# 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 : 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).

Lemma comm_right : forall x y z:U, cong (op x (op y z)) (op x (op z y)).

Lemma comm_left : forall x y z:U, cong (op (op x y) z) (op (op y x) z).

Lemma perm_right : forall x y z:U, cong (op (op x y) z) (op (op x z) y).

Lemma perm_left : forall x y z:U, cong (op x (op y z)) (op y (op x z)).

Lemma op_rotate : forall x y z t:U, cong (op x (op y z)) (op z (op x y)).

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).

End Axiomatisation.