Properties of the disjoint sum _+_ of types.

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{-# OPTIONS --safe --without-K #-}

module MLTT.Plus-Properties where

open import MLTT.Plus
open import MLTT.Negation
open import MLTT.Id
open import MLTT.Empty
open import MLTT.Unit
open import MLTT.Unit-Properties

+-commutative : {A : 𝓀 Μ‡ } {B : π“₯ Μ‡ } β†’ A + B β†’ B + A
+-commutative = cases inr inl

+disjoint : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } {x : X} {y : Y} β†’ Β¬ (inl x ＝ inr y)
+disjoint {𝓀} {π“₯} {X} {Y} p = πŸ™-is-not-𝟘 q
 where
  f : X + Y β†’ 𝓀₀ Μ‡
  f (inl x) = πŸ™
  f (inr y) = 𝟘

  q : πŸ™ ＝ 𝟘
  q = ap f p

+disjoint' : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } {x : X} {y : Y} β†’ Β¬ (inr y ＝ inl x)
+disjoint' p = +disjoint (p ⁻¹)

inl-lc : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } {x x' : X} β†’ inl {𝓀} {π“₯} {X} {Y} x ＝ inl x' β†’ x ＝ x'
inl-lc refl = refl

inr-lc : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } {y y' : Y} β†’ inr {𝓀} {π“₯} {X} {Y} y ＝ inr y' β†’ y ＝ y'
inr-lc refl = refl

equality-cases : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } {A : 𝓦 Μ‡ } (z : X + Y)
               β†’ ((x : X) β†’ z ＝ inl x β†’ A) β†’ ((y : Y) β†’ z ＝ inr y β†’ A) β†’ A
equality-cases (inl x) f g = f x refl
equality-cases (inr y) f g = g y refl

Cases-equality-l : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } {A : 𝓦 Μ‡ } (f : X β†’ A) (g : Y β†’ A)
                 β†’ (z : X + Y) (x : X) β†’ z ＝ inl x β†’ Cases z f g ＝ f x
Cases-equality-l f g .(inl x) x refl = refl

Cases-equality-r : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } {A : 𝓦 Μ‡ } (f : X β†’ A) (g : Y β†’ A)
                 β†’ (z : X + Y) (y : Y) β†’ z ＝ inr y β†’ Cases z f g ＝ g y
Cases-equality-r f g .(inr y) y refl = refl

Left-fails-gives-right-holds : {P : 𝓀 Μ‡ } {Q : π“₯ Μ‡ } β†’ P + Q β†’ Β¬ P β†’ Q
Left-fails-gives-right-holds (inl p) u = 𝟘-elim (u p)
Left-fails-gives-right-holds (inr q) u = q

Right-fails-gives-left-holds : {P : 𝓀 Μ‡ } {Q : π“₯ Μ‡ } β†’ P + Q β†’ Β¬ Q β†’ P
Right-fails-gives-left-holds (inl p) u = p
Right-fails-gives-left-holds (inr q) u = 𝟘-elim (u q)

open import MLTT.Sigma
open import Notation.General

inl-preservation : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } (f : X + πŸ™ {𝓦}  β†’ Y + πŸ™ {𝓣})
                 β†’ f (inr ⋆) ＝ inr ⋆
                 β†’ left-cancellable f
                 β†’ (x : X) β†’ Ξ£ y κž‰ Y , f (inl x) ＝ inl y
inl-preservation {𝓀} {π“₯} {𝓦} {𝓣} {X} {Y} f p l x = Ξ³ x (f (inl x)) refl
 where
  Ξ³ : (x : X) (z : Y + πŸ™) β†’ f (inl x) ＝ z β†’ Ξ£ y κž‰ Y , z ＝ inl y
  Ξ³ x (inl y) q = y , refl
  Ξ³ x (inr ⋆) q = 𝟘-elim (+disjoint (l r))
   where
    r : f (inl x) ＝ f (inr ⋆)
    r = q βˆ™ p ⁻¹

+functor : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } {A : 𝓦 Μ‡ } {B : 𝓣 Μ‡ }
         β†’ (X β†’ A) β†’ (Y β†’ B) β†’ X + Y β†’ A + B
+functor f g (inl x) = inl (f x)
+functor f g (inr y) = inr (g y)

+functorβ‚‚ : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } {Z : 𝓦 Μ‡ } {X' : 𝓀' Μ‡ } {Y' : π“₯' Μ‡ } {Z' : 𝓦' Μ‡ }
          β†’ (X β†’ X') β†’ (Y β†’ Y') β†’ (Z β†’ Z') β†’ X + Y + Z β†’ X' + Y' + Z'
+functorβ‚‚ f g h = +functor f (+functor g h)

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