SMTCoq

SMTCoq is a Coq tool that checks proof witnesses coming from external SAT and SMT solvers.

It relies on a certified checker for such witnesses. On top of it, vernacular commands and tactics to interface with the SAT solver zChaff and the SMT solvers veriT and CVC4 are provided. It is designed in a modular way allowing to extend it easily to other solvers.

SMTCoq also provides an extracted version of the checker, that can be run outside Coq.

The current stable version is version 1.3.

Installation

See the INSTALL.md file for instructions.

License

SMTCoq is released under the CeCILL-C license; see LICENSE for more details.

Use

Examples are given in the file examples/Example.v. They are meant to be easily re-usable for your own usage.

Overview

The SMTCoq module can be used in Coq files via the Require Import SMTCoq. command. For each supported solver, it provides:

• a vernacular command to check answers: XXX_Checker "problem_file" "witness_file" returns true only if witness_file contains a zChaff proof of the unsatisfiability of the problem stated in problem_file;

• a vernacular command to safely import theorems: XXX_Theorem theo "problem_file" "witness_file" produces a Coq term teo whose type is the theorem stated in problem_file if witness_file is a proof of the unsatisfiability of it, and fails otherwise.

• safe tactics to try to solve a Coq goal using the chosen solver.

The SMTCoq checker can also be extracted to OCaml and then used independently from Coq.

We now give more details for each solver, and explanations on extraction.

zChaff

Compile and install zChaff as explained in the installation instructions. In the following, we consider that the command zchaff is in your PATH variable environment.

Checking zChaff answers of unsatisfiability and importing theorems

To check the result given by zChaff on an unsatisfiable dimacs file file.cnf:

• Produce a zChaff proof witness: zchaff file.cnf. This command produces a proof witness file named resolve_trace.

• In a Coq file file.v, put:

Require Import SMTCoq.
Zchaff_Checker "file.cnf" "resolve_trace".

• Compile file.v: coqc file.v. If it returns true then zChaff indeed proved that the problem was unsatisfiable.

• You can also produce Coq theorems from zChaff proof witnesses: the commands

Require Import SMTCoq.
Zchaff_Theorem theo "file.cnf" "resolve_trace".

will produce a Coq term theo whose type is the theorem stated in file.cnf.

zChaff as a Coq decision procedure

The zchaff tactic can be used to solve any goal of the form:

forall l, b1 = b2

where l is a list of Booleans (that can be concrete terms).

veriT

Compile and install veriT as explained in the installation instructions. In the following, we consider that the command veriT is in your PATH variable environment.

Checking veriT answers of unsatisfiability and importing theorems

To check the result given by veriT on an unsatisfiable SMT-LIB2 file file.smt2:

• Produce a veriT proof witness:

veriT --proof-prune --proof-merge --proof-with-sharing --cnf-definitional --disable-e --disable-ackermann --input=smtlib2 --proof=file.log file.smt2

This command produces a proof witness file named file.log.

• In a Coq file file.v, put:

Require Import SMTCoq.
Section File.
  Verit_Checker "file.smt2" "file.log".
End File.

• Compile file.v: coqc file.v. If it returns true then veriT indeed proved that the problem was unsatisfiable.

• You can also produce Coq theorems from veriT proof witnesses: the commands

Require Import SMTCoq.
Section File.
  Verit_Theorem theo "file.smt2" "file.log".
End File.

will produce a Coq term theo whose type is the theorem stated in file.smt2.

The theories that are currently supported are QF_UF, QF_LIA, QF_IDL and their combinations.

veriT as a Coq decision procedure

The verit_bool tactic can be used to solve any goal of the form:

forall l, b1 = b2

where l is a list of Booleans. Those Booleans can be any concrete terms. The theories that are currently supported are QF_UF, QF_LIA, QF_IDL and their combinations.

Tactic for goals live in Prop

The verit tactic applies to Coq goals of type Prop: it first converts the goal to a term of type bool (thanks to the reflect predicate of SSReflect), it then calls the reification tactic verit_bool (which applies only to Boolean goals), and it finally converts the goals back to Prop, if not solved.

CVC4

Compile and install CVC4 as explained in the installation instructions.

Checking CVC4 answers of unsatisfiability and importing theorems

To check the result given by CVC4 on an unsatisfiable SMT-LIB2 file name.smt2 (in ..smtcoq/src/lfsc/tests directory):

• Produce a CVC4 proof witness; run:

cvc4 --dump-proof --no-simplification --fewer-preprocessing-holes --no-bv-eq --no-bv-ineq --no-bv-algebraic name.smt2 > name.lfsc

This set of commands produces a proof witness file named name.lfsc.

• In a Coq file name.v, put:

Require Import SMTCoq Bool List.
Import ListNotations BVList.BITVECTOR_LIST FArray.
Local Open Scope list_scope.
Local Open Scope farray_scope.
Local Open Scope bv_scope.

Section File.
  Lfsc_Checker "name.smt2" "name.lfsc".
End File.

• Compile name.v: coqc -R ../../ SMTCoq name.v. If it returns true then CVC4 indeed proved that the problem was unsatisfiable.

NB: Use cvc4tocoq script in src/lfsc/tests to automatize above steps.

• Ex: ./cvc4tocoq name.smt2, similary returned true amounts to correct unsatisfiability proof of the problem by CVC4.

CVC4 as a Coq decision procedure

The cvc4_bool tactic can be used to solve any goal of the form:

forall l, b1 = b2

where l is a list of Booleans. Those Booleans can be any concrete terms. The theories that are currently supported are QF_UF, QF_LIA, QF_IDL, QF_BV, QF_A and their combinations.

Tactic for goals live in Prop

The cvc4 tactic applies to Coq goals of type Prop: it first converts the goal to a term of type bool (thanks to the reflect predicate of SSReflect), then calls the reification tactic cvc4_bool (which applies only to Boolean goals), and it finally converts any unsolved subgoals returned by CVC4 back to Prop.

The smt tactic

The final notable outcome is the smt tactic that has the combined effect of the cvc4 and verit tactics: it first converts the goal to a term of type bool (thanks to the reflect predicate of SSReflect), it then calls either of the cvc4_bool and verit_bool tactics, and it finally converts any unsolved subgoals back to Prop.