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(* -*- coding: utf-8 -*- *)
(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * Copyright INRIA, CNRS and contributors *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** Standard functions and combinators.
Proofs about them require functional extensionality and can be found
in [Combinators].
Author: Matthieu Sozeau
Institution: LRI, CNRS UMR 8623 - University Paris Sud
*)
(** The polymorphic identity function is defined in [Datatypes]. *)
Arguments id {A} x.
(** Function composition. *)
Definition compose {A B C} (g : B -> C) (f : A -> B) :=
fun x : A => g (f x).
#[global]
Hint Unfold compose : core.
Declare Scope program_scope.
Notation " g ∘ f " := (compose g f)
(at level 40, left associativity) : program_scope.
Local Open Scope program_scope.
(** The non-dependent function space between [A] and [B]. *)
Definition arrow (A B : Type) := A -> B.
(** Logical implication. *)
Definition impl (A B : Prop) : Prop := A -> B.
(** The constant function [const a] always returns [a]. *)
Definition const {A B} (a : A) := fun _ : B => a.
(** The [flip] combinator reverses the first two arguments of a function. *)
Definition flip {A B C} (f : A -> B -> C) x y := f y x.
(** Application as a combinator. *)
Definition apply {A B} (f : A -> B) (x : A) := f x.
(** Curryfication of [prod] is defined in [Logic.Datatypes]. *)
Arguments prod_curry_subdef {A B C} f p.
Arguments prod_uncurry_subdef {A B C} f x y.
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