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 :=
  | F : symb
  | T : symb
  | _0_1 : symb
  | _if_2 : symb
  | a : symb
  | and : symb
  | andt : symb
  | append : symb
  | calls : symb
  | case0 : symb
  | case1 : symb
  | case2 : symb
  | case4 : symb
  | case5 : symb
  | case6 : symb
  | case8 : symb
  | case9 : symb
  | cons : symb
  | delete : symb
  | element : symb
  | empty : symb
  | eq : symb
  | eqc : symb
  | eqs : symb
  | eqt : symb
  | equal : symb
  | excl : symb
  | excllock : symb
  | false : symb
  | gen_modtageq : symb
  | gen_tag : symb
  | imp : symb
  | int : symb
  | istops : symb
  | lock : symb
  | locker : symb
  | locker2_add_pending : symb
  | locker2_adduniq : symb
  | locker2_check_available : symb
  | locker2_check_availables : symb
  | locker2_claim_lock : symb
  | locker2_map_add_pending : symb
  | locker2_map_claim_lock : symb
  | locker2_map_promote_pending : symb
  | locker2_obtainable : symb
  | locker2_obtainables : symb
  | locker2_promote_pending : symb
  | locker2_release_lock : symb
  | locker2_remove_pending : symb
  | mcrlrecord : symb
  | member : symb
  | nil : symb
  | nocalls : symb
  | not : symb
  | ok : symb
  | or : symb
  | pending : symb
  | pid : symb
  | pops : symb
  | push : symb
  | push1 : symb
  | pushs : symb
  | record_extract : symb
  | record_new : symb
  | record_update : symb
  | record_updates : symb
  | release : symb
  | request : symb
  | resource : symb
  | s : symb
  | stack : symb
  | subtract : symb
  | tag : symb
  | tops : symb
  | true : symb
  | tuple : symb
  | tuplenil : symb
  | undefined : 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.F => 0
  | M.T => 0
  | M._0_1 => 0
  | M._if_2 => 3
  | M.a => 0
  | M.and => 2
  | M.andt => 2
  | M.append => 2
  | M.calls => 3
  | M.case0 => 3
  | M.case1 => 4
  | M.case2 => 3
  | M.case4 => 3
  | M.case5 => 4
  | M.case6 => 4
  | M.case8 => 4
  | M.case9 => 4
  | M.cons => 2
  | M.delete => 2
  | M.element => 2
  | M.empty => 0
  | M.eq => 2
  | M.eqc => 2
  | M.eqs => 2
  | M.eqt => 2
  | M.equal => 2
  | M.excl => 0
  | M.excllock => 0
  | M.false => 0
  | M.gen_modtageq => 2
  | M.gen_tag => 1
  | M.imp => 2
  | M.int => 1
  | M.istops => 2
  | M.lock => 0
  | M.locker => 0
  | M.locker2_add_pending => 3
  | M.locker2_adduniq => 2
  | M.locker2_check_available => 2
  | M.locker2_check_availables => 2
  | M.locker2_claim_lock => 3
  | M.locker2_map_add_pending => 3
  | M.locker2_map_claim_lock => 3
  | M.locker2_map_promote_pending => 2
  | M.locker2_obtainable => 2
  | M.locker2_obtainables => 2
  | M.locker2_promote_pending => 2
  | M.locker2_release_lock => 2
  | M.locker2_remove_pending => 2
  | M.mcrlrecord => 0
  | M.member => 2
  | M.nil => 0
  | M.nocalls => 0
  | M.not => 1
  | M.ok => 0
  | M.or => 2
  | M.pending => 0
  | M.pid => 1
  | M.pops => 1
  | M.push => 3
  | M.push1 => 6
  | M.pushs => 2
  | M.record_extract => 3
  | M.record_new => 1
  | M.record_update => 4
  | M.record_updates => 3
  | M.release => 0
  | M.request => 0
  | M.resource => 0
  | M.s => 1
  | M.stack => 2
  | M.subtract => 2
  | M.tag => 0
  | M.tops => 1
  | M.true => 0
  | M.tuple => 2
  | M.tuplenil => 1
  | M.undefined => 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 F := F0 M.F Vnil.
  Definition T := F0 M.T Vnil.
  Definition _0_1 := F0 M._0_1 Vnil.
  Definition _if_2 x3 x2 x1 := F0 M._if_2 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition a := F0 M.a Vnil.
  Definition and x2 x1 := F0 M.and (Vcons x2 (Vcons x1 Vnil)).
  Definition andt x2 x1 := F0 M.andt (Vcons x2 (Vcons x1 Vnil)).
  Definition append x2 x1 := F0 M.append (Vcons x2 (Vcons x1 Vnil)).
  Definition calls x3 x2 x1 := F0 M.calls (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition case0 x3 x2 x1 := F0 M.case0 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition case1 x4 x3 x2 x1 := F0 M.case1 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition case2 x3 x2 x1 := F0 M.case2 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition case4 x3 x2 x1 := F0 M.case4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition case5 x4 x3 x2 x1 := F0 M.case5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition case6 x4 x3 x2 x1 := F0 M.case6 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition case8 x4 x3 x2 x1 := F0 M.case8 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition case9 x4 x3 x2 x1 := F0 M.case9 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition cons x2 x1 := F0 M.cons (Vcons x2 (Vcons x1 Vnil)).
  Definition delete x2 x1 := F0 M.delete (Vcons x2 (Vcons x1 Vnil)).
  Definition element x2 x1 := F0 M.element (Vcons x2 (Vcons x1 Vnil)).
  Definition empty := F0 M.empty Vnil.
  Definition eq x2 x1 := F0 M.eq (Vcons x2 (Vcons x1 Vnil)).
  Definition eqc x2 x1 := F0 M.eqc (Vcons x2 (Vcons x1 Vnil)).
  Definition eqs x2 x1 := F0 M.eqs (Vcons x2 (Vcons x1 Vnil)).
  Definition eqt x2 x1 := F0 M.eqt (Vcons x2 (Vcons x1 Vnil)).
  Definition equal x2 x1 := F0 M.equal (Vcons x2 (Vcons x1 Vnil)).
  Definition excl := F0 M.excl Vnil.
  Definition excllock := F0 M.excllock Vnil.
  Definition false := F0 M.false Vnil.
  Definition gen_modtageq x2 x1 := F0 M.gen_modtageq (Vcons x2 (Vcons x1 Vnil)).
  Definition gen_tag x1 := F0 M.gen_tag (Vcons x1 Vnil).
  Definition imp x2 x1 := F0 M.imp (Vcons x2 (Vcons x1 Vnil)).
  Definition int x1 := F0 M.int (Vcons x1 Vnil).
  Definition istops x2 x1 := F0 M.istops (Vcons x2 (Vcons x1 Vnil)).
  Definition lock := F0 M.lock Vnil.
  Definition locker := F0 M.locker Vnil.
  Definition locker2_add_pending x3 x2 x1 := F0 M.locker2_add_pending (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition locker2_adduniq x2 x1 := F0 M.locker2_adduniq (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_check_available x2 x1 := F0 M.locker2_check_available (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_check_availables x2 x1 := F0 M.locker2_check_availables (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_claim_lock x3 x2 x1 := F0 M.locker2_claim_lock (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition locker2_map_add_pending x3 x2 x1 := F0 M.locker2_map_add_pending (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition locker2_map_claim_lock x3 x2 x1 := F0 M.locker2_map_claim_lock (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition locker2_map_promote_pending x2 x1 := F0 M.locker2_map_promote_pending (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_obtainable x2 x1 := F0 M.locker2_obtainable (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_obtainables x2 x1 := F0 M.locker2_obtainables (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_promote_pending x2 x1 := F0 M.locker2_promote_pending (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_release_lock x2 x1 := F0 M.locker2_release_lock (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_remove_pending x2 x1 := F0 M.locker2_remove_pending (Vcons x2 (Vcons x1 Vnil)).
  Definition mcrlrecord := F0 M.mcrlrecord Vnil.
  Definition member x2 x1 := F0 M.member (Vcons x2 (Vcons x1 Vnil)).
  Definition nil := F0 M.nil Vnil.
  Definition nocalls := F0 M.nocalls Vnil.
  Definition not x1 := F0 M.not (Vcons x1 Vnil).
  Definition ok := F0 M.ok Vnil.
  Definition or x2 x1 := F0 M.or (Vcons x2 (Vcons x1 Vnil)).
  Definition pending := F0 M.pending Vnil.
  Definition pid x1 := F0 M.pid (Vcons x1 Vnil).
  Definition pops x1 := F0 M.pops (Vcons x1 Vnil).
  Definition push x3 x2 x1 := F0 M.push (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition push1 x6 x5 x4 x3 x2 x1 := F0 M.push1 (Vcons x6 (Vcons x5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))))).
  Definition pushs x2 x1 := F0 M.pushs (Vcons x2 (Vcons x1 Vnil)).
  Definition record_extract x3 x2 x1 := F0 M.record_extract (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition record_new x1 := F0 M.record_new (Vcons x1 Vnil).
  Definition record_update x4 x3 x2 x1 := F0 M.record_update (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition record_updates x3 x2 x1 := F0 M.record_updates (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition release := F0 M.release Vnil.
  Definition request := F0 M.request Vnil.
  Definition resource := F0 M.resource Vnil.
  Definition s x1 := F0 M.s (Vcons x1 Vnil).
  Definition stack x2 x1 := F0 M.stack (Vcons x2 (Vcons x1 Vnil)).
  Definition subtract x2 x1 := F0 M.subtract (Vcons x2 (Vcons x1 Vnil)).
  Definition tag := F0 M.tag Vnil.
  Definition tops x1 := F0 M.tops (Vcons x1 Vnil).
  Definition true := F0 M.true Vnil.
  Definition tuple x2 x1 := F0 M.tuple (Vcons x2 (Vcons x1 Vnil)).
  Definition tuplenil x1 := F0 M.tuplenil (Vcons x1 Vnil).
  Definition undefined := F0 M.undefined Vnil.
End S0.

Definition E :=
   @nil (@ATrs.rule s0).

