Mercurial > hg > Members > atton > agda > systemT

annotate int.agda @ 5:a3cf5cb2b7d3

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Auto proof sum-assoc
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Wed, 21 May 2014 14:52:44 +0900
parents 6b1230883bfa
children db4c6d435f23
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7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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1 open import systemT
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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2 open import Relation.Binary.PropositionalEquality
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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3 open ≡-Reasoning
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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4
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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5 module int where
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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6
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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7 _+_ : Int -> Int -> Int
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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8 n + O = n
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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9 n + (S m) = S (n + m)
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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10
4
6b1230883bfa mini fixes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
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11 left-increment : (n m : Int) -> (S n) + m ≡ S (n + m)
6b1230883bfa mini fixes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
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12 left-increment n O = refl
6b1230883bfa mini fixes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
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13 left-increment n (S m) = cong S (left-increment n m)
3
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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14
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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15 double : Int -> Int
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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16 double O = O
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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17 double (S n) = S (S (double n))
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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18
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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19 sum-sym : (n : Int) (m : Int) -> n + m ≡ m + n
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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20 sum-sym O O = refl
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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21 sum-sym O (S m) = cong S (sum-sym O m)
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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22 sum-sym (S n) O = cong S (sum-sym n O)
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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23 sum-sym (S n) (S m) = begin
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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24 (S n) + (S m)
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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25 ≡⟨ refl ⟩
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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26 S ((S n) + m)
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6b1230883bfa mini fixes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
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27 ≡⟨ cong S (left-increment n m)⟩
3
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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28 S (S (n + m))
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
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29 ≡⟨ cong (\x -> S (S x)) (sum-sym n m) ⟩
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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30 S (S (m + n))
4
6b1230883bfa mini fixes
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
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31 ≡⟨ sym (cong S (left-increment m n)) ⟩
3
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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32 S ((S m) + n)
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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33 ≡⟨ refl ⟩
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
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34 (S m) + (S n)
7138e79615b3 Proof add-sym
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
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35 ∎
5
a3cf5cb2b7d3 Auto proof sum-assoc
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
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36
a3cf5cb2b7d3 Auto proof sum-assoc
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
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37 sum-assoc : (x y z : Int) -> x + (y + z) ≡ (x + y) + z
a3cf5cb2b7d3 Auto proof sum-assoc
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
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38 sum-assoc O O O = refl
a3cf5cb2b7d3 Auto proof sum-assoc
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
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39 sum-assoc O O (S z) = cong S (sum-assoc O O z)
a3cf5cb2b7d3 Auto proof sum-assoc
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
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40 sum-assoc O (S y) O = refl
a3cf5cb2b7d3 Auto proof sum-assoc
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
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41 sum-assoc O (S y) (S z) = cong S (sum-assoc O (S y) z)
a3cf5cb2b7d3 Auto proof sum-assoc
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
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42 sum-assoc (S x) O O = refl
a3cf5cb2b7d3 Auto proof sum-assoc
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
43 sum-assoc (S x) O (S z) = cong S (sum-assoc (S x) O z)
a3cf5cb2b7d3 Auto proof sum-assoc
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
44 sum-assoc (S x) (S y) O = refl
a3cf5cb2b7d3 Auto proof sum-assoc
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
45 sum-assoc (S x) (S y) (S z) = cong S (sum-assoc (S x) (S y) z)