34 lines
973 B
Agda
34 lines
973 B
Agda
{-# OPTIONS --without-K #-}
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module CircleThing2 where
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open import Agda.Primitive renaming (Set to Type)
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open import elementary-number-theory.integers
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open import elementary-number-theory.natural-numbers
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open import foundation-core.identity-types
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open import foundation.univalence
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open import synthetic-homotopy-theory.circle
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infix 10 _≡_
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_≡_ = _＝_
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loops-to-ℤ : base-𝕊¹ ≡ base-𝕊¹ → ℤ
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loops-to-ℤ p = ?
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ℤ-to-loops : ℤ → base-𝕊¹ ≡ base-𝕊¹
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ℤ-to-loops = ind-ℤ
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(λ _ → base-𝕊¹ ≡ base-𝕊¹)
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(inv loop-𝕊¹)
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neg-ver
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refl
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(loop-𝕊¹)
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pos-ver
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where
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pos-ver : ℕ → base-𝕊¹ ≡ base-𝕊¹ → base-𝕊¹ ≡ base-𝕊¹
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pos-ver zero-ℕ p = refl
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pos-ver (succ-ℕ n) p = loop-𝕊¹ ∙ pos-ver n p
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neg-ver : ℕ → base-𝕊¹ ≡ base-𝕊¹ → base-𝕊¹ ≡ base-𝕊¹
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neg-ver zero-ℕ p = refl
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neg-ver (succ-ℕ n) p = (inv loop-𝕊¹) ∙ neg-ver n p
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