Mercurial > hg > Papers > 2021 > soto-prosym
comparison Paper/src/Reasoning.agda.replaced @ 0:c59202657321
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| author | soto <soto@cr.ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 02 Nov 2021 06:55:58 +0900 |
| parents | |
| children | 339fb67b4375 |
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| -1:000000000000 | 0:c59202657321 |
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| 1 open import Relation.Binary.PropositionalEquality | |
| 2 open import nat | |
| 3 open import nat_add | |
| 4 open @$\equiv$@-Reasoning | |
| 5 | |
| 6 module nat_add_sym_reasoning where | |
| 7 | |
| 8 addToRight : (n m : Nat) @$\rightarrow$@ S (n + m) @$\equiv$@ n + (S m) | |
| 9 addToRight O m = refl | |
| 10 addToRight (S n) m = cong S (addToRight n m) | |
| 11 | |
| 12 addSym : (n m : Nat) @$\rightarrow$@ n + m @$\equiv$@ 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) @$\equiv$@@$\langle$@ refl @$\rangle$@ | |
| 18 S (n + S m) @$\equiv$@@$\langle$@ cong S (addSym n (S m)) @$\rangle$@ | |
| 19 S ((S m) + n) @$\equiv$@@$\langle$@ addToRight (S m) n @$\rangle$@ | |
| 20 S (m + S n) @$\equiv$@@$\langle$@ refl @$\rangle$@ | |
| 21 (S m) + (S n) @$\blacksquare$@ |