added quest2solutions.agda

1FundamentalGroup/Preambles/P2.agda

@@ -5,4 +5,4 @@ open import Cubical.Data.Int using (ℤ ; pos ; negsuc ; -_) public
 open import Cubical.Foundations.Isomorphism public
 open import Cubical.Foundations.Prelude renaming (subst to endPt) public
 open import Cubical.HITs.S1 using (S¹ ; base ; loop) public
-open import 1FundamentalGroup.Quest1 public
+open import 1FundamentalGroup.Quest1Solutions public

1FundamentalGroup/Quest0Solutions.agda

@@ -1,12 +1,6 @@
+-- ignore
 module 1FundamentalGroup.Quest0Solutions where
-
-open import Cubical.Data.Empty
-open import Cubical.Data.Unit renaming ( Unit to ⊤ )
-open import Cubical.Data.Bool
-open import Cubical.Foundations.Prelude renaming ( subst to endPt )
-open import Cubical.Foundations.Isomorphism renaming ( Iso to _≅_ )
-open import Cubical.Foundations.Path
-open import Cubical.HITs.S1
+open import 1FundamentalGroup.Preambles.P0
 
 Refl : base ≡ base
 Refl = λ i → base

1FundamentalGroup/Quest2.agda

@@ -2,40 +2,27 @@
 module 1FundamentalGroup.Quest2 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
+{-
+The definition of sucℤ goes here.
+-}
 
-predℤ : ℤ → ℤ
-predℤ (pos zero) = negsuc zero
-predℤ (pos (suc n)) = pos n
-predℤ (negsuc n) = negsuc (suc n)
+{-
+The definition of predℤ goes here.
+-}
 
-sucℤIso : Iso ℤ ℤ
-sucℤIso = iso sucℤ predℤ s r where
+{-
+The definition of sucℤIso goes here.
+-}
 
-  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
+{-
+The definition of sucℤPath goes here.
+-}
 
 helix : S¹ → Type
-helix base = ℤ
-helix (loop i) = sucℤPath i
+helix = {!!}
 
 spinCountBase : base ≡ base → ℤ
-spinCountBase p = endPt helix p (pos zero)
+spinCountBase = ?
 
 spinCount : (x : S¹) → base ≡ x → helix x
-spinCount x p = endPt helix p (pos zero)
+spinCount = ?

1FundamentalGroup/Quest2Solutions.agda

@@ -0,0 +1,41 @@
+-- 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 : Iso 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
+
+spinCountBase : base ≡ base → ℤ
+spinCountBase p = endPt helix p (pos zero)
+
+spinCount : (x : S¹) → base ≡ x → helix x
+spinCount x p = endPt helix p (pos zero)