Added templates and some code done

0Trinitarianism/Preambles/P0.agda

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+module 0Trinitarianism.Preambles.P0 where
+
+open import Cubical.Foundations.Prelude hiding (_∨_) public

0Trinitarianism/Preambles/P1.agda

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+module 0Trinitarianism.Preambles.P1 where
+
+open import Cubical.Foundations.Prelude public
+open import Cubical.Data.Unit public renaming (Unit to ⊤)
+open import Cubical.Data.Empty public using (⊥)
+open import Cubical.Data.Nat public hiding (isEven)

0Trinitarianism/Preambles/P2.agda

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+module 0Trinitarianism.Preambles.P2 where
+
+open import Cubical.Foundations.Prelude public
+open import Cubical.Data.Unit public renaming (Unit to ⊤)
+open import Cubical.Data.Empty public using (⊥)
+open import Cubical.Data.Nat public hiding (isEven)

0Trinitarianism/Preambles/P3.agda

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+module 0Trinitarianism.Preambles.P3 where
+
+open import Cubical.Foundations.Prelude public
+open import Cubical.Data.Nat public hiding (_+_ ; isEven)
+open import 0Trinitarianism.Quest1 public
+open import Cubical.Data.Empty public using (⊥)

0Trinitarianism/Preambles/P4.agda

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+module 0Trinitarianism.Preambles.P4 where
+
+open import Cubical.Foundations.Prelude using
+  ( Level ; Type ; _≡_ ; J ; JRefl ; refl ; i1 ; i0 ; I ; cong) public
+open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) public

0Trinitarianism/Preambles/P5.agda

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+module 0Trinitarianism.Preambles.P5 where
+
+open import Cubical.Foundations.Prelude renaming
+  (funExt to libFunExt ;
+   sym to libSym ;
+   _∎ to lib_∎ ;
+   _∙_ to lib_∙_ ;
+   fst to libFst ;
+   snd to libSnd
+  ) hiding ( step-≡ ) public
+open import Cubical.HITs.S1 using ( S¹ ; base ; loop ) public
+open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) public
+open import Cubical.Foundations.Path public
+open import 0Trinitarianism.Quest4Solutions public
+open import 1FundamentalGroup.Quest0Solutions public
+open import Cubical.Data.Bool public
+
+pathToFun≡transport : {u : Level} {A B : Type u} (p : A ≡ B) (x : A)
+  → pathToFun p x ≡ transport p x
+pathToFun≡transport {u} {A} = J (λ B p → (x : A) → pathToFun p x ≡ transport p x)
+  λ x →
+      pathToFun refl x
+    ≡⟨ pathToFunReflx x ⟩
+      x
+    ≡⟨ sym (transportRefl x) ⟩
+      transport refl x ∎
+
+PathPIsoPathD : {u : Level} {A B : Type u} (p : A ≡ B) (x : A) (y : B) →
+  (PathP (λ i → p i) x y) ≅ (pathToFun p x ≡ y)
+PathPIsoPathD {u} {A} {B} p x =
+  subst (λ b → (y : B) → (PathP (λ i → p i) x y) ≅ (b ≡ y))
+    (sym (pathToFun≡transport p x))
+    (PathPIsoPath _ x)

0Trinitarianism/Quest0.agda

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+module 0Trinitarianism.Quest0 where
+open import 0Trinitarianism.Preambles.P0
+
+data ⊤ : Type where
+  tt : ⊤
+
+TrueToTrue : ⊤ → ⊤
+TrueToTrue = λ x → x
+
+TrueToTrue' : ⊤ → ⊤
+TrueToTrue' tt = tt
+
+data ⊥ : Type where
+
+explosion : ⊥ → ⊤
+explosion ()
+
+data ℕ : Type where
+  zero : ℕ
+  suc : ℕ → ℕ

0Trinitarianism/Quest1.agda

@@ -0,0 +1,21 @@
+module 0Trinitarianism.Quest1 where
+
+open import 0Trinitarianism.Preambles.P1
+
+isEven : ℕ → Type
+isEven zero = ⊤
+isEven (suc zero) = ⊥
+isEven (suc (suc n)) = isEven n
+
+{-
+This is a comment block.
+Remove this comment block and formulate
+'there exists an even natural' here.
+-}
+
+_×_ : Type → Type → Type
+A × C = Σ A (λ a → C)
+
+div2 : Σ ℕ isEven → ℕ
+div2 (zero , snd₁) = zero
+div2 (suc (suc fst₁) , snd₁) = suc (div2 (fst₁ , snd₁))

