mirror of
https://github.com/Z3Prover/z3
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2224 lines
76 KiB
Java
2224 lines
76 KiB
Java
/*++
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Copyright (c) 2012 Microsoft Corporation
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Module Name:
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Program.java
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Abstract:
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Z3 Java API: Example program
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Author:
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Christoph Wintersteiger (cwinter) 2012-11-27
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Notes:
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--*/
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import java.util.*;
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import com.microsoft.z3.*;
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class JavaExample
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{
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@SuppressWarnings("serial")
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class TestFailedException extends Exception
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{
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public TestFailedException()
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{
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super("Check FAILED");
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}
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};
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// / Create axiom: function f is injective in the i-th argument.
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// / <remarks>
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// / The following axiom is produced:
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// / <code>
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// / forall (x_0, ..., x_n) finv(f(x_0, ..., x_i, ..., x_{n-1})) = x_i
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// / </code>
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// / Where, <code>finv</code>is a fresh function declaration.
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public BoolExpr InjAxiom(Context ctx, FuncDecl f, int i) throws Z3Exception
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{
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Sort[] domain = f.Domain();
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int sz = f.DomainSize();
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if (i >= sz)
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{
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System.out.println("failed to create inj axiom");
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return null;
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}
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/* declare the i-th inverse of f: finv */
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Sort finv_domain = f.Range();
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Sort finv_range = domain[i];
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FuncDecl finv = ctx.MkFuncDecl("f_fresh", finv_domain, finv_range);
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/* allocate temporary arrays */
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Expr[] xs = new Expr[sz];
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Symbol[] names = new Symbol[sz];
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Sort[] types = new Sort[sz];
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/* fill types, names and xs */
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for (int j = 0; j < sz; j++)
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{
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types[j] = domain[j];
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names[j] = ctx.MkSymbol("x_" + Integer.toString(j));
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xs[j] = ctx.MkBound(j, types[j]);
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}
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Expr x_i = xs[i];
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/* create f(x_0, ..., x_i, ..., x_{n-1}) */
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Expr fxs = f.Apply(xs);
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/* create f_inv(f(x_0, ..., x_i, ..., x_{n-1})) */
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Expr finv_fxs = finv.Apply(fxs);
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/* create finv(f(x_0, ..., x_i, ..., x_{n-1})) = x_i */
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Expr eq = ctx.MkEq(finv_fxs, x_i);
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/* use f(x_0, ..., x_i, ..., x_{n-1}) as the pattern for the quantifier */
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Pattern p = ctx.MkPattern(new Expr[] { fxs });
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/* create & assert quantifier */
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BoolExpr q = ctx.MkForall(types, /* types of quantified variables */
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names, /* names of quantified variables */
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eq, 1, new Pattern[] { p } /* patterns */, null, null, null);
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return q;
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}
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// / Create axiom: function f is injective in the i-th argument.
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// / <remarks>
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// / The following axiom is produced:
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// / <code>
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// / forall (x_0, ..., x_n) finv(f(x_0, ..., x_i, ..., x_{n-1})) = x_i
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// / </code>
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// / Where, <code>finv</code>is a fresh function declaration.
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public BoolExpr InjAxiomAbs(Context ctx, FuncDecl f, int i)
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throws Z3Exception
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{
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Sort[] domain = f.Domain();
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int sz = f.DomainSize();
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if (i >= sz)
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{
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System.out.println("failed to create inj axiom");
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return null;
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}
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/* declare the i-th inverse of f: finv */
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Sort finv_domain = f.Range();
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Sort finv_range = domain[i];
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FuncDecl finv = ctx.MkFuncDecl("f_fresh", finv_domain, finv_range);
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/* allocate temporary arrays */
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Expr[] xs = new Expr[sz];
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/* fill types, names and xs */
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for (int j = 0; j < sz; j++)
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{
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xs[j] = ctx.MkConst("x_" + Integer.toString(j), domain[j]);
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}
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Expr x_i = xs[i];
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/* create f(x_0, ..., x_i, ..., x_{n-1}) */
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Expr fxs = f.Apply(xs);
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/* create f_inv(f(x_0, ..., x_i, ..., x_{n-1})) */
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Expr finv_fxs = finv.Apply(fxs);
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/* create finv(f(x_0, ..., x_i, ..., x_{n-1})) = x_i */
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Expr eq = ctx.MkEq(finv_fxs, x_i);
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/* use f(x_0, ..., x_i, ..., x_{n-1}) as the pattern for the quantifier */
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Pattern p = ctx.MkPattern(new Expr[] { fxs });
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/* create & assert quantifier */
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BoolExpr q = ctx.MkForall(xs, /* types of quantified variables */
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eq, /* names of quantified variables */
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1, new Pattern[] { p } /* patterns */, null, null, null);
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return q;
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}
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// / Assert the axiom: function f is commutative.
