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* add coproduct and either * address cr issues * stylistic improvements on either as coproduct
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|---|---|---|
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| \iffalse | ||
| SPDX-License-Identifier: AGPL-3.0-only | ||
|
|
||
| This file is part of `idris-ct` Category Theory in Idris library. | ||
|
|
||
| Copyright (C) 2019 Stichting Statebox <https://statebox.nl> | ||
|
|
||
| This program is free software: you can redistribute it and/or modify | ||
| it under the terms of the GNU Affero General Public License as published by | ||
| the Free Software Foundation, either version 3 of the License, or | ||
| (at your option) any later version. | ||
|
|
||
| This program is distributed in the hope that it will be useful, | ||
| but WITHOUT ANY WARRANTY; without even the implied warranty of | ||
| MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | ||
| GNU Affero General Public License for more details. | ||
|
|
||
| You should have received a copy of the GNU Affero General Public License | ||
| along with this program. If not, see <https://www.gnu.org/licenses/>. | ||
| \fi | ||
|
|
||
| > module CoLimits.CoProduct | ||
| > | ||
| > import Basic.Category | ||
| > | ||
| > %access public export | ||
| > %default total | ||
| > %auto_implicits off | ||
| > | ||
| > record CommutingMorphism | ||
| > (cat : Category) | ||
| > (a : obj cat) (b : obj cat) (carrier : obj cat) (c : obj cat) | ||
| > (inl : mor cat a carrier) (inr : mor cat b carrier) | ||
| > (f : mor cat a c) (g : mor cat b c) | ||
| > where | ||
| > constructor MkCommutingMorphism | ||
| > challenger : mor cat carrier c | ||
| > commutativityLeft : compose cat a carrier c inl challenger = f | ||
| > commutativityRight : compose cat b carrier c inr challenger = g | ||
| > | ||
| > record CoProduct | ||
| > (cat : Category) | ||
| > (a : obj cat) (b : obj cat) | ||
| > where | ||
| > constructor MkCoProduct | ||
| > carrier: obj cat | ||
| > inl: mor cat a carrier | ||
| > inr: mor cat b carrier | ||
| > exists: | ||
| > (c : obj cat) | ||
| > -> (f : mor cat a c) | ||
| > -> (g : mor cat b c) | ||
| > -> CommutingMorphism cat a b carrier c inl inr f g | ||
| > unique: | ||
| > (c : obj cat) | ||
| > -> (f : mor cat a c) | ||
| > -> (g : mor cat b c) | ||
| > -> (h : CommutingMorphism cat a b carrier c inl inr f g) | ||
| > -> challenger h = challenger (exists c f g) |
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| Original file line number | Diff line number | Diff line change |
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| \iffalse | ||
| SPDX-License-Identifier: AGPL-3.0-only | ||
|
|
||
| This file is part of `idris-ct` Category Theory in Idris library. | ||
|
|
||
| Copyright (C) 2019 Stichting Statebox <https://statebox.nl> | ||
|
|
||
| This program is free software: you can redistribute it and/or modify | ||
| it under the terms of the GNU Affero General Public License as published by | ||
| the Free Software Foundation, either version 3 of the License, or | ||
| (at your option) any later version. | ||
|
|
||
| This program is distributed in the hope that it will be useful, | ||
| but WITHOUT ANY WARRANTY; without even the implied warranty of | ||
| MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | ||
| GNU Affero General Public License for more details. | ||
|
|
||
| You should have received a copy of the GNU Affero General Public License | ||
| along with this program. If not, see <https://www.gnu.org/licenses/>. | ||
| \fi | ||
|
|
||
| > module Idris.EitherAsCoProduct | ||
| > | ||
| > import Idris.TypesAsCategoryExtensional | ||
| > | ||
| > import Basic.Category | ||
| > import CoLimits.CoProduct | ||
| > | ||
| > applyLeftOrRight : | ||
| > (a, b, c : Type) | ||
| > -> (f : ExtensionalTypeMorphism a c) | ||
| > -> (g : ExtensionalTypeMorphism b c) | ||
| > -> ExtensionalTypeMorphism (Either a b) c | ||
| > applyLeftOrRight a b c (MkExtensionalTypeMorphism f) (MkExtensionalTypeMorphism g) = | ||
| > (MkExtensionalTypeMorphism (either f g)) | ||
| > | ||
| > leftCompose : | ||
| > (a, b, c : Type) | ||
| > -> (f : ExtensionalTypeMorphism a c) | ||
| > -> (g : ExtensionalTypeMorphism b c) | ||
| > -> extCompose a (Either a b) c (MkExtensionalTypeMorphism Left) (applyLeftOrRight a b c f g) = f | ||
| > leftCompose a b c (MkExtensionalTypeMorphism f) (MkExtensionalTypeMorphism g) = Refl | ||
| > | ||
| > rightCompose : | ||
| > (a, b, c : Type) | ||
| > -> (f : ExtensionalTypeMorphism a c) | ||
| > -> (g : ExtensionalTypeMorphism b c) | ||
| > -> extCompose b (Either a b) c (MkExtensionalTypeMorphism Right) (applyLeftOrRight a b c f g) = g | ||
| > rightCompose a b c (MkExtensionalTypeMorphism f) (MkExtensionalTypeMorphism g) = Refl | ||
| > | ||
| > applyExtWith : | ||
| > (x : a) | ||
| > -> (f : ExtensionalTypeMorphism a b) | ||
| > -> b | ||
| > applyExtWith x f = apply (func f) x | ||
| > | ||
| > unique : | ||
| > (f : ExtensionalTypeMorphism a c) | ||
| > -> (g : ExtensionalTypeMorphism b c) | ||
| > -> (h : ExtensionalTypeMorphism (Either a b) c) | ||
| > -> (commutativityLeft : extCompose a (Either a b) c (MkExtensionalTypeMorphism Left) h = f) | ||
| > -> (commutativityRight: extCompose b (Either a b) c (MkExtensionalTypeMorphism Right) h = g) | ||
| > -> h = applyLeftOrRight a b c f g | ||
| > unique (MkExtensionalTypeMorphism f) | ||
| > (MkExtensionalTypeMorphism g) | ||
| > (MkExtensionalTypeMorphism h) | ||
| > commutativityLeft | ||
| > commutativityRight = | ||
| > funExt(\x => | ||
| > case x of | ||
| > Left l => cong {f = applyExtWith l} commutativityLeft | ||
| > Right r => cong {f = applyExtWith r} commutativityRight | ||
| > ) | ||
| > | ||
| > eitherToCoProduct : (a, b : Type) -> CoProduct Idris.TypesAsCategoryExtensional.typesAsCategoryExtensional a b | ||
| > eitherToCoProduct a b = MkCoProduct | ||
| > (Either a b) | ||
| > (MkExtensionalTypeMorphism Left) | ||
| > (MkExtensionalTypeMorphism Right) | ||
| > (\c, f, g => MkCommutingMorphism (applyLeftOrRight a b c f g) (leftCompose a b c f g) (rightCompose a b c f g)) | ||
| > (\c, f, g, h => unique f g (challenger h) (commutativityLeft h) (commutativityRight h)) |