Mercurial > hg > Papers > 2020 > soto-midterm
comparison src/Reasoning.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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0:b919985837a3 | 1:73127e0ab57c |
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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) ∎ |