Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# 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).
#[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 {A B C} f p.
Arguments prod_uncurry {A B C} f x y.