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comparison Paper/src/Reasoning.agda @ 0:14a0e409d574

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author soto <soto@cr.ie.u-ryukyu.ac.jp>
date Sun, 24 Apr 2022 23:13:44 +0900
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1 open import Relation.Binary.PropositionalEquality
2 open import nat
3 open import nat_add
4 open ≡-Reasoning
5
6 module nat_add_sym_reasoning where
7
8 addToRight : (n m : Nat) -> S (n + m) ≡ n + (S m)
9 addToRight O m = refl
10 addToRight (S n) m = cong S (addToRight n m)
11
12 addSym : (n m : Nat) -> n + m ≡ m + n
13 addSym O O = refl
14 addSym O (S m) = cong S (addSym O m)
15 addSym (S n) O = cong S (addSym n O)
16 addSym (S n) (S m) = begin
17 (S n) + (S m) ≡⟨ refl ⟩
18 S (n + S m) ≡⟨ cong S (addSym n (S m)) ⟩
19 S ((S m) + n) ≡⟨ addToRight (S m) n ⟩
20 S (m + S n) ≡⟨ refl ⟩
21 (S m) + (S n) ∎