and (V1 0) (V1 1))) (S1.hand (V1 0) (V1 1)) :: R1 (S1.hand (S1.tt) (V1 0)) (S1.hactivate (V1 0)) :: nil) :: nil. (* 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.__) => nil | (int_symb M.__) => (3%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (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) => nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.U12) => nil | (int_symb M.U12) => nil | (hd_symb M.isNeList) => nil | (int_symb M.isNeList) => nil | (hd_symb M.activate) => (3%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.activate) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => nil | (hd_symb M.isList) => nil | (int_symb M.isList) => (2%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => nil | (hd_symb M.U32) => nil | (int_symb M.U32) => nil | (hd_symb M.isQid) => nil | (int_symb M.isQid) => 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.U51) => nil | (int_symb M.U51) => nil | (hd_symb M.U52) => nil | (int_symb M.U52) => nil | (hd_symb M.U53) => nil | (int_symb M.U53) => nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: nil | (hd_symb M.U62) => nil | (int_symb M.U62) => nil | (hd_symb M.U71) => nil | (int_symb M.U71) => nil | (hd_symb M.U72) => nil | (int_symb M.U72) => nil | (hd_symb M.isNePal) => nil | (int_symb M.isNePal) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.and) => (2%Z, (Vcons 0 (Vcons 0 Vnil))) :: (3%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.and) => (3%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isPalListKind) => nil | (int_symb M.isPalListKind) => (1%Z, (Vcons 0 Vnil)) :: (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.n__nil) => nil | (int_symb M.n__nil) => (1%Z, Vnil) :: nil | (hd_symb M.n____) => nil | (int_symb M.n____) => (3%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.n__isPalListKind) => nil | (int_symb M.n__isPalListKind) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.n__and) => nil | (int_symb M.n__and) => (3%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isPal) => nil | (int_symb M.isPal) => nil | (hd_symb M.n__a) => nil | (int_symb M.n__a) => (1%Z, Vnil) :: nil | (hd_symb M.n__e) => nil | (int_symb M.n__e) => (1%Z, Vnil) :: nil | (hd_symb M.n__i) => nil | (int_symb M.n__i) => nil | (hd_symb M.n__o) => nil | (int_symb M.n__o) => nil | (hd_symb M.n__u) => nil | (int_symb M.n__u) => (1%Z, Vnil) :: nil | (hd_symb M.a) => nil | (int_symb M.a) => (2%Z, Vnil) :: nil | (hd_symb M.e) => nil | (int_symb M.e) => (2%Z, Vnil) :: nil | (hd_symb M.i) => nil | (int_symb M.i) => nil | (hd_symb M.o) => nil | (int_symb M.o) => nil | (hd_symb M.u) => nil | (int_symb M.u) => (1%Z, Vnil) :: nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS3. Module PI3 := PolyInt PIS3. (* 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.__) => nil | (int_symb M.__) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (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 1 Vnil))) :: nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.U12) => nil | (int_symb M.U12) => nil | (hd_symb M.isNeList) => nil | (int_symb M.isNeList) => (1%Z, (Vcons 0 Vnil)) :: (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.activate) => nil | (int_symb M.activate) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => nil | (int_symb M.U21) => (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U22) => nil | (int_symb M.U22) => nil | (hd_symb M.isList) => nil | (int_symb M.isList) => (1%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U23) => nil | (int_symb M.U23) => 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) => nil | (hd_symb M.isQid) => nil | (int_symb M.isQid) => nil | (hd_symb M.U41) => nil | (int_symb M.U41) => (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U42) => nil | (int_symb M.U42) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U51) => nil | (int_symb M.U51) => (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (hd_symb M.U52) => nil | (int_symb M.U52) => nil | (hd_symb M.U53) => nil | (int_symb M.U53) => nil | (hd_symb M.U61) => nil | (int_symb M.U61) => nil | (hd_symb M.U62) => nil | (int_symb M.U62) => nil | (hd_symb M.U71) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.U71) => nil | (hd_symb M.U72) => nil | (int_symb M.U72) => nil | (hd_symb M.isNePal) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.isNePal) => nil | (hd_symb M.and) => nil | (int_symb M.and) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isPalListKind) => nil | (int_symb M.isPalListKind) => nil | (hd_symb M.n__nil) => nil | (int_symb M.n__nil) => nil | (hd_symb M.n____) => nil | (int_symb M.n____) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.n__isPalListKind) => nil | (int_symb M.n__isPalListKind) => nil | (hd_symb M.n__and) => nil | (int_symb M.n__and) => (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isPal) => (3%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.isPal) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.n__a) => nil | (int_symb M.n__a) => nil | (hd_symb M.n__e) => nil | (int_symb M.n__e) => nil | (hd_symb M.n__i) => nil | (int_symb M.n__i) => nil | (hd_symb M.n__o) => nil | (int_symb M.n__o) => nil | (hd_symb M.n__u) => nil | (int_symb M.n__u) => nil | (hd_symb M.a) => nil | (int_symb M.a) => nil | (hd_symb M.e) => nil | (int_symb M.e) => nil | (hd_symb M.i) => nil | (int_symb M.i) => nil | (hd_symb M.o) => nil | (int_symb M.o) => nil | (hd_symb M.u) => nil | (int_symb M.u) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS4. Module PI4 := PolyInt PIS4. (* graph decomposition 3 *) Definition cs3 : list (list (@ATrs.rule s1)) := ( R1 (S1.hU71 (S1.tt) (V1 0)) (S1.hisNePal (S1.activate (V1 0))) :: 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.