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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
Require PolyList.
Parameter in_list : (list nat*nat)->nat->Prop.
Definition not_in_list : (list nat*nat)->nat->Prop
:= [l,n]~(in_list l n).
(* Hints Unfold not_in_list. *)
Axiom lem1 : (l1,l2:(list nat*nat))(n:nat)
(not_in_list (app l1 l2) n)->(not_in_list l1 n).
Axiom lem2 : (l1,l2:(list nat*nat))(n:nat)
(not_in_list (app l1 l2) n)->(not_in_list l2 n).
Axiom lem3 : (l:(list nat*nat))(n,p,q:nat)
(not_in_list (cons (p,q) l) n)->(not_in_list l n).
Axiom lem4 : (l1,l2:(list nat*nat))(n:nat)
(not_in_list l1 n)->(not_in_list l2 n)->(not_in_list (app l1 l2) n).
Hints Resolve lem1 lem2 lem3 lem4: essai.
Goal (l:(list nat*nat))(n,p,q:nat)
(not_in_list (cons (p,q) l) n)->(not_in_list l n).
Intros.
EAuto with essai.
Save.
(* Example from Nicolas Magaud on coq-club - Jul 2000 *)
Definition Nat: Set := nat.
Parameter S':Nat ->Nat.
Parameter plus':Nat -> Nat ->Nat.
Lemma simpl_plus_l_rr1:
((n0:Nat) ((m, p:Nat) (plus' n0 m)=(plus' n0 p) ->m=p) ->
(m, p:Nat) (S' (plus' n0 m))=(S' (plus' n0 p)) ->m=p) ->
(n:Nat) ((m, p:Nat) (plus' n m)=(plus' n p) ->m=p) ->
(m, p:Nat) (S' (plus' n m))=(S' (plus' n p)) ->m=p.
Intros.
EAuto. (* does EApply H *)
Qed.
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