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// / <remarks>
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// / This example uses the SMT-LIB parser to simplify the axiom
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// construction.
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// / </remarks>
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private BoolExpr CommAxiom(Context ctx, FuncDecl f) throws Exception
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{
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Sort t = f.Range();
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Sort[] dom = f.Domain();
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if (dom.length != 2 || !t.Equals(dom[0]) || !t.Equals(dom[1]))
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{
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System.out.println(Integer.toString(dom.length) + " "
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+ dom[0].toString() + " " + dom[1].toString() + " "
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+ t.toString());
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throw new Exception(
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"function must be binary, and argument types must be equal to return type");
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}
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String bench = "(benchmark comm :formula (forall (x " + t.Name()
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+ ") (y " + t.Name() + ") (= (" + f.Name() + " x y) ("
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+ f.Name() + " y x))))";
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ctx.ParseSMTLIBString(bench, new Symbol[] { t.Name() },
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new Sort[] { t }, new Symbol[] { f.Name() },
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new FuncDecl[] { f });
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return ctx.SMTLIBFormulas()[0];
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}
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// / "Hello world" example: create a Z3 logical context, and delete it.
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public void SimpleExample() throws Z3Exception
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{
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System.out.println("SimpleExample");
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{
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Context ctx = new Context();
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/* do something with the context */
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/* be kind to dispose manually and not wait for the GC. */
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ctx.Dispose();
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}
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}
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Model Check(Context ctx, BoolExpr f, Status sat) throws Z3Exception,
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TestFailedException
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{
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Solver s = ctx.MkSolver();
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s.Assert(f);
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if (s.Check() != sat)
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throw new TestFailedException();
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if (sat == Status.SATISFIABLE)
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return s.Model();
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else
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return null;
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}
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void SolveTactical(Context ctx, Tactic t, Goal g, Status sat)
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throws Z3Exception, TestFailedException
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{
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Solver s = ctx.MkSolver(t);
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System.out.println("\nTactical solver: " + s);
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for (BoolExpr a : g.Formulas())
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s.Assert(a);
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System.out.println("Solver: " + s);
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if (s.Check() != sat)
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throw new TestFailedException();
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}
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ApplyResult ApplyTactic(Context ctx, Tactic t, Goal g) throws Z3Exception
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{
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System.out.println("\nGoal: " + g);
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ApplyResult res = t.Apply(g);
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System.out.println("Application result: " + res);
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Status q = Status.UNKNOWN;
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for (Goal sg : res.Subgoals())
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if (sg.IsDecidedSat())
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q = Status.SATISFIABLE;
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else if (sg.IsDecidedUnsat())
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q = Status.UNSATISFIABLE;
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switch (q)
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{
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case UNKNOWN:
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System.out.println("Tactic result: Undecided");
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break;
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case SATISFIABLE:
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System.out.println("Tactic result: SAT");
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break;
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case UNSATISFIABLE:
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System.out.println("Tactic result: UNSAT");
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break;
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}
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return res;
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}
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void Prove(Context ctx, BoolExpr f) throws Z3Exception, TestFailedException
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{
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BoolExpr[] assumptions = new BoolExpr[0];
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Prove(ctx, f, assumptions);
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}
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void Prove(Context ctx, BoolExpr f, BoolExpr assumption)
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throws Z3Exception, TestFailedException
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{
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BoolExpr[] assumptions = { assumption };
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Prove(ctx, f, assumptions);
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}
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void Prove(Context ctx, BoolExpr f, BoolExpr[] assumptions)
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throws Z3Exception, TestFailedException
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{
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System.out.println("Proving: " + f);
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Solver s = ctx.MkSolver();
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for (BoolExpr a : assumptions)
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s.Assert(a);
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s.Assert(ctx.MkNot(f));
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Status q = s.Check();
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switch (q)
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{
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case UNKNOWN:
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System.out.println("Unknown because: " + s.ReasonUnknown());
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break;
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case SATISFIABLE:
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throw new TestFailedException();
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case UNSATISFIABLE:
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System.out.println("OK, proof: " + s.Proof());
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break;
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}
