Remove bad lemma for showing univalence
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@ -127,67 +127,6 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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Univalent[Contr] : Set _
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Univalent[Contr] = ∀ A → isContr (Σ[ X ∈ Object ] A ≅ X)
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private
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module _ (A : Object)
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-- It may be that we need something weaker than this, in that there
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-- may be some other lemmas available to us.
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-- For instance, `need0` should be available to us when we prove `need1`.
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(need0 : (s : Σ Object (A ≅_)) → (open Σ s renaming (fst to Y) using ()) → A ≡ Y)
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(need2 : (iso : A ≅ A)
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→ (open Σ iso renaming (fst to f ; snd to iso-f))
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→ (open Σ iso-f renaming (fst to f~ ; snd to areInv))
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→ (identity , identity) ≡ (f , f~)
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) where
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c : Σ Object (A ≅_)
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c = A , idIso A
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module _ (y : Σ Object (A ≅_)) where
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open Σ y renaming (fst to Y ; snd to isoY)
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q : A ≡ Y
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q = need0 y
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-- Some error with primComp
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isoAY : A ≅ Y
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isoAY = {!idToIso A Y q!}
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lem : PathP (λ i → A ≅ q i) (idIso A) isoY
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lem = d* isoAY
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where
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D : (Y : Object) → (A ≡ Y) → Set _
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D Y p = (A≅Y : A ≅ Y) → PathP (λ i → A ≅ p i) (idIso A) A≅Y
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d : D A refl
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d A≅Y i = a0 i , a1 i , a2 i
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where
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open Σ A≅Y renaming (fst to f ; snd to iso-f)
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open Σ iso-f renaming (fst to f~ ; snd to areInv)
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aaa : (identity , identity) ≡ (f , f~)
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aaa = need2 A≅Y
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a0 : identity ≡ f
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a0 i = fst (aaa i)
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a1 : identity ≡ f~
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a1 i = snd (aaa i)
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-- we do have this!
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-- I just need to rearrange the proofs a bit.
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postulate
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prop : ∀ {A B} (fg : Arrow A B × Arrow B A) → isProp (IsInverseOf (fst fg) (snd fg))
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a2 : PathP (λ i → IsInverseOf (a0 i) (a1 i)) isIdentity areInv
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a2 = lemPropF prop aaa
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d* : D Y q
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d* = pathJ D d Y q
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p : (A , idIso A) ≡ (Y , isoY)
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p i = q i , lem i
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univ-lem : isContr (Σ Object (A ≅_))
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univ-lem = c , p
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univalence-lemma
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: (∀ {A} → (s : Σ Object (_≅_ A)) → A ≡ fst s)
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→ (∀ {A} → (iso : A ≅ A) → (identity , identity) ≡ (fst iso , fst (snd iso)))
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→ Univalent[Contr]
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univalence-lemma s u A = univ-lem A s u
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-- From: Thierry Coquand <Thierry.Coquand@cse.gu.se>
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-- Date: Wed, Mar 21, 2018 at 3:12 PM
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--
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@ -198,103 +198,6 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
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-- alt : isContr (Σ Object (X ≡_))
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-- alt = (X , refl) , alt'
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univalent' : Univalent[Contr]
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univalent' = univalence-lemma p q
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where
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module _ {𝕏 : Object} where
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open Σ 𝕏 renaming (fst to X ; snd to x0x1)
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open Σ x0x1 renaming (fst to x0 ; snd to x1)
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-- x0 : X → A in ℂ
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-- x1 : X → B in ℂ
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module _ (𝕐-isoY : Σ Object (𝕏 ≅_)) where
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open Σ 𝕐-isoY renaming (fst to 𝕐 ; snd to isoY)
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open Σ 𝕐 renaming (fst to Y ; snd to y0y1)
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open Σ y0y1 renaming (fst to y0 ; snd to y1)
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open Σ isoY renaming (fst to 𝓯 ; snd to iso-𝓯)
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open Σ iso-𝓯 renaming (fst to 𝓯~ ; snd to inv-𝓯)
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open Σ 𝓯 renaming (fst to f ; snd to inv-f)
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open Σ 𝓯~ renaming (fst to f~ ; snd to inv-f~)
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open Σ inv-𝓯 renaming (fst to left ; snd to right)
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-- y0 : Y → A in ℂ
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-- y1 : Y → B in ℂ
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-- f : X → Y in ℂ
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-- inv-f : ℂ [ y0 ∘ f ] ≡ x0 × ℂ [ y1 ∘ f ] ≡ x1
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-- left : 𝓯~ ∘ 𝓯 ≡ identity
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-- left~ : 𝓯 ∘ 𝓯~ ≡ identity
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isoℂ : X ℂ.≅ Y
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isoℂ
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= f
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, f~
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, ( begin
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ℂ [ f~ ∘ f ] ≡⟨ (λ i → fst (left i)) ⟩
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ℂ.identity ∎
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)
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, ( begin
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ℂ [ f ∘ f~ ] ≡⟨ (λ i → fst (right i)) ⟩
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ℂ.identity ∎
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)
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p0 : X ≡ Y
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p0 = ℂ.iso-to-id isoℂ
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-- I note `left2` and right2` here as a reminder.
