Library Coq.Program.Basics

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).

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.