Definition R :=
   R0 (S0.or S0.T S0.T)
      S0.T
:: R0 (S0.or S0.F S0.T)
      S0.T
:: R0 (S0.or S0.T S0.F)
      S0.T
:: R0 (S0.or S0.F S0.F)
      S0.F
:: R0 (S0.and S0.T (V0 0))
      (V0 0)
:: R0 (S0.and (V0 0) S0.T)
      (V0 0)
:: R0 (S0.and S0.F (V0 0))
      S0.F
:: R0 (S0.and (V0 0) S0.F)
      S0.F
:: R0 (S0.imp S0.T (V0 0))
      (V0 0)
:: R0 (S0.imp S0.F (V0 0))
      S0.T
:: R0 (S0.not S0.T)
      S0.F
:: R0 (S0.not S0.F)
      S0.T
:: R0 (S0._if_2 S0.T (V0 0) (V0 1))
      (V0 0)
:: R0 (S0._if_2 S0.F (V0 0) (V0 1))
      (V0 1)
:: R0 (S0.eq S0.T S0.T)
      S0.T
:: R0 (S0.eq S0.F S0.F)
      S0.T
:: R0 (S0.eq S0.T S0.F)
      S0.F
:: R0 (S0.eq S0.F S0.T)
      S0.F
:: R0 (S0.eqt S0.nil S0.undefined)
      S0.F
:: R0 (S0.eqt S0.nil (S0.pid (V0 0)))
      S0.F
:: R0 (S0.eqt S0.nil (S0.int (V0 0)))
      S0.F
:: R0 (S0.eqt S0.nil (S0.cons (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.nil (S0.tuple (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.nil (S0.tuplenil (V0 0)))
      S0.F
:: R0 (S0.eqt S0.a S0.nil)
      S0.F
:: R0 (S0.eqt S0.a S0.a)
      S0.T
:: R0 (S0.eqt S0.a S0.excl)
      S0.F
:: R0 (S0.eqt S0.a S0.false)
      S0.F
:: R0 (S0.eqt S0.a S0.lock)
      S0.F
:: R0 (S0.eqt S0.a S0.locker)
      S0.F
:: R0 (S0.eqt S0.a S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt S0.a S0.ok)
      S0.F
:: R0 (S0.eqt S0.a S0.pending)
      S0.F
:: R0 (S0.eqt S0.a S0.release)
      S0.F
:: R0 (S0.eqt S0.a S0.request)
      S0.F
:: R0 (S0.eqt S0.a S0.resource)
      S0.F
:: R0 (S0.eqt S0.a S0.tag)
      S0.F
:: R0 (S0.eqt S0.a S0.true)
      S0.F
:: R0 (S0.eqt S0.a S0.undefined)
      S0.F
:: R0 (S0.eqt S0.a (S0.pid (V0 0)))
      S0.F
:: R0 (S0.eqt S0.a (S0.int (V0 0)))
      S0.F
:: R0 (S0.eqt S0.a (S0.cons (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.a (S0.tuple (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.a (S0.tuplenil (V0 0)))
      S0.F
:: R0 (S0.eqt S0.excl S0.nil)
      S0.F
:: R0 (S0.eqt S0.excl S0.a)
      S0.F
:: R0 (S0.eqt S0.excl S0.excl)
      S0.T
:: R0 (S0.eqt S0.excl S0.false)
      S0.F
:: R0 (S0.eqt S0.excl S0.lock)
      S0.F
:: R0 (S0.eqt S0.excl S0.locker)
      S0.F
:: R0 (S0.eqt S0.excl S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt S0.excl S0.ok)
      S0.F
:: R0 (S0.eqt S0.excl S0.pending)
      S0.F
:: R0 (S0.eqt S0.excl S0.release)
      S0.F
:: R0 (S0.eqt S0.excl S0.request)
      S0.F
:: R0 (S0.eqt S0.excl S0.resource)
      S0.F
:: R0 (S0.eqt S0.excl S0.tag)
      S0.F
:: R0 (S0.eqt S0.excl S0.true)
      S0.F
:: R0 (S0.eqt S0.excl S0.undefined)
      S0.F
:: R0 (S0.eqt S0.excl (S0.pid (V0 0)))
      S0.F
:: R0 (S0.eqt S0.excl (S0.eqt S0.false (S0.int (V0 0))))
      S0.F
:: R0 (S0.eqt S0.false (S0.cons (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.false (S0.tuple (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.false (S0.tuplenil (V0 0)))
      S0.F
:: R0 (S0.eqt S0.lock S0.nil)
      S0.F
:: R0 (S0.eqt S0.lock S0.a)
      S0.F
:: R0 (S0.eqt S0.lock S0.excl)
      S0.F
:: R0 (S0.eqt S0.lock S0.false)
      S0.F
:: R0 (S0.eqt S0.lock S0.lock)
      S0.T
:: R0 (S0.eqt S0.lock S0.locker)
      S0.F
:: R0 (S0.eqt S0.lock S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt S0.lock S0.ok)
      S0.F
:: R0 (S0.eqt S0.lock S0.pending)
      S0.F
:: R0 (S0.eqt S0.lock S0.release)
      S0.F
:: R0 (S0.eqt S0.lock S0.request)
      S0.F
:: R0 (S0.eqt S0.lock S0.resource)
      S0.F
:: R0 (S0.eqt S0.lock S0.tag)
      S0.F
:: R0 (S0.eqt S0.lock S0.true)
      S0.F
:: R0 (S0.eqt S0.lock S0.undefined)
      S0.F
:: R0 (S0.eqt S0.lock (S0.pid (V0 0)))
      S0.F
:: R0 (S0.eqt S0.lock (S0.int (V0 0)))
      S0.F
:: R0 (S0.eqt S0.lock (S0.cons (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.lock (S0.tuple (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.lock (S0.tuplenil (V0 0)))
      S0.F
:: R0 (S0.eqt S0.locker S0.nil)
      S0.F
:: R0 (S0.eqt S0.locker S0.a)
      S0.F
:: R0 (S0.eqt S0.locker S0.excl)
      S0.F
:: R0 (S0.eqt S0.locker S0.false)
      S0.F
:: R0 (S0.eqt S0.locker S0.lock)
      S0.F
:: R0 (S0.eqt S0.locker S0.locker)
      S0.T
:: R0 (S0.eqt S0.locker S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt S0.locker S0.ok)
      S0.F
:: R0 (S0.eqt S0.locker S0.pending)
      S0.F
:: R0 (S0.eqt S0.locker S0.release)
      S0.F
:: R0 (S0.eqt S0.locker S0.request)
      S0.F
:: R0 (S0.eqt S0.locker S0.resource)
      S0.F
:: R0 (S0.eqt S0.locker S0.tag)
      S0.F
:: R0 (S0.eqt S0.locker S0.true)
      S0.F
:: R0 (S0.eqt S0.locker S0.undefined)
      S0.F
:: R0 (S0.eqt S0.locker (S0.pid (V0 0)))
      S0.F
:: R0 (S0.eqt S0.locker (S0.int (V0 0)))
      S0.F
:: R0 (S0.eqt S0.locker (S0.cons (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.locker (S0.tuple (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.locker (S0.tuplenil (V0 0)))
      S0.F
:: R0 (S0.eqt S0.mcrlrecord S0.nil)
      S0.F
:: R0 (S0.eqt S0.mcrlrecord S0.a)
      S0.F
:: R0 (S0.eqt S0.mcrlrecord S0.excl)
      S0.F
:: R0 (S0.eqt S0.mcrlrecord S0.false)
      S0.F
:: R0 (S0.eqt S0.mcrlrecord S0.lock)
      S0.F
:: R0 (S0.eqt S0.mcrlrecord S0.locker)
      S0.F
:: R0 (S0.eqt S0.mcrlrecord S0.mcrlrecord)
      S0.T
:: R0 (S0.eqt S0.mcrlrecord S0.ok)
      S0.F
:: R0 (S0.eqt S0.mcrlrecord S0.pending)
      S0.F
:: R0 (S0.eqt S0.mcrlrecord S0.release)
      S0.F
:: R0 (S0.eqt S0.mcrlrecord S0.request)
      S0.F
:: R0 (S0.eqt S0.mcrlrecord S0.resource)
      S0.F
:: R0 (S0.eqt S0.ok S0.resource)
      S0.F
:: R0 (S0.eqt S0.ok S0.tag)
      S0.F
:: R0 (S0.eqt S0.ok S0.true)
      S0.F
:: R0 (S0.eqt S0.ok S0.undefined)
      S0.F
:: R0 (S0.eqt S0.ok (S0.pid (V0 0)))
      S0.F
:: R0 (S0.eqt S0.ok (S0.int (V0 0)))
      S0.F
:: R0 (S0.eqt S0.ok (S0.cons (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.ok (S0.tuple (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.ok (S0.tuplenil (V0 0)))
      S0.F
:: R0 (S0.eqt S0.pending S0.nil)
      S0.F
:: R0 (S0.eqt S0.pending S0.a)
      S0.F
:: R0 (S0.eqt S0.pending S0.excl)
      S0.F
:: R0 (S0.eqt S0.pending S0.false)
      S0.F
:: R0 (S0.eqt S0.pending S0.lock)
      S0.F
:: R0 (S0.eqt S0.pending S0.locker)
      S0.F
:: R0 (S0.eqt S0.pending S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt S0.pending S0.ok)
      S0.F
:: R0 (S0.eqt S0.pending S0.pending)
      S0.T
:: R0 (S0.eqt S0.pending S0.release)
      S0.F
:: R0 (S0.eqt S0.pending S0.request)
      S0.F
:: R0 (S0.eqt S0.pending S0.resource)
      S0.F
:: R0 (S0.eqt S0.pending S0.tag)
      S0.F
:: R0 (S0.eqt S0.pending S0.true)
      S0.F
:: R0 (S0.eqt S0.pending S0.undefined)
      S0.F
:: R0 (S0.eqt S0.pending (S0.pid (V0 0)))
      S0.F
:: R0 (S0.eqt S0.pending (S0.int (V0 0)))
      S0.F
:: R0 (S0.eqt S0.pending (S0.cons (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.pending (S0.tuple (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.pending (S0.tuplenil (V0 0)))
      S0.F
:: R0 (S0.eqt S0.release S0.nil)
      S0.F
:: R0 (S0.eqt S0.release S0.a)
      S0.F
:: R0 (S0.eqt S0.release S0.excl)
      S0.F
:: R0 (S0.eqt S0.release S0.false)
      S0.F
:: R0 (S0.eqt S0.release S0.lock)
      S0.F
:: R0 (S0.eqt S0.release S0.locker)
      S0.F
:: R0 (S0.eqt S0.release S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt S0.release S0.ok)
      S0.F
:: R0 (S0.eqt S0.request S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt S0.request S0.ok)
      S0.F
:: R0 (S0.eqt S0.request S0.pending)
      S0.F
:: R0 (S0.eqt S0.request S0.release)
      S0.F
:: R0 (S0.eqt S0.request S0.request)
      S0.T
:: R0 (S0.eqt S0.request S0.resource)
      S0.F
:: R0 (S0.eqt S0.request S0.tag)
      S0.F
:: R0 (S0.eqt S0.request S0.true)
      S0.F
:: R0 (S0.eqt S0.request S0.undefined)
      S0.F
:: R0 (S0.eqt S0.request (S0.pid (V0 0)))
      S0.F
:: R0 (S0.eqt S0.request (S0.int (V0 0)))
      S0.F
:: R0 (S0.eqt S0.request (S0.cons (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.request (S0.tuple (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.request (S0.tuplenil (V0 0)))
      S0.F
:: R0 (S0.eqt S0.resource S0.nil)
      S0.F
:: R0 (S0.eqt S0.resource S0.a)
      S0.F
:: R0 (S0.eqt S0.resource S0.excl)
      S0.F
:: R0 (S0.eqt S0.resource S0.false)
      S0.F
:: R0 (S0.eqt S0.resource S0.lock)
      S0.F
:: R0 (S0.eqt S0.resource S0.locker)
      S0.F
:: R0 (S0.eqt S0.resource S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt S0.resource S0.ok)
      S0.F
:: R0 (S0.eqt S0.resource S0.pending)
      S0.F
:: R0 (S0.eqt S0.resource S0.release)
      S0.F
:: R0 (S0.eqt S0.resource S0.request)
      S0.F
:: R0 (S0.eqt S0.resource S0.resource)
      S0.T
:: R0 (S0.eqt S0.resource S0.tag)
      S0.F
:: R0 (S0.eqt S0.resource S0.true)
      S0.F
:: R0 (S0.eqt S0.resource S0.undefined)
      S0.F
:: R0 (S0.eqt S0.resource (S0.pid (V0 0)))
      S0.F
:: R0 (S0.eqt S0.resource (S0.int (V0 0)))
      S0.F
:: R0 (S0.eqt S0.resource (S0.cons (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.resource (S0.tuple (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.resource (S0.tuplenil (V0 0)))
