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Michael Zhang 2024-04-22 03:28:04 +00:00
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@ -259,28 +259,8 @@ A ≃ B = Σ[ f ∈ (A → B) ] isequiv f
### Theorem 2.8.1
```
-- open import Function.HalfAdjointEquivalence
-- _is-⋆ : (x : 𝟙) → x ≡ ⋆
-- ⋆ is-⋆ = refl
--
-- theorem281-helper2 : {x : 𝟙} → (p : x ≡ x) → p ≡ refl
-- theorem281-helper2 {x} p =
-- let a = sym (x is-⋆) ∙ p ∙ (x is-⋆) in
-- J (λ the what → p ≡ the) ? ?
--
-- theorem281-helper : {x y : 𝟙} → (p : x ≡ y) → (x is-⋆) ∙ sym (y is-⋆) ≡ p
-- theorem281-helper {x} p =
-- J (λ the what → (x is-⋆) ∙ sym (the is-⋆) ≡ what) p (theorem281-helper2 _)
--
-- theorem281 : (x y : 𝟙) → (x ≡ y) ≃ 𝟙
-- theorem281 x y = record
-- { to = λ _ → ⋆
-- ; from = λ _ → (x is-⋆) ∙ sym (y is-⋆)
-- ; left-inverse-of = theorem281-helper
-- -- λ z → J (λ the what → (x is-⋆) ∙ sym (the is-⋆) ≡ what) z ?
-- ; right-inverse-of = λ z → sym (z is-⋆)
-- ; left-right = ?
-- }
theorem2∙8∙1 : (x y : 𝟙) → (x ≡ y) ≃ 𝟙
theorem2∙8∙1 = {! !}
```
## 2.9 Π-types and the function extensionality axiom