1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
|
import { Add, Multiply, Succ, _0, _1, _3 } from './nat';
import { Op, Rewrite, ChainRewrites, Sym, Equation, Refl, VerifyEquation, assert } from './util';
export namespace addition {
// Identity
export type Identity<A extends Op> = Rewrite<Add<A, _0>, A>;
export type IdentityR<A extends Op> = Rewrite<Add<_0, A>, A>;
export type SumSucc<A extends Op, B extends Op> = Rewrite<Add<Succ<A>, B>, Succ<Add<A, B>>>
export type SumSuccR<A extends Op, B extends Op> = Rewrite<Add<A, Succ<B>>, Succ<Add<A, B>>>
// Commutativity
export type Commutativity<A extends Op, B extends Op> = Rewrite<Add<A, B>, Add<B, A>>;
export interface Comm_Base_Proof<B extends Op> {
type: 'rewrite',
left: Equation<Add<_0, B>, Add<B, _0>>; // 0 + b = b + 0
right: ChainRewrites<[
IdentityR<B>, // b = b + 0
Sym<Identity<B>>, // b + 0 = b + 0
Refl<Add<B, _0>>, // true
], this['left']>;
};
export interface Comm_Inductive_Proof<A extends Op, B extends Op> {
type: 'rewrite',
left: Equation<Add<Succ<A>, B>, Add<B, Succ<A>>>; // succ(a) + b = b + succ(a)
right: ChainRewrites<[
SumSucc<A, B>, // succ(a + b) = b + succ(a)
Commutativity<A, B>, // succ(b + a) = b + succ(a)
SumSuccR<B, A>, // succ(b + a) = succ(b + a)
Refl<Succ<Add<B, A>>>, // true
], this['left']>;
};
export namespace spec {
export type commutativity = [
assert<VerifyEquation<Comm_Base_Proof<'B'>>>,
assert<VerifyEquation<Comm_Inductive_Proof<'A', 'B'>>>,
]
}
// Associativity
export type Associativity<A extends Op, B extends Op, C extends Op> = Rewrite<Add<Add<A, B>, C>, Add<A, Add<B, C>>>;
export interface Assoc_Base_Proof<B extends Op, C extends Op> {
type: 'rewrite',
left: Equation<Add<Add<_0, B>, C>, Add<_0, Add<B, C>>>; // (0 + b) + c = 0 + (b + c)
right: ChainRewrites<[
Commutativity<_0, B>, // (b + 0) + c = 0 + (b + c)
Identity<B>, // b + c = 0 + (b + c)
IdentityR<Add<B, C>>, // b + c = b + c
Refl<Add<B, C>>, // true
], this['left']>;
};
export interface Assoc_Inductive_Proof<A extends Op, B extends Op, C extends Op> {
type: 'rewrite',
left: Equation<Add<Add<Succ<A>, B>, C>, Add<Succ<A>, Add<B, C>>>; // (succ(a) + b) + c = succ(a) + (b + c)
right: ChainRewrites<[
SumSucc<A, B>, // succ(a + b) + c = succ(a) + (b + c)
SumSucc<A, Add<B, C>>, // succ(a + b) + c = succ(a + (b + c))
SumSucc<Add<A, B>, C>, // succ((a + b) + c) = succ(a + (b + c))
Associativity<A, B, C>, // succ(a + (b + c)) = succ(a + (b + c))
Refl<Succ<Add<A, Add<B, C>>>>, // true
], this['left']>;
};
export namespace spec {
export type associativity = [
assert<VerifyEquation<Assoc_Base_Proof<'A', 'B'>>>,
assert<VerifyEquation<Assoc_Inductive_Proof<'A', 'B', 'C'>>>,
]
}
}
export namespace multiplication {
// Identity
export type Identity<A extends Op> = Rewrite<Multiply<A, _1>, A>;
export type IdentityR<A extends Op> = Rewrite<Multiply<_1, A>, A>;
export type MulSucc<A extends Op, B extends Op> = Rewrite<Multiply<Succ<A>, B>, Add<B, Multiply<A, B>>>;
export type MulSuccR<A extends Op, B extends Op> = Rewrite<Multiply<A, Succ<B>>, Add<A, Multiply<A, B>>>;
// Commutativity
export type Commutativity<A extends Op, B extends Op> = Rewrite<Multiply<A, B>, Multiply<B, A>>;
export interface Comm_Base_Proof<B extends Op> {
type: 'rewrite',
left: Equation<Multiply<_1, B>, Multiply<B, _1>>; // 1 * b = b * 1
