Investigations on graph-theoretical constructions in Homotopy type theory

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

module foundations.UnivalenceIdEquiv where
open import foundations.TransportLemmas
open import foundations.EquivalenceType

open import foundations.HomotopyType

open import foundations.EquivalenceType
open import foundations.QuasiinverseType
open import foundations.QuasiinverseLemmas
open import foundations.UnivalenceAxiom
open import foundations.UnivalenceTransport
open import foundations.HLevelLemmas

The identity equivalence creates the trivial path.

ua-id
  : ∀ {ℓ : Level} {A : Type ℓ}
  --------------------
  → ua idEqv ≡ refl A

ua-id {A = A} =
  begin
    ua idEqv
      ≡⟨ ap ua (sameEqv (refl id)) ⟩
    ua (idtoeqv (refl A))
      ≡⟨ ua-η (refl A) ⟩
    refl A
  ∎

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