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Fix non-termination in mod rewriter for symbolic modulus (#10105)

Browse source 
Combining `mod0`/`div0` quantifier axioms with a mod-idempotency
quantifier caused Z3 to loop forever. The core issue was that
`mk_mod_core` in `arith_rewriter.cpp` only handled rewrite rules for
*numeral* moduli, leaving two gaps for symbolic `y`:

1. `mod(a + k*y, y)` was not reduced to `mod(a, y)`, so `(not (= (mod (+
a b) b) (mod a b)))` stayed unreduced and caused the nlsat solver to
spin.
2. The E-matching pattern `(mod (mod x y) y)` fired on every new term it
produced, creating an unbounded chain of nested `mod` expressions.

```lisp
; Previously non-terminating, now returns unsat immediately
(assert (forall ((x Int)) (! (= (mod0 x 0) 0) :pattern ((mod0 x 0)))))
(assert (forall ((x Int)) (! (= (div0 x 0) 0) :pattern ((div0 x 0)))))
(assert (forall ((x Int) (y Int))
  (! (= (mod (mod x y) y) (mod x y)) :pattern ((mod (mod x y) y)))))
(assert (not (= (mod (+ a b) b) (mod a b))))
(check-sat)
```

## Changes

- **`src/ast/rewriter/arith_rewriter.cpp` — symbolic summand
elimination**: In `mk_mod_core`, when the modulus is a non-numeral
integer and the dividend is an `add`, strip any summand equal to the
modulus or an integer multiple of it. Soundness: `k*0 = 0` for all `k`,
so the rule holds even at `y = 0`. This immediately collapses the
reported formula to `false`.

- **`src/ast/rewriter/arith_rewriter.cpp` — symbolic idempotency via
ite**: Extend the existing `mod(mod(x,y), y) → mod(x,y)` rule
(previously numeral-only) to symbolic `y` by rewriting to `ite(y=0,
mod(mod(x,0),0), mod(x,y))`. The `y=0` branch uses a numeral divisor,
which is excluded by the `!v2.is_zero()` guard, halting the E-matching
chain.

- **`src/test/arith_rewriter.cpp`**: Regression tests for `mod(a+y, y) =
mod(a,y)`, `mod(a+2y, y) = mod(a,y)`, and `mod(mod(a,3),3) = mod(a,3)`.

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
This commit is contained in:
Copilot 2026-07-13 09:20:03 -07:00 • committed by GitHub
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commit 038b367d68
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2 changed files with 72 additions and 3 deletions
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19
src/test/arith_rewriter.cpp
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@ -83,4 +83,23 @@ void tst_arith_rewriter() {
rw(fml);
std::cout << "consecutive product (minus) >= 0: " << mk_pp(fml, m) << "\n";
ENSURE(m.is_true(fml));
// Issue #7403: mod (a + y) y should simplify to mod a y for symbolic y
// i.e. (= (mod (+ I S) S) (mod I S)) should reduce to true
fml = parse_int_fml(m, "(= (mod (+ I S) S) (mod I S))");
rw(fml);
std::cout << "mod (a+y) y = mod a y: " << mk_pp(fml, m) << "\n";
ENSURE(m.is_true(fml));
// mod (a + 2*y) y should simplify to mod a y (multiple of modulus dropped)
fml = parse_int_fml(m, "(= (mod (+ I (* 2 S)) S) (mod I S))");
rw(fml);
std::cout << "mod (a+2y) y = mod a y: " << mk_pp(fml, m) << "\n";
ENSURE(m.is_true(fml));
// mod (mod a b) b should simplify for non-zero numeral b
fml = parse_int_fml(m, "(= (mod (mod I 3) 3) (mod I 3))");
rw(fml);
std::cout << "mod (mod a 3) 3 = mod a 3: " << mk_pp(fml, m) << "\n";
ENSURE(m.is_true(fml));
}