73151f1960
Adds a new lemma pattern to nla_grobner::propagate_quotients that
derives a modular-residue constraint from polynomial divisibility,
filling a gap between quotient1-5 (model-value-driven case splits) and
the polynomials Grobner actually produces on Skolem-encoded mod
arithmetic.
Pattern
-------
For a polynomial p with all-integer free variables and a linear monomial
c_v * v (single integer var), the pattern computes M = gcd(|c_i/c_v|)
over the other monomials and K = c0/c_v for the constant term. When both
are integers, dividing p by c_v gives
v + M*Q + K = 0 with Q an integer
so v ≡ -K (mod M). The pattern emits the sound disjunctive lemma
(v < 0) ∨ (v ≥ M) ∨ (v = target)
where target = (-K) mod M ∈ [0, M-1]. This encodes "v ∈ target + M·Z" in
a form the LP / SAT layer can refute against current bounds.
Motivation
----------
QF_UFNIA verification benchmarks over fixed-prime modular arithmetic
(e.g. zk applications using the BabyBear prime 2013265921) regularly
produce basis polynomials of the form
-p*v_div + p*(v_a * v_b) - v_mod = 0
where v_mod is the result of (mod (* v_a v_b) p). The polynomial sits in
the Grobner basis but none of quotient1-5 fires: they all require
specific model-value alignments (r_value == 0, |v_value| > |r_value|,
etc.) that don't hold when all variables in scope are similarly sized
integers in [0, p). The proof spins on interval-tightening lemmas
without ever extracting the modular conclusion.
The author of propagate_quotients flagged this gap with the comment
\"other division lemmas are possible\" preceding the fall-through \"no
lemmas found\" CTRACE. This patch supplies one.
Soundness
---------
The lemma is sound regardless of v's LP bounds — the bound-negation
disjuncts (v < 0) and (v ≥ M) make the disjunction unconditionally true
under the polynomial identity, with v = target as the canonical residue
in [0, M-1]. M is derived from the polynomial's coefficient gcd, not
from any LP-side bound.
Validated under smt.arith.validate=true on the mod-factor-propagation
reproducers (PR #9235 follow-up), zk verifier benchmarks, and a broader
QF_UFNIA sample — 50+ files total, zero validate_conflict() assertion
violations.
Performance
-----------
A model-value gate (skip emission when v's current value already
satisfies one of the disjuncts) prevents the pattern from
short-circuiting the propagate_quotients || propagate_gcd_test ||
propagate_eqs || propagate_factorization || propagate_linear_equations
chain with redundant emissions. Without the gate, a single (v, M,
target) triple can re-emit each Grobner round and starve the downstream
propagators — observed in regression testing as thousands of identical
emissions on a small benchmark, turning a sub-second closure into a
timeout.
On six small mod-factor-propagation reproducers, the patch closes four
cases that previously timed out at 30 s (~1 s typical under the
Grobner-ramped config: smt.arith.nl.gr_q=50,
smt.arith.nl.grobner_eqs_growth=50,
smt.arith.nl.grobner_exp_delay=false, smt.arith.nl.grobner_frequency=1).
The two remaining timeouts in that set are attributable to different
gaps (Boolean-disjunction propagation, and the multi-bounded-mod-result
polynomial shape that needs Grobner over Z/pZ), not to mod_residue
itself.
Diagnostics
-----------
TRACE under the existing 'grobner' tag emits one line per lemma
emission, recording v, M, c_v, c0, and target.
---------
Co-authored-by: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Co-authored-by: Nikolaj Bjorner <nbjorner@microsoft.com>
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| a3 | ||
| agentics | ||
| cmake | ||
| codeql/custom_queries | ||
| contrib | ||
| doc | ||
| docker | ||
| examples | ||
| noarch | ||
| resources | ||
| scripts | ||
| src | ||
| .bazelrc | ||
| .clang-format | ||
| .dockerignore | ||
| .gitattributes | ||
| .gitignore | ||
| BUILD.bazel | ||
| build_z3.bat | ||
| CMakeLists.txt | ||
| configure | ||
| LICENSE.txt | ||
| MODULE.bazel | ||
| README-CMake.md | ||
| README.md | ||
| RELEASE_NOTES.md | ||
| Z3-AGENT.md | ||
| z3.pc.cmake.in | ||
| z3guide.jpeg | ||
Z3
Z3 is a theorem prover from Microsoft Research. It is licensed under the MIT license. Windows binary distributions include C++ runtime redistributables
If you are not familiar with Z3, you can start here.
