6.4.2
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src/CubicalHott/Lemma6-4-2.agda
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src/CubicalHott/Lemma6-4-2.agda
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{-# OPTIONS --cubical #-}
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module CubicalHott.Lemma6-4-2 where
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open import Cubical.Foundations.Prelude
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open import Cubical.HITs.S1
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open import Cubical.Relation.Nullary
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open import CubicalHott.Lemma6-4-1
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lemma : Σ ((x : S¹) → x ≡ x) (λ f → ¬ (f ≡ λ x _ → x))
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lemma = f , p where
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f : (x : S¹) → x ≡ x
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f base = loop
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f (loop i) j =
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let u = λ k → λ where
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(i = i0) → loop (j ∨ ~ k)
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(i = i1) → loop (j ∧ k)
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(j = i0) → loop (i ∨ ~ k)
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(j = i1) → loop (i ∧ k)
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in hcomp u base
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p : ¬ f ≡ (λ x _ → x)
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p q = loop≢refl (funExt⁻ q base)
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