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Migrate systemT from atton/agda/systemT (13:5a81867278af)
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 02 Nov 2014 09:40:54 +0900
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0:8a5f4ebdd34d 1:fe247f476ecb
1 open import systemT
2 open import Relation.Binary.PropositionalEquality
3 open ≡-Reasoning
4
5 module int where
6
7 double : Int -> Int
8 double O = O
9 double (S n) = S (S (double n))
10
11
12 infixl 30 _+_
13 _+_ : Int -> Int -> Int
14 n + O = n
15 n + (S m) = S (n + m)
16
17 left-add-zero : (n : Int) -> O + n ≡ n
18 left-add-zero O = refl
19 left-add-zero (S n) = cong S (left-add-zero n)
20
21 left-add-one : (n : Int) -> (S n) ≡ S O + n
22 left-add-one O = refl
23 left-add-one (S n) = cong S (left-add-one n)
24
25 left-increment : (n m : Int) -> (S n) + m ≡ S (n + m)
26 left-increment n O = refl
27 left-increment n (S m) = cong S (left-increment n m)
28
29 sum-sym : (x : Int) (y : Int) -> x + y ≡ y + x
30 sum-sym O O = refl
31 sum-sym O (S y) = cong S (sum-sym O y)
32 sum-sym (S x) O = cong S (sum-sym x O)
33 sum-sym (S x) (S y) = begin
34 (S x) + (S y)
35 ≡⟨ refl ⟩
36 S ((S x) + y)
37 ≡⟨ cong S (sum-sym (S x) y) ⟩
38 S (y + (S x))
39 ≡⟨ (sym (left-increment y (S x))) ⟩
40 (S y) + (S x)
41 ∎
42
43 sum-assoc : (x y z : Int) -> x + (y + z) ≡ (x + y) + z
44 sum-assoc O O O = refl
45 sum-assoc O O (S z) = cong S (sum-assoc O O z)
46 sum-assoc O (S y) O = refl
47 sum-assoc O (S y) (S z) = cong S (sum-assoc O (S y) z)
48 sum-assoc (S x) O O = refl
49 sum-assoc (S x) O (S z) = cong S (sum-assoc (S x) O z)
50 sum-assoc (S x) (S y) O = refl
51 sum-assoc (S x) (S y) (S z) = cong S (sum-assoc (S x) (S y) z)
52
53
54 infixl 40 _*_
55 _*_ : Int -> Int -> Int
56 n * O = O
57 n * (S O) = n
58 n * (S m) = n + (n * m)
59
60 right-mult-zero : (n : Int) -> n * O ≡ O
61 right-mult-zero n = refl
62
63 right-mult-one : (n : Int) -> n * (S O) ≡ n
64 right-mult-one n = refl
65
66 right-mult-distr-one : (n m : Int) -> n * (S m) ≡ n + (n * m)
67 right-mult-distr-one O O = refl
68 right-mult-distr-one O (S m) = refl
69 right-mult-distr-one (S n) O = refl
70 right-mult-distr-one (S n) (S m) = refl
71
72
73 left-mult-zero : (n : Int) -> O * n ≡ O
74 left-mult-zero O = refl
75 left-mult-zero (S n) = begin
76 O * (S n)
77 ≡⟨ right-mult-distr-one O n ⟩
78 O + (O * n)
79 ≡⟨ sum-sym O (O * n) ⟩
80 (O * n) + O
81 ≡⟨ refl ⟩
82 (O * n)
83 ≡⟨ left-mult-zero n ⟩
84 O
85 ∎
86
87 left-mult-one : (n : Int) -> (S O) * n ≡ n
88 left-mult-one O = refl
89 left-mult-one (S n) = begin
90 (S O) * S n
91 ≡⟨ right-mult-distr-one (S O) n ⟩
92 (S O) + ((S O) * n)
93 ≡⟨ cong (_+_ (S O)) (left-mult-one n) ⟩
94 (S O) + n
95 ≡⟨ sum-sym (S O) n ⟩
96 n + (S O)
97 ≡⟨ refl ⟩
98 S n
99 ∎
100
101
102 left-mult-distr-one : (n m : Int) -> (S n) * m ≡ m + (n * m)
103 left-mult-distr-one O O = refl
104 left-mult-distr-one O (S m) = begin
105 (S O) * S m
106 ≡⟨ left-mult-one (S m) ⟩
107 S m
108 ≡⟨ refl ⟩
109 S m + O
110 ≡⟨ cong (_+_ (S m)) (sym (left-mult-zero (S m))) ⟩
111 S m + (O * S m)
112 ∎
113 left-mult-distr-one (S n) O = refl
114 left-mult-distr-one (S n) (S m) = begin
115 S (S n) * S m
116 ≡⟨ right-mult-distr-one (S (S n)) m ⟩
117 (S (S n)) + ((S (S n)) * m)
118 ≡⟨ cong (\x -> (S (S n)) + x) (left-mult-distr-one (S n) m) ⟩
119 S (S n) + (m + S n * m)
120 ≡⟨ cong (\x -> x + (m + S n * m)) (left-add-one (S n)) ⟩
121 (S O) + (S n) + (m + S n * m)
122 ≡⟨ sum-assoc ((S O) + (S n)) m (S n * m) ⟩
123 (S O) + (S n) + m + S n * m
124 ≡⟨ cong (\x -> x + m + S n * m) (sum-sym (S O) (S n)) ⟩
125 ((((S n) + (S O)) + m) + S n * m)
126 ≡⟨ cong (\x -> x + (S n * m)) (sym (sum-assoc (S n) (S O) m))⟩
127 (((S n) + ((S O) + m)) + S n * m)
128 ≡⟨ cong (\x -> (S n + x + S n * m)) (sym (left-add-one m)) ⟩
129 ((S n) + (S m) + S n * m)
130 ≡⟨ cong (\x -> x + (S n * m)) (sum-sym (S n) (S m)) ⟩
131 (S m) + (S n) + (S n * m)
132 ≡⟨ sym (sum-assoc (S m) (S n) (S n * m)) ⟩
133 (S m) + ((S n) + ((S n) * m))
134 ≡⟨ cong (\x -> (S m) + x ) (sym (right-mult-distr-one (S n) m )) ⟩
135 S m + S n * S m
136 ∎
137
138
139 mult-sym : (n m : Int) -> n * m ≡ m * n
140 mult-sym n O = begin
141 n * O
142 ≡⟨ refl ⟩
143 O
144 ≡⟨ sym (left-mult-zero n) ⟩
145 O * n
146 ∎
147 mult-sym n (S m) = begin
148 n * (S m)
149 ≡⟨ right-mult-distr-one n m ⟩
150 n + (n * m)
151 ≡⟨ cong (\x -> n + x ) (mult-sym n m) ⟩
152 n + (m * n)
153 ≡⟨ sym (left-mult-distr-one m n) ⟩
154 (S m) * n
155 ∎