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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2017 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* 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 : Type.
Variable op : U -> U -> U.
Variable cong : U -> U -> Prop.
Hypothesis op_comm : forall x y:U, cong (op x y) (op y x).
Hypothesis op_ass : forall x y z:U, cong (op (op x y) z) (op x (op y z)).
Hypothesis cong_left : forall x y z:U, cong x y -> cong (op x z) (op y z).
Hypothesis cong_right : forall x y z:U, cong x y -> cong (op z x) (op z y).
Hypothesis cong_trans : forall x y z:U, cong x y -> cong y z -> cong x z.
Hypothesis cong_sym : forall 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 :
forall 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 : forall 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 : forall 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 : forall 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 : forall 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 : forall 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 :
forall 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.
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