Pre-built binaries for stable and nightly releases are available here.
Z3 can be built using Visual Studio, a Makefile, using CMake, using vcpkg, or using Bazel. It provides bindings for several programming languages.
See the release notes for notes on various stable releases of Z3.
Build status
Pull Request & Push Workflows
| WASM Build | Windows Build | CI | OCaml Binding |
|---|---|---|---|
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Scheduled Workflows
| Open Bugs | Android Build | Pyodide Build | Nightly Build | Cross Build |
|---|---|---|---|---|
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| MSVC Static | MSVC Clang-CL | Build Z3 Cache | Code Coverage | Memory Safety | Mark PRs Ready |
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Manual & Release Workflows
| Documentation | Release Build | WASM Release | NuGet Build |
|---|---|---|---|
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Specialized Workflows
| Nightly Validation | Copilot Setup | Agentics Maintenance |
|---|---|---|
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Agentic Workflows
| A3 Python | API Coherence | Code Simplifier | Release Notes | Workflow Suggestion |
|---|---|---|---|---|
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| Academic Citation | Build Warning Fixer | Code Conventions | CSA Report | Issue Backlog |
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| Memory Safety Report | Ostrich Benchmark | QF-S Benchmark | Tactic-to-Simplifier | ZIPT Code Reviewer |
|---|---|---|---|---|
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Building Z3 on Windows using Visual Studio Command Prompt
For 32-bit builds, start with:
python scripts/mk_make.py
or instead, for a 64-bit build:
python scripts/mk_make.py -x
then run:
cd build
nmake
Z3 uses C++20. The recommended version of Visual Studio is therefore VS2019 or later.
Security Features (MSVC): When building with Visual Studio/MSVC, a couple of security features are enabled by default for Z3:
- Control Flow Guard (
/guard:cf) - enabled by default to detect attempts to compromise your code by preventing calls to locations other than function entry points, making it more difficult for attackers to execute arbitrary code through control flow redirection - Address Space Layout Randomization (
/DYNAMICBASE) - enabled by default for memory layout randomization, required by the/GUARD:CFlinker option - These can be disabled using
python scripts/mk_make.py --no-guardcf(Python build) orcmake -DZ3_ENABLE_CFG=OFF(CMake build) if needed
Building Z3 using make and GCC/Clang
Execute:
python scripts/mk_make.py
cd build
make
sudo make install
Note by default g++ is used as C++ compiler if it is available. If you
prefer to use Clang, change the mk_make.py invocation to:
CXX=clang++ CC=clang python scripts/mk_make.py
Note that Clang < 3.7 does not support OpenMP.
You can also build Z3 for Windows using Cygwin and the Mingw-w64 cross-compiler. In that case, make sure to use Cygwin's own Python and not some Windows installation of Python.
For a 64-bit build (from Cygwin64), configure Z3's sources with
CXX=x86_64-w64-mingw32-g++ CC=x86_64-w64-mingw32-gcc AR=x86_64-w64-mingw32-ar python scripts/mk_make.py
A 32-bit build should work similarly (but is untested); the same is true for 32/64 bit builds from within Cygwin32.
By default, it will install z3 executables at PREFIX/bin, libraries at
PREFIX/lib, and include files at PREFIX/include, where the PREFIX
installation prefix is inferred by the mk_make.py script. It is usually
/usr for most Linux distros, and /usr/local for FreeBSD and macOS. Use
the --prefix= command-line option to change the install prefix. For example:
python scripts/mk_make.py --prefix=/home/leo
cd build
make
make install
To uninstall Z3, use
sudo make uninstall
To clean Z3, you can delete the build directory and run the mk_make.py script again.