0Trinitarianism/Quest2.agda

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+module 0Trinitarianism.Quest2 where
+
+open import 0Trinitarianism.Preambles.P2
+
+isEven : ℕ → Type
+isEven zero = ⊤
+isEven (suc zero) = ⊥
+isEven (suc (suc n)) = isEven n
+
+{-
+This is a comment block.
+Remove this comment block and formulate
+'there exists an even natural' here.
+-}
+
+_×_ : Type → Type → Type
+A × C = Σ A (λ a → C)
+
+div2 : Σ ℕ isEven → ℕ
+div2 (zero , snd₁) = zero
+div2 (suc (suc fst₁) , snd₁) = suc (div2 ( fst₁ , snd₁))
+
+private
+  postulate
+    A B C : Type
+
+uncurry : (A → B → C) → (A × B → C)
+uncurry f x = f (fst x) (snd x)
+
+curry : (A × B → C) → (A → B → C)
+curry f a b = f (a , b)

0Trinitarianism/Quest3.agda

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+module 0Trinitarianism.Quest3 where
+
+open import 0Trinitarianism.Preambles.P3
+
+_+_ : ℕ → ℕ → ℕ
+n + zero = n
+n + suc m = suc (n + m)
+
+_+'_ : ℕ → ℕ → ℕ
+zero +' m = m
+suc n +' m = suc (n +' m)
+
+{-
+Write here your proof that the sum of
+even naturals is even.
+-}
+
+SumOfEven : (x y : Σ ℕ isEven) → isEven (x .fst + y .fst)
+SumOfEven x@(fst₁ , snd₁) (zero , snd₂) = snd₁
+SumOfEven x@(fst₁ , snd₁) (suc (suc fst₂) , snd₂) = SumOfEven x ( fst₂ , snd₂)
+
+data _⊕_ (A B : Type) : Type where
+  inl : A → A ⊕ B
+  inr : B → A ⊕ B
+
+_⇔_ : Type → Type → Type
+A ⇔ B = (A → B) × (B → A)
+
+ex : {A : Type} → (A ⇔ ⊥) ⇔ (A → ⊥)
+ex .fst = λ z → z .fst
+ex .snd = λ x →  x , λ ()
+
+¬ : Type → Type
+¬ A = A → ⊥
+
+proof : (x : ℕ) → (isEven x) ⊕ (¬ (isEven x))
+proof zero = inl tt
+proof (suc zero) = inr (λ ())
+proof (suc (suc x)) = proof x

0Trinitarianism/Quest4.agda

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+module 0Trinitarianism.Quest4 where
+
+open import 0Trinitarianism.Preambles.P4
+
+data Id {A : Type} : (x y : A) → Type where
+  rfl : {x : A} → Id x x
+
+private
+  variable
+    A : Type
+    x y z w : A
+
+sym : Id x y → Id y x
+sym rfl = rfl
+
+trans : Id x y → Id y z → Id x z
+trans rfl rfl = rfl
+
+_*_ : Id x y → Id y z → Id x z
+_*_ = trans
+
+rfl* : (p : Id x y) → Id (rfl * p) p
+rfl* rfl = rfl
+
+*rfl : (p : Id x y) → Id (p * rfl) p
+*rfl rfl = rfl
+
+*sym : (p : Id x y) → Id (sym p * p) rfl
+*sym rfl = rfl
+
+sym* : (p : Id x y) → Id (p * sym p) rfl
+sym* rfl = rfl
+
+assoc : (p : Id x y) (q : Id y z) (r : Id z w) → Id ((p * q) * r) (p * (q * r))
+assoc rfl rfl rfl = rfl
+
+outOfId : (M : (y : A) → Id x y → Type) → M x rfl → {y : A} → (p : Id x y) → M y p
+outOfId M x rfl = x

0Trinitarianism/Quest5.agda

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+module 0Trinitarianism.Quest5 where
+
+open import 0Trinitarianism.Preambles.P5

1FundamentalGroup/Preambles/P0.agda

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+module 1FundamentalGroup.Preambles.P0 where
+
+open import Cubical.Data.Empty using (⊥) public
+open import Cubical.Data.Unit renaming ( Unit to ⊤ ) public
+open import Cubical.Data.Bool renaming ( elim to Bool-elim ) public
+open import Cubical.Foundations.Prelude
+     renaming ( subst to endPt
+              ; transport to pathToFun
+              ) public
+open import Cubical.Foundations.Isomorphism renaming ( Iso to _≅_ ) public
+open import Cubical.Foundations.Path public
+open import Cubical.HITs.S1 renaming ( elim to S¹-elim ) public

