Added templates and some code done

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