__) => nil | (int_symb M.__) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.nil) => nil | (int_symb M.nil) => (3%Z, Vnil) :: nil | (hd_symb M.U11) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.U11) => nil | (hd_symb M.tt) => nil | (int_symb M.tt) => nil | (hd_symb M.U12) => nil | (int_symb M.U12) => nil | (hd_symb M.isNeList) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.isNeList) => nil | (hd_symb M.activate) => nil | (int_symb M.activate) => (1%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.U21) => (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.U21) => nil | (hd_symb M.U22) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.U22) => nil | (hd_symb M.isList) => (2%Z, (Vcons 1 Vnil)) :: nil | (int_symb M.isList) => nil | (hd_symb M.U23) => nil | (int_symb M.U23) => nil | (hd_symb M.U31) => nil | (int_symb M.U31) => nil | (hd_symb M.U32) => nil | (int_symb M.U32) => nil | (hd_symb M.isQid) => nil | (int_symb M.isQid) => nil | (hd_symb M.U41) => (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))) :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))) :: nil | (int_symb M.U41) => nil | (hd_symb M.U42) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.U42) => nil | (hd_symb M.U43) => nil | (int_symb M.U43) => nil | (hd_symb M.U51) => (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 | (int_symb M.U51) => nil | (hd_symb M.U52) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (int_symb M.U52) => nil | (hd_symb M.U53) => nil | (int_symb M.U53) => nil | (hd_symb M.U61) => nil | (int_symb M.U61) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.U62) => nil | (int_symb M.U62) => (1%Z, (Vcons 0 Vnil)) :: nil | (hd_symb M.U71) => nil | (int_symb M.U71) => nil | (hd_symb M.U72) => nil | (int_symb M.U72) => nil | (hd_symb M.isNePal) => nil | (int_symb M.isNePal) => (2%Z, (Vcons 0 Vnil)) :: (2%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.and) => nil | (int_symb M.and) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isPalListKind) => nil | (int_symb M.isPalListKind) => nil | (hd_symb M.n__nil) => nil | (int_symb M.n__nil) => (3%Z, Vnil) :: nil | (hd_symb M.n____) => nil | (int_symb M.n____) => (1%Z, (Vcons 0 (Vcons 0 Vnil))) :: (2%Z, (Vcons 1 (Vcons 0 Vnil))) :: (1%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.n__isPalListKind) => nil | (int_symb M.n__isPalListKind) => nil | (hd_symb M.n__and) => nil | (int_symb M.n__and) => (2%Z, (Vcons 0 (Vcons 1 Vnil))) :: nil | (hd_symb M.isPal) => nil | (int_symb M.isPal) => (3%Z, (Vcons 1 Vnil)) :: nil | (hd_symb M.n__a) => nil | (int_symb M.n__a) => nil | (hd_symb M.n__e) => nil | (int_symb M.n__e) => nil | (hd_symb M.n__i) => nil | (int_symb M.n__i) => nil | (hd_symb M.n__o) => nil | (int_symb M.n__o) => nil | (hd_symb M.n__u) => nil | (int_symb M.n__u) => nil | (hd_symb M.a) => nil | (int_symb M.a) => nil | (hd_symb M.e) => nil | (int_symb M.e) => nil | (hd_symb M.i) => nil | (int_symb M.i) => nil | (hd_symb M.o) => nil | (int_symb M.o) => nil | (hd_symb M.u) => nil | (int_symb M.u) => nil end. Lemma trsInt_wm : forall f, pweak_monotone (trsInt f). Proof. pmonotone. Qed. End PIS5. Module PI5 := PolyInt PIS5. (* graph decomposition 4 *) Definition cs4 : list (list (@ATrs.rule s1)) := ( R1 (S1.hisNeList (S1.n____ (V1 0) (V1 1))) (S1.hU51 (S1.and (S1.isPalListKind (S1.activate (V1 0))) (S1.n__isPalListKind (S1.activate (V1 1)))) (S1.activate (V1 0)) (S1.activate (V1 1))) :: nil) :: ( R1 (S1.hU42 (S1.tt) (V1 0)) (S1.hisNeList (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hU41 (S1.tt) (V1 0) (V1 1)) (S1.hU42 (S1.isList (S1.activate (V1 0))) (S1.activate (V1 1))) :: nil) :: ( R1 (S1.hU11 (S1.tt) (V1 0)) (S1.hisNeList (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hisList (V1 0)) (S1.hU11 (S1.isPalListKind (S1.activate (V1 0))) (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hU41 (S1.tt) (V1 0) (V1 1)) (S1.hisList (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hU21 (S1.tt) (V1 0) (V1 1)) (S1.hisList (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hU22 (S1.tt) (V1 0)) (S1.hisList (S1.activate (V1 0))) :: nil) :: ( R1 (S1.hU21 (S1.tt) (V1 0) (V1 1)) (S1.hU22 (S1.isList (S1.activate (V1 0))) (S1.activate (V1 1))) :: nil) :: ( R1 (S1.hU52 (S1.tt) (V1 0)) (S1.hisList (S1.activate (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. left. co_scc. left. co_scc. left. co_scc. right. PI1.prove_termination. termination_trivial. left. co_scc. right. PI2.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. left. co_scc. right. PI3.prove_termination. termination_trivial. 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) 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. 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) cs4; 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. Qed.