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proofs/extended_euclidean_bounds/helpers/essential_asserts__b_gt_0__b_eq_a.h
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// --- This file is distributed under the MIT Open Source License, as detailed | |
// in the file "LICENSE.TXT" in the root of this repository --- | |
// This file is a trimmed copy of extended_euclidean__b_gt_0__b_eq_a.h. | |
// It contains only the essential assertion results, without any of the proofs. | |
#ifndef ESSENTIAL_ASSERTS__B_GT_0__B_EQ_A | |
#define ESSENTIAL_ASSERTS__B_GT_0__B_EQ_A 1 | |
#ifndef NDEBUG | |
# include "assert_helper_gcd.h" | |
#endif | |
#include <assert.h> | |
#include <limits> | |
#if defined(assert_precondition) | |
# error "assert_precondition was already defined" | |
#endif | |
// assert aliases will help self-document the code | |
#define assert_precondition assert | |
template <typename T> | |
void essential_asserts__b_gt_0__b_eq_a(T a, T b, T* pGcd, T* pX, T* pY) | |
{ | |
static_assert(std::numeric_limits<T>::is_integer, ""); | |
static_assert(std::numeric_limits<T>::is_signed, ""); | |
assert_precondition(b == a); | |
assert_precondition(b > 0); | |
assert(gcd(a,b) >= 1); | |
T x0 = 1, y0 = 0, a0 = a; | |
T x1 = 0, y1 = 1, a1 = b; | |
// The peeled-out first iteration: | |
{ | |
T q = a0/a1; | |
assert(1 <= q && q <= a0); | |
T a2 = a0 - q*a1; | |
assert(0 < (q*a1) && (q*a1) <= a0); | |
assert(0 <= a2 && a2 < a1); | |
if (a2 != 0) assert(q <= a0/2); | |
T x2 = x0 - q*x1; | |
assert(abs(q*x1) <= abs(x2)); | |
assert(abs(x1) <= abs(x2)); | |
T y2 = y0 - q*y1; | |
assert(abs(q*y1) <= abs(y2)); | |
assert(abs(y1) <= abs(y2)); | |
x0=x1; y0=y1; a0=a1; | |
x1=x2; y1=y2; a1=a2; | |
} | |
assert(abs(x1) == b/gcd(a,b)); | |
assert(abs(y1) == a/gcd(a,b)); | |
assert(abs(x0) <= (b/gcd(a,b))/2); | |
assert(y0 == 1); | |
assert(a0 == gcd(a,b)); | |
assert(a*x0 + b*y0 == gcd(a,b)); | |
*pX = x0; | |
*pY = y0; | |
*pGcd = a0; | |
} | |
#undef assert_precondition | |
#endif |