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 ).

Global Hint Rewrite @compose_id_left @compose_id_right : core.
Global Hint Rewrite <- @compose_assoc : core.

flip is involutive.

Lemma flip_flip : forall A B C, @flip A B C ∘ flip = id.

uncurry and curry are each others inverses.

Lemma curry_uncurry : forall A B C, @curry A B C ∘ uncurry = id.

Lemma uncurry_curry : forall A B C, @uncurry A B C ∘ curry = id.

#[deprecated(since = "8.15", note = "Use curry_uncurry instead.")]
Notation prod_uncurry_curry := curry_uncurry.
#[deprecated(since = "8.15", note = "Use uncurry_curry instead.")]
Notation prod_curry_uncurry := uncurry_curry.