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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.