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}
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void Disprove(Context ctx, BoolExpr f) throws Z3Exception,
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TestFailedException
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{
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BoolExpr[] a = {};
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Disprove(ctx, f, a);
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}
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void Disprove(Context ctx, BoolExpr f, BoolExpr assumption)
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throws Z3Exception, TestFailedException
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{
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BoolExpr[] a = { assumption };
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Disprove(ctx, f, a);
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}
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void Disprove(Context ctx, BoolExpr f, BoolExpr[] assumptions)
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throws Z3Exception, TestFailedException
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{
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System.out.println("Disproving: " + f);
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Solver s = ctx.MkSolver();
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for (BoolExpr a : assumptions)
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s.Assert(a);
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s.Assert(ctx.MkNot(f));
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Status q = s.Check();
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switch (q)
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{
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case UNKNOWN:
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System.out.println("Unknown because: " + s.ReasonUnknown());
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break;
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case SATISFIABLE:
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System.out.println("OK, model: " + s.Model());
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break;
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case UNSATISFIABLE:
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throw new TestFailedException();
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}
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}
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void ModelConverterTest(Context ctx) throws Z3Exception,
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TestFailedException
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{
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System.out.println("ModelConverterTest");
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ArithExpr xr = (ArithExpr) ctx.MkConst(ctx.MkSymbol("x"),
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ctx.MkRealSort());
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ArithExpr yr = (ArithExpr) ctx.MkConst(ctx.MkSymbol("y"),
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ctx.MkRealSort());
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Goal g4 = ctx.MkGoal(true, false, true);
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g4.Assert(ctx.MkGt(xr, ctx.MkReal(10, 1)));
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g4.Assert(ctx.MkEq(yr,
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ctx.MkAdd(new ArithExpr[] { xr, ctx.MkReal(1, 1) })));
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g4.Assert(ctx.MkGt(yr, ctx.MkReal(1, 1)));
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ApplyResult ar = ApplyTactic(ctx, ctx.MkTactic("simplify"), g4);
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if (ar.NumSubgoals() == 1
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&& (ar.Subgoals()[0].IsDecidedSat() || ar.Subgoals()[0]
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.IsDecidedUnsat()))
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throw new TestFailedException();
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ar = ApplyTactic(ctx, ctx.AndThen(ctx.MkTactic("simplify"),
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ctx.MkTactic("solve-eqs"), null), g4);
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if (ar.NumSubgoals() == 1
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&& (ar.Subgoals()[0].IsDecidedSat() || ar.Subgoals()[0]
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.IsDecidedUnsat()))
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throw new TestFailedException();
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Solver s = ctx.MkSolver();
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for (BoolExpr e : ar.Subgoals()[0].Formulas())
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s.Assert(e);
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Status q = s.Check();
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System.out.println("Solver says: " + q);
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System.out.println("Model: \n" + s.Model());
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System.out.println("Converted Model: \n"
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+ ar.ConvertModel(0, s.Model()));
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if (q != Status.SATISFIABLE)
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throw new TestFailedException();
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}
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// / A simple array example.
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void ArrayExample1(Context ctx) throws Z3Exception, TestFailedException
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{
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System.out.println("ArrayExample1");
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Goal g = ctx.MkGoal(true, false, false);
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ArraySort asort = ctx.MkArraySort(ctx.IntSort(), ctx.MkBitVecSort(32));
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ArrayExpr aex = (ArrayExpr) ctx.MkConst(ctx.MkSymbol("MyArray"), asort);
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Expr sel = ctx.MkSelect(aex, ctx.MkInt(0));
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g.Assert(ctx.MkEq(sel, ctx.MkBV(42, 32)));
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Symbol xs = ctx.MkSymbol("x");
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IntExpr xc = (IntExpr) ctx.MkConst(xs, ctx.IntSort());
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Symbol fname = ctx.MkSymbol("f");
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Sort[] domain = { ctx.IntSort() };
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FuncDecl fd = ctx.MkFuncDecl(fname, domain, ctx.IntSort());
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Expr[] fargs = { ctx.MkConst(xs, ctx.IntSort()) };
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IntExpr fapp = (IntExpr) ctx.MkApp(fd, fargs);
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g.Assert(ctx.MkEq(ctx.MkAdd(new ArithExpr[] { xc, fapp }),
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ctx.MkInt(123)));
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Solver s = ctx.MkSolver();
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for (BoolExpr a : g.Formulas())
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s.Assert(a);
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System.out.println("Solver: " + s);
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Status q = s.Check();
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System.out.println("Status: " + q);
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if (q != Status.SATISFIABLE)
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throw new TestFailedException();
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System.out.println("Model = " + s.Model());
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System.out.println("Interpretation of MyArray:\n"
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+ s.Model().FuncInterp(aex.FuncDecl()));
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System.out
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.println("Interpretation of x:\n" + s.Model().ConstInterp(xc));
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System.out.println("Interpretation of f:\n" + s.Model().FuncInterp(fd));
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System.out.println("Interpretation of MyArray as Term:\n"
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+ s.Model().FuncInterp(aex.FuncDecl()));
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}
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// / Prove <tt>store(a1, i1, v1) = store(a2, i2, v2) implies (i1 = i3 or i2
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// = i3 or select(a1, i3) = select(a2, i3))</tt>.
