-- ignore
module 1FundamentalGroup.Quest2Solutions where
open import 1FundamentalGroup.Preambles.P2

-- sucℤ : ℤ → ℤ
-- sucℤ (pos n) = pos (suc n)
-- sucℤ (negsuc zero) = pos zero
-- sucℤ (negsuc (suc n)) = negsuc n

-- predℤ : ℤ → ℤ
-- predℤ (pos zero) = negsuc zero
-- predℤ (pos (suc n)) = pos n
-- predℤ (negsuc n) = negsuc (suc n)

-- sucℤIso : ℤ ≅ ℤ
-- sucℤIso = iso sucℤ predℤ s r where

--   s : section sucℤ predℤ
--   s (pos zero) = refl
--   s (pos (suc n)) = refl
--   s (negsuc zero) = refl
--   s (negsuc (suc n)) = refl

--   r : retract sucℤ predℤ
--   r (pos zero) = refl
--   r (pos (suc n)) = refl
--   r (negsuc zero) = refl
--   r (negsuc (suc n)) = refl

-- sucℤPath : ℤ ≡ ℤ
-- sucℤPath = isoToPath sucℤIso

-- helix : S¹ → Type
-- helix base = ℤ
-- helix (loop i) = sucℤPath i

-- windingNumberBase : base ≡ base → ℤ
-- windingNumberBase p = endPt helix p (pos zero)

-- windingNumber : (x : S¹) → base ≡ x → helix x
-- windingNumber x p = endPt helix p (pos zero)