generalized lem3 and made progress for Sets univalence
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@ -138,14 +138,10 @@ module _ (ℓ : Level) where
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iso : (p ≡ q) Eqv.≅ (proj₁ p ≡ proj₁ q)
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iso = f , g , inv
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lem3 : {Q : A → Set ℓb}
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lem3 : ∀ {ℓc} {Q : A → Set (ℓc ⊔ ℓb)}
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→ ((a : A) → P a ≃ Q a) → Σ A P ≃ Σ A Q
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lem3 {Q} eA = res
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lem3 {Q = Q} eA = res
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where
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P→Q : ∀ {a} → P a ≡ Q a
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P→Q = ua (eA _)
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Q→P : ∀ {a} → Q a ≡ P a
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Q→P = sym P→Q
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f : Σ A P → Σ A Q
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f (a , pA) = a , _≃_.eqv (eA a) pA
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g : Σ A Q → Σ A P
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@ -226,7 +222,7 @@ module _ (ℓ : Level) where
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-- lem3 and the equivalence from lem4
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step0 : Σ (A → B) isIso ≃ Σ (A → B) (isEquiv A B)
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step0 = lem3 (λ f → sym≃ (lem4 sA sB f))
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step0 = lem3 {ℓc = lzero} (λ f → sym≃ (lem4 sA sB f))
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-- univalence
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step1 : Σ (A → B) (isEquiv A B) ≃ (A ≡ B)
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step1 = hh ⊙ h
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@ -297,8 +293,11 @@ module _ (ℓ : Level) where
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--
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-- is contractible, which implies univalence.
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eq1 : ∀ hA → (Σ[ hB ∈ Object ] hA ≅ hB) ≡ (Σ[ hB ∈ Object ] hA ≡ hB)
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eq1 hA = (ua (lem3 (\ hB → sym≃ thr)))
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univalent[Contr] : ∀ hA → isContr (Σ[ hB ∈ Object ] hA ≅ hB)
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univalent[Contr] hA = {!!} , {!!}
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univalent[Contr] hA = subst {P = isContr} (sym (eq1 hA)) {!!}
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univalent : Univalent
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univalent = from[Contr] univalent[Contr]
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