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// / <remarks>This example demonstrates how to use the array
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// theory.</remarks>
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public void ArrayExample2(Context ctx) throws Z3Exception,
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TestFailedException
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{
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System.out.println("ArrayExample2");
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Sort int_type = ctx.IntSort();
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Sort array_type = ctx.MkArraySort(int_type, int_type);
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ArrayExpr a1 = (ArrayExpr) ctx.MkConst("a1", array_type);
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ArrayExpr a2 = ctx.MkArrayConst("a2", int_type, int_type);
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Expr i1 = ctx.MkConst("i1", int_type);
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Expr i2 = ctx.MkConst("i2", int_type);
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Expr i3 = ctx.MkConst("i3", int_type);
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Expr v1 = ctx.MkConst("v1", int_type);
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Expr v2 = ctx.MkConst("v2", int_type);
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Expr st1 = ctx.MkStore(a1, i1, v1);
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Expr st2 = ctx.MkStore(a2, i2, v2);
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Expr sel1 = ctx.MkSelect(a1, i3);
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Expr sel2 = ctx.MkSelect(a2, i3);
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/* create antecedent */
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BoolExpr antecedent = ctx.MkEq(st1, st2);
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/*
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* create consequent: i1 = i3 or i2 = i3 or select(a1, i3) = select(a2,
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* i3)
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*/
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BoolExpr consequent = ctx.MkOr(new BoolExpr[] { ctx.MkEq(i1, i3),
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ctx.MkEq(i2, i3), ctx.MkEq(sel1, sel2) });
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/*
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* prove store(a1, i1, v1) = store(a2, i2, v2) implies (i1 = i3 or i2 =
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* i3 or select(a1, i3) = select(a2, i3))
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*/
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BoolExpr thm = ctx.MkImplies(antecedent, consequent);
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System.out
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.println("prove: store(a1, i1, v1) = store(a2, i2, v2) implies (i1 = i3 or i2 = i3 or select(a1, i3) = select(a2, i3))");
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System.out.println(thm);
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Prove(ctx, thm);
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}
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// / Show that <code>distinct(a_0, ... , a_n)</code> is
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// / unsatisfiable when <code>a_i</code>'s are arrays from boolean to
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// / boolean and n > 4.
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// / <remarks>This example also shows how to use the <code>distinct</code>
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// construct.</remarks>
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public void ArrayExample3(Context ctx) throws Z3Exception,
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TestFailedException
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{
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System.out.println("ArrayExample3");
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for (int n = 2; n <= 5; n++)
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{
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System.out.println("n = " + Integer.toString(n));
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Sort bool_type = ctx.MkBoolSort();
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Sort array_type = ctx.MkArraySort(bool_type, bool_type);
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Expr[] a = new Expr[n];
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/* create arrays */
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for (int i = 0; i < n; i++)
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{
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a[i] = ctx.MkConst("array_" + Integer.toString(i), array_type);
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}
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/* assert distinct(a[0], ..., a[n]) */
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BoolExpr d = ctx.MkDistinct(a);
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System.out.println(d);
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/* context is satisfiable if n < 5 */
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Model model = Check(ctx, d, n < 5 ? Status.SATISFIABLE
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: Status.UNSATISFIABLE);
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if (n < 5)
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{
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for (int i = 0; i < n; i++)
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{
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System.out.println(a[i].toString() + " = "
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+ model.Evaluate(a[i], false));
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}
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}
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}
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}
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// / Sudoku solving example.
