Progress on univalence
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@ -344,25 +344,25 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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pq : Arrow a b ≡ Arrow a' b'
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pq i = Arrow (p i) (q i)
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U : ∀ b'' → b ≡ b'' → Set _
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U b'' q' = coe (λ i → Arrow a (q' i)) f ≡ fst (idToIso _ _ q') <<< f <<< (fst (snd (idToIso _ _ refl)))
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u : coe (λ i → Arrow a b) f ≡ fst (idToIso _ _ refl) <<< f <<< (fst (snd (idToIso _ _ refl)))
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u = begin
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coe refl f ≡⟨ id-coe ⟩
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f ≡⟨ sym leftIdentity ⟩
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identity <<< f ≡⟨ sym rightIdentity ⟩
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identity <<< f <<< identity ≡⟨ cong (λ φ → identity <<< f <<< φ) lem ⟩
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identity <<< f <<< (fst (snd (idToIso _ _ refl))) ≡⟨ cong (λ φ → φ <<< f <<< (fst (snd (idToIso _ _ refl)))) lem ⟩
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fst (idToIso _ _ refl) <<< f <<< (fst (snd (idToIso _ _ refl))) ∎
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where
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lem : ∀ {x} → PathP (λ _ → Arrow x x) identity (fst (idToIso x x refl))
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lem = sym (subst-neutral {P = λ x → Arrow x x})
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U : ∀ b'' → b ≡ b'' → Set _
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U b'' q' = coe (λ i → Arrow a (q' i)) f ≡ fst (idToIso _ _ q') <<< f <<< (fst (snd (idToIso _ _ refl)))
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u : coe (λ i → Arrow a b) f ≡ fst (idToIso _ _ refl) <<< f <<< (fst (snd (idToIso _ _ refl)))
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u = begin
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coe refl f ≡⟨ id-coe ⟩
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f ≡⟨ sym leftIdentity ⟩
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identity <<< f ≡⟨ sym rightIdentity ⟩
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identity <<< f <<< identity ≡⟨ cong (λ φ → identity <<< f <<< φ) lem ⟩
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identity <<< f <<< (fst (snd (idToIso _ _ refl))) ≡⟨ cong (λ φ → φ <<< f <<< (fst (snd (idToIso _ _ refl)))) lem ⟩
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fst (idToIso _ _ refl) <<< f <<< (fst (snd (idToIso _ _ refl))) ∎
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where
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lem : ∀ {x} → PathP (λ _ → Arrow x x) identity (fst (idToIso x x refl))
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lem = sym subst-neutral
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D : ∀ a'' → a ≡ a'' → Set _
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D a'' p' = coe (λ i → Arrow (p' i) (q i)) f ≡ fst (idToIso b b' q) <<< f <<< (fst (snd (idToIso _ _ p')))
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D : ∀ a'' → a ≡ a'' → Set _
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D a'' p' = coe (λ i → Arrow (p' i) (q i)) f ≡ fst (idToIso b b' q) <<< f <<< (fst (snd (idToIso _ _ p')))
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d : coe (λ i → Arrow a (q i)) f ≡ fst (idToIso b b' q) <<< f <<< (fst (snd (idToIso _ _ refl)))
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d = pathJ U u b' q
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d : coe (λ i → Arrow a (q i)) f ≡ fst (idToIso b b' q) <<< f <<< (fst (snd (idToIso _ _ refl)))
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d = pathJ U u b' q
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9-1-9 : coe pq f ≡ q* <<< f <<< p~
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9-1-9 = pathJ D d a' p
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@ -258,22 +258,45 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
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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 : ∀ {x} → ℂ.idToIso _ _ (ℂ.isoToId x) ≡ x
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lem {x} i = snd ℂ.inverse-from-to-iso' i x
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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 ≡⟨ {!!} ⟩
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coe r-arrow xa
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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 {P = λ x → ℂ.Arrow x x}) ⟩
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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 , {!h!} , {!!})
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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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, lemA (fst inv-f)
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, lemA (snd inv-f)
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@ -283,14 +306,28 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
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let
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iso : X ℂ.≅ Y
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iso = fst f , fst f~ , cong fst inv-f , cong fst inv-f~
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helper : PathP (λ i → ℂ.Arrow (ℂ.isoToId {!!} i) A) xa ya
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helper = {!!}
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in iso , helper , {!!}})
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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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, 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.propIsomorphism _ _ _)))
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(Σ≡ {!ℂ!} (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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