29 lines
757 B
Agda
29 lines
757 B
Agda
|
module hwk1 where
|
|||
|
|
|
|||
|
|
open import Data.Nat
|
|||
|
|
open import Relation.Binary.PropositionalEquality as Eq
|
|||
|
|
open Eq.≡-Reasoning
|
|||
|
|
|
|||
|
|
natrec : {C : Set} → (a : ℕ) → (d : C) → (e : ℕ → C → C) → C
|
|||
|
|
natrec zero d e = d
|
|||
|
|
natrec (suc b) d e = e b (natrec b d e)
|
|||
|
|
|
|||
|
|
⊕ : ℕ → ℕ → ℕ
|
|||
|
|
⊕ x y = natrec x y (λ u v → suc v)
|
|||
|
|
|
|||
|
|
problem2-lemma : (x : ℕ) → natrec x 0 (λ u v → suc v) ≡ x
|
|||
|
|
problem2-lemma zero = refl
|
|||
|
|
problem2-lemma (suc x) = cong suc (problem2-lemma x)
|
|||
|
|
|
|||
|
|
-- (λ u v → suc v) x (natrec x 0 (λ u v → suc v))
|
|||
|
|
-- suc (natrec x 0 (λ u v → suc v))
|
|||
|
|
|
|||
|
|
problem2 : (x : ℕ) → ⊕ x 0 ≡ x
|
|||
|
|
problem2 x = problem2-lemma x
|
|||
|
|
-- begin
|
|||
|
|
-- ⊕ x 0
|
|||
|
|
-- ≡⟨⟩
|
|||
|
|
-- natrec x 0 (λ u v → suc v)
|
|||
|
|
-- ≡⟨ {! !} ⟩
|
|||
|
|
-- x
|
|||
|
|
-- ∎
|