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void SudokuExample(Context ctx) throws Z3Exception, TestFailedException
|
|
{
|
|
System.out.println("SudokuExample");
|
|
|
|
// 9x9 matrix of integer variables
|
|
IntExpr[][] X = new IntExpr[9][];
|
|
for (int i = 0; i < 9; i++)
|
|
{
|
|
X[i] = new IntExpr[9];
|
|
for (int j = 0; j < 9; j++)
|
|
X[i][j] = (IntExpr) ctx.MkConst(
|
|
ctx.MkSymbol("x_" + (i + 1) + "_" + (j + 1)),
|
|
ctx.IntSort());
|
|
}
|
|
|
|
// each cell contains a value in {1, ..., 9}
|
|
BoolExpr[][] cells_c = new BoolExpr[9][];
|
|
for (int i = 0; i < 9; i++)
|
|
{
|
|
cells_c[i] = new BoolExpr[9];
|
|
for (int j = 0; j < 9; j++)
|
|
cells_c[i][j] = ctx.MkAnd(new BoolExpr[] {
|
|
ctx.MkLe(ctx.MkInt(1), X[i][j]),
|
|
ctx.MkLe(X[i][j], ctx.MkInt(9)) });
|
|
}
|
|
|
|
// each row contains a digit at most once
|
|
BoolExpr[] rows_c = new BoolExpr[9];
|
|
for (int i = 0; i < 9; i++)
|
|
rows_c[i] = ctx.MkDistinct(X[i]);
|
|
|
|
// each column contains a digit at most once
|
|
BoolExpr[] cols_c = new BoolExpr[9];
|
|
for (int j = 0; j < 9; j++)
|
|
cols_c[j] = ctx.MkDistinct(X[j]);
|
|
|
|
// each 3x3 square contains a digit at most once
|
|
BoolExpr[][] sq_c = new BoolExpr[3][];
|
|
for (int i0 = 0; i0 < 3; i0++)
|
|
{
|
|
sq_c[i0] = new BoolExpr[3];
|
|
for (int j0 = 0; j0 < 3; j0++)
|
|
{
|
|
IntExpr[] square = new IntExpr[9];
|
|
for (int i = 0; i < 3; i++)
|
|
for (int j = 0; j < 3; j++)
|
|
square[3 * i + j] = X[3 * i0 + i][3 * j0 + j];
|
|
sq_c[i0][j0] = ctx.MkDistinct(square);
|
|
}
|
|
}
|
|
|
|
BoolExpr sudoku_c = ctx.MkTrue();
|
|
for (BoolExpr[] t : cells_c)
|
|
sudoku_c = ctx.MkAnd(new BoolExpr[] { ctx.MkAnd(t), sudoku_c });
|
|
sudoku_c = ctx.MkAnd(new BoolExpr[] { ctx.MkAnd(rows_c), sudoku_c });
|
|
sudoku_c = ctx.MkAnd(new BoolExpr[] { ctx.MkAnd(cols_c), sudoku_c });
|
|
for (BoolExpr[] t : sq_c)
|
|
sudoku_c = ctx.MkAnd(new BoolExpr[] { ctx.MkAnd(t), sudoku_c });
|
|
|
|
// sudoku instance, we use '0' for empty cells
|
|
int[][] instance = { { 0, 0, 0, 0, 9, 4, 0, 3, 0 },
|
|
{ 0, 0, 0, 5, 1, 0, 0, 0, 7 }, { 0, 8, 9, 0, 0, 0, 0, 4, 0 },
|
|
{ 0, 0, 0, 0, 0, 0, 2, 0, 8 }, { 0, 6, 0, 2, 0, 1, 0, 5, 0 },
|
|
{ 1, 0, 2, 0, 0, 0, 0, 0, 0 }, { 0, 7, 0, 0, 0, 0, 5, 2, 0 },
|
|
{ 9, 0, 0, 0, 6, 5, 0, 0, 0 }, { 0, 4, 0, 9, 7, 0, 0, 0, 0 } };
|
|
|
|
BoolExpr instance_c = ctx.MkTrue();
|
|
for (int i = 0; i < 9; i++)
|
|
for (int j = 0; j < 9; j++)
|
|
instance_c = ctx
|
|
.MkAnd(new BoolExpr[] {
|
|
instance_c,
|
|
(BoolExpr) ctx.MkITE(
|
|
ctx.MkEq(ctx.MkInt(instance[i][j]),
|
|
ctx.MkInt(0)),
|
|
ctx.MkTrue(),
|
|
ctx.MkEq(X[i][j],
|
|
ctx.MkInt(instance[i][j]))) });
|
|
|
|
Solver s = ctx.MkSolver();
|
|
s.Assert(sudoku_c);
|
|
s.Assert(instance_c);
|
|
|
|
if (s.Check() == Status.SATISFIABLE)
|
|
{