Building Z3 using CMake
Z3 has a build system using CMake. Read the README-CMake.md file for details. It is recommended for most build tasks, except for building OCaml bindings.
Building Z3 using vcpkg
vcpkg is a full platform package manager. To install Z3 with vcpkg, execute:
git clone https://github.com/microsoft/vcpkg.git
./bootstrap-vcpkg.bat # For powershell
./bootstrap-vcpkg.sh # For bash
./vcpkg install z3
Building Z3 using Bazel
Z3 can be built using Bazel. This is known to work on Ubuntu with Clang (but may work elsewhere with other compilers):
bazel build //...
Dependencies
Z3 itself has only few dependencies. It uses C++ runtime libraries, including pthreads for multi-threading. It is optionally possible to use GMP for multi-precision integers, but Z3 contains its own self-contained multi-precision functionality. Python is required to build Z3. Building Java, .NET, OCaml and Julia APIs requires installing relevant toolchains.
Z3 bindings
Z3 has bindings for various programming languages.
.NET
You can install a NuGet package for the latest release Z3 from nuget.org.
Use the --dotnet command line flag with mk_make.py to enable building these.
See examples/dotnet for examples.
C
These are always enabled.
See examples/c for examples.
C++
These are always enabled.
See examples/c++ for examples.
Java
Use the --java command line flag with mk_make.py to enable building these.
For IDE setup instructions (Eclipse, IntelliJ IDEA, Visual Studio Code) and troubleshooting, see the Java IDE Setup Guide.
See examples/java for examples.
Go
Use the --go command line flag with mk_make.py to enable building these. Note that Go bindings use CGO and require a Go toolchain (Go 1.20 or later) to build.
With CMake, use the -DZ3_BUILD_GO_BINDINGS=ON option.
See examples/go for examples and src/api/go/README.md for complete API documentation.
OCaml
Use the --ml command line flag with mk_make.py to enable building these.
See examples/ml for examples.
Python
You can install the Python wrapper for Z3 for the latest release from pypi using the command:
pip install z3-solver
Use the --python command line flag with mk_make.py to enable building these.
Note that it is required on certain platforms that the Python package directory
(site-packages on most distributions and dist-packages on Debian-based
distributions) live under the install prefix. If you use a non-standard prefix
you can use the --pypkgdir option to change the Python package directory
used for installation. For example:
python scripts/mk_make.py --prefix=/home/leo --python --pypkgdir=/home/leo/lib/python-2.7/site-packages
If you do need to install to a non-standard prefix, a better approach is to use
a Python virtual environment
and install Z3 there. Python packages also work for Python3.
Under Windows, recall to build inside the Visual C++ native command build environment.
Note that the build/python/z3 directory should be accessible from where Python is used with Z3
and it requires libz3.dll to be in the path.
virtualenv venv
source venv/bin/activate
python scripts/mk_make.py --python
cd build
make
make install
# You will find Z3 and the Python bindings installed in the virtual environment
venv/bin/z3 -h
...
python -c 'import z3; print(z3.get_version_string())'
...
See examples/python for examples.
Julia
The Julia package Z3.jl wraps the C API of Z3. A previous version of it wrapped the C++ API: Information about updating and building the Julia bindings can be found in src/api/julia.
WebAssembly / TypeScript / JavaScript
A WebAssembly build with associated TypeScript typings is published on npm as z3-solver. Information about building these bindings can be found in src/api/js.
Smalltalk (Pharo / Smalltalk/X)
Project MachineArithmetic provides a Smalltalk interface to Z3's C API. For more information, see MachineArithmetic/README.md.
AIX
Build settings for AIX are described here.
System Overview
Interfaces
-
Default input format is SMTLIB2
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Other native foreign function interfaces:
-
Python API (also available in pydoc format)
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C
-
OCaml
-
Smalltalk (supports Pharo and Smalltalk/X)
Power Tools
- The Axiom Profiler currently developed by ETH Zurich