lean2/library/init/function.lean
2015-09-30 17:41:44 -07:00

162 lines
5.6 KiB
Text
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura, Jeremy Avigad, Haitao Zhang
General operations on functions.
-/
prelude
import init.prod init.funext init.logic
namespace function
variables {A : Type} {B : Type} {C : Type} {D : Type} {E : Type}
definition compose [reducible] [unfold-full] (f : B → C) (g : A → B) : A → C :=
λx, f (g x)
definition compose_right [reducible] [unfold-full] (f : B → B → B) (g : A → B) : B → A → B :=
λ b a, f b (g a)
definition compose_left [reducible] [unfold-full] (f : B → B → B) (g : A → B) : A → B → B :=
λ a b, f (g a) b
definition id [reducible] [unfold-full] (a : A) : A :=
a
definition on_fun [reducible] [unfold-full] (f : B → B → C) (g : A → B) : A → A → C :=
λx y, f (g x) (g y)
definition combine [reducible] [unfold-full] (f : A → B → C) (op : C → D → E) (g : A → B → D)
: A → B → E :=
λx y, op (f x y) (g x y)
definition const [reducible] [unfold-full] (B : Type) (a : A) : B → A :=
λx, a
definition dcompose [reducible] [unfold-full] {B : A → Type} {C : Π {x : A}, B x → Type}
(f : Π {x : A} (y : B x), C y) (g : Πx, B x) : Πx, C (g x) :=
λx, f (g x)
definition swap [reducible] [unfold-full] {C : A → B → Type} (f : Πx y, C x y) : Πy x, C x y :=
λy x, f x y
definition app [reducible] {B : A → Type} (f : Πx, B x) (x : A) : B x :=
f x
definition curry [reducible] [unfold-full] : (A × B → C) → A → B → C :=
λ f a b, f (a, b)
definition uncurry [reducible] [unfold 5] : (A → B → C) → (A × B → C) :=
λ f p, match p with (a, b) := f a b end
theorem curry_uncurry (f : A → B → C) : curry (uncurry f) = f :=
rfl
theorem uncurry_curry (f : A × B → C) : uncurry (curry f) = f :=
funext (λ p, match p with (a, b) := rfl end)
infixr ` ∘ ` := compose
infixr ` ∘' `:60 := dcompose
infixl ` on `:1 := on_fun
infixr ` $ `:1 := app
notation f ` -[` op `]- ` g := combine f op g
lemma left_id (f : A → B) : id ∘ f = f := rfl
lemma right_id (f : A → B) : f ∘ id = f := rfl
theorem compose.assoc (f : C → D) (g : B → C) (h : A → B) : (f ∘ g) ∘ h = f ∘ (g ∘ h) := rfl
theorem compose.left_id (f : A → B) : id ∘ f = f := rfl
theorem compose.right_id (f : A → B) : f ∘ id = f := rfl
theorem compose_const_right (f : B → C) (b : B) : f ∘ (const A b) = const A (f b) := rfl
definition injective [reducible] (f : A → B) : Prop := ∀ ⦃a₁ a₂⦄, f a₁ = f a₂ → a₁ = a₂
theorem injective_compose {g : B → C} {f : A → B} (Hg : injective g) (Hf : injective f) :
injective (g ∘ f) :=
take a₁ a₂, assume Heq, Hf (Hg Heq)
definition surjective [reducible] (f : A → B) : Prop := ∀ b, ∃ a, f a = b
theorem surjective_compose {g : B → C} {f : A → B} (Hg : surjective g) (Hf : surjective f) :
surjective (g ∘ f) :=
take c,
obtain b (Hb : g b = c), from Hg c,
obtain a (Ha : f a = b), from Hf b,
exists.intro a (eq.trans (congr_arg g Ha) Hb)
definition bijective (f : A → B) := injective f ∧ surjective f
theorem bijective_compose {g : B → C} {f : A → B} (Hg : bijective g) (Hf : bijective f) :
bijective (g ∘ f) :=
obtain Hginj Hgsurj, from Hg,
obtain Hfinj Hfsurj, from Hf,
and.intro (injective_compose Hginj Hfinj) (surjective_compose Hgsurj Hfsurj)
-- g is a left inverse to f
definition left_inverse (g : B → A) (f : A → B) : Prop := ∀x, g (f x) = x
definition id_of_left_inverse {g : B → A} {f : A → B} : left_inverse g f → g ∘ f = id :=
assume h, funext h
definition has_left_inverse (f : A → B) : Prop := ∃ finv : B → A, left_inverse finv f
-- g is a right inverse to f
definition right_inverse (g : B → A) (f : A → B) : Prop := left_inverse f g
definition id_of_righ_inverse {g : B → A} {f : A → B} : right_inverse g f → f ∘ g = id :=
assume h, funext h
definition has_right_inverse (f : A → B) : Prop := ∃ finv : B → A, right_inverse finv f
theorem injective_of_left_inverse {g : B → A} {f : A → B} : left_inverse g f → injective f :=
assume h, take a b, assume faeqfb,
calc a = g (f a) : by rewrite h
... = g (f b) : faeqfb
... = b : by rewrite h
theorem injective_of_has_left_inverse {f : A → B} : has_left_inverse f → injective f :=
assume h, obtain (finv : B → A) (inv : left_inverse finv f), from h,
injective_of_left_inverse inv
theorem right_inverse_of_injective_of_left_inverse {f : A → B} {g : B → A}
(injf : injective f) (lfg : left_inverse f g) :
right_inverse f g :=
take x,
have H : f (g (f x)) = f x, from lfg (f x),
injf H
theorem surjective_of_has_right_inverse {f : A → B} : has_right_inverse f → surjective f :=
assume h, take b,
obtain (finv : B → A) (inv : right_inverse finv f), from h,
let a : A := finv b in
have h : f a = b, from calc
f a = (f ∘ finv) b : rfl
... = id b : by rewrite inv
... = b : rfl,
exists.intro a h
theorem left_inverse_of_surjective_of_right_inverse {f : A → B} {g : B → A}
(surjf : surjective f) (rfg : right_inverse f g) :
left_inverse f g :=
take y,
obtain x (Hx : f x = y), from surjf y,
calc
f (g y) = f (g (f x)) : Hx
... = f x : rfg
... = y : Hx
theorem injective_id : injective (@id A) := take a₁ a₂ H, H
theorem surjective_id : surjective (@id A) := take a, exists.intro a rfl
theorem bijective_id : bijective (@id A) := and.intro injective_id surjective_id
end function
-- copy reducible annotations to top-level
export [reduce-hints] [unfold-hints] function