Created
February 23, 2024 18:39
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This version of + sucks
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| open import Agda.Builtin.Equality | |
| open import Agda.Builtin.Nat using (Nat; suc; zero) | |
| cong : (f : Nat → Nat) → ∀ {a b} → a ≡ b → f a ≡ f b | |
| cong f refl = refl | |
| trans : ∀ {a b c : Nat} → a ≡ b → b ≡ c → a ≡ c | |
| trans refl p = p | |
| sym : ∀ {a b : Nat} → a ≡ b → b ≡ a | |
| sym refl = refl | |
| _+_ : Nat → Nat → Nat | |
| a + zero = a | |
| a + suc b = suc (b + a) | |
| +-comm : ∀ a b → a + b ≡ b + a | |
| +-suc : ∀ a b → suc a + b ≡ suc (b + a) | |
| +-zero : ∀ a → zero + a ≡ a | |
| +-comm zero b = +-zero b | |
| +-comm (suc a) b = trans (+-suc a b) (cong suc (+-comm b a)) | |
| +-zero zero = refl | |
| +-zero (suc a) = refl | |
| +-suc a zero = cong suc (sym (+-zero a)) | |
| +-suc a (suc b) = cong suc (sym (+-suc b a)) |
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