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-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-
-(*i $Id: Permut.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*)
-
-(* G. Huet 1-9-95 *)
-
-(** We consider a Set [U], given with a commutative-associative operator [op],
-    and a congruence [cong]; we show permutation lemmas *)
-
-Section Axiomatisation.
-
-Variable U: Set.
-
-Variable op: U -> U -> U.
-
-Variable cong : U -> U -> Prop.
-
-Hypothesis op_comm : (x,y:U)(cong (op x y) (op y x)).
-Hypothesis op_ass  : (x,y,z:U)(cong (op (op x y) z) (op x (op y z))).
-
-Hypothesis cong_left :  (x,y,z:U)(cong x y)->(cong (op x z) (op y z)).
-Hypothesis cong_right : (x,y,z:U)(cong x y)->(cong (op z x) (op z y)).
-Hypothesis cong_trans : (x,y,z:U)(cong x y)->(cong y z)->(cong x z).
-Hypothesis cong_sym : (x,y:U)(cong x y)->(cong y x).
-
-(** Remark. we do not need: [Hypothesis cong_refl : (x:U)(cong x x)]. *)
-
-Lemma cong_congr :
- (x,y,z,t:U)(cong x y)->(cong z t)->(cong (op x z) (op y t)).
-Proof.
-Intros; Apply cong_trans with (op y z).
-Apply cong_left; Trivial.
-Apply cong_right; Trivial.
-Qed.
-
-Lemma comm_right : (x,y,z:U)(cong (op x (op y z)) (op x (op z y))).
-Proof.
-Intros; Apply cong_right; Apply op_comm.
-Qed.
-
-Lemma comm_left : (x,y,z:U)(cong (op (op x y) z) (op (op y x) z)).
-Proof.
-Intros; Apply cong_left; Apply op_comm.
-Qed.
-
-Lemma perm_right : (x,y,z:U)(cong (op (op x y) z) (op (op x z) y)).
-Proof.
-Intros.
-Apply cong_trans with (op x (op y z)).
-Apply op_ass.
-Apply cong_trans with (op x (op z y)). 
-Apply cong_right; Apply op_comm.
-Apply cong_sym; Apply op_ass.
-Qed.
-
-Lemma perm_left : (x,y,z:U)(cong (op x (op y z)) (op y (op x z))).
-Proof.
-Intros.
-Apply cong_trans with (op (op x y) z).
-Apply cong_sym; Apply op_ass.
-Apply cong_trans with (op (op y x) z).
-Apply cong_left; Apply op_comm.
-Apply op_ass.
-Qed.
-
-Lemma op_rotate : (x,y,z,t:U)(cong (op x (op y z)) (op z (op x y))).
-Proof.
-Intros; Apply cong_trans with (op (op x y) z).
-Apply cong_sym; Apply op_ass.
-Apply op_comm.
-Qed.
-
-(* Needed for treesort ... *)
-Lemma twist : (x,y,z,t:U)
-   (cong (op x (op (op y z) t)) (op (op y (op x t)) z)).
-Proof.
-Intros.
-Apply cong_trans with (op x (op (op y t) z)).
-Apply cong_right; Apply perm_right.
-Apply cong_trans with (op (op x (op y t)) z).
-Apply cong_sym; Apply op_ass.
-Apply cong_left; Apply perm_left.
-Qed.
-
-End Axiomatisation.