holy shit solved leg3
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@ -41,33 +41,69 @@ module _ {A∙ @ (A , a₀) : Pointed ℓ} {B∙ @ (B , b₀) : Pointed ℓ} (f
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P : A → Type ℓ
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P a = f a ≡ b₀
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fw : Σ A (λ a → (a ≡ a₀) × (f a ≡ b₀)) → f a₀ ≡ b₀
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fw (a , p , feq) = subst P p feq
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fw (a , p , feq) = cong f (sym p) ∙ feq
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gw : f a₀ ≡ b₀ → Σ A (λ a → (a ≡ a₀) × (f a ≡ b₀))
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gw p = a₀ , refl , p
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fg : section fw gw
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fg p = substRefl {B = λ a → f a ≡ b₀} p
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gf : retract fw gw
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-- p : f a₀ ≡ b₀
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fg p = sym (lUnit p)
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-- fw (gw p) ≡ p
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-- fw (a₀ , refl , p) ≡ p
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-- subst (λ a' → f a' ≡ b₀) refl p ≡ p
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test : (feq : f a₀ ≡ b₀) → subst P refl feq ≡ feq
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test feq = substRefl {B = P} feq
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module _ (a : A) (p : a ≡ a₀) (feq : f a ≡ b₀) where
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-- bottomFace : Square (subst P p feq) (subst P refl feq) (λ i → f (p (~ i))) refl
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-- bottomFace i j = subst {! !} {! !} {! !} {! !}
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-- cong f (sym refl) ∙ p ≡ p
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-- fg p = substRefl {B = λ a → f a ≡ b₀} p
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gf : retract fw gw
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gf (a , p , feq) i = p (~ i) , (λ j → p (~ i ∨ j)) , λ j → sq i j where
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-- gw (fw (a , p , feq)) ≡ (a , p , feq)
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-- gw (cong f (sym p) ∙ feq) ≡ (a , p , feq)
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-- a₀ , refl , (cong f (sym p) ∙ feq) ≡ (a , p , feq)
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postulate
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-- TODO: FINISH THIS
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topFace : Square (λ i → f (p (~ i))) (λ i → b₀) (subst P p feq) feq
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-- topFace = {! !}
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-- (cong f (λ i → p (~ i)) ∙ feq) (cong f (λ i → a) ∙ feq)
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gf (a , p , feq) i = p (~ i) , (λ j → p (~ i ∨ j)) , λ j → topFace a p feq j i
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sq : PathP (λ i → f (p (~ i)) ≡ b₀) ((λ i → f (p (~ i))) ∙ feq) feq
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sq i j =
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let u = λ k → λ where
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(i = i0) → compPath-filler (cong f (sym p)) feq k j
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(i = i1) → feq (j ∧ k)
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(j = i0) → f (p (~ i))
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(j = i1) → feq k
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in hcomp u (cong f (sym p) (i ∨ j))
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-- -- p : f a₀ ≡ b₀
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-- -- fw (gw p) ≡ p
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-- -- fw (a₀ , refl , p) ≡ p
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-- -- subst (λ a' → f a' ≡ b₀) refl p ≡ p
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-- module _ (a : A) (p : a ≡ a₀) (feq : f a ≡ b₀) where
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-- -- bottomFace : Square (subst P p feq) (subst P refl feq) (λ i → f (p (~ i))) refl
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-- -- bottomFace i j = subst {! !} {! !} {! !} {! !}
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-- test : PathP (λ i → a ≡ p (~ i)) p refl
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-- test i j = p (~ i ∧ j)
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-- test2 : f a₀ ≡ b₀
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-- test2 = cong f (sym p) ∙ feq
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-- test3 : f a ≡ b₀
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-- test3 = refl ∙ feq
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-- bottomFace : PathP (λ i → f (p (~ i)) ≡ b₀) (cong f (sym p) ∙ feq) (refl ∙ feq)
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-- bottomFace = congP (λ i q → cong f (sym q) ∙ feq) test
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-- topFace : PathP (λ i → f (p (~ i)) ≡ b₀) (subst P p feq) feq
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-- topFace i j = hcomp u (bottomFace i j) where
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-- u = λ k → λ where
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-- (i = i0) → {! !}
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-- (i = i1) → {! !}
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-- (j = i0) → f (p (~ i))
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-- (j = i1) → b₀
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-- gf (a , p , feq) i = p (~ i) , (λ j → p (~ i ∨ j)) , λ j → {! !}
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-- topFace a p feq i j
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-- gw (fw (a , p , feq)) ≡ (a , p , feq)
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-- gw (subst (λ a' → f a' ≡ b₀) p feq) ≡ (a , p , feq)
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-- (a₀ , refl , p) ≡ (a , p , feq)
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eq : subst P refl f₀ ≡ f₀
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eq = substRefl {B = P} f₀
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eq : refl ∙ f₀ ≡ f₀
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eq = sym (lUnit f₀)
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-- eq = substRefl {B = P} f₀
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leg4 : (f a₀ ≡ b₀) , f₀ ≃∙ Ω B∙
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leg4 = isoToEquiv (iso fw gw fg gf) , eq where
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