Succesfully apply path-induction.
Now all that's left to do is prove the original proposition in a heterogenous equality
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@ -147,7 +147,7 @@ module _ {ℓa} {ℓb} {ℂ : RawCategory ℓa ℓb} where
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ident : (λ _ → IsIdentity 𝟙) [ X.ident ≡ Y.ident ]
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ident = X.propIsIdentity X.ident Y.ident
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-- A version of univalence indexed by the identity proof.
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-- Not of course that since it's defined where `RawCategory ℂ` has been opened
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-- Note of course that since it's defined where `RawCategory ℂ` has been opened
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-- this is specialized to that category.
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Univ : IsIdentity 𝟙 → Set _
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Univ idnt = {A B : Y.Raw.Object} →
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@ -156,8 +156,15 @@ module _ {ℓa} {ℓb} {ℂ : RawCategory ℓa ℓb} where
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done : x ≡ y
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U : ∀ {a : IsIdentity 𝟙} → (λ _ → IsIdentity 𝟙) [ X.ident ≡ a ] → (b : Univ a) → Set _
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U eqwal bbb = (λ i → Univ (eqwal i)) [ X.univalent ≡ bbb ]
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P : (y : IsIdentity 𝟙)
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→ (λ _ → IsIdentity 𝟙) [ X.ident ≡ y ] → Set _
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P y eq = ∀ (b' : Univ y) → U eq b'
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helper : ∀ (b' : Univ X.ident)
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→ (λ _ → Univ X.ident) [ X.univalent ≡ b' ]
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helper univ = {!!}
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foo = pathJ P helper Y.ident ident
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eqUni : U ident Y.univalent
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eqUni = {!!}
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eqUni = foo Y.univalent
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IC.assoc (done i) = X.propIsAssociative X.assoc Y.assoc i
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IC.ident (done i) = ident i
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IC.arrowIsSet (done i) = X.propArrowIsSet X.arrowIsSet Y.arrowIsSet i
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