      S0.F
:: R0 (S0.eqt S0.tag S0.nil)
      S0.F
:: R0 (S0.eqt S0.tag S0.a)
      S0.F
:: R0 (S0.eqt S0.tag S0.excl)
      S0.F
:: R0 (S0.eqt S0.tag S0.false)
      S0.F
:: R0 (S0.eqt S0.tag S0.lock)
      S0.F
:: R0 (S0.eqt S0.tag S0.locker)
      S0.F
:: R0 (S0.eqt S0.tag S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt S0.tag S0.ok)
      S0.F
:: R0 (S0.eqt S0.tag S0.pending)
      S0.F
:: R0 (S0.eqt S0.tag S0.release)
      S0.F
:: R0 (S0.eqt S0.tag S0.request)
      S0.F
:: R0 (S0.eqt S0.tag S0.resource)
      S0.F
:: R0 (S0.eqt S0.tag S0.tag)
      S0.T
:: R0 (S0.eqt S0.tag S0.true)
      S0.F
:: R0 (S0.eqt S0.tag S0.undefined)
      S0.F
:: R0 (S0.eqt S0.tag (S0.pid (V0 0)))
      S0.F
:: R0 (S0.eqt S0.tag (S0.int (V0 0)))
      S0.F
:: R0 (S0.eqt S0.tag (S0.cons (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.tag (S0.tuple (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.tag (S0.tuplenil (V0 0)))
      S0.F
:: R0 (S0.eqt S0.true S0.nil)
      S0.F
:: R0 (S0.eqt S0.true S0.a)
      S0.F
:: R0 (S0.eqt S0.true S0.excl)
      S0.F
:: R0 (S0.eqt S0.true S0.false)
      S0.F
:: R0 (S0.eqt S0.true S0.lock)
      S0.F
:: R0 (S0.eqt S0.true S0.locker)
      S0.F
:: R0 (S0.eqt S0.true S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt S0.true S0.ok)
      S0.F
:: R0 (S0.eqt S0.true S0.pending)
      S0.F
:: R0 (S0.eqt S0.true S0.release)
      S0.F
:: R0 (S0.eqt S0.true S0.request)
      S0.F
:: R0 (S0.eqt S0.true S0.resource)
      S0.F
:: R0 (S0.eqt S0.true S0.tag)
      S0.F
:: R0 (S0.eqt S0.true S0.true)
      S0.T
:: R0 (S0.eqt S0.true S0.undefined)
      S0.F
:: R0 (S0.eqt S0.true (S0.pid (V0 0)))
      S0.F
:: R0 (S0.eqt S0.true (S0.int (V0 0)))
      S0.F
:: R0 (S0.eqt S0.true (S0.cons (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.true (S0.tuple (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqt S0.true (S0.tuplenil (V0 0)))
      S0.F
:: R0 (S0.eqt S0.undefined S0.nil)
      S0.F
:: R0 (S0.eqt S0.undefined S0.a)
      S0.F
:: R0 (S0.eqt S0.undefined (S0.tuplenil (V0 0)))
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.nil)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.a)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.excl)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.false)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.lock)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.locker)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.ok)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.pending)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.release)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.request)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.resource)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.tag)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.true)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) S0.undefined)
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) (S0.pid (V0 1)))
      (S0.eqt (V0 0) (V0 1))
:: R0 (S0.eqt (S0.pid (V0 0)) (S0.int (V0 1)))
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) (S0.cons (V0 1) (V0 2)))
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) (S0.tuple (V0 1) (V0 2)))
      S0.F
:: R0 (S0.eqt (S0.pid (V0 0)) (S0.tuplenil (V0 1)))
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.nil)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.a)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.excl)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.false)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.lock)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.locker)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.ok)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.pending)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.release)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.request)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.resource)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.tag)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.true)
      S0.F
:: R0 (S0.eqt (S0.int (V0 0)) S0.undefined)
      S0.F
:: R0 (S0.eqt (S0.cons (V0 0) (V0 1)) S0.resource)
      S0.F
:: R0 (S0.eqt (S0.cons (V0 0) (V0 1)) S0.tag)
      S0.F
:: R0 (S0.eqt (S0.cons (V0 0) (V0 1)) S0.true)
      S0.F
:: R0 (S0.eqt (S0.cons (V0 0) (V0 1)) S0.undefined)
      S0.F
:: R0 (S0.eqt (S0.cons (V0 0) (V0 1)) (S0.pid (V0 2)))
      S0.F
:: R0 (S0.eqt (S0.cons (V0 0) (V0 1)) (S0.int (V0 2)))
      S0.F
:: R0 (S0.eqt (S0.cons (V0 0) (V0 1)) (S0.cons (V0 2) (V0 3)))
      (S0.and (S0.eqt (V0 0) (V0 2)) (S0.eqt (V0 1) (V0 3)))
:: R0 (S0.eqt (S0.cons (V0 0) (V0 1)) (S0.tuple (V0 2) (V0 3)))
      S0.F
:: R0 (S0.eqt (S0.cons (V0 0) (V0 1)) (S0.tuplenil (V0 2)))
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.nil)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.a)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.excl)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.false)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.lock)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.locker)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.ok)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.pending)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.release)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.request)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.resource)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.tag)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.true)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) S0.undefined)
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) (S0.pid (V0 2)))
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) (S0.int (V0 2)))
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) (S0.cons (V0 2) (V0 3)))
      S0.F
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) (S0.tuple (V0 2) (V0 3)))
      (S0.and (S0.eqt (V0 0) (V0 2)) (S0.eqt (V0 1) (V0 3)))
:: R0 (S0.eqt (S0.tuple (V0 0) (V0 1)) (S0.tuplenil (V0 2)))
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.nil)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.a)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.excl)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.false)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.lock)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.locker)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.mcrlrecord)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.ok)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.pending)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.release)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.request)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.resource)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.tag)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.true)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) S0.undefined)
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) (S0.pid (V0 1)))
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) (S0.int (V0 1)))
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) (S0.cons (V0 1) (V0 2)))
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) (S0.tuple (V0 1) (V0 2)))
      S0.F
:: R0 (S0.eqt (S0.tuplenil (V0 0)) (S0.tuplenil (V0 1)))
      (S0.eqt (V0 0) (V0 1))
:: R0 (S0.element (S0.int (S0.s S0._0_1)) (S0.tuplenil (V0 0)))
      (V0 0)
:: R0 (S0.element (S0.int (S0.s S0._0_1)) (S0.tuple (V0 0) (V0 1)))
      (V0 0)
:: R0 (S0.element (S0.int (S0.s (S0.s (V0 0)))) (S0.tuple (V0 1) (V0 2)))
      (S0.element (S0.int (S0.s (V0 0))) (V0 2))
:: R0 (S0.record_new S0.lock)
      (S0.tuple S0.mcrlrecord (S0.tuple S0.lock (S0.tuple S0.undefined (S0.tuple S0.nil (S0.tuplenil S0.nil)))))
:: R0 (S0.record_extract (S0.tuple S0.mcrlrecord (S0.tuple S0.lock (S0.tuple (V0 0) (S0.tuple (V0 1) (S0.tuplenil (V0 2)))))) S0.lock S0.resource)
      (S0.tuple S0.mcrlrecord (S0.tuple S0.lock (S0.tuple (V0 0) (S0.tuple (V0 1) (S0.tuplenil (V0 2))))))
:: R0 (S0.record_update (S0.tuple S0.mcrlrecord (S0.tuple S0.lock (S0.tuple (V0 0) (S0.tuple (V0 1) (S0.tuplenil (V0 2)))))) S0.lock S0.pending (V0 3))
      (S0.tuple S0.mcrlrecord (S0.tuple S0.lock (S0.tuple (V0 0) (S0.tuple (V0 1) (S0.tuplenil (V0 3))))))
:: R0 (S0.record_updates (V0 0) (V0 1) S0.nil)
      (V0 0)
:: R0 (S0.record_updates (V0 0) (V0 1) (S0.cons (S0.tuple (V0 2) (S0.tuplenil (V0 3))) (V0 4)))
      (S0.record_updates (S0.record_update (V0 0) (V0 1) (V0 2) (V0 3)) (V0 1) (V0 4))
:: R0 (S0.locker2_map_promote_pending S0.nil (V0 0))
      S0.nil
:: R0 (S0.locker2_map_promote_pending (S0.cons (V0 0) (V0 1)) (V0 2))
      (S0.cons (S0.locker2_promote_pending (V0 0) (V0 2)) (S0.locker2_map_promote_pending (V0 1) (V0 2)))
:: R0 (S0.locker2_map_claim_lock S0.nil (V0 0) (V0 1))
      S0.nil
:: R0 (S0.locker2_map_claim_lock (S0.cons (V0 0) (V0 1)) (V0 2) (V0 3))
