cat/src/Cat/Equality.agda

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{-# OPTIONS --cubical #-}
-- Defines equality-principles for data-types from the standard library.
module Cat.Equality where
open import Level
open import Cubical
-- _[_≡_] = PathP
module Equality where
module Data where
module Product where
open import Data.Product
module _ {a b : Level} {A : Set a} {B : A Set b} {a b : Σ A B}
(proj₁≡ : (λ _ A) [ proj₁ a proj₁ b ])
(proj₂≡ : (λ i B (proj₁≡ i)) [ proj₂ a proj₂ b ]) where
Σ≡ : a b
proj₁ (Σ≡ i) = proj₁≡ i
proj₂ (Σ≡ i) = proj₂≡ i
-- Remark 2.7.1: This theorem:
--
-- (x , u) ≡ (x , v) → u ≡ v
--
-- does *not* hold! We can only conclude that there *exists* `p : x ≡ x`
-- such that
--
-- p* u ≡ v
-- thm : isSet A → (∀ {a} → isSet (B a)) → isSet (Σ A B)
-- thm sA sB (x , y) (x' , y') p q = res
-- where
-- x≡x'0 : x ≡ x'
-- x≡x'0 = λ i → proj₁ (p i)
-- x≡x'1 : x ≡ x'
-- x≡x'1 = λ i → proj₁ (q i)
-- someP : x ≡ x'
-- someP = {!!}
-- tricky : {!y!} ≡ y'
-- tricky = {!!}
-- -- res' : (λ _ → Σ A B) [ (x , y) ≡ (x' , y') ]
-- res' : ({!!} , {!!}) ≡ ({!!} , {!!})
-- res' = {!!}
-- res : p ≡ q
-- res i = {!res'!}