right: ChainRewrites<[
IdentityR<B>, // b = b * 1
Identity<B>, // b = b
Refl<B>, // true
], this['left']>;
};
export interface Comm_Inductive_Proof<A extends Op, B extends Op> {
type: 'rewrite',
left: Equation<Multiply<Succ<A>, B>, Multiply<B, Succ<A>>>; // succ(a)*b = b*succ(a)
right: ChainRewrites<[
MulSucc<A, B>, // b + a*b = b*succ(a)
MulSuccR<B, A>, // b + a*b = b + b*a
Commutativity<B, A>, // b + a*b = b + a*b
Refl<Add<B, Multiply<A, B>>>, // true
], this['left']>;
};
export namespace spec {
export type commutativity = [
assert<VerifyEquation<Comm_Base_Proof<'B'>>>,
assert<VerifyEquation<Comm_Inductive_Proof<'A', 'B'>>>,
]
}
// Distributivity
export type Distributivity<A extends Op, B extends Op, C extends Op> =
Rewrite<Multiply<Add<A, B>, C>, Add<Multiply<A, C>, Multiply<B, C>>>;
export interface Dist_Base_Proof<B extends Op, C extends Op> {
type: 'rewrite',
left: Equation<Multiply<Add<_1, B>, C>, Add<Multiply<_1, C>, Multiply<B, C>>>; // (1 + b)*c = 1*c + b*c
right: ChainRewrites<[
IdentityR<C>, // (1 + b)*c = c + b*c
addition.SumSucc<_0, B>, // succ(0 + b)*c = c + b*c
addition.IdentityR<B>, // succ(b)*c = c + b*c
MulSucc<B, C>, // c + b*c = c + b*c
Refl<Add<C, Multiply<B, C>>>, // true
], this['left']>;
};
export interface Dist_Inductive_Proof<A extends Op, B extends Op, C extends Op> {
type: 'rewrite',
left: Equation<Multiply<Add<Succ<A>, B>, C>, Add<Multiply<Succ<A>, C>, Multiply<B, C>>>; // (succ(a) + b)*c = succ(a)*c + b*c
right: ChainRewrites<[
addition.SumSucc<A, B>, // succ(a + b)*c = succ(a)*c + b*c
MulSucc<Add<A, B>, C>, // c + (a + b)*c = succ(a)*c + b*c
MulSucc<A, C>, // c + (a + b)*c = (c + a*c) + b*c
Distributivity<A, B, C>, // c + (a*c + b*c) = (c + a*c) + b*c
addition.Associativity<C, Multiply<A, C>, Multiply<B, C>>, // (c + a*c) + b*c = (c + a*c) + b*c
Refl<Add<C, Add<Multiply<A, C>, Multiply<B, C>>>>, // true
], this['left']>;
};
export namespace spec {
export type distributivity = [
assert<VerifyEquation<Dist_Base_Proof<'B', 'C'>>>,
assert<VerifyEquation<Dist_Inductive_Proof<'A', 'B', 'C'>>>,
]
}
// Associativity
export type Associativity<A extends Op, B extends Op, C extends Op> =
Rewrite<Multiply<Multiply<A, B>, C>, Multiply<A, Multiply<B, C>>>;
export interface Assoc_Base_Proof<B extends Op, C extends Op> {
type: 'rewrite',
left: Equation<Multiply<Multiply<_1, B>, C>, Multiply<_1, Multiply<B, C>>>; // (1*b)*c = 1*(b*c)
right: ChainRewrites<[
IdentityR<B>, // b*c = 1*(b*c)
IdentityR<Multiply<B, C>>, // b*c = b*c
Refl<Multiply<B, C>>, // true
], this['left']>;
};
export interface Assoc_Inductive_Proof<A extends Op, B extends Op, C extends Op> {
type: 'rewrite',
left: Equation<Multiply<Multiply<Succ<A>, B>, C>, Multiply<Succ<A>, Multiply<B, C>>>; // (succ(a)*b)*c = succ(a)*(b*c)
right: ChainRewrites<[
MulSucc<A, B>, // (b + a*b)*c = succ(a) * (b*c)
Distributivity<B, Multiply<A, B>, C>, // b*c + (a*b)*c = succ(a) * (b*c)
MulSucc<A, Multiply<B, C>>, // b*c + (a*b)*c = b*c + a*(b*c)
Associativity<A, B, C>, // b*c + a*(b*c) = b*c + a*(b*c)
Refl<Add<Multiply<B, C>, Multiply<A, Multiply<B, C>>>>, // true
], this['left']>;
};
export namespace spec {
export type associativity = [
assert<VerifyEquation<Assoc_Base_Proof<'B', 'C'>>>,
assert<VerifyEquation<Assoc_Inductive_Proof<'A', 'B', 'C'>>>,
]
}
}
|
