{-# OPTIONS --without-K --rewriting --overlapping-instances --instance-search-depth=3 --lossy-unification #-}

open import lib.Basics
open import 2Grp
open import Hmtpy2Grp
open import KFunctor
open import Delooping
open import SqKLoop
open import LoopK-hom

module KLoop-ptr-comp-aux3 where

module KLPC-aux3 {i j k} {X : Type i} {Y : Type j} {Z : Type k}
  {{ηX : has-level  X}} {{ηY : has-level  Y}} {{ηZ : has-level  Z}}
  (f : X → Y) (g : Y → Z) (x₀ : X) (x : x₀ == x₀) where

  open import KLoop-ptr-comp-defs f g x₀ x

  β-pts-red1-aux : {c₁ c₂ : K₂ (y₀ == y₀) (Loop2Grp y₀)} {p₁ p₂ : c₁ == c₂} (τ : p₁ == p₂)
    →
    ap (ap g) (! (ap (ap (K₂-rec-y₀ x₀ y₀)) τ)) ∙
    ∘-ap g (K₂-rec-y₀ x₀ y₀) p₁ ∙
    ap (λ q → q) (ap (ap (fst ((g , idp) ⊙∘ K₂-rec-hom y₀ idf2G))) τ)
      ==
    ∘-ap g (K₂-rec-y₀ x₀ y₀) p₂
  β-pts-red1-aux {p₁ = idp} idp = idp

  abstract
    β-pts-red1 : 
      ap (ap g) (! (ap (ap (K₂-rec-y₀ x₀ y₀)) (K₂-map-β-pts (Loop2Grp-map-str (f , idp)) x))) ◃∙
      ap (ap g) (! (ap-∘ (K₂-rec-y₀ x₀ y₀) (K₂-map (Loop2Grp-map-str (f , idp))) (loop (x₀ == x₀) x))) ◃∙
      ap (λ q → q) (ap (λ q → q) (∘-ap g (λ z → fst (K₂-rec-hom y₀ idf2G ⊙∘ K₂-map⊙ (Loop2Grp-map-str (f , idp))) z) (loop (x₀ == x₀) x))) ◃∙
      ap (λ q → q) (ap-∘ (fst ((g , idp) ⊙∘ K₂-rec-hom y₀ idf2G)) (fst (K₂-map⊙ (Loop2Grp-map-str (f , idp)))) (loop (x₀ == x₀) x)) ◃∙
      ap (λ q → q) (ap (ap (fst ((g , idp) ⊙∘ K₂-rec-hom y₀ idf2G))) (K₂-map-β-pts (Loop2Grp-map-str (f , idp)) x)) ◃∎
        =ₛ
      ∘-ap g (K₂-rec-y₀ x₀ y₀) (loop (y₀ == y₀) (ap f x)) ◃∎
    β-pts-red1 =
      ap (ap g) (! (ap (ap (K₂-rec-y₀ x₀ y₀)) (K₂-map-β-pts (Loop2Grp-map-str (f , idp)) x))) ◃∙
      ap (ap g) (! (ap-∘ (K₂-rec-y₀ x₀ y₀) (K₂-map (Loop2Grp-map-str (f , idp))) (loop (x₀ == x₀) x))) ◃∙
      ap (λ q → q) (ap (λ q → q) (∘-ap g (λ z → fst (K₂-rec-hom y₀ idf2G ⊙∘ K₂-map⊙ (Loop2Grp-map-str (f , idp))) z) (loop (x₀ == x₀) x))) ◃∙
      ap (λ q → q) (ap-∘ (fst ((g , idp) ⊙∘ K₂-rec-hom y₀ idf2G)) (fst (K₂-map⊙ (Loop2Grp-map-str (f , idp)))) (loop (x₀ == x₀) x)) ◃∙
      ap (λ q → q) (ap (ap (fst ((g , idp) ⊙∘ K₂-rec-hom y₀ idf2G))) (K₂-map-β-pts (Loop2Grp-map-str (f , idp)) x)) ◃∎
        =ₛ⟨  &  & lemma (loop (x₀ == x₀) x) ⟩
      ap (ap g) (! (ap (ap (K₂-rec-y₀ x₀ y₀)) (K₂-map-β-pts (Loop2Grp-map-str (f , idp)) x))) ◃∙
      ∘-ap g (K₂-rec-y₀ x₀ y₀) (ap (K₂-map (Loop2Grp-map-str (f , idp))) (loop (x₀ == x₀) x)) ◃∙
      ap (λ q → q) (ap (ap (fst ((g , idp) ⊙∘ K₂-rec-hom y₀ idf2G))) (K₂-map-β-pts (Loop2Grp-map-str (f , idp)) x)) ◃∎
        =ₛ₁⟨ β-pts-red1-aux (K₂-map-β-pts (Loop2Grp-map-str (f , idp)) x) ⟩
      ∘-ap g (K₂-rec-y₀ x₀ y₀) (loop (y₀ == y₀) (ap f x)) ◃∎ ∎ₛ
        where
          lemma : {b : _} (p : base (x₀ == x₀) == b) →
            ap (ap g) (! (ap-∘ (K₂-rec-y₀ x₀ y₀) (K₂-map (Loop2Grp-map-str (f , idp))) p)) ◃∙
            ap (λ q → q) (ap (λ q → q) (∘-ap g (λ z → fst (K₂-rec-hom y₀ idf2G ⊙∘ K₂-map⊙ (Loop2Grp-map-str (f , idp))) z) p)) ◃∙
            ap (λ q → q) (ap-∘ (fst ((g , idp) ⊙∘ K₂-rec-hom y₀ idf2G)) (fst (K₂-map⊙ (Loop2Grp-map-str (f , idp)))) p) ◃∎
              =ₛ
            ∘-ap g (K₂-rec-y₀ x₀ y₀) (ap (K₂-map (Loop2Grp-map-str (f , idp))) p) ◃∎
          lemma idp = =ₛ-in idp