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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 | /*@ Import Lean
import Algorithms.huffman_encoding.lean.helper_lib
open scoped SimpleC
*/
/*
* Huffman encoding: minimum weighted path length of a binary prefix code.
*
* Repeatedly merging the two smallest live weights constructs a Huffman tree.
* The sum of all merge weights is that tree's weighted path length and is
* minimum among all full binary prefix-code trees over the input frequencies.
*
* This verification-oriented implementation uses a fresh selection scan for
* each of the two minima. Removing a selected item is done by moving the last
* live item into its slot, so only the prefix work[0..active) is live.
*/
/*@ Extern Coq
(HuffmanInputBounded : list Z -> Prop)
(HuffmanOptimalCost : list Z -> Z -> Prop)
(HuffmanScratchFinal : list Z -> list Z -> Prop)
(HuffmanProgress : list Z -> list Z -> Z -> Z -> Prop)
(HuffmanMinScan : list Z -> Z -> Z -> Prop)
(HuffmanFirstHeld : list Z -> list Z -> Z -> Z -> Z -> Prop)
(HuffmanPairReady : list Z -> list Z -> Z -> Z -> Z -> Z -> Prop)
*/
int huffman_cost(int *weights, int n, int *work)
/*@ With (weights_l : list Z)
Require
1 <= n && n <= 8 &&
Zlength(weights_l) == n &&
HuffmanInputBounded(weights_l) &&
IntArray::full(weights, n, weights_l) *
IntArray::undef_full(work, n)
Ensure
exists work_l,
Zlength(work_l) == n &&
HuffmanOptimalCost(weights_l, __return) &&
HuffmanScratchFinal(weights_l, work_l) &&
0 <= __return && __return <= 56000 &&
IntArray::full(weights, n, weights_l) *
IntArray::full(work, n, work_l)
*/
{
/*@ Inv Assert
exists copied remaining,
weights == weights@pre && work == work@pre && n == n@pre &&
1 <= n@pre && n@pre <= 8 &&
Zlength(weights_l) == n@pre &&
HuffmanInputBounded(weights_l) &&
weights_l == app(copied, remaining) &&
Zlength(copied) == i &&
0 <= i && i <= n@pre &&
IntArray::full(weights, n@pre, weights_l) *
IntArray::seg(work, 0, i, copied) *
IntArray::undef_seg(work, i, n@pre)
*/
for (int i = 0; i < n; ++i) {
work[i] = weights[i];
}
int active = n;
int total = 0;
/*@ Inv Assert
exists work_l,
weights == weights@pre && work == work@pre && n == n@pre &&
1 <= n@pre && n@pre <= 8 &&
Zlength(weights_l) == n@pre &&
Zlength(work_l) == n@pre &&
HuffmanInputBounded(weights_l) &&
1 <= active && active <= n@pre &&
0 <= total && total <= 56000 &&
(forall (k : Z),
0 <= k && k < active =>
1 <= Znth(k, work_l, 0) && Znth(k, work_l, 0) <= 8000) &&
HuffmanProgress(weights_l, work_l, active, total) &&
IntArray::full(weights, n@pre, weights_l) *
IntArray::full(work, n@pre, work_l)
*/
while (active > 1) {
int first = 0;
/*@ Inv Assert
exists work_l,
weights == weights@pre && work == work@pre && n == n@pre &&
1 <= n@pre && n@pre <= 8 &&
Zlength(weights_l) == n@pre &&
Zlength(work_l) == n@pre &&
HuffmanInputBounded(weights_l) &&
2 <= active && active <= n@pre &&
1 <= i && i <= active && i <= n@pre &&
0 <= first && first < i && first < n@pre &&
0 <= total && total <= 56000 &&
(forall (k : Z),
0 <= k && k < active =>
