89 lines
2.2 KiB
Agda
89 lines
2.2 KiB
Agda
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{-# OPTIONS --cubical #-}
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module Lemma641 where
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open import Level
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open import Cubical.Foundations.Prelude
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using (_≡_; refl; _∙_; _≡⟨⟩_; _≡⟨_⟩_; _∎; cong; sym; fst; snd; _,_; ~_)
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open import Cubical.Data.Empty as ⊥
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open import Cubical.Foundations.Equiv using (isEquiv; equiv-proof)
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open import Relation.Nullary using (¬_)
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open import Relation.Binary.Core using (Rel)
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_≢_ : {ℓ : Level} {A : Set ℓ} → Rel A ℓ
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x ≢ y = ¬ x ≡ y
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data S¹ : Set where
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base : S¹
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loop : base ≡ base
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data Bool : Set where
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true : Bool
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false : Bool
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bool-id : Bool → Bool
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bool-id true = true
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bool-id false = false
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bool-id≡bool : (b : Bool) → bool-id b ≡ b
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bool-id≡bool true _ = true
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bool-id≡bool false _ = false
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bool-id-is-equiv : isEquiv bool-id
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bool-id-is-equiv .equiv-proof y .fst = ( y , bool-id≡bool y )
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bool-id-is-equiv .equiv-proof y .snd (y′ , p) i = let b = p (~ i) in (? , bool-id≡bool y)
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bool-flip : Bool → Bool
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bool-flip true = false
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bool-flip false = true
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f : {A : Set} (x : A) → (p : x ≡ x) → S¹ → A
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f x p base = x
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f x p (loop i) = p i
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refl-base : base ≡ base
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refl-base = refl
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refl-x : {A : Set} → (x : A) → x ≡ x
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refl-x _ = refl
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-- p : x ≡ x
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-- p : PathP (λ _ → A) x x
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-- p : I → A
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-- f : S¹ → A
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-- loop : I → S¹
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-- λ i → f (loop i) : I → A
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-- f : S¹ → A
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-- refl-base : base ≡ base
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-- refl-base : I → S¹
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-- λ i → f (refl-base i) : I → A
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-- refl-x : x ≡ x
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-- refl-x : I → A
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-- p ≡ refl : I → (I → A)
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types-arent-sets : {A : Set} (x : A) → (p : x ≡ x) → p ≡ refl → ⊥
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types-arent-sets x p p≡refl = ?
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-- This is the consequence of loop ≡ refl, which says that for any p : x ≡ x, it
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-- also equals refl. It is then used with a proof that p ≡ refl → ⊥
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consequence : {A : Set} (x : A) → (p : x ≡ x) → loop ≡ refl → p ≡ refl
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consequence x p loop≡refl = p
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≡⟨ refl ⟩
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(λ i → f x p (loop i))
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≡⟨ cong (λ l i → f x p (l i)) loop≡refl ⟩
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(λ i → f x p (refl-base i))
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≡⟨ refl ⟩
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refl-x x
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∎
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lemma641 : loop ≢ refl
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lemma641 x = ?
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-- https://serokell.io/blog/playing-with-negation
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-- f : {A : Set} (x : A) → (p : x ≡ x) → (S¹ → A)
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