100 lines
3.3 KiB
Text
100 lines
3.3 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: algebra.function
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Author: Leonardo de Moura
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General operations on functions.
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-/
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namespace function
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variables {A : Type} {B : Type} {C : Type} {D : Type} {E : Type}
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definition compose [reducible] (f : B → C) (g : A → B) : A → C :=
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λx, f (g x)
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definition compose_right [reducible] (f : B → B → B) (g : A → B) : B → A → B :=
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λ b a, f b (g a)
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definition compose_left [reducible] (f : B → B → B) (g : A → B) : A → B → B :=
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λ a b, f (g a) b
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definition id [reducible] (a : A) : A :=
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a
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definition on_fun [reducible] (f : B → B → C) (g : A → B) : A → A → C :=
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λx y, f (g x) (g y)
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definition combine [reducible] (f : A → B → C) (op : C → D → E) (g : A → B → D) : A → B → E :=
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λx y, op (f x y) (g x y)
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definition const [reducible] (B : Type) (a : A) : B → A :=
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λx, a
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definition dcompose [reducible] {B : A → Type} {C : Π {x : A}, B x → Type}
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(f : Π {x : A} (y : B x), C y) (g : Πx, B x) : Πx, C (g x) :=
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λx, f (g x)
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definition flip [reducible] {C : A → B → Type} (f : Πx y, C x y) : Πy x, C x y :=
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λy x, f x y
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definition app [reducible] {B : A → Type} (f : Πx, B x) (x : A) : B x :=
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f x
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definition curry [reducible] : (A × B → C) → A → B → C :=
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λ f a b, f (a, b)
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definition uncurry [reducible] : (A → B → C) → (A × B → C) :=
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λ f p, match p with (a, b) := f a b end
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theorem curry_uncurry (f : A → B → C) : curry (uncurry f) = f :=
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rfl
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theorem uncurry_curry (f : A × B → C) : uncurry (curry f) = f :=
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funext (λ p, match p with (a, b) := rfl end)
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precedence `∘'`:60
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precedence `on`:1
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precedence `$`:1
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infixr ∘ := compose
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infixr ∘' := dcompose
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infixl on := on_fun
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infixr $ := app
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notation f `-[` op `]-` g := combine f op g
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lemma left_inv_eq {finv : B → A} {f : A → B} (linv : finv ∘ f = id) : ∀ x, finv (f x) = x :=
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take x, show (finv ∘ f) x = x, by rewrite linv
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lemma right_inv_eq {finv : B → A} {f : A → B} (rinv : f ∘ finv = id) : ∀ x, f (finv x) = x :=
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take x, show (f ∘ finv) x = x, by rewrite rinv
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definition injective (f : A → B) : Prop := ∀ a₁ a₂, f a₁ = f a₂ → a₁ = a₂
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definition surjective (f : A → B) : Prop := ∀ b, ∃ a, f a = b
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definition has_left_inverse (f : A → B) : Prop := ∃ finv : B → A, finv ∘ f = id
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definition has_right_inverse (f : A → B) : Prop := ∃ finv : B → A, f ∘ finv = id
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lemma injective_of_has_left_inverse {f : A → B} : has_left_inverse f → injective f :=
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assume h, take a b, assume faeqfb,
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obtain (finv : B → A) (inv : finv ∘ f = id), from h,
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calc a = finv (f a) : by rewrite (left_inv_eq inv)
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... = finv (f b) : faeqfb
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... = b : by rewrite (left_inv_eq inv)
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lemma surjective_of_has_right_inverse {f : A → B} : has_right_inverse f → surjective f :=
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assume h, take b,
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obtain (finv : B → A) (inv : f ∘ finv = id), from h,
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let a : A := finv b in
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have h : f a = b, from calc
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f a = (f ∘ finv) b : rfl
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... = id b : by rewrite (right_inv_eq inv)
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... = b : rfl,
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exists.intro a h
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end function
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-- copy reducible annotations to top-level
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export [reduce-hints] function
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