66 lines
1.9 KiB
Text
66 lines
1.9 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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Author: Leonardo de Moura
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General operations on functions.
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-/
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prelude
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import init.reserved_notation .types
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open prod
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namespace function
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variables {A B C D E : Type}
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definition compose [reducible] [unfold-full] (f : B → C) (g : A → B) : A → C :=
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λx, f (g x)
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definition compose_right [reducible] [unfold-full] (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] [unfold-full] (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] [unfold-full] (a : A) : A :=
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a
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definition on_fun [reducible] [unfold-full] (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] [unfold-full] (f : A → B → C) (op : C → D → E) (g : A → B → D)
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: A → B → E :=
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λx y, op (f x y) (g x y)
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definition const [reducible] [unfold-full] (B : Type) (a : A) : B → A :=
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λx, a
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definition dcompose [reducible] [unfold-full] {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] [unfold-full] {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] [unfold-full] {B : A → Type} (f : Πx, B x) (x : A) : B x :=
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f x
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definition curry [reducible] [unfold-full] : (A × B → C) → A → B → C :=
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λ f a b, f (a, b)
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definition uncurry [reducible] [unfold 5] : (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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infixr ` ∘ ` := compose
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infixr ` ∘' `:60 := dcompose
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infixl ` on `:1 := on_fun
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infixr ` $ `:1 := app
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notation f ` -[` op `]- ` g := combine f op g
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end function
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-- copy reducible annotations to top-level
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export [reduce-hints] [unfold-hints] function
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