# Library Coq.Program.Combinators

# Proofs about standard combinators, exports functional extensionality.

Author: Matthieu Sozeau Institution: LRI, CNRS UMR 8623 - University Paris Sud
Require Import Coq.Program.Basics.
Require Export FunctionalExtensionality.
Open Scope program_scope.
Composition has id for neutral element and is associative.
Lemma compose_id_left : forall A B (f : A -> B), id ∘ f = f.
Lemma compose_id_right : forall A B (f : A -> B), f ∘ id = f.
Lemma compose_assoc : forall A B C D (f : A -> B) (g : B -> C) (h : C -> D),
h ∘ g ∘ f = h ∘ (g ∘ f).
Hint Rewrite @compose_id_left @compose_id_right : core.
Hint Rewrite <- @compose_assoc : core.
flip is involutive.
Lemma flip_flip : forall A B C, @flip A B C ∘ flip = id.
prod_curry and prod_uncurry are each others inverses.
Lemma prod_uncurry_curry : forall A B C, @prod_uncurry A B C ∘ prod_curry = id.
Lemma prod_curry_uncurry : forall A B C, @prod_curry A B C ∘ prod_uncurry = id.