litex-lang 0.9.85-beta

A simple formal proof language and verifier, learnable in 2 hours
Documentation 
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
172
173
174
175
176
177
178
179

// Set unions, intersections, subset, finite_set count, etc.

pub const BUILTIN_ENV_CODE_FOR_SET_OPERATORS: &str = r#"

prop in_intersect_is_in_both(z set, A set, B set):
    $in(z, A)
    $in(z, B)

prop in_set_minus_is_in_first_operand(z set, A set, B set):
    $in(z, A)

prop in_set_minus_is_not_in_second_operand(z set, A set, B set):
    not $in(z, B)

prop in_cup_via_member_set(z set, F set, Y set):
    $in(z, cup(F))

prop subset_of_finite_set_is_finite(A set, B finite_set):
    $is_finite_set(A)

know:
    forall z set, A set, B set:
        $in(z, A)
        =>:
            $in(z, union(A, B))

    forall z set, A set, B set:
        $in(z, B)
        =>:
            $in(z, union(A, B))

    forall z set, A set, B set:
        $in(z, union(A, B))
        =>:
            $in(z, A) or $in(z, B)

    forall z set, A set, B set:
        $in(z, A)
        $in(z, B)
        =>:
            $in(z, intersect(A, B))

    forall z set, A set, B set:
        $in(z, intersect(A, B))
        =>:
            $in_intersect_is_in_both(z, A, B)

    forall z set, A set, B set:
        not $in(z, A)
        =>:
            not $in(z, intersect(A, B))

    forall z set, A set, B set:
        not $in(z, B)
        =>:
            not $in(z, intersect(A, B))

    forall A, B set:
        intersect(A, B) $subset A

    forall A, B set:
        intersect(A, B) $subset B

    forall A, B set:
        A $subset union(A, B)

    forall A, B set:
        B $subset union(A, B)

    forall A, B set:
        union(A, B) = union(B, A)

    forall A, B set:
        intersect(A, B) = intersect(B, A)

    forall A, B, C set:
        union(union(A, B), C) = union(A, union(B, C))

    forall A, B, C set:
        intersect(intersect(A, B), C) = intersect(A, intersect(B, C))

    forall A, B set:
        union(A, intersect(A, B)) = A

    forall A, B set:
        intersect(A, union(A, B)) = A

    forall A set:
        union(A, A) = A

    forall A set:
        intersect(A, A) = A

    forall A set:
        union(A, {}) = A

    forall A set:
        intersect(A, {}) = {}

    forall A, B, C set:
        intersect(A, union(B, C)) = union(intersect(A, B), intersect(A, C))

    forall A, B, C set:
        union(A, intersect(B, C)) = intersect(union(A, B), union(A, C))

    forall z set, A set, B set:
        $in(z, A)
        not $in(z, B)
        =>:
            $in(z, set_minus(A, B))

    forall z set, A set, B set:
        $in(z, set_minus(A, B))
        =>:
            $in_set_minus_is_in_first_operand(z, A, B)

    forall z set, A set, B set:
        $in(z, set_minus(A, B))
        =>:
            $in_set_minus_is_not_in_second_operand(z, A, B)

    forall A, B set:
        set_minus(A, B) $subset A

    forall A, B set:
        set_diff(A, B) = union(set_minus(A, B), set_minus(B, A))

    forall z set, F set, Y set:
        $in(Y, F)
        $in(z, Y)
        =>:
            $in_cup_via_member_set(z, F, Y)

    forall A, B finite_set:
        $is_finite_set(union(A, B))
        $is_finite_set(intersect(A, B))
        $is_finite_set(set_minus(A, B))
        $is_finite_set(set_diff(A, B))

    forall A finite_set:
        count(A) $in N

    forall A set, B finite_set:
        A $subset B
        =>:
            $subset_of_finite_set_is_finite(A, B)

    forall A finite_set, B set:
        B $subset A
        =>:
            $is_finite_set(B)
            $is_finite_set(set_minus(A, B))
            count(set_minus(A, B)) = count(A) - count(B)

    forall A, B finite_set:
        count(union(A, B)) = count(A) + count(B) - count(intersect(A, B))
        count(A) = count(intersect(A, B)) + count(set_minus(A, B))
        count(B) = count(intersect(A, B)) + count(set_minus(B, A))
        count(set_minus(A, B)) = count(A) - count(intersect(A, B))
        count(set_minus(B, A)) = count(B) - count(intersect(A, B))
        count(set_diff(A, B)) = count(set_minus(A, B)) + count(set_minus(B, A))

    forall A, B finite_set:
        A $subset B
        =>:
            count(A) <= count(B)

    forall A, B finite_set:
        A $superset B
        =>:
            count(A) >= count(B)

    forall A, B finite_set:
        count(intersect(A, B)) <= count(A)
        count(intersect(A, B)) <= count(B)
        count(set_minus(A, B)) <= count(A)
        count(union(A, B)) <= count(A) + count(B)
        count(set_diff(A, B)) <= count(A) + count(B)
"#;