      (S0.cons (S0.locker2_claim_lock (V0 0) (V0 2) (V0 3)) (S0.locker2_map_claim_lock (V0 1) (V0 2) (V0 3)))
:: R0 (S0.locker2_map_add_pending S0.nil (V0 0) (V0 1))
      S0.nil
:: R0 (S0.locker2_promote_pending (V0 0) (V0 1))
      (S0.case0 (V0 1) (V0 0) (S0.record_extract (V0 0) S0.lock S0.pending))
:: R0 (S0.case0 (V0 0) (V0 1) (S0.cons (V0 0) (V0 3)))
      (S0.record_updates (V0 1) S0.lock (S0.cons (S0.tuple S0.excl (S0.tuplenil (V0 0))) (S0.cons (S0.tuple S0.pending (S0.tuplenil (V0 3))) S0.nil)))
:: R0 (S0.case0 (V0 0) (V0 1) (V0 2))
      (V0 1)
:: R0 (S0.locker2_remove_pending (V0 0) (V0 1))
      (S0.record_updates (V0 0) S0.lock (S0.cons (S0.tuple S0.pending (S0.tuplenil (S0.subtract (S0.record_extract (V0 0) S0.lock S0.pending) (S0.cons (V0 1) S0.nil)))) S0.nil))
:: R0 (S0.locker2_add_pending (V0 0) (V0 1) (V0 2))
      (S0.case1 (V0 2) (V0 1) (V0 0) (S0.member (S0.record_extract (V0 0) S0.lock S0.resource) (V0 1)))
:: R0 (S0.case1 (V0 0) (V0 1) (V0 2) S0.true)
      (S0.record_updates (V0 2) S0.lock (S0.cons (S0.tuple S0.pending (S0.tuplenil (S0.append (S0.record_extract (V0 2) S0.lock S0.pending) (S0.cons (V0 0) S0.nil)))) S0.nil))
:: R0 (S0.case1 (V0 0) (V0 1) (V0 2) S0.false)
      (V0 2)
:: R0 (S0.locker2_release_lock (V0 0) (V0 1))
      (S0.case2 (V0 1) (V0 0) (S0.gen_modtageq (V0 1) (S0.record_extract (V0 0) S0.lock S0.excl)))
:: R0 (S0.case2 (V0 0) (V0 1) S0.true)
      (S0.record_updates (V0 1) S0.lock (S0.cons (S0.tuple S0.excllock S0.excl) S0.nil))
:: R0 (S0.case4 (V0 0) (V0 1) (V0 2))
      S0.false
:: R0 (S0.locker2_obtainables S0.nil (V0 0))
      S0.true
:: R0 (S0.locker2_obtainables (S0.cons (V0 0) (V0 1)) (V0 2))
      (S0.case5 (V0 2) (V0 1) (V0 0) (S0.member (V0 2) (S0.record_extract (V0 0) S0.lock S0.pending)))
:: R0 (S0.case5 (V0 0) (V0 1) (V0 2) S0.true)
      (S0.andt (S0.locker2_obtainable (V0 2) (V0 0)) (S0.locker2_obtainables (V0 1) (V0 0)))
:: R0 (S0.case5 (V0 0) (V0 1) (V0 2) S0.false)
      (S0.locker2_obtainables (V0 1) (V0 0))
:: R0 (S0.locker2_check_available (V0 0) S0.nil)
      S0.false
:: R0 (S0.locker2_check_available (V0 0) (S0.cons (V0 1) (V0 2)))
      (S0.case6 (V0 2) (V0 1) (V0 0) (S0.equal (V0 0) (S0.record_extract (V0 1) S0.lock S0.resource)))
:: R0 (S0.case6 (V0 0) (V0 1) (V0 2) S0.true)
      (S0.andt (S0.equal (S0.record_extract (V0 1) S0.lock S0.excl) S0.nil) (S0.equal (S0.record_extract (V0 1) S0.lock S0.pending) S0.nil))
:: R0 (S0.case6 (V0 0) (V0 1) (V0 2) S0.false)
      (S0.locker2_check_available (V0 2) (V0 0))
:: R0 (S0.locker2_check_availables S0.nil (V0 0))
      S0.true
:: R0 (S0.locker2_check_availables (S0.cons (V0 0) (V0 1)) (V0 2))
      (S0.andt (S0.locker2_check_available (V0 0) (V0 2)) (S0.locker2_check_availables (V0 1) (V0 2)))
:: R0 (S0.locker2_adduniq S0.nil (V0 0))
      (V0 0)
:: R0 (S0.append (S0.cons (V0 0) (V0 1)) (V0 2))
      (S0.cons (V0 0) (S0.append (V0 1) (V0 2)))
:: R0 (S0.subtract (V0 0) S0.nil)
      (V0 0)
:: R0 (S0.subtract (V0 0) (S0.cons (V0 1) (V0 2)))
      (S0.subtract (S0.delete (V0 1) (V0 0)) (V0 2))
:: R0 (S0.delete (V0 0) S0.nil)
      S0.nil
:: R0 (S0.delete (V0 0) (S0.cons (V0 1) (V0 2)))
      (S0.case8 (V0 2) (V0 1) (V0 0) (S0.equal (V0 0) (V0 1)))
:: R0 (S0.case8 (V0 0) (V0 1) (V0 2) S0.true)
      (V0 0)
:: R0 (S0.case8 (V0 0) (V0 1) (V0 2) S0.false)
      (S0.cons (V0 1) (S0.delete (V0 2) (V0 0)))
:: R0 (S0.gen_tag (V0 0))
      (S0.tuple (V0 0) (S0.tuplenil S0.tag))
:: R0 (S0.gen_modtageq (V0 0) (V0 1))
      (S0.equal (V0 0) (V0 1))
:: R0 (S0.member (V0 0) S0.nil)
      S0.false
:: R0 (S0.member (V0 0) (S0.cons (V0 1) (V0 2)))
      (S0.case9 (V0 2) (V0 1) (V0 0) (S0.equal (V0 0) (V0 1)))
:: R0 (S0.case9 (V0 0) (V0 1) (V0 2) S0.true)
      S0.true
:: R0 (S0.case9 (V0 0) (V0 1) (V0 2) S0.false)
      (S0.member (V0 2) (V0 0))
:: R0 (S0.eqs S0.empty S0.empty)
      S0.T
:: R0 (S0.eqs S0.empty (S0.stack (V0 0) (V0 1)))
      S0.F
:: R0 (S0.eqs (S0.stack (V0 0) (V0 1)) S0.empty)
      S0.F
:: R0 (S0.eqs (S0.stack (V0 0) (V0 1)) (S0.stack (V0 2) (V0 3)))
      (S0.and (S0.eqt (V0 0) (V0 2)) (S0.eqs (V0 1) (V0 3)))
:: R0 (S0.pushs (V0 0) (V0 1))
      (S0.stack (V0 0) (V0 1))
:: R0 (S0.pops (S0.stack (V0 0) (V0 1)))
      (V0 1)
:: R0 (S0.tops (S0.stack (V0 0) (V0 1)))
      (V0 0)
:: R0 (S0.istops (V0 0) S0.empty)
      S0.F
:: R0 (S0.istops (V0 0) (S0.stack (V0 1) (V0 2)))
      (S0.eqt (V0 0) (V0 1))
:: R0 (S0.eqc S0.nocalls S0.nocalls)
      S0.T
:: R0 (S0.eqc S0.nocalls (S0.calls (V0 0) (V0 1) (V0 2)))
      S0.F
:: R0 (S0.eqc (S0.calls (V0 0) (V0 1) (V0 2)) S0.nocalls)
      S0.F
:: R0 (S0.eqc (S0.calls (V0 0) (V0 1) (V0 2)) (S0.calls (V0 3) (V0 4) (V0 5)))
      (S0.and (S0.eqt (V0 0) (V0 3)) (S0.and (S0.eqs (V0 1) (V0 4)) (S0.eqc (V0 2) (V0 5))))
:: R0 (S0.push (V0 0) (V0 1) S0.nocalls)
      (S0.calls (V0 0) (S0.stack (V0 1) S0.empty) S0.nocalls)
:: R0 (S0.push (V0 0) (V0 1) (S0.calls (V0 2) (V0 3) (V0 4)))
      (S0.push1 (V0 0) (V0 1) (V0 2) (V0 3) (V0 4) (S0.eqt (V0 0) (V0 2)))
:: R0 (S0.push1 (V0 0) (V0 1) (V0 2) (V0 3) (V0 4) S0.T)
      (S0.calls (V0 2) (S0.pushs (V0 1) (V0 3)) (V0 4))
:: @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 hF := F1 (hd_symb s1_p M.F) Vnil.
  Definition F := F1 (int_symb s1_p M.F) Vnil.
  Definition hT := F1 (hd_symb s1_p M.T) Vnil.
  Definition T := F1 (int_symb s1_p M.T) 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 h_if_2 x3 x2 x1 := F1 (hd_symb s1_p M._if_2) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition _if_2 x3 x2 x1 := F1 (int_symb s1_p M._if_2) (Vcons x3 (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 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 handt x2 x1 := F1 (hd_symb s1_p M.andt) (Vcons x2 (Vcons x1 Vnil)).
  Definition andt x2 x1 := F1 (int_symb s1_p M.andt) (Vcons x2 (Vcons x1 Vnil)).
  Definition happend x2 x1 := F1 (hd_symb s1_p M.append) (Vcons x2 (Vcons x1 Vnil)).
  Definition append x2 x1 := F1 (int_symb s1_p M.append) (Vcons x2 (Vcons x1 Vnil)).
  Definition hcalls x3 x2 x1 := F1 (hd_symb s1_p M.calls) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition calls x3 x2 x1 := F1 (int_symb s1_p M.calls) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hcase0 x3 x2 x1 := F1 (hd_symb s1_p M.case0) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition case0 x3 x2 x1 := F1 (int_symb s1_p M.case0) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hcase1 x4 x3 x2 x1 := F1 (hd_symb s1_p M.case1) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition case1 x4 x3 x2 x1 := F1 (int_symb s1_p M.case1) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition hcase2 x3 x2 x1 := F1 (hd_symb s1_p M.case2) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition case2 x3 x2 x1 := F1 (int_symb s1_p M.case2) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hcase4 x3 x2 x1 := F1 (hd_symb s1_p M.case4) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition case4 x3 x2 x1 := F1 (int_symb s1_p M.case4) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hcase5 x4 x3 x2 x1 := F1 (hd_symb s1_p M.case5) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition case5 x4 x3 x2 x1 := F1 (int_symb s1_p M.case5) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition hcase6 x4 x3 x2 x1 := F1 (hd_symb s1_p M.case6) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition case6 x4 x3 x2 x1 := F1 (int_symb s1_p M.case6) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition hcase8 x4 x3 x2 x1 := F1 (hd_symb s1_p M.case8) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition case8 x4 x3 x2 x1 := F1 (int_symb s1_p M.case8) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition hcase9 x4 x3 x2 x1 := F1 (hd_symb s1_p M.case9) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition case9 x4 x3 x2 x1 := F1 (int_symb s1_p M.case9) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 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 hdelete x2 x1 := F1 (hd_symb s1_p M.delete) (Vcons x2 (Vcons x1 Vnil)).
  Definition delete x2 x1 := F1 (int_symb s1_p M.delete) (Vcons x2 (Vcons x1 Vnil)).
  Definition helement x2 x1 := F1 (hd_symb s1_p M.element) (Vcons x2 (Vcons x1 Vnil)).
  Definition element x2 x1 := F1 (int_symb s1_p M.element) (Vcons x2 (Vcons x1 Vnil)).
  Definition hempty := F1 (hd_symb s1_p M.empty) Vnil.
  Definition empty := F1 (int_symb s1_p M.empty) Vnil.
  Definition heq x2 x1 := F1 (hd_symb s1_p M.eq) (Vcons x2 (Vcons x1 Vnil)).
  Definition eq x2 x1 := F1 (int_symb s1_p M.eq) (Vcons x2 (Vcons x1 Vnil)).
  Definition heqc x2 x1 := F1 (hd_symb s1_p M.eqc) (Vcons x2 (Vcons x1 Vnil)).
  Definition eqc x2 x1 := F1 (int_symb s1_p M.eqc) (Vcons x2 (Vcons x1 Vnil)).
  Definition heqs x2 x1 := F1 (hd_symb s1_p M.eqs) (Vcons x2 (Vcons x1 Vnil)).
  Definition eqs x2 x1 := F1 (int_symb s1_p M.eqs) (Vcons x2 (Vcons x1 Vnil)).
  Definition heqt x2 x1 := F1 (hd_symb s1_p M.eqt) (Vcons x2 (Vcons x1 Vnil)).
  Definition eqt x2 x1 := F1 (int_symb s1_p M.eqt) (Vcons x2 (Vcons x1 Vnil)).
  Definition hequal x2 x1 := F1 (hd_symb s1_p M.equal) (Vcons x2 (Vcons x1 Vnil)).
