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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import List.
Parameter in_list : list (nat * nat) -> nat -> Prop.
Definition not_in_list (l : list (nat * nat)) (n : nat) : Prop :=
~ in_list l n.
(* Hints Unfold not_in_list. *)
Axiom
lem1 :
forall (l1 l2 : list (nat * nat)) (n : nat),
not_in_list (l1 ++ l2) n -> not_in_list l1 n.
Axiom
lem2 :
forall (l1 l2 : list (nat * nat)) (n : nat),
not_in_list (l1 ++ l2) n -> not_in_list l2 n.
Axiom
lem3 :
forall (l : list (nat * nat)) (n p q : nat),
not_in_list ((p, q) :: l) n -> not_in_list l n.
Axiom
lem4 :
forall (l1 l2 : list (nat * nat)) (n : nat),
not_in_list l1 n -> not_in_list l2 n -> not_in_list (l1 ++ l2) n.
Hint Resolve lem1 lem2 lem3 lem4: essai.
Goal
forall (l : list (nat * nat)) (n p q : nat),
not_in_list ((p, q) :: l) n -> not_in_list l n.
intros.
eauto with essai.
Qed.
(* 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 :
(forall n0 : Nat,
(forall m p : Nat, plus' n0 m = plus' n0 p -> m = p) ->
forall m p : Nat, S' (plus' n0 m) = S' (plus' n0 p) -> m = p) ->
forall n : Nat,
(forall m p : Nat, plus' n m = plus' n p -> m = p) ->
forall m p : Nat, S' (plus' n m) = S' (plus' n p) -> m = p.
intros.
eauto. (* does EApply H *)
Qed.
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