1FundamentalGroup/Preambles/P1.agda

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+module 1FundamentalGroup.Preambles.P1 where
+
+open import Cubical.HITs.S1 using (S¹ ; base ; loop) public
+open import Cubical.Data.Nat using (ℕ ; suc ; zero) public
+open import Cubical.Data.Int using (ℤ ; pos ; negsuc ; -_) public
+open import Cubical.Data.Empty public
+open import Cubical.Foundations.Prelude
+     renaming ( subst to endPt
+              ; transport to pathToFun
+              ) public
+open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) public
+open import 1FundamentalGroup.Quest0Solutions using ( Refl ; Refl≢loop ) public

1FundamentalGroup/Preambles/P2.agda

@@ -0,0 +1,19 @@
+module 1FundamentalGroup.Preambles.P2 where
+
+open import Cubical.Data.Nat public
+open import Cubical.Data.Int using (ℤ ; pos ; negsuc ; -_) public
+open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) public
+open import Cubical.Data.Empty using (⊥) public
+open import Cubical.Data.Unit renaming (Unit to ⊤) public
+open import Cubical.Foundations.Prelude
+     renaming ( subst to endPt
+              ; transport to pathToFun
+              ) public
+open import Cubical.HITs.S1 using (S¹ ; base ; loop) public
+open import 1FundamentalGroup.Quest1Solutions public
+
+refl∙refl : {A : Type} {a : A} → refl ∙ refl ≡ refl {x = a}
+refl∙refl {a = a} = sym (λ i j → compPath-filler (refl {x = a}) refl i j)
+
+symRefl : {A : Type} {a : A} → sym refl ≡ refl {x = a}
+symRefl = refl

1FundamentalGroup/Preambles/P3.agda

@@ -0,0 +1,33 @@
+module 1FundamentalGroup.Preambles.P3 where
+
+open import Cubical.Foundations.Prelude public
+  renaming (transport to pathToFun ;
+            transportRefl to pathToFunRefl ;
+            subst to endPt) public
+open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) public
+open import Cubical.Foundations.GroupoidLaws
+  renaming (lCancel to sym∙ ; rCancel to ∙sym ; lUnit to Refl∙ ; rUnit to ∙Refl) public
+open import Cubical.Foundations.Path public
+open import Cubical.Data.Int using (ℤ ; isSetℤ) public
+open import Cubical.Data.Nat public
+open import Cubical.HITs.S1 using ( S¹ ; base ; loop ) public
+open import 1FundamentalGroup.Quest1Solutions public
+
+open ℤ public
+
+PathD : {A0 A1 : Type} (A : A0 ≡ A1) (x : A0) (y : A1) → Type
+PathD A x y = pathToFun A x ≡ y
+
+endPtRefl : {A : Type} {x : A} (B : A → Type) → endPt B (refl {x = x}) ≡ λ b → b
+endPtRefl {x = x} B = funExt (λ b → substRefl {B = B} b)
+
+outOfS¹P : (B : S¹ → Type) (b : B base) → PathP (λ i → B (loop i)) b b → (x : S¹) → B x
+outOfS¹P B b p base = b
+outOfS¹P B b p (loop i) = p i
+
+outOfS¹D : (B : S¹ → Type) (b : B base) → PathD (λ i → B (loop i)) b b → (x : S¹) → B x
+outOfS¹D B b p x = outOfS¹P B b (_≅_.inv (PathPIsoPath (λ i → B (loop i)) b b) p) x
+
+outOfS¹DBase : (B : S¹ → Type) (b : B base)
+  (p : PathD (λ i → B (loop i)) b b) → outOfS¹D B b p base ≡ b
+outOfS¹DBase B b p = refl