  Definition equal x2 x1 := F1 (int_symb s1_p M.equal) (Vcons x2 (Vcons x1 Vnil)).
  Definition hexcl := F1 (hd_symb s1_p M.excl) Vnil.
  Definition excl := F1 (int_symb s1_p M.excl) Vnil.
  Definition hexcllock := F1 (hd_symb s1_p M.excllock) Vnil.
  Definition excllock := F1 (int_symb s1_p M.excllock) Vnil.
  Definition hfalse := F1 (hd_symb s1_p M.false) Vnil.
  Definition false := F1 (int_symb s1_p M.false) Vnil.
  Definition hgen_modtageq x2 x1 := F1 (hd_symb s1_p M.gen_modtageq) (Vcons x2 (Vcons x1 Vnil)).
  Definition gen_modtageq x2 x1 := F1 (int_symb s1_p M.gen_modtageq) (Vcons x2 (Vcons x1 Vnil)).
  Definition hgen_tag x1 := F1 (hd_symb s1_p M.gen_tag) (Vcons x1 Vnil).
  Definition gen_tag x1 := F1 (int_symb s1_p M.gen_tag) (Vcons x1 Vnil).
  Definition himp x2 x1 := F1 (hd_symb s1_p M.imp) (Vcons x2 (Vcons x1 Vnil)).
  Definition imp x2 x1 := F1 (int_symb s1_p M.imp) (Vcons x2 (Vcons x1 Vnil)).
  Definition hint x1 := F1 (hd_symb s1_p M.int) (Vcons x1 Vnil).
  Definition int x1 := F1 (int_symb s1_p M.int) (Vcons x1 Vnil).
  Definition histops x2 x1 := F1 (hd_symb s1_p M.istops) (Vcons x2 (Vcons x1 Vnil)).
  Definition istops x2 x1 := F1 (int_symb s1_p M.istops) (Vcons x2 (Vcons x1 Vnil)).
  Definition hlock := F1 (hd_symb s1_p M.lock) Vnil.
  Definition lock := F1 (int_symb s1_p M.lock) Vnil.
  Definition hlocker := F1 (hd_symb s1_p M.locker) Vnil.
  Definition locker := F1 (int_symb s1_p M.locker) Vnil.
  Definition hlocker2_add_pending x3 x2 x1 := F1 (hd_symb s1_p M.locker2_add_pending) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition locker2_add_pending x3 x2 x1 := F1 (int_symb s1_p M.locker2_add_pending) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hlocker2_adduniq x2 x1 := F1 (hd_symb s1_p M.locker2_adduniq) (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_adduniq x2 x1 := F1 (int_symb s1_p M.locker2_adduniq) (Vcons x2 (Vcons x1 Vnil)).
  Definition hlocker2_check_available x2 x1 := F1 (hd_symb s1_p M.locker2_check_available) (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_check_available x2 x1 := F1 (int_symb s1_p M.locker2_check_available) (Vcons x2 (Vcons x1 Vnil)).
  Definition hlocker2_check_availables x2 x1 := F1 (hd_symb s1_p M.locker2_check_availables) (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_check_availables x2 x1 := F1 (int_symb s1_p M.locker2_check_availables) (Vcons x2 (Vcons x1 Vnil)).
  Definition hlocker2_claim_lock x3 x2 x1 := F1 (hd_symb s1_p M.locker2_claim_lock) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition locker2_claim_lock x3 x2 x1 := F1 (int_symb s1_p M.locker2_claim_lock) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hlocker2_map_add_pending x3 x2 x1 := F1 (hd_symb s1_p M.locker2_map_add_pending) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition locker2_map_add_pending x3 x2 x1 := F1 (int_symb s1_p M.locker2_map_add_pending) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hlocker2_map_claim_lock x3 x2 x1 := F1 (hd_symb s1_p M.locker2_map_claim_lock) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition locker2_map_claim_lock x3 x2 x1 := F1 (int_symb s1_p M.locker2_map_claim_lock) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hlocker2_map_promote_pending x2 x1 := F1 (hd_symb s1_p M.locker2_map_promote_pending) (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_map_promote_pending x2 x1 := F1 (int_symb s1_p M.locker2_map_promote_pending) (Vcons x2 (Vcons x1 Vnil)).
  Definition hlocker2_obtainable x2 x1 := F1 (hd_symb s1_p M.locker2_obtainable) (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_obtainable x2 x1 := F1 (int_symb s1_p M.locker2_obtainable) (Vcons x2 (Vcons x1 Vnil)).
  Definition hlocker2_obtainables x2 x1 := F1 (hd_symb s1_p M.locker2_obtainables) (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_obtainables x2 x1 := F1 (int_symb s1_p M.locker2_obtainables) (Vcons x2 (Vcons x1 Vnil)).
  Definition hlocker2_promote_pending x2 x1 := F1 (hd_symb s1_p M.locker2_promote_pending) (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_promote_pending x2 x1 := F1 (int_symb s1_p M.locker2_promote_pending) (Vcons x2 (Vcons x1 Vnil)).
  Definition hlocker2_release_lock x2 x1 := F1 (hd_symb s1_p M.locker2_release_lock) (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_release_lock x2 x1 := F1 (int_symb s1_p M.locker2_release_lock) (Vcons x2 (Vcons x1 Vnil)).
  Definition hlocker2_remove_pending x2 x1 := F1 (hd_symb s1_p M.locker2_remove_pending) (Vcons x2 (Vcons x1 Vnil)).
  Definition locker2_remove_pending x2 x1 := F1 (int_symb s1_p M.locker2_remove_pending) (Vcons x2 (Vcons x1 Vnil)).
  Definition hmcrlrecord := F1 (hd_symb s1_p M.mcrlrecord) Vnil.
  Definition mcrlrecord := F1 (int_symb s1_p M.mcrlrecord) Vnil.
  Definition hmember x2 x1 := F1 (hd_symb s1_p M.member) (Vcons x2 (Vcons x1 Vnil)).
  Definition member x2 x1 := F1 (int_symb s1_p M.member) (Vcons x2 (Vcons x1 Vnil)).
  Definition hnil := F1 (hd_symb s1_p M.nil) Vnil.
  Definition nil := F1 (int_symb s1_p M.nil) Vnil.
  Definition hnocalls := F1 (hd_symb s1_p M.nocalls) Vnil.
  Definition nocalls := F1 (int_symb s1_p M.nocalls) Vnil.
  Definition hnot x1 := F1 (hd_symb s1_p M.not) (Vcons x1 Vnil).
  Definition not x1 := F1 (int_symb s1_p M.not) (Vcons x1 Vnil).
  Definition hok := F1 (hd_symb s1_p M.ok) Vnil.
  Definition ok := F1 (int_symb s1_p M.ok) Vnil.
  Definition hor x2 x1 := F1 (hd_symb s1_p M.or) (Vcons x2 (Vcons x1 Vnil)).
  Definition or x2 x1 := F1 (int_symb s1_p M.or) (Vcons x2 (Vcons x1 Vnil)).
  Definition hpending := F1 (hd_symb s1_p M.pending) Vnil.
  Definition pending := F1 (int_symb s1_p M.pending) Vnil.
  Definition hpid x1 := F1 (hd_symb s1_p M.pid) (Vcons x1 Vnil).
  Definition pid x1 := F1 (int_symb s1_p M.pid) (Vcons x1 Vnil).
  Definition hpops x1 := F1 (hd_symb s1_p M.pops) (Vcons x1 Vnil).
  Definition pops x1 := F1 (int_symb s1_p M.pops) (Vcons x1 Vnil).
  Definition hpush x3 x2 x1 := F1 (hd_symb s1_p M.push) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition push x3 x2 x1 := F1 (int_symb s1_p M.push) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hpush1 x6 x5 x4 x3 x2 x1 := F1 (hd_symb s1_p M.push1) (Vcons x6 (Vcons x5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))))).
  Definition push1 x6 x5 x4 x3 x2 x1 := F1 (int_symb s1_p M.push1) (Vcons x6 (Vcons x5 (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))))).
  Definition hpushs x2 x1 := F1 (hd_symb s1_p M.pushs) (Vcons x2 (Vcons x1 Vnil)).
  Definition pushs x2 x1 := F1 (int_symb s1_p M.pushs) (Vcons x2 (Vcons x1 Vnil)).
  Definition hrecord_extract x3 x2 x1 := F1 (hd_symb s1_p M.record_extract) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition record_extract x3 x2 x1 := F1 (int_symb s1_p M.record_extract) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hrecord_new x1 := F1 (hd_symb s1_p M.record_new) (Vcons x1 Vnil).
  Definition record_new x1 := F1 (int_symb s1_p M.record_new) (Vcons x1 Vnil).
  Definition hrecord_update x4 x3 x2 x1 := F1 (hd_symb s1_p M.record_update) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition record_update x4 x3 x2 x1 := F1 (int_symb s1_p M.record_update) (Vcons x4 (Vcons x3 (Vcons x2 (Vcons x1 Vnil)))).
  Definition hrecord_updates x3 x2 x1 := F1 (hd_symb s1_p M.record_updates) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition record_updates x3 x2 x1 := F1 (int_symb s1_p M.record_updates) (Vcons x3 (Vcons x2 (Vcons x1 Vnil))).
  Definition hrelease := F1 (hd_symb s1_p M.release) Vnil.
  Definition release := F1 (int_symb s1_p M.release) Vnil.
  Definition hrequest := F1 (hd_symb s1_p M.request) Vnil.
  Definition request := F1 (int_symb s1_p M.request) Vnil.
  Definition hresource := F1 (hd_symb s1_p M.resource) Vnil.
  Definition resource := F1 (int_symb s1_p M.resource) 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 hstack x2 x1 := F1 (hd_symb s1_p M.stack) (Vcons x2 (Vcons x1 Vnil)).
  Definition stack x2 x1 := F1 (int_symb s1_p M.stack) (Vcons x2 (Vcons x1 Vnil)).
  Definition hsubtract x2 x1 := F1 (hd_symb s1_p M.subtract) (Vcons x2 (Vcons x1 Vnil)).
  Definition subtract x2 x1 := F1 (int_symb s1_p M.subtract) (Vcons x2 (Vcons x1 Vnil)).
  Definition htag := F1 (hd_symb s1_p M.tag) Vnil.
  Definition tag := F1 (int_symb s1_p M.tag) Vnil.
  Definition htops x1 := F1 (hd_symb s1_p M.tops) (Vcons x1 Vnil).
  Definition tops x1 := F1 (int_symb s1_p M.tops) (Vcons x1 Vnil).
  Definition htrue := F1 (hd_symb s1_p M.true) Vnil.
  Definition true := F1 (int_symb s1_p M.true) Vnil.
  Definition htuple x2 x1 := F1 (hd_symb s1_p M.tuple) (Vcons x2 (Vcons x1 Vnil)).
  Definition tuple x2 x1 := F1 (int_symb s1_p M.tuple) (Vcons x2 (Vcons x1 Vnil)).
  Definition htuplenil x1 := F1 (hd_symb s1_p M.tuplenil) (Vcons x1 Vnil).
  Definition tuplenil x1 := F1 (int_symb s1_p M.tuplenil) (Vcons x1 Vnil).
  Definition hundefined := F1 (hd_symb s1_p M.undefined) Vnil.
  Definition undefined := F1 (int_symb s1_p M.undefined) Vnil.
End S1.

