Investigations on graph-theoretical constructions in Homotopy type theory

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

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

open import foundations.HomotopyType
open import foundations.HomotopyLemmas

open import foundations.HalfAdjointType
open import foundations.QuasiinverseType
open import foundations.QuasiinverseLemmas

module EquivalenceReasoning where

_≃⟨⟩_
  : ∀ {ℓ₁ ℓ₂} (A : Type ℓ₁) {B : Type ℓ₂}
  → A ≃ B
  -------
  → A ≃ B

_ ≃⟨⟩ e = e
infixr  _≃⟨⟩_

_≃⟨by-def⟩_
  : ∀ {ℓ₁ ℓ₂} (A : Type ℓ₁) {B : Type ℓ₂}
  → A ≃ B
  -------
  → A ≃ B

_ ≃⟨by-def⟩ e = e
infixr  _≃⟨by-def⟩_

_≃⟨_⟩_
  : ∀ {ℓ₁ ℓ₂ ℓ₃ : Level} (A : Type ℓ₁){B : Type ℓ₂} {C : Type ℓ₃}
  → A ≃ B → B ≃ C
  ---------------
  → A ≃ C

_ ≃⟨ e₁ ⟩ e₂ = ≃-trans e₁ e₂

infixr  _≃⟨_⟩_

_≃∎
  : ∀ {ℓ : Level} (A : Type ℓ)
  → A ≃ A

_≃∎ = λ A → idEqv {A = A}
infix   _≃∎

begin≃_
  : ∀ {ℓ₁ ℓ₂ : Level} {A : Type ℓ₁}{B : Type ℓ₂}
  → A ≃ B
  -------
  → A ≃ B

begin≃_ e = e
infix   begin≃_

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