297d50378d
define embedding, (split) surjection, retraction, existential quantifier, 'or' connective also add a whole bunch of theorems about these definitions still has two sorry's which can be solved after #564 is closed
57 lines
1.4 KiB
Text
57 lines
1.4 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: init.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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prelude
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import init.reserved_notation
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namespace function
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variables {A B C D 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 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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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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end function
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-- copy reducible annotations to top-level
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export [reduce-hints] function
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