(* graph decomposition 1 *)

Definition cs1 : list (list (@ATrs.rule s1)) :=

   (  R1 (S1.hpush1 (V1 0) (V1 1) (V1 2) (V1 3) (V1 4) (S1.T))
         (S1.hpushs (V1 1) (V1 3))
   :: nil)

:: (  R1 (S1.hpush (V1 0) (V1 1) (S1.calls (V1 2) (V1 3) (V1 4)))
         (S1.hpush1 (V1 0) (V1 1) (V1 2) (V1 3) (V1 4) (S1.eqt (V1 0) (V1 2)))
   :: nil)

:: (  R1 (S1.heqc (S1.calls (V1 0) (V1 1) (V1 2)) (S1.calls (V1 3) (V1 4) (V1 5)))
         (S1.hand (S1.eqs (V1 1) (V1 4)) (S1.eqc (V1 2) (V1 5)))
   :: nil)

:: (  R1 (S1.heqc (S1.calls (V1 0) (V1 1) (V1 2)) (S1.calls (V1 3) (V1 4) (V1 5)))
         (S1.hand (S1.eqt (V1 0) (V1 3)) (S1.and (S1.eqs (V1 1) (V1 4)) (S1.eqc (V1 2) (V1 5))))
   :: nil)

:: (  R1 (S1.heqs (S1.stack (V1 0) (V1 1)) (S1.stack (V1 2) (V1 3)))
         (S1.hand (S1.eqt (V1 0) (V1 2)) (S1.eqs (V1 1) (V1 3)))
   :: nil)

:: (  R1 (S1.hmember (V1 0) (S1.cons (V1 1) (V1 2)))
         (S1.hcase9 (V1 2) (V1 1) (V1 0) (S1.equal (V1 0) (V1 1)))
   :: nil)

:: (  R1 (S1.hcase9 (V1 0) (V1 1) (V1 2) (S1.false))
         (S1.hmember (V1 2) (V1 0))
   :: nil)

:: (  R1 (S1.hdelete (V1 0) (S1.cons (V1 1) (V1 2)))
         (S1.hcase8 (V1 2) (V1 1) (V1 0) (S1.equal (V1 0) (V1 1)))
   :: nil)

:: (  R1 (S1.hcase8 (V1 0) (V1 1) (V1 2) (S1.false))
         (S1.hdelete (V1 2) (V1 0))
   :: nil)

:: (  R1 (S1.hsubtract (V1 0) (S1.cons (V1 1) (V1 2)))
         (S1.hdelete (V1 1) (V1 0))
   :: nil)

:: (  R1 (S1.hsubtract (V1 0) (S1.cons (V1 1) (V1 2)))
         (S1.hsubtract (S1.delete (V1 1) (V1 0)) (V1 2))
   :: nil)

:: (  R1 (S1.happend (S1.cons (V1 0) (V1 1)) (V1 2))
         (S1.happend (V1 1) (V1 2))
   :: nil)

:: (  R1 (S1.hcase6 (V1 0) (V1 1) (V1 2) (S1.true))
         (S1.hrecord_extract (V1 1) (S1.lock) (S1.pending))
   :: nil)

:: (  R1 (S1.hcase6 (V1 0) (V1 1) (V1 2) (S1.true))
         (S1.hrecord_extract (V1 1) (S1.lock) (S1.excl))
   :: nil)

:: (  R1 (S1.hlocker2_check_available (V1 0) (S1.cons (V1 1) (V1 2)))
         (S1.hrecord_extract (V1 1) (S1.lock) (S1.resource))
   :: nil)

:: (  R1 (S1.hlocker2_check_available (V1 0) (S1.cons (V1 1) (V1 2)))
         (S1.hcase6 (V1 2) (V1 1) (V1 0) (S1.equal (V1 0) (S1.record_extract (V1 1) (S1.lock) (S1.resource))))
   :: nil)

:: (  R1 (S1.hlocker2_check_availables (S1.cons (V1 0) (V1 1)) (V1 2))
         (S1.hlocker2_check_available (V1 0) (V1 2))
   :: nil)

:: (  R1 (S1.hlocker2_check_availables (S1.cons (V1 0) (V1 1)) (V1 2))
         (S1.hlocker2_check_availables (V1 1) (V1 2))
   :: nil)

:: (  R1 (S1.hcase6 (V1 0) (V1 1) (V1 2) (S1.false))
         (S1.hlocker2_check_available (V1 2) (V1 0))
   :: nil)

:: (  R1 (S1.hlocker2_obtainables (S1.cons (V1 0) (V1 1)) (V1 2))
         (S1.hrecord_extract (V1 0) (S1.lock) (S1.pending))
   :: nil)

:: (  R1 (S1.hlocker2_obtainables (S1.cons (V1 0) (V1 1)) (V1 2))
         (S1.hmember (V1 2) (S1.record_extract (V1 0) (S1.lock) (S1.pending)))
   :: nil)

:: (  R1 (S1.hcase5 (V1 0) (V1 1) (V1 2) (S1.true))
         (S1.hlocker2_obtainables (V1 1) (V1 0))
   :: R1 (S1.hlocker2_obtainables (S1.cons (V1 0) (V1 1)) (V1 2))
         (S1.hcase5 (V1 2) (V1 1) (V1 0) (S1.member (V1 2) (S1.record_extract (V1 0) (S1.lock) (S1.pending))))
   :: R1 (S1.hcase5 (V1 0) (V1 1) (V1 2) (S1.false))
         (S1.hlocker2_obtainables (V1 1) (V1 0))
   :: nil)

:: (  R1 (S1.hlocker2_release_lock (V1 0) (V1 1))
         (S1.hrecord_extract (V1 0) (S1.lock) (S1.excl))
   :: nil)

:: (  R1 (S1.hlocker2_release_lock (V1 0) (V1 1))
         (S1.hgen_modtageq (V1 1) (S1.record_extract (V1 0) (S1.lock) (S1.excl)))
   :: nil)

:: (  R1 (S1.hcase1 (V1 0) (V1 1) (V1 2) (S1.true))
         (S1.hrecord_extract (V1 2) (S1.lock) (S1.pending))
   :: nil)

:: (  R1 (S1.hcase1 (V1 0) (V1 1) (V1 2) (S1.true))
         (S1.happend (S1.record_extract (V1 2) (S1.lock) (S1.pending)) (S1.cons (V1 0) (S1.nil)))
   :: nil)

:: (  R1 (S1.hlocker2_add_pending (V1 0) (V1 1) (V1 2))
         (S1.hrecord_extract (V1 0) (S1.lock) (S1.resource))
   :: nil)

:: (  R1 (S1.hlocker2_add_pending (V1 0) (V1 1) (V1 2))
         (S1.hmember (S1.record_extract (V1 0) (S1.lock) (S1.resource)) (V1 1))
   :: nil)

:: (  R1 (S1.hlocker2_remove_pending (V1 0) (V1 1))
         (S1.hrecord_extract (V1 0) (S1.lock) (S1.pending))
   :: nil)

:: (  R1 (S1.hlocker2_remove_pending (V1 0) (V1 1))
         (S1.hsubtract (S1.record_extract (V1 0) (S1.lock) (S1.pending)) (S1.cons (V1 1) (S1.nil)))
   :: nil)

:: (  R1 (S1.hlocker2_promote_pending (V1 0) (V1 1))
         (S1.hrecord_extract (V1 0) (S1.lock) (S1.pending))
   :: nil)

:: (  R1 (S1.hlocker2_map_claim_lock (S1.cons (V1 0) (V1 1)) (V1 2) (V1 3))
         (S1.hlocker2_map_claim_lock (V1 1) (V1 2) (V1 3))
   :: nil)