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left2 : PathP
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(λ i → ℂ [ x0 ∘ fst (left i) ] ≡ x0 × ℂ [ x1 ∘ fst (left i) ] ≡ x1)
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(snd (𝓯~ ∘ 𝓯)) (snd identity)
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left2 i = snd (left i)
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right2 : PathP
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(λ i → ℂ [ y0 ∘ fst (right i) ] ≡ y0 × ℂ [ y1 ∘ fst (right i) ] ≡ y1)
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(snd (𝓯 ∘ 𝓯~)) (snd identity)
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right2 i = snd (right i)
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-- My idea:
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--
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-- x0, x1 and y0 and y1 are product arrows as in the diagram
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--
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-- X
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-- ↙ ↘
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-- A ⇣ ⇡ B
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-- ↖ ↗
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-- Y (All hail unicode)
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--
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-- The dotted lines indicate the unique product arrows. Since they are
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-- unique they necessarily be each others inverses. Alas, more than
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-- this we must show that they are actually (heterogeneously)
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-- identical as in `p1`:
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p1 : PathP (λ i → ℂ.Arrow (p0 i) A × ℂ.Arrow (p0 i) B) x0x1 y0y1
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p1 = {!!}
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where
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-- This, however, should probably somehow follow from them being
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-- inverses on objects that are propositionally equal cf. `p0`.
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helper : {A B : Object} {f : Arrow A B} {g : Arrow B A}
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→ IsInverseOf f g
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→ (p : A ≡ B)
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→ PathP (λ i → Arrow (p i) (p (~ i))) f g
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helper = {!!}
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p : (X , x0x1) ≡ (Y , y0y1)
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p i = p0 i , {!!}
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module _ (iso : 𝕏 ≅ 𝕏) where
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open Σ iso renaming (fst to 𝓯 ; snd to inv-𝓯)
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open Σ inv-𝓯 renaming (fst to 𝓯~) using ()
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open Σ 𝓯 renaming (fst to f ; snd to inv-f)
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open Σ 𝓯~ renaming (fst to f~ ; snd to inv-f~)
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q0' : ℂ.identity ≡ f
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q0' i = {!!}
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prop : ∀ x → isProp (ℂ [ x0 ∘ x ] ≡ x0 × ℂ [ x1 ∘ x ] ≡ x1)
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prop x = propSig
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( ℂ.arrowsAreSets (ℂ [ x0 ∘ x ]) x0)
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(λ _ → ℂ.arrowsAreSets (ℂ [ x1 ∘ x ]) x1)
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q0'' : PathP (λ i → ℂ [ x0 ∘ q0' i ] ≡ x0 × ℂ [ x1 ∘ q0' i ] ≡ x1) (snd identity) inv-f
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q0'' = lemPropF prop q0'
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q0 : identity ≡ 𝓯
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q0 i = q0' i , q0'' i
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q1' : ℂ.identity ≡ f~
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q1' = {!!}
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q1'' : PathP (λ i → (ℂ [ x0 ∘ q1' i ]) ≡ x0 × (ℂ [ x1 ∘ q1' i ]) ≡ x1) (snd identity) inv-f~
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q1'' = lemPropF prop q1'
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q1 : identity ≡ 𝓯~
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q1 i = q1' i , {!!}
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q : (identity , identity) ≡ (𝓯 , 𝓯~)
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q i = q0 i , q1 i
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univalent : Univalent
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univalent {X , x} {Y , y} = {!res!}
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where
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