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Add mod-factor-propagation lemma to NLA divisions solver (#9235)

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When a monic x*y has a factor x with mod(x, p) = 0 (fixed), propagate
mod(x*y, p) = 0. This enables Z3 to prove divisibility properties like
x mod p = 0 => (x*y) mod p = 0, which previously timed out even for
p = 2. The lemma fires in the NLA divisions check and allows Gröbner
basis and LIA to subsequently derive distributivity of div over addition.

Extends division tuples from (q, x, y) to (q, x, y, r) to track the
mod lpvar. Also registers bounded divisions from the mod internalization
path in theory_lra, not just the idiv path.

Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
This commit is contained in:
Arie 2026-04-05 20:34:11 -04:00 • committed by GitHub
parent 8d7ed66ebf
commit 477a1d817d
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9 changed files with 169 additions and 42 deletions
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81
src/math/lp/nla_divisions.cpp
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@ -18,26 +18,26 @@ Description:
namespace nla {
void divisions::add_idivision(lpvar q, lpvar x, lpvar y) {
if (x == null_lpvar || y == null_lpvar || q == null_lpvar)
void divisions::add_idivision(lpvar q, lpvar x, lpvar y, lpvar r) {
if (x == null_lpvar || y == null_lpvar || q == null_lpvar || r == null_lpvar)
return;
m_idivisions.push_back({q, x, y});
m_idivisions.push_back({q, x, y, r});
m_core.trail().push(push_back_vector(m_idivisions));
}
void divisions::add_rdivision(lpvar q, lpvar x, lpvar y) {
if (x == null_lpvar || y == null_lpvar || q == null_lpvar)
void divisions::add_rdivision(lpvar q, lpvar x, lpvar y, lpvar r) {
if (x == null_lpvar || y == null_lpvar || q == null_lpvar || r == null_lpvar)
return;
m_rdivisions.push_back({ q, x, y });
m_rdivisions.push_back({ q, x, y, r });
m_core.trail().push(push_back_vector(m_rdivisions));
}
void divisions::add_bounded_division(lpvar q, lpvar x, lpvar y) {
if (x == null_lpvar || y == null_lpvar || q == null_lpvar)
void divisions::add_bounded_division(lpvar q, lpvar x, lpvar y, lpvar r) {
if (x == null_lpvar || y == null_lpvar || q == null_lpvar || r == null_lpvar)
return;
if (m_core.lra.column_has_term(x) || m_core.lra.column_has_term(y) || m_core.lra.column_has_term(q))
return;
m_bounded_divisions.push_back({ q, x, y });
m_bounded_divisions.push_back({ q, x, y, r });
m_core.trail().push(push_back_vector(m_bounded_divisions));
}
@ -111,7 +111,7 @@ namespace nla {
return false;
};
for (auto const & [r, x, y] : m_idivisions) {
for (auto const & [r, x, y, md] : m_idivisions) {
if (!c.is_relevant(r))
continue;
auto xval = c.val(x);
@ -120,7 +120,7 @@ namespace nla {
// idiv semantics
if (!xval.is_int() || !yval.is_int() || yval == 0 || rval == div(xval, yval))
continue;
for (auto const& [q2, x2, y2] : m_idivisions) {
for (auto const& [q2, x2, y2, md2] : m_idivisions) {
if (q2 == r)
continue;
if (!c.is_relevant(q2))
@ -133,7 +133,7 @@ namespace nla {
}
}
for (auto const& [r, x, y] : m_rdivisions) {
for (auto const& [r, x, y, md] : m_rdivisions) {
if (!c.is_relevant(r))
continue;
auto xval = c.val(x);
@ -142,7 +142,7 @@ namespace nla {
// / semantics
if (yval == 0 || rval == xval / yval)
continue;
for (auto const& [q2, x2, y2] : m_rdivisions) {
for (auto const& [q2, x2, y2, md2] : m_rdivisions) {
if (q2 == r)
continue;
if (!c.is_relevant(q2))
@ -154,7 +154,8 @@ namespace nla {
return;
}
}
check_mod_mult();
}
// if p is bounded, q a value, r = eval(p):
@ -163,11 +164,11 @@ namespace nla {
void divisions::check_bounded_divisions() {
core& c = m_core;
unsigned offset = c.random(), sz = m_bounded_divisions.size();
unsigned offset = c.random(), sz = m_bounded_divisions.size();
for (unsigned j = 0; j < sz; ++j) {
unsigned i = (offset + j) % sz;
auto [q, x, y] = m_bounded_divisions[i];
auto [q, x, y, r] = m_bounded_divisions[i];
if (!c.is_relevant(q))
continue;
auto xv = c.val(x);
@ -188,9 +189,9 @@ namespace nla {
rational lo = yv * div_v;
if (xv > hi) {
lemma_builder lemma(c, "y = yv & x <= yv * div(xv, yv) + yv - 1 => div(p, y) <= div(xv, yv)");
lemma |= ineq(y, llc::NE, yv);
lemma |= ineq(x, llc::GT, hi);
lemma |= ineq(q, llc::LE, div_v);
lemma |= ineq(y, llc::NE, yv);
lemma |= ineq(x, llc::GT, hi);
lemma |= ineq(q, llc::LE, div_v);
return;
}
if (xv < lo) {
@ -201,5 +202,45 @@ namespace nla {
return;
}
}
}
}
// mod(factor, p) = 0 => mod(factor * k, p) = 0
// For each division (q, x, y, r) where x is a monic m = f1 * f2 * ... * fk,
// if some factor fi has mod(fi, p) = 0 (fixed), then mod(x, p) = 0.
void divisions::check_mod_mult() {
core& c = m_core;
unsigned offset = c.random(), sz = m_bounded_divisions.size();
for (unsigned j = 0; j < sz; ++j) {
unsigned i = (offset + j) % sz;
auto [q, x, y, r] = m_bounded_divisions[i];
if (!c.is_relevant(q))
continue;
if (c.var_is_fixed_to_zero(r))
continue;
if (c.val(r).is_zero())
continue;
if (!c.is_monic_var(x))
continue;
auto yv = c.val(y);
if (yv <= 0 || !yv.is_int())
continue;
auto const& m = c.emons()[x];
for (lpvar f : m.vars()) {
for (auto const& [q2, x2, y2, r2] : m_bounded_divisions) {
if (x2 != f)
continue;
if (c.val(y2) != yv)
continue;
if (!c.var_is_fixed_to_zero(r2))
continue;
// mod(factor, p) = 0 => mod(product, p) = 0
lemma_builder lemma(c, "mod(factor, p) = 0 => mod(factor * k, p) = 0");
lemma |= ineq(r2, llc::NE, 0);
lemma |= ineq(r, llc::EQ, 0);
return;
}
}
}
}
}