:: (  R1 (S1.hrecord_updates (V1 0) (V1 1) (S1.cons (S1.tuple (V1 2) (S1.tuplenil (V1 3))) (V1 4)))
         (S1.hrecord_update (V1 0) (V1 1) (V1 2) (V1 3))
   :: nil)

:: (  R1 (S1.hrecord_updates (V1 0) (V1 1) (S1.cons (S1.tuple (V1 2) (S1.tuplenil (V1 3))) (V1 4)))
         (S1.hrecord_updates (S1.record_update (V1 0) (V1 1) (V1 2) (V1 3)) (V1 1) (V1 4))
   :: nil)

:: (  R1 (S1.hcase2 (V1 0) (V1 1) (S1.true))
         (S1.hrecord_updates (V1 1) (S1.lock) (S1.cons (S1.tuple (S1.excllock) (S1.excl)) (S1.nil)))
   :: nil)

:: (  R1 (S1.hlocker2_release_lock (V1 0) (V1 1))
         (S1.hcase2 (V1 1) (V1 0) (S1.gen_modtageq (V1 1) (S1.record_extract (V1 0) (S1.lock) (S1.excl))))
   :: nil)

:: (  R1 (S1.hcase1 (V1 0) (V1 1) (V1 2) (S1.true))
         (S1.hrecord_updates (V1 2) (S1.lock) (S1.cons (S1.tuple (S1.pending) (S1.tuplenil (S1.append (S1.record_extract (V1 2) (S1.lock) (S1.pending)) (S1.cons (V1 0) (S1.nil))))) (S1.nil)))
   :: nil)

:: (  R1 (S1.hlocker2_add_pending (V1 0) (V1 1) (V1 2))
         (S1.hcase1 (V1 2) (V1 1) (V1 0) (S1.member (S1.record_extract (V1 0) (S1.lock) (S1.resource)) (V1 1)))
   :: nil)

:: (  R1 (S1.hlocker2_remove_pending (V1 0) (V1 1))
         (S1.hrecord_updates (V1 0) (S1.lock) (S1.cons (S1.tuple (S1.pending) (S1.tuplenil (S1.subtract (S1.record_extract (V1 0) (S1.lock) (S1.pending)) (S1.cons (V1 1) (S1.nil))))) (S1.nil)))
   :: nil)

:: (  R1 (S1.hcase0 (V1 0) (V1 1) (S1.cons (V1 0) (V1 3)))
         (S1.hrecord_updates (V1 1) (S1.lock) (S1.cons (S1.tuple (S1.excl) (S1.tuplenil (V1 0))) (S1.cons (S1.tuple (S1.pending) (S1.tuplenil (V1 3))) (S1.nil))))
   :: nil)

:: (  R1 (S1.hlocker2_promote_pending (V1 0) (V1 1))
         (S1.hcase0 (V1 1) (V1 0) (S1.record_extract (V1 0) (S1.lock) (S1.pending)))
   :: nil)

:: (  R1 (S1.hlocker2_map_promote_pending (S1.cons (V1 0) (V1 1)) (V1 2))
         (S1.hlocker2_promote_pending (V1 0) (V1 2))
   :: nil)

:: (  R1 (S1.hlocker2_map_promote_pending (S1.cons (V1 0) (V1 1)) (V1 2))
         (S1.hlocker2_map_promote_pending (V1 1) (V1 2))
   :: nil)

:: (  R1 (S1.helement (S1.int (S1.s (S1.s (V1 0)))) (S1.tuple (V1 1) (V1 2)))
         (S1.helement (S1.int (S1.s (V1 0))) (V1 2))
   :: nil)

:: (  R1 (S1.heqt (S1.tuple (V1 0) (V1 1)) (S1.tuple (V1 2) (V1 3)))
         (S1.hand (S1.eqt (V1 0) (V1 2)) (S1.eqt (V1 1) (V1 3)))
   :: nil)

:: (  R1 (S1.heqt (S1.cons (V1 0) (V1 1)) (S1.cons (V1 2) (V1 3)))
         (S1.hand (S1.eqt (V1 0) (V1 2)) (S1.eqt (V1 1) (V1 3)))
   :: nil)

:: (  R1 (S1.heqt (S1.cons (V1 0) (V1 1)) (S1.cons (V1 2) (V1 3)))
         (S1.heqt (V1 0) (V1 2))
   :: R1 (S1.heqt (S1.pid (V1 0)) (S1.pid (V1 1)))
         (S1.heqt (V1 0) (V1 1))
   :: R1 (S1.heqt (S1.cons (V1 0) (V1 1)) (S1.cons (V1 2) (V1 3)))
         (S1.heqt (V1 1) (V1 3))
   :: R1 (S1.heqt (S1.tuple (V1 0) (V1 1)) (S1.tuple (V1 2) (V1 3)))
         (S1.heqt (V1 0) (V1 2))
   :: R1 (S1.heqt (S1.tuple (V1 0) (V1 1)) (S1.tuple (V1 2) (V1 3)))
         (S1.heqt (V1 1) (V1 3))
   :: R1 (S1.heqt (S1.tuplenil (V1 0)) (S1.tuplenil (V1 1)))
         (S1.heqt (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.hpush (V1 0) (V1 1) (S1.calls (V1 2) (V1 3) (V1 4)))
         (S1.heqt (V1 0) (V1 2))
   :: nil)

:: (  R1 (S1.heqc (S1.calls (V1 0) (V1 1) (V1 2)) (S1.calls (V1 3) (V1 4) (V1 5)))
         (S1.heqt (V1 0) (V1 3))
   :: nil)

:: (  R1 (S1.histops (V1 0) (S1.stack (V1 1) (V1 2)))
         (S1.heqt (V1 0) (V1 1))
   :: nil)

:: (  R1 (S1.heqs (S1.stack (V1 0) (V1 1)) (S1.stack (V1 2) (V1 3)))
         (S1.heqt (V1 0) (V1 2))
   :: nil)

:: (  R1 (S1.heqs (S1.stack (V1 0) (V1 1)) (S1.stack (V1 2) (V1 3)))
         (S1.heqs (V1 1) (V1 3))
   :: nil)

:: (  R1 (S1.heqc (S1.calls (V1 0) (V1 1) (V1 2)) (S1.calls (V1 3) (V1 4) (V1 5)))
         (S1.heqs (V1 1) (V1 4))
   :: nil)

:: (  R1 (S1.heqc (S1.calls (V1 0) (V1 1) (V1 2)) (S1.calls (V1 3) (V1 4) (V1 5)))
         (S1.heqc (V1 2) (V1 5))
   :: 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.or) =>
         nil
    | (int_symb M.or) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M.eqt) =>
         nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (2%Z, (Vcons 0 (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.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.case1) =>
         nil
    | (int_symb M.case1) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (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.case2) =>
         nil
    | (int_symb M.case2) =>
         (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.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.push1) =>
         nil
    | (int_symb M.push1) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))))
      :: 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.or) =>
         nil
    | (int_symb M.or) =>
         nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M.eqt) =>
         nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (2%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.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (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))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.case1) =>
         nil
    | (int_symb M.case1) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (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.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.push1) =>
         nil
    | (int_symb M.push1) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 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.or) =>
         nil
    | (int_symb M.or) =>
         nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M.eqt) =>
         nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (2%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.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (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))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.case1) =>
         nil
    | (int_symb M.case1) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         nil
    | (hd_symb M.locker2_check_availables) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.locker2_check_availables) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (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.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.push1) =>
         nil
    | (int_symb M.push1) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 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.or) =>
         nil
    | (int_symb M.or) =>
         nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M.eqt) =>
         nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.case1) =>
         nil
    | (int_symb M.case1) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case5) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (int_symb M.case5) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (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.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (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.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         (3%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))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.push1) =>
         nil
    | (int_symb M.push1) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: nil
    end.

  Lemma trsInt_wm : forall f, pweak_monotone (trsInt f).
  Proof.
    pmonotone.
  Qed.

End PIS4.

Module PI4 := PolyInt PIS4.

(* 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.or) =>
         nil
    | (int_symb M.or) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.eqt) =>
         nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (3%Z, (Vcons 0 (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.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.case1) =>
         nil
    | (int_symb M.case1) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (3%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.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.locker2_obtainables) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case5) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (int_symb M.case5) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         nil
    | (hd_symb M.push1) =>
         nil
    | (int_symb M.push1) =>
         nil
    end.

  Lemma trsInt_wm : forall f, pweak_monotone (trsInt f).
  Proof.
    pmonotone.
  Qed.

End PIS5.

Module PI5 := PolyInt PIS5.

(* graph decomposition 2 *)

Definition cs2 : list (list (@ATrs.rule s1)) :=

   (  R1 (S1.hcase5 (V1 0) (V1 1) (V1 2) (S1.false))
         (S1.hlocker2_obtainables (V1 1) (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.or) =>
         nil
    | (int_symb M.or) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.eqt) =>
         nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (int_symb M.locker2_map_claim_lock) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (1%Z, (Vcons 0 (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.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (3%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.case1) =>
         nil
    | (int_symb M.case1) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (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.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         (3%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.push1) =>
         nil
    | (int_symb M.push1) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))))
      :: nil
    end.

  Lemma trsInt_wm : forall f, pweak_monotone (trsInt f).
  Proof.
    pmonotone.
  Qed.

End PIS6.

Module PI6 := PolyInt PIS6.

(* 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.or) =>
         nil
    | (int_symb M.or) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M.eqt) =>
         nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (2%Z, (Vcons 0 (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.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.case1) =>
         nil
    | (int_symb M.case1) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (2%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.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (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.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         nil
    | (hd_symb M.push1) =>
         nil
    | (int_symb M.push1) =>
         nil
    end.

  Lemma trsInt_wm : forall f, pweak_monotone (trsInt f).
  Proof.
    pmonotone.
  Qed.

End PIS7.

Module PI7 := PolyInt PIS7.

(* 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.or) =>
         nil
    | (int_symb M.or) =>
         nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M.eqt) =>
         nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         (3%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (2%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.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (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))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.case1) =>
         nil
    | (int_symb M.case1) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (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.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.push1) =>
         nil
    | (int_symb M.push1) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: nil
    end.

  Lemma trsInt_wm : forall f, pweak_monotone (trsInt f).
  Proof.
    pmonotone.
  Qed.

End PIS8.

Module PI8 := PolyInt PIS8.

