Construct the morphism for equivalence 2
I must still show that they are inverses.
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@ -327,6 +327,7 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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asTypeIso : TypeIsomorphism (idToIso A B)
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asTypeIso = toIso _ _ univalent
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-- FIXME Rename
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inverse-from-to-iso' : AreInverses (idToIso A B) isoToId
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inverse-from-to-iso' = snd iso
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@ -377,6 +378,62 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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where
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lem : p~ <<< p* ≡ identity
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lem = fst (snd (snd (idToIso _ _ p)))
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module _ {A B X : Object} (iso : A ≅ B) where
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private
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p : A ≡ B
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p = isoToId iso
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p-dom : Arrow A X ≡ Arrow B X
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p-dom = cong (λ x → Arrow x X) p
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p-cod : Arrow X A ≡ Arrow X B
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p-cod = cong (λ x → Arrow X x) p
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lem : ∀ {A B} {x : A ≅ B} → idToIso A B (isoToId x) ≡ x
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lem {x = x} i = snd inverse-from-to-iso' i x
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open Σ iso renaming (fst to ι) using ()
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open Σ (snd iso) renaming (fst to ι~ ; snd to inv)
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coe-dom : {f : Arrow A X} → coe p-dom f ≡ f <<< ι~
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coe-dom {f} = begin
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coe p-dom f
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≡⟨ 9-1-9 p refl f ⟩
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fst (idToIso _ _ refl) <<< f <<< fst (snd (idToIso _ _ p))
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≡⟨ cong (λ φ → φ <<< f <<< fst (snd (idToIso _ _ p))) subst-neutral ⟩
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identity <<< f <<< fst (snd (idToIso _ _ p))
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≡⟨ cong (λ φ → identity <<< f <<< φ) (cong (λ x → (fst (snd x))) lem) ⟩
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identity <<< f <<< ι~
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≡⟨ cong (_<<< ι~) leftIdentity ⟩
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f <<< ι~ ∎
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coe-cod : {f : Arrow X A} → coe p-cod f ≡ ι <<< f
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coe-cod {f} = begin
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coe p-cod f
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≡⟨ 9-1-9 refl p f ⟩
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fst (idToIso _ _ p) <<< f <<< fst (snd (idToIso _ _ refl))
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≡⟨ cong (λ φ → fst (idToIso _ _ p) <<< f <<< φ) subst-neutral ⟩
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fst (idToIso _ _ p) <<< f <<< identity
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≡⟨ cong (λ φ → φ <<< f <<< identity) (cong fst lem) ⟩
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ι <<< f <<< identity
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≡⟨ sym isAssociative ⟩
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ι <<< (f <<< identity)
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≡⟨ cong (ι <<<_) rightIdentity ⟩
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ι <<< f ∎
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module _ {f : Arrow A X} {g : Arrow B X} (q : PathP (λ i → p-dom i) f g) where
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domain-twist : g ≡ f <<< ι~
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domain-twist = begin
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g ≡⟨ sym (coe-lem q) ⟩
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coe p-dom f ≡⟨ coe-dom ⟩
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f <<< ι~ ∎
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-- This can probably also just be obtained from the above my taking the
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-- symmetric isomorphism.
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domain-twist0 : f ≡ g <<< ι
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domain-twist0 = begin
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f ≡⟨ sym rightIdentity ⟩
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f <<< identity ≡⟨ cong (f <<<_) (sym (fst inv)) ⟩
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f <<< (ι~ <<< ι) ≡⟨ isAssociative ⟩
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f <<< ι~ <<< ι ≡⟨ cong (_<<< ι) (sym domain-twist) ⟩
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g <<< ι ∎
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-- | All projections are propositions.
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module Propositionality where
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@ -238,7 +238,7 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
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-- I have `φ p` in scope, but surely `p` and `x` are the same - though
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-- perhaps not definitonally.
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, (λ{ (iso , x) → ℂ.isoToId iso , x})
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, funExt (λ{ (p , q , r) → Σ≡ (sym iso-id-inv) (toPathP {A = λ i → {!!}} {!!})})
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, funExt (λ{ (p , q , r) → Σ≡ (sym iso-id-inv) {!!}})
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, funExt (λ x → Σ≡ (sym id-iso-inv) {!!})
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step2
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: Σ (X ℂ.≅ Y) (λ iso
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@ -249,55 +249,9 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
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)
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≈ ((X , xa , xb) ≅ (Y , ya , yb))
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step2
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= ( λ{ ((f , f~ , inv-f) , p , q)
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→
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let
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r : X ≡ Y
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r = ℂ.isoToId (f , f~ , inv-f)
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r-arrow : (ℂ.Arrow X A) ≡ (ℂ.Arrow Y A)
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r-arrow i = ℂ.Arrow (r i) A
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t : coe r-arrow xa ≡ ℂ.identity ℂ.<<< xa ℂ.<<< f~
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t = {!? ≡ ?!}
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lem : ∀ {A B} {x : A ℂ.≅ B} → ℂ.idToIso A B (ℂ.isoToId x) ≡ x
