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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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24
src/smt/theory_lra.cpp
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@ -449,25 +449,43 @@ class theory_lra::imp {
internalize_term(to_app(n));
internalize_term(to_app(n1));
internalize_term(to_app(n2));
internalize_term(to_app(mod));
theory_var q = mk_var(n);
theory_var x = mk_var(n1);
theory_var y = mk_var(n2);
m_nla->add_idivision(register_theory_var_in_lar_solver(q), register_theory_var_in_lar_solver(x), register_theory_var_in_lar_solver(y));
theory_var rv = mk_var(mod);
m_nla->add_idivision(register_theory_var_in_lar_solver(q), register_theory_var_in_lar_solver(x), register_theory_var_in_lar_solver(y), register_theory_var_in_lar_solver(rv));
}
if (a.is_numeral(n2) && a.is_bounded(n1)) {
ensure_nla();
internalize_term(to_app(n));
internalize_term(to_app(n1));
internalize_term(to_app(n2));
internalize_term(to_app(mod));
theory_var q = mk_var(n);
theory_var x = mk_var(n1);
theory_var y = mk_var(n2);
m_nla->add_bounded_division(register_theory_var_in_lar_solver(q), register_theory_var_in_lar_solver(x), register_theory_var_in_lar_solver(y));
theory_var rv = mk_var(mod);
m_nla->add_bounded_division(register_theory_var_in_lar_solver(q), register_theory_var_in_lar_solver(x), register_theory_var_in_lar_solver(y), register_theory_var_in_lar_solver(rv));
}
}
else if (a.is_mod(n, n1, n2)) {
if (!a.is_numeral(n2, r) || r.is_zero()) found_underspecified(n);
if (!ctx().relevancy()) mk_idiv_mod_axioms(n1, n2);
if (!ctx().relevancy()) mk_idiv_mod_axioms(n1, n2);
if (a.is_numeral(n2) && !r.is_zero()) {
ensure_nla();
app_ref div(a.mk_idiv(n1, n2), m);
ctx().internalize(div, false);
internalize_term(to_app(div));
internalize_term(to_app(n1));
internalize_term(to_app(n2));
internalize_term(t);
theory_var q = mk_var(div);
theory_var x = mk_var(n1);
theory_var y = mk_var(n2);
theory_var rv = mk_var(n);
m_nla->add_bounded_division(register_theory_var_in_lar_solver(q), register_theory_var_in_lar_solver(x), register_theory_var_in_lar_solver(y), register_theory_var_in_lar_solver(rv));
}
}
else if (a.is_rem(n, n1, n2)) {
if (!a.is_numeral(n2, r) || r.is_zero()) found_underspecified(n);