(* 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.or) =>
         nil
    | (int_symb M.or) =>
         nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M.eqt) =>
         nil
    | (int_symb M.eqt) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         (1%Z, (Vcons 0 Vnil))
      :: (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: 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.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         (1%Z, Vnil)
      :: nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (int_symb M.element) =>
         (1%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._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (3%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.case1) =>
         nil
    | (int_symb M.case1) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (3%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.eqc) =>
         nil
    | (int_symb M.eqc) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         (2%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.push) =>
         nil
    | (int_symb M.push) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.push1) =>
         nil
    | (int_symb M.push1) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))))
      :: 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.or) =>
         nil
    | (int_symb M.or) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.eqt) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         (3%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.case1) =>
         nil
    | (int_symb M.case1) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (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.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.push1) =>
         nil
    | (int_symb M.push1) =>
         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.or) =>
         nil
    | (int_symb M.or) =>
         nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M.eqt) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (1%Z, (Vcons 1 (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._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.case1) =>
         nil
    | (int_symb M.case1) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (2%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.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (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.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.push1) =>
         nil
    | (int_symb M.push1) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: 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.or) =>
         nil
    | (int_symb M.or) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M.eqt) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (1%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (2%Z, (Vcons 0 (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.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (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.case1) =>
         nil
    | (int_symb M.case1) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (2%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.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (2%Z, (Vcons 0 Vnil))
      :: (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (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.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         nil
    | (hd_symb M.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         (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.push1) =>
         nil
    | (int_symb M.push1) =>
         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.or) =>
         nil
    | (int_symb M.or) =>
         nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (3%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M.eqt) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (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.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         (2%Z, Vnil)
      :: nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         (3%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (2%Z, (Vcons 0 (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.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.case1) =>
         nil
    | (int_symb M.case1) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (3%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.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         (1%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (1%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         nil
    | (hd_symb M.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         (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.push1) =>
         nil
    | (int_symb M.push1) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))))
      :: 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.or) =>
         nil
    | (int_symb M.or) =>
         nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M.eqt) =>
         nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         nil
    | (hd_symb M.cons) =>
         nil
    | (int_symb M.cons) =>
         (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.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (2%Z, (Vcons 0 (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.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.case1) =>
         nil
    | (int_symb M.case1) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case2) =>
         nil
    | (int_symb M.case2) =>
         (2%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.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (2%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         nil
    | (hd_symb M.eqs) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.eqc) =>
         nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.push1) =>
         nil
    | (int_symb M.push1) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 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.or) =>
         nil
    | (int_symb M.or) =>
         nil
    | (hd_symb M.T) =>
         nil
    | (int_symb M.T) =>
         nil
    | (hd_symb M.F) =>
         nil
    | (int_symb M.F) =>
         nil
    | (hd_symb M.and) =>
         nil
    | (int_symb M.and) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.imp) =>
         nil
    | (int_symb M.imp) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.not) =>
         nil
    | (int_symb M.not) =>
         (2%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M._if_2) =>
         nil
    | (int_symb M._if_2) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.eq) =>
         nil
    | (int_symb M.eq) =>
         nil
    | (hd_symb M.eqt) =>
         nil
    | (int_symb M.eqt) =>
         nil
    | (hd_symb M.nil) =>
         nil
    | (int_symb M.nil) =>
         nil
    | (hd_symb M.undefined) =>
         nil
    | (int_symb M.undefined) =>
         nil
    | (hd_symb M.pid) =>
         nil
    | (int_symb M.pid) =>
         nil
    | (hd_symb M.int) =>
         nil
    | (int_symb M.int) =>
         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.tuple) =>
         nil
    | (int_symb M.tuple) =>
         (2%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.tuplenil) =>
         nil
    | (int_symb M.tuplenil) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.a) =>
         nil
    | (int_symb M.a) =>
         nil
    | (hd_symb M.excl) =>
         nil
    | (int_symb M.excl) =>
         nil
    | (hd_symb M.false) =>
         nil
    | (int_symb M.false) =>
         nil
    | (hd_symb M.lock) =>
         nil
    | (int_symb M.lock) =>
         nil
    | (hd_symb M.locker) =>
         nil
    | (int_symb M.locker) =>
         nil
    | (hd_symb M.mcrlrecord) =>
         nil
    | (int_symb M.mcrlrecord) =>
         nil
    | (hd_symb M.ok) =>
         nil
    | (int_symb M.ok) =>
         nil
    | (hd_symb M.pending) =>
         nil
    | (int_symb M.pending) =>
         nil
    | (hd_symb M.release) =>
         nil
    | (int_symb M.release) =>
         nil
    | (hd_symb M.request) =>
         nil
    | (int_symb M.request) =>
         nil
    | (hd_symb M.resource) =>
         nil
    | (int_symb M.resource) =>
         nil
    | (hd_symb M.tag) =>
         nil
    | (int_symb M.tag) =>
         nil
    | (hd_symb M.true) =>
         nil
    | (int_symb M.true) =>
         nil
    | (hd_symb M.element) =>
         nil
    | (int_symb M.element) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.s) =>
         nil
    | (int_symb M.s) =>
         nil
    | (hd_symb M._0_1) =>
         nil
    | (int_symb M._0_1) =>
         nil
    | (hd_symb M.record_new) =>
         nil
    | (int_symb M.record_new) =>
         (1%Z, (Vcons 0 Vnil))
      :: nil
    | (hd_symb M.record_extract) =>
         nil
    | (int_symb M.record_extract) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.record_update) =>
         nil
    | (int_symb M.record_update) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 Vnil)))))
      :: nil
    | (hd_symb M.record_updates) =>
         nil
    | (int_symb M.record_updates) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_map_promote_pending) =>
         nil
    | (int_symb M.locker2_map_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_promote_pending) =>
         nil
    | (int_symb M.locker2_promote_pending) =>
         (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_map_claim_lock) =>
         nil
    | (int_symb M.locker2_map_claim_lock) =>
         nil
    | (hd_symb M.locker2_claim_lock) =>
         nil
    | (int_symb M.locker2_claim_lock) =>
         nil
    | (hd_symb M.locker2_map_add_pending) =>
         nil
    | (int_symb M.locker2_map_add_pending) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.case0) =>
         nil
    | (int_symb M.case0) =>
         (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.locker2_remove_pending) =>
         nil
    | (int_symb M.locker2_remove_pending) =>
         (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.subtract) =>
         nil
    | (int_symb M.subtract) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_add_pending) =>
         nil
    | (int_symb M.locker2_add_pending) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.case1) =>
         nil
    | (int_symb M.case1) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.member) =>
         nil
    | (int_symb M.member) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.append) =>
         nil
    | (int_symb M.append) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_release_lock) =>
         nil
    | (int_symb M.locker2_release_lock) =>
         (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.case2) =>
         nil
    | (int_symb M.case2) =>
         (1%Z, (Vcons 1 (Vcons 0 (Vcons 0 Vnil))))
      :: (1%Z, (Vcons 0 (Vcons 1 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.gen_modtageq) =>
         nil
    | (int_symb M.gen_modtageq) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.excllock) =>
         nil
    | (int_symb M.excllock) =>
         nil
    | (hd_symb M.case4) =>
         nil
    | (int_symb M.case4) =>
         (3%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: nil
    | (hd_symb M.locker2_obtainables) =>
         nil
    | (int_symb M.locker2_obtainables) =>
         nil
    | (hd_symb M.case5) =>
         nil
    | (int_symb M.case5) =>
         nil
    | (hd_symb M.andt) =>
         nil
    | (int_symb M.andt) =>
         nil
    | (hd_symb M.locker2_obtainable) =>
         nil
    | (int_symb M.locker2_obtainable) =>
         nil
    | (hd_symb M.locker2_check_available) =>
         nil
    | (int_symb M.locker2_check_available) =>
         (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case6) =>
         nil
    | (int_symb M.case6) =>
         (3%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (3%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.equal) =>
         nil
    | (int_symb M.equal) =>
         nil
    | (hd_symb M.locker2_check_availables) =>
         nil
    | (int_symb M.locker2_check_availables) =>
         (3%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: nil
    | (hd_symb M.locker2_adduniq) =>
         nil
    | (int_symb M.locker2_adduniq) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.delete) =>
         nil
    | (int_symb M.delete) =>
         (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.case8) =>
         nil
    | (int_symb M.case8) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.gen_tag) =>
         nil
    | (int_symb M.gen_tag) =>
         (2%Z, (Vcons 0 Vnil))
      :: (3%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.case9) =>
         nil
    | (int_symb M.case9) =>
         (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 Vnil)))))
      :: nil
    | (hd_symb M.eqs) =>
         nil
    | (int_symb M.eqs) =>
         nil
    | (hd_symb M.empty) =>
         nil
    | (int_symb M.empty) =>
         nil
    | (hd_symb M.stack) =>
         nil
    | (int_symb M.stack) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pushs) =>
         nil
    | (int_symb M.pushs) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.pops) =>
         nil
    | (int_symb M.pops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.tops) =>
         nil
    | (int_symb M.tops) =>
         (1%Z, (Vcons 1 Vnil))
      :: nil
    | (hd_symb M.istops) =>
         nil
    | (int_symb M.istops) =>
         (2%Z, (Vcons 0 (Vcons 0 Vnil)))
      :: (3%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (2%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (hd_symb M.eqc) =>
         (1%Z, (Vcons 1 (Vcons 0 Vnil)))
      :: (1%Z, (Vcons 0 (Vcons 1 Vnil)))
      :: nil
    | (int_symb M.eqc) =>
         nil
    | (hd_symb M.nocalls) =>
         nil
    | (int_symb M.nocalls) =>
         nil
    | (hd_symb M.calls) =>
         nil
    | (int_symb M.calls) =>
         (1%Z, (Vcons 0 (Vcons 0 (Vcons 0 Vnil))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.push) =>
         nil
    | (int_symb M.push) =>
         (2%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))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 1 Vnil))))
      :: nil
    | (hd_symb M.push1) =>
         nil
    | (int_symb M.push1) =>
         (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (2%Z, (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (2%Z, (Vcons 0 (Vcons 1 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 Vnil)))))))
      :: (2%Z, (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 0 (Vcons 1 (Vcons 0 Vnil)))))))
      :: nil
    end.

  Lemma trsInt_wm : forall f, pweak_monotone (trsInt f).
  Proof.
    pmonotone.
  Qed.

End PIS15.

Module PI15 := PolyInt PIS15.

(* 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.
right. PI1.prove_termination.
termination_trivial.
right. PI2.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI3.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI4.prove_termination.
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) cs2; 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.
right. PI6.prove_termination.
termination_trivial.
left. co_scc.
right. PI7.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.
right. PI8.prove_termination.
termination_trivial.
right. PI9.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
right. PI10.prove_termination.
PI11.prove_termination.
PI12.prove_termination.
PI13.prove_termination.
termination_trivial.
left. co_scc.
left. co_scc.
left. co_scc.
left. co_scc.
right. PI14.prove_termination.
termination_trivial.
left. co_scc.
right. PI15.prove_termination.
termination_trivial.
Qed.