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lem = λ{ {x = x} i → snd ℂ.inverse-from-to-iso' i x}
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lem' : ∀ {A B} {x : A ≡ B} → ℂ.isoToId (ℂ.idToIso _ _ x) ≡ x
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lem' = λ{ {x = x} i → fst ℂ.inverse-from-to-iso' i x}
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h : ya ≡ xa ℂ.<<< f~
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h = begin
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ya ≡⟨ sym (coe-lem p) ⟩
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coe r-arrow xa
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≡⟨ ℂ.9-1-9 r refl xa ⟩
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fst (ℂ.idToIso _ _ refl) ℂ.<<< xa ℂ.<<< fst (snd (ℂ.idToIso _ _ r))
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≡⟨ cong (λ φ → φ ℂ.<<< xa ℂ.<<< fst (snd (ℂ.idToIso _ _ r))) subst-neutral ⟩
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ℂ.identity ℂ.<<< xa ℂ.<<< fst (snd (ℂ.idToIso _ _ r))
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≡⟨ cong (λ φ → ℂ.identity ℂ.<<< xa ℂ.<<< φ) (cong (λ x → (fst (snd x))) lem) ⟩
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ℂ.identity ℂ.<<< xa ℂ.<<< f~
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≡⟨ cong (ℂ._<<< f~) ℂ.leftIdentity ⟩
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xa ℂ.<<< f~ ∎
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complicated : Y ≡ X
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complicated = (λ z → sym (ℂ.isoToId (f , f~ , inv-f)) z)
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in
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( f , (begin
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ℂ [ ya ∘ f ] ≡⟨⟩
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ya ℂ.<<< f ≡⟨ sym ℂ.leftIdentity ⟩
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ℂ.identity ℂ.<<< (ya ℂ.<<< f)
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≡⟨ ℂ.isAssociative ⟩
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ℂ.identity ℂ.<<< ya ℂ.<<< f
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≡⟨ cong (λ φ → φ ℂ.<<< ya ℂ.<<< f) (sym subst-neutral) ⟩
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fst (ℂ.idToIso _ _ refl) ℂ.<<< ya ℂ.<<< f
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≡⟨ cong (λ φ → fst (ℂ.idToIso _ _ refl) ℂ.<<< ya ℂ.<<< φ)
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(begin
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f ≡⟨ {!cong (λ x → (fst (snd x))) ((lem {x = f~ , f , swap inv-f}))!} ⟩
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-- f ≡⟨ cong fst (sym (lem {f , f~ , inv-f})) ⟩
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-- fst ( ℂ.idToIso X Y ( ℂ.isoToId (f , f~ , inv-f))) ≡⟨ {!fst inv-f!} ⟩
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fst (snd (ℂ.idToIso Y X (λ z → sym (ℂ.isoToId (f , f~ , inv-f)) z))) ∎)
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⟩
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_ ℂ.<<< ya ℂ.<<< fst (snd (ℂ.idToIso _ _ _))
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≡⟨ sym (ℂ.9-1-9 (sym r) refl ya) ⟩
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coe (sym r-arrow) ya
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≡⟨ coe-lem (λ i → p (~ i)) ⟩
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xa ∎) , {!!})
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, ( f~ , sym h , {!!})
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= ( λ{ (iso@(f , f~ , inv-f) , p , q)
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→ ( f , sym (ℂ.domain-twist0 iso p) , sym (ℂ.domain-twist0 iso q))
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, ( f~ , sym (ℂ.domain-twist iso p) , sym (ℂ.domain-twist iso q))
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, lemA (fst inv-f)
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, lemA (snd inv-f)
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}
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@ -308,26 +262,26 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
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iso = fst f , fst f~ , cong fst inv-f , cong fst inv-f~
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p : X ≡ Y
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p = ℂ.isoToId iso
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lem : xa ≡ _
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lem = begin
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xa ≡⟨ sym (fst (snd f)) ⟩
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ya ℂ.<<< fst f ≡⟨ sym ℂ.leftIdentity ⟩
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ℂ.identity ℂ.<<< (ya ℂ.<<< fst f) ≡⟨ ℂ.isAssociative ⟩
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ℂ.identity ℂ.<<< ya ℂ.<<< fst f ≡⟨ {!sym (ℂ.9-1-9 ? refl ya)!} ⟩
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_ ℂ.<<< ya ℂ.<<< _ ≡⟨ sym (ℂ.9-1-9 (sym p) refl ya) ⟩
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coe (λ i → ℂ.Arrow (p (~ i)) A) ya ∎
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lem0 = begin
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xa ≡⟨ {!!} ⟩
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coe _ xa ≡⟨ {!!} ⟩
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ya ℂ.<<< fst f ∎
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helper : PathP (λ i → ℂ.Arrow (p i) A) xa ya
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helper = {!snd f!}
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in iso , ? , {!!}})
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pA : ℂ.Arrow X A ≡ ℂ.Arrow Y A
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pA = cong (λ x → ℂ.Arrow x A) p
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pB : ℂ.Arrow X B ≡ ℂ.Arrow Y B
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pB = cong (λ x → ℂ.Arrow x B) p
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k0 = begin
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coe pB xb ≡⟨ ℂ.coe-dom iso ⟩
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xb ℂ.<<< fst f~ ≡⟨ snd (snd f~) ⟩
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yb ∎
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k1 = begin
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coe pA xa ≡⟨ ℂ.coe-dom iso ⟩
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xa ℂ.<<< fst f~ ≡⟨ fst (snd f~) ⟩
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ya ∎
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helper : PathP (λ i → pA i) xa ya
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helper = coe-lem-inv k1
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in iso , coe-lem-inv k1 , coe-lem-inv k0})
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, record
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{ fst = funExt (λ x → lemSig
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(λ x → propSig prop0 (λ _ → prop1))
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_ _
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(Σ≡ {!ℂ!} (ℂ.propIsomorphism _ _ _)))
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(Σ≡ {!!} (ℂ.propIsomorphism _ _ _)))
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; snd = funExt (λ{ (f , _) → lemSig propIsomorphism _ _ {!refl!}})
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}
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where
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