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

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+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