File size: 6,493 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 | /*@ Import Lean
import Algorithms.lucas_theorem.lean.helper_lib
open scoped SimpleC
*/
/*
* Compute C(n + m, n) modulo a prime p with Lucas' theorem.
*
* The problem guarantees
*
* 1 <= n, m, p <= 100000
*
* and that p is prime. For this verification example, all arithmetic used by
* the program is additionally assumed to stay within the signed int range.
*
* Input/output handling is intentionally left to the caller. The function
* lucas_theorem is the algorithmic entry point for one test case.
*/
/*@ Extern Coq
(ModularPower : Z -> Z -> Z -> Z -> Prop)
*/
/*@ Extern Coq
(PrimeForLucas : Z -> Prop)
(DigitBinomialMachineSafe : Z -> Z -> Z -> Prop)
(BinomialDigitResidue : Z -> Z -> Z -> Z -> Prop)
(LucasMachineSafe : Z -> Z -> Z -> Prop)
(LucasBinomialResidue : Z -> Z -> Z -> Z -> Prop)
*/
/*@ Extern Coq
(DigitProductProgress : Z -> Z -> Z -> Z -> Z -> Z -> Prop)
(LucasProgress : Z -> Z -> Z -> Z -> Z -> Z -> Prop)
*/
int modular_power(int base, int exponent, int modulus)
/*@ Require
0 <= base && base < modulus &&
0 <= exponent && 2 <= modulus && emp
Ensure
0 <= __return && __return < modulus &&
ModularPower(base, exponent, modulus, __return) && emp
*/;
/*
* Compute C(upper, lower) modulo prime, where
* 0 <= lower <= upper < prime.
*
* Because every factor in lower! is nonzero modulo prime, Fermat's little
* theorem gives lower!^(prime - 2) as its modular inverse.
*/
int binomial_digit_mod_prime(int upper, int lower, int prime)
/*@ Require
PrimeForLucas(prime) &&
0 <= lower && lower <= upper && upper < prime &&
2 <= prime && prime <= 100000 &&
DigitBinomialMachineSafe(upper, lower, prime) && emp
Ensure
0 <= __return && __return < prime@pre &&
BinomialDigitResidue(upper@pre, lower@pre, prime@pre, __return) && emp
*/
{
if (lower > upper) {
return 0;
}
if (lower > upper - lower) {
lower = upper - lower;
}
int numerator = 1;
int denominator = 1;
/*@ Inv Assert
upper == upper@pre && prime == prime@pre &&
PrimeForLucas(prime@pre) &&
0 <= lower@pre && lower@pre <= upper@pre &&
upper@pre < prime@pre &&
2 <= prime@pre && prime@pre <= 100000 &&
((lower == lower@pre &&
lower@pre <= upper@pre - lower@pre) ||
(lower == upper@pre - lower@pre &&
lower@pre > upper@pre - lower@pre)) &&
0 <= lower && lower <= upper@pre - lower &&
1 <= i && i <= lower + 1 &&
0 <= numerator && numerator < prime@pre &&
0 <= denominator && denominator < prime@pre &&
DigitBinomialMachineSafe(upper@pre, lower@pre, prime@pre) &&
DigitProductProgress(upper@pre, lower, prime@pre,
i, numerator, denominator) && emp
*/
for (int i = 1; i <= lower; ++i) {
int factor = upper - lower + i;
int numerator_product = numerator * factor;
int denominator_product = denominator * i;
numerator = numerator_product % prime;
denominator = denominator_product % prime;
}
/*@ Assert
upper == upper@pre && prime == prime@pre &&
PrimeForLucas(prime@pre) &&
0 <= lower@pre && lower@pre <= upper@pre &&
upper@pre < prime@pre &&
2 <= prime@pre && prime@pre <= 100000 &&
((lower == lower@pre &&
lower@pre <= upper@pre - lower@pre) ||
(lower == upper@pre - lower@pre &&
lower@pre > upper@pre - lower@pre)) &&
0 <= lower && lower <= upper@pre - lower &&
0 <= numerator && numerator < prime@pre &&
0 <= denominator && denominator < prime@pre &&
0 <= prime@pre - 2 &&
DigitBinomialMachineSafe(upper@pre, lower@pre, prime@pre) &&
DigitProductProgress(upper@pre, lower, prime@pre,
lower + 1, numerator, denominator) && emp
*/
int inverse = modular_power(denominator, prime - 2, prime);
int answer = numerator * inverse;
return answer % prime;
}
int lucas_theorem(int n, int m, int prime)
/*@ Require
1 <= n && n <= 100000 &&
1 <= m && m <= 100000 &&
2 <= prime && prime <= 100000 &&
PrimeForLucas(prime) &&
LucasMachineSafe(n, m, prime) && emp
Ensure
0 <= __return && __return < prime@pre &&
LucasBinomialResidue(n@pre, m@pre, prime@pre, __return) && emp
*/
{
int upper = n + m;
int lower = n;
int result = 1;
/*@ Inv Assert
n == n@pre && m == m@pre && prime == prime@pre &&
1 <= n@pre && n@pre <= 100000 &&
1 <= m@pre && m@pre <= 100000 &&
2 <= prime@pre && prime@pre <= 100000 &&
PrimeForLucas(prime@pre) &&
LucasMachineSafe(n@pre, m@pre, prime@pre) &&
0 <= lower && lower <= upper &&
upper <= n@pre + m@pre && n@pre + m@pre <= 200000 &&
0 <= result && result < prime@pre &&
LucasProgress(n@pre + m@pre, n@pre, prime@pre,
upper, lower, result) && emp
*/
while (upper > 0 || lower > 0) {
int upper_digit = upper % prime;
int lower_digit = lower % prime;
if (lower_digit > upper_digit) {
return 0;
}
/*@ Assert
n == n@pre && m == m@pre && prime == prime@pre &&
1 <= n@pre && n@pre <= 100000 &&
1 <= m@pre && m@pre <= 100000 &&
2 <= prime@pre && prime@pre <= 100000 &&
PrimeForLucas(prime@pre) &&
LucasMachineSafe(n@pre, m@pre, prime@pre) &&
0 < upper && 0 <= lower && lower <= upper &&
upper <= n@pre + m@pre && n@pre + m@pre <= 200000 &&
upper_digit == upper % prime@pre &&
lower_digit == lower % prime@pre &&
0 <= lower_digit && lower_digit <= upper_digit &&
upper_digit < prime@pre &&
0 <= result && result < prime@pre &&
DigitBinomialMachineSafe(
upper_digit, lower_digit, prime@pre) &&
LucasProgress(n@pre + m@pre, n@pre, prime@pre,
upper, lower, result) && emp
*/
int digit_binomial =
binomial_digit_mod_prime(upper_digit, lower_digit, prime);
int product = result * digit_binomial;
result = product % prime;
upper /= prime;
lower /= prime;
}
return result;
}
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