1 <= Znth(k, work_l, 0) && Znth(k, work_l, 0) <= 8000) &&
HuffmanProgress(weights_l, work_l, active, total) &&
HuffmanMinScan(work_l, i, first) &&
IntArray::full(weights, n@pre, weights_l) *
IntArray::full(work, n@pre, work_l)
*/
for (int i = 1; i < active; ++i) {
/*@ 0 <= i && i < n@pre */
if (work[i] < work[first]) {
first = i;
}
}
int x = work[first];
--active;
/*@ 0 <= active && active < n@pre */
work[first] = work[active];
/*@ Assert
exists work_l,
weights == weights@pre && work == work@pre && n == n@pre &&
1 <= n@pre && n@pre <= 8 &&
Zlength(weights_l) == n@pre &&
Zlength(work_l) == n@pre &&
HuffmanInputBounded(weights_l) &&
1 <= active && active < n@pre &&
0 <= first && first < n@pre &&
1 <= x && x <= 8000 &&
0 <= total && total <= 56000 &&
(forall (k : Z),
0 <= k && k < active =>
1 <= Znth(k, work_l, 0) && Znth(k, work_l, 0) <= 8000) &&
HuffmanFirstHeld(weights_l, work_l, active, x, total) &&
IntArray::full(weights, n@pre, weights_l) *
IntArray::full(work, n@pre, work_l)
*/
int second = 0;
/*@ Inv Assert
exists work_l,
weights == weights@pre && work == work@pre && n == n@pre &&
1 <= n@pre && n@pre <= 8 &&
Zlength(weights_l) == n@pre &&
Zlength(work_l) == n@pre &&
HuffmanInputBounded(weights_l) &&
1 <= active && active < n@pre &&
1 <= i && i <= active && i < n@pre &&
0 <= first && first < n@pre &&
0 <= second && second < i && second < n@pre &&
1 <= x && x <= 8000 &&
0 <= total && total <= 56000 &&
(forall (k : Z),
0 <= k && k < active =>
1 <= Znth(k, work_l, 0) && Znth(k, work_l, 0) <= 8000) &&
HuffmanFirstHeld(weights_l, work_l, active, x, total) &&
HuffmanMinScan(work_l, i, second) &&
IntArray::full(weights, n@pre, weights_l) *
IntArray::full(work, n@pre, work_l)
*/
for (int i = 1; i < active; ++i) {
if (work[i] < work[second]) {
second = i;
}
}
int y = work[second];
--active;
/*@ 0 <= active && active < n@pre */
work[second] = work[active];
/*@ Assert
exists work_l,
weights == weights@pre && work == work@pre && n == n@pre &&
1 <= n@pre && n@pre <= 8 &&
Zlength(weights_l) == n@pre &&
Zlength(work_l) == n@pre &&
HuffmanInputBounded(weights_l) &&
0 <= active && active <= n@pre - 2 &&
0 <= first && first < n@pre &&
0 <= second && second < n@pre &&
1 <= x && x <= 8000 &&
1 <= y && y <= 8000 &&
x + y <= 8000 &&
0 <= total && total + x + y <= 56000 &&
(forall (k : Z),
0 <= k && k < active =>
1 <= Znth(k, work_l, 0) && Znth(k, work_l, 0) <= 8000) &&
HuffmanPairReady(weights_l, work_l, active, x, y, total) &&
IntArray::full(weights, n@pre, weights_l) *
IntArray::full(work, n@pre, work_l)
*/
int merged = x + y;
total += merged;
work[active] = merged;
++active;
}
/*@ Assert
exists final_work_l,
weights == weights@pre && work == work@pre && n == n@pre &&
1 <= n@pre && n@pre <= 8 &&
Zlength(weights_l) == n@pre &&
Zlength(final_work_l) == n@pre &&
active == 1 &&
0 <= total && total <= 56000 &&
HuffmanOptimalCost(weights_l, total) &&
HuffmanScratchFinal(weights_l, final_work_l) &&
IntArray::full(weights, n@pre, weights_l) *
IntArray::full(work, n@pre, final_work_l)
*/
return total;
}
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