1FundamentalGroup/Quest0.agda

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+-- ignore
+module 1FundamentalGroup.Quest0 where
+open import 1FundamentalGroup.Preambles.P0
+
+Refl : base ≡ base
+Refl = {!!}
+
+Flip : Bool → Bool
+Flip x = {!!}
+
+flipIso : Bool ≅ Bool
+flipIso = {!!}
+
+flipPath : Bool ≡ Bool
+flipPath = {!!}
+
+doubleCover : S¹ → Type
+doubleCover x = {!!}
+
+endPtOfTrue : base ≡ base → doubleCover base
+endPtOfTrue p = {!!}
+
+Refl≢loop : Refl ≡ loop → ⊥
+Refl≢loop p = {!!}
+
+------------------- Side Quest - Empty -------------------------
+
+{-
+-- This is a comment box,
+-- remove the {- and -} to do the side quests
+
+toEmpty : (A : Type) → Type
+toEmpty A = {!!}
+
+pathEmpty : (A : Type) → Type₁
+pathEmpty A = {!!}
+
+isoEmpty : (A : Type) → Type
+isoEmpty A = {!!}
+
+outOf⊥ : (A : Type) → ⊥ → A
+outOf⊥ A ()
+
+toEmpty→isoEmpty : (A : Type) → toEmpty A → isoEmpty A
+toEmpty→isoEmpty A = {!!}
+
+isoEmpty→pathEmpty : (A : Type) → isoEmpty A → pathEmpty A
+isoEmpty→pathEmpty A = {!!}
+
+pathEmpty→toEmpty : (A : Type) → pathEmpty A → toEmpty A
+pathEmpty→toEmpty A = {!!}
+-}
+
+------------------- Side Quests - true≢false --------------------
+
+{-
+-- This is a comment box,
+-- remove the {- and -} to do the side quests
+
+true≢false' : true ≡ false → ⊥
+true≢false' = {!!}
+
+-}
+

1FundamentalGroup/Quest1.agda

@@ -0,0 +1,35 @@
+-- ignore
+module 1FundamentalGroup.Quest1 where
+open import 1FundamentalGroup.Preambles.P1
+
+
+loopSpace : (A : Type) (a : A) → Type
+loopSpace A a = a ≡ a
+
+loop_times : ℤ → loopSpace S¹ base
+loop n times = {!!}
+
+{-
+The definition of sucℤ goes here.
+-}
+
+{-
+The definition of predℤ goes here.
+-}
+
+{-
+The definition of sucℤIso goes here.
+-}
+
+{-
+The definition of sucℤPath goes here.
+-}
+
+helix : S¹ → Type
+helix = {!!}
+
+windingNumberBase : base ≡ base → ℤ
+windingNumberBase = {!!}
+
+windingNumber : (x : S¹) → base ≡ x → helix x
+windingNumber = {!!}

1FundamentalGroup/Quest1SideQuests/Sn.agda

@@ -0,0 +1,23 @@
+module 1FundamentalGroup.Quest1SideQuests.Sn where
+
+open import Cubical.Data.Nat
+open import Cubical.Data.Empty
+open import Cubical.Data.Unit renaming (Unit to ⊤)
+open import Cubical.Data.Bool
+open import Cubical.Foundations.Prelude
+
+
+data susp (X : Type) : Type where
+  north : {!!}
+  south : {!!}
+  merid : {!!}
+
+Sphere : ℕ → Type
+Sphere = {!!}
+
+Disk : (n : ℕ) → Type
+Disk zero = {!!}
+Disk (suc n) = {!!}
+
+SphereToDisk : {n : ℕ} → Sphere n → Disk n
+SphereToDisk {n} s = {!!}

1FundamentalGroup/Quest2.agda

@@ -0,0 +1,27 @@
+-- ignore
+module 1FundamentalGroup.Quest2 where
+open import 1FundamentalGroup.Preambles.P2
+
+isSet→LoopSpace≡⊤ : {A : Type} (x : A) → isSet A → (x ≡ x) ≡ ⊤
+isSet→LoopSpace≡⊤ = {!!}
+
+data _⊔_ (A B : Type) : Type where
+
+     inl : A → A ⊔ B
+     inr : B → A ⊔ B
+
+{-
+
+Your definition of ℤ≡ℕ⊔ℕ goes here.
+
+Your definition of ⊔NoConfusion goes here.
+
+Your definition of Path≡⊔NoConfusion goes here.
+
+Your definition of isSet⊔NoConfusion goes here.
+
+Your definition of isSet⊔ goes here.
+
+Your definition of isSetℤ goes here.
+
+-}

1FundamentalGroup/Quest3.agda

@@ -0,0 +1,3 @@
+module 1FundamentalGroup.Quest3 where
+
+open import 1FundamentalGroup.Preambles.P3

Heated-Gaming.agda-lib

@@ -0,0 +1,9 @@
+name: Heated-Gaming
+include: .
+depend: cubical
+flags:
+  --cubical
+  --guardedness
+  --no-import-sorts
+  -W noUnsupportedIndexedMatch
+  --allow-unsolved-metas