Mercurial > hg > Papers > 2020 > soto-midterm
diff src/agda-term.agda @ 1:73127e0ab57c
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author | soto@cr.ie.u-ryukyu.ac.jp |
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date | Tue, 08 Sep 2020 18:38:08 +0900 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/agda-term.agda Tue Sep 08 18:38:08 2020 +0900 @@ -0,0 +1,28 @@ +module agda-term where + +open import Data.Nat.Base +open import Relation.Binary.PropositionalEquality + ++zero : {y : ℕ} → y + zero ≡ y ++zero {zero} = refl ++zero {suc y} = cong (λ yy → suc yy) (+zero {y}) + ++-suc : {x y : ℕ} → x + suc y ≡ suc (x + y) ++-suc {zero} {y} = refl ++-suc {suc x} {y} = cong suc (+-suc {x} {y}) + ++-comm : (x y : ℕ) → x + y ≡ y + x ++-comm zero y rewrite (+zero {y}) = refl ++-comm (suc x) y = let open ≡-Reasoning in + begin + suc (x + y) ≡⟨⟩ + suc (x + y) ≡⟨ cong suc (+-comm x y) ⟩ + suc (y + x) ≡⟨ sym (+-suc {y} {x}) ⟩ + y + suc x ∎ + ++-come : (x y : ℕ) → x + y ≡ y + x ++-come zero y rewrite (+zero {y}) = refl ++-come (suc x) y + rewrite (cong suc (+-come x y)) | sym (+-suc {y} {x}) = refl + +