Left cancellable maps.

The definition is given in UF-Base. Here we prove things about them.

\begin{code}

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

module UF-LeftCancellable where

open import SpartanMLTT
open import UF-Base
open import UF-Subsingletons
open import UF-Retracts
open import UF-Equiv

left-cancellable-reflects-is-prop : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } (f : X β†’ Y)
                                 β†’ left-cancellable f β†’ is-prop Y β†’ is-prop X
left-cancellable-reflects-is-prop f lc i x x' = lc (i (f x) (f x'))

section-lc : {X : 𝓀 Μ‡ } {A : π“₯ Μ‡ } (s : X β†’ A) β†’ is-section s β†’ left-cancellable s
section-lc {𝓀} {π“₯} {X} {Y} s (r , rs) {x} {y} p = (rs x)⁻¹ βˆ™ ap r p βˆ™ rs y

is-equiv-lc : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } (f : X β†’ Y) β†’ is-equiv f β†’ left-cancellable f
is-equiv-lc f (_ , hasr) = section-lc f hasr

left-cancellable-closed-under-∘ : {X : 𝓀 Μ‡ } {Y : π“₯ Μ‡ } {Z : 𝓦 Μ‡ } (f : X β†’ Y) (g : Y β†’ Z)
                                β†’ left-cancellable f β†’ left-cancellable g β†’ left-cancellable (g ∘ f)
left-cancellable-closed-under-∘ f g lcf lcg = lcf ∘ lcg

NatΞ£-lc : {X : 𝓀 Μ‡ } {A : X β†’ π“₯ Μ‡ } {B : X β†’ 𝓦 Μ‡ } (f : Nat A B)
        β†’ ((x : X) β†’ left-cancellable(f x))
        β†’ left-cancellable (NatΞ£ f)
NatΞ£-lc {𝓀} {π“₯} {𝓦} {X} {A} {B} f flc {x , a} {x' , a'} p = to-Ξ£-≑ (ap pr₁ p , Ξ³)
 where
  Ξ³ : transport A (ap pr₁ p) a ≑ a'
  Ξ³ = flc x' (f x' (transport A (ap pr₁ p) a) β‰‘βŸ¨ nat-transport f (ap pr₁ p) ⟩
              transport B (ap pr₁ p) (f x a)  β‰‘βŸ¨ from-Ξ£-≑' p ⟩
              f x' a'                         ∎)

NatΞ -lc : {X : 𝓀 Μ‡ } {A : X β†’ π“₯ Μ‡ } {B : X β†’ 𝓦 Μ‡ } (f : Nat A B)
        β†’ ((x : X) β†’ left-cancellable(f x))
        β†’ {g g' : Ξ  A} β†’ NatΞ  f g ≑ NatΞ  f g' β†’ g ∼ g'
NatΞ -lc f flc {g} {g'} p x = flc x (happly p x)

\end{code}