## Summary
Fixes the divergence in issue #7464: formulas involving `mod`/`div` by a
**variable** divisor could send `smt.arith.solver=6` into a
non-terminating nonlinear search.
Minimal reproducer (UNSAT, previously timed out; now solved in <0.5s):
```smt2
(declare-fun V () Int)
(declare-fun n () Int)
(declare-fun l () Int)
(assert (and (> V 0) (= 0 (mod n 2)) (= (div n 2) (div n l)) (= 0 (mod (div n l) V))))
(assert (distinct 0 (mod n V)))
(check-sat)
```
## Root cause
A variable-divisor `mod n V` is axiomatized by the Euclidean identity
`n = V*(n div V) + (n mod V)`. The `V*(n div V)` term is nonlinear, so
arith.solver=6
hands the problem to the nlsat/Gröbner branch, which branches on values
of `V` with no
termination bound and diverges.
## Fix
Add a **linear divisibility closure** lemma in `nla_divisions`:
> `mod(a, y) = 0 & x = c*a` (c an integer constant) ⟹ `mod(x, y) = 0`.
The emitted clause
```
(x - c*a != 0) \/ (mod(a, y) != 0) \/ (mod(x, y) = 0)
```
is a **tautology for every integer `c`**, so mining a candidate `c =
val(x)/val(a)` from
the current model can never be unsound. It is only emitted when all
three literals are
false in the current model, so the clause is a genuine
conflict/propagation and always
makes progress. This lets the theory refute the instance directly
instead of entering the
divergent nonlinear branch.
Variable-divisor `mod` terms were previously **not registered** in nla
at all; they are now
registered into a new `m_divisibility` list in `theory_lra`, so the
reasoner can pair a
violated `mod(x, y)` with a satisfied `mod(a, y)` of the same divisor.
## Changes
- `src/math/lp/nla_divisions.{h,cpp}` — new `m_divisibility` list
`{r=mod, x=dividend, y=divisor}`, `add_divisibility(...)`, and
`check_linear_divisibility()`; invoked from `divisions::check()`.
- `src/math/lp/nla_core.h`, `src/math/lp/nla_solver.{h,cpp}` —
forwarding of `add_divisibility`.
- `src/smt/theory_lra.cpp` — register variable-divisor `mod` into the
divisibility list.
## Validation
- `min.smt2` → `unsat` in 0.46s, minimized core → 0.15s (were timeouts).
- Soundness: 350 differential fuzz formulas (arith.solver=6 vs
arith.solver=2), **0 mismatches**.
- Spot checks correct (divisor-3 variant → unsat; non-divisible variants
→ sat).
Co-authored-by: Copilot <223556219+Copilot@users.noreply.github.com>
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>
* rename ul_pair to column
Signed-off-by: Lev Nachmanson <levnach@hotmail.com>
* t
Signed-off-by: Lev Nachmanson <levnach@hotmail.com>
* simple test passed
* remove an assert
* relax an assertion
* remove an obsolete function
Signed-off-by: Lev Nachmanson <levnach@hotmail.com>
* access a term by the term column
* remove the column index from colunm.h
* remove an unused method
* remove debug code
* fix the build of lp_tst
Signed-off-by: Lev Nachmanson <levnach@hotmail.com>
---------
Signed-off-by: Lev Nachmanson <levnach@hotmail.com>
Co-authored-by: Lev Nachmanson <levnach@hotmail.com>
this update enables new incremental linear axioms based on division terms.
It also consolidates some of the backtracking state in nla_core / emons to use stack traces instead of custom backtracking state.
- convert reduce-args to a simplifier. Currently exposed as reduce-args2 tactic until the old tactic code gets removed.
- bug fixes in model_reconstruction trail
- allow multiple defs to be added with same pool of removed formulas
- fix tracking of function symbols instead of expressions to filter replay
- add nla_divisions to track (cheap) divisibility lemmas.
-
