Library Coq.Sets.Permut

``` ```
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 : 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. ```