{-# OPTIONS --without-K --rewriting --lossy-unification #-}

open import lib.Basics
open import 2Grp
open import Codes

module Decode-aux3a where

module _ {i} {G : Type i} {{η : CohGrp G}} where

  open import Delooping G

  module Long-aux8 (z : G) (p₁ p₂ : base == base)
    (q₂ : loop (coe (ap fst (ap codes p₁)) z) ∙ p₂ == loop (coe (ap fst (ap codes p₂)) (coe (ap fst (ap codes p₁)) z)))
    where

    private
      τ =
        ap (λ v → loop v ∙ p₂)
          (transp-coe p₁ z ∙ ap (λ q → coe q z) (ap-∘ fst codes p₁) ∙ idp) ◃∙
        ! (ap loop (transp-coe p₂ (coe (ap fst (ap codes p₁)) z) ∙
            ap (λ q → coe q (coe (ap fst (ap codes p₁)) z))
              (ap-∘ fst codes p₂) ∙ idp) ∙
          ! q₂) ◃∙
        ! (ap (λ v → loop (transport codes-fst p₂ v))
            (transp-coe p₁ z ∙ ap (λ q → coe q z) (ap-∘ fst codes p₁) ∙ idp)) ◃∎

    long-aux8-aux0 :
      ! (transport (λ v → loop (transport codes-fst p₂ v) == loop v ∙ p₂)
        (! (transp-coe p₁ z ∙ ap (λ q → coe q z) (ap-∘ fst codes p₁) ∙ idp))
        (ap loop (transp-coe p₂ (coe (ap fst (ap codes p₁)) z) ∙
          ap (λ q → coe q (coe (ap fst (ap codes p₁)) z))
            (ap-∘ fst codes p₂) ∙ idp) ∙
        ! q₂)) ◃∎
        =ₛ
      τ
    long-aux8-aux0 =
      transp-pth!-!◃
        (transp-coe p₁ z ∙ ap (λ q → coe q z) (ap-∘ fst codes p₁) ∙ idp)
        (ap loop (transp-coe p₂ (coe (ap fst (ap codes p₁)) z) ∙
          ap (λ q → coe q (coe (ap fst (ap codes p₁)) z))
            (ap-∘ fst codes p₂) ∙ idp) ∙
        ! q₂)
         
    long-aux8-aux1 =
      τ
        =ₛ⟨  &  &
          !-∙-!
            (ap loop (transp-coe p₂ (coe (ap fst (ap codes p₁)) z) ∙
              ap (λ q → coe q (coe (ap fst (ap codes p₁)) z))
              (ap-∘ fst codes p₂) ∙ idp))
            q₂ ⟩
      ap (λ v → loop v ∙ p₂)
        (transp-coe p₁ z ∙ ap (λ q → coe q z) (ap-∘ fst codes p₁) ∙ idp) ◃∙
      q₂ ◃∙
      ! (ap loop (transp-coe p₂ (coe (ap fst (ap codes p₁)) z) ∙
        ap (λ q → coe q (coe (ap fst (ap codes p₁)) z))
          (ap-∘ fst codes p₂) ∙ idp)) ◃∙
      ! (ap (λ v → loop (transport codes-fst p₂ v))
          (transp-coe p₁ z ∙ ap (λ q → coe q z) (ap-∘ fst codes p₁) ∙ idp)) ◃∎ ∎ₛ