File size: 5,839 Bytes
bff4ec7 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 | /*@ Import Lean
import Algorithms.minimal_representation.lean.helper_lib
open scoped SimpleC
*/
/*
* Find the lexicographically smallest cyclic rotation of an integer sequence.
*
* The sequence is copied twice into b so that every cyclic rotation is
* represented by one contiguous segment. The two-candidate scan eliminates
* at least one possible starting position after every mismatch, and therefore
* runs in linear time.
*
* The verification case limits n to 1000. The caller provides n initialized
* elements in a, room for 2 * n elements in b, and room for n elements in out.
* The function writes the smallest rotation to out and returns its zero-based
* starting position in a.
*/
/*@ Extern Coq
(MRRotation : list Z -> Z -> list Z)
(MRRotationPrefixEq : list Z -> Z -> Z -> Z -> Prop)
(MRRotationEq : list Z -> Z -> Z -> Prop)
(MRFirstMinimalRotationAt : list Z -> Z -> Prop)
(MRCandidateState : list Z -> Z -> Z -> Z -> Prop)
*/
int minimal_representation(int *a, int n, int *b, int *out)
/*@ With (l : list Z) (best : Z)
Require
1 <= n && n <= 1000 &&
Zlength(l) == n &&
0 <= best && best < n &&
MRFirstMinimalRotationAt(l, best) &&
IntArray::full(a, n, l) *
IntArray::undef_full(b, 2 * n) *
IntArray::undef_full(out, n)
Ensure
exists bl,
__return == best &&
MRFirstMinimalRotationAt(l, __return) &&
IntArray::full(a, n, l) *
IntArray::full(b, 2 * n, bl) *
IntArray::full(out, n, MRRotation(l, __return))
*/
{
int p = 0;
/*@ Inv Assert
a == a@pre && n == n@pre && b == b@pre && out == out@pre &&
1 <= n@pre && n@pre <= 1000 &&
Zlength(l) == n@pre &&
0 <= best && best < n@pre &&
0 <= p && p <= n@pre &&
MRFirstMinimalRotationAt(l, best) &&
IntArray::full(a@pre, n@pre, l) *
IntArray::seg(b@pre, 0, p, sublist(0, p, l)) *
IntArray::undef_seg(b@pre, p, n@pre) *
IntArray::seg(b@pre, n@pre, n@pre + p, sublist(0, p, l)) *
IntArray::undef_seg(b@pre, n@pre + p, 2 * n@pre) *
IntArray::undef_full(out@pre, n@pre)
*/
while (p < n) {
b[p] = a[p];
b[n + p] = a[p];
++p;
}
/*@ Assert
a == a@pre && n == n@pre && b == b@pre && out == out@pre &&
1 <= n@pre && n@pre <= 1000 &&
Zlength(l) == n@pre &&
p == n@pre &&
0 <= best && best < n@pre &&
MRFirstMinimalRotationAt(l, best) &&
IntArray::full(a@pre, n@pre, l) *
IntArray::full(b@pre, 2 * n@pre, app(l, l)) *
IntArray::undef_full(out@pre, n@pre)
*/
int i = 0;
int j = 1;
int k = 0;
/*@ Inv Assert
a == a@pre && n == n@pre && b == b@pre && out == out@pre &&
1 <= n@pre && n@pre <= 1000 &&
Zlength(l) == n@pre &&
p == n@pre &&
0 <= best && best < n@pre &&
0 <= i && i < 2 * n@pre &&
0 <= j && j < 2 * n@pre &&
i != j &&
(i < n@pre || j < n@pre) &&
0 <= k && k < n@pre &&
MRFirstMinimalRotationAt(l, best) &&
MRCandidateState(l, best, i, j) &&
IntArray::full(a@pre, n@pre, l) *
IntArray::full(b@pre, 2 * n@pre, app(l, l)) *
IntArray::undef_full(out@pre, n@pre)
*/
while (i < n && j < n) {
k = 0;
/*@ Inv Assert
a == a@pre && n == n@pre && b == b@pre && out == out@pre &&
1 <= n@pre && n@pre <= 1000 &&
Zlength(l) == n@pre &&
p == n@pre &&
0 <= best && best < n@pre &&
0 <= i && i < n@pre &&
0 <= j && j < n@pre &&
i != j &&
0 <= k && k <= n@pre &&
MRFirstMinimalRotationAt(l, best) &&
MRCandidateState(l, best, i, j) &&
MRRotationPrefixEq(l, i, j, k) &&
IntArray::full(a@pre, n@pre, l) *
IntArray::full(b@pre, 2 * n@pre, app(l, l)) *
IntArray::undef_full(out@pre, n@pre)
*/
while (k < n && b[i + k] == b[j + k]) {
++k;
}
if (k == n) {
break;
}
if (b[i + k] > b[j + k]) {
i = i + k + 1;
if (i == j) {
++i;
}
}
else {
j = j + k + 1;
if (i == j) {
++j;
}
}
}
/*@ Assert
a == a@pre && n == n@pre && b == b@pre && out == out@pre &&
1 <= n@pre && n@pre <= 1000 &&
Zlength(l) == n@pre &&
p == n@pre &&
0 <= best && best < n@pre &&
0 <= i && i < 2 * n@pre &&
0 <= j && j < 2 * n@pre &&
i != j &&
(i < n@pre || j < n@pre) &&
0 <= k && k <= n@pre &&
(i < j => i == best) &&
(i >= j => j == best) &&
MRFirstMinimalRotationAt(l, best) &&
IntArray::full(a@pre, n@pre, l) *
IntArray::full(b@pre, 2 * n@pre, app(l, l)) *
IntArray::undef_full(out@pre, n@pre)
*/
if (i < j) {
p = i;
}
else {
p = j;
}
k = 0;
/*@ Inv Assert
a == a@pre && n == n@pre && b == b@pre && out == out@pre &&
1 <= n@pre && n@pre <= 1000 &&
Zlength(l) == n@pre &&
0 <= best && best < n@pre &&
p == best &&
0 <= p && p < n@pre &&
0 <= i && i < 2 * n@pre &&
0 <= j && j < 2 * n@pre &&
i != j &&
0 <= k && k <= n@pre &&
MRFirstMinimalRotationAt(l, best) &&
IntArray::full(a@pre, n@pre, l) *
IntArray::full(b@pre, 2 * n@pre, app(l, l)) *
IntArray::seg(out@pre, 0, k, sublist(0, k, MRRotation(l, p))) *
IntArray::undef_seg(out@pre, k, n@pre)
*/
while (k < n) {
out[k] = b[p + k];
++k;
}
return p;
}
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