42 lines
983 B
Agda
42 lines
983 B
Agda
-- ignore
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module 1FundamentalGroup.Quest2Solutions where
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open import 1FundamentalGroup.Preambles.P2
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sucℤ : ℤ → ℤ
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sucℤ (pos n) = pos (suc n)
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sucℤ (negsuc zero) = pos zero
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sucℤ (negsuc (suc n)) = negsuc n
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predℤ : ℤ → ℤ
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predℤ (pos zero) = negsuc zero
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predℤ (pos (suc n)) = pos n
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predℤ (negsuc n) = negsuc (suc n)
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sucℤIso : Iso ℤ ℤ
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sucℤIso = iso sucℤ predℤ s r where
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s : section sucℤ predℤ
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s (pos zero) = refl
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s (pos (suc n)) = refl
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s (negsuc zero) = refl
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s (negsuc (suc n)) = refl
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r : retract sucℤ predℤ
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r (pos zero) = refl
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r (pos (suc n)) = refl
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r (negsuc zero) = refl
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r (negsuc (suc n)) = refl
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sucℤPath : ℤ ≡ ℤ
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sucℤPath = isoToPath sucℤIso
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helix : S¹ → Type
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helix base = ℤ
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helix (loop i) = sucℤPath i
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windingNumberBase : base ≡ base → ℤ
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windingNumberBase p = endPt helix p (pos zero)
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windingNumber : (x : S¹) → base ≡ x → helix x
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windingNumber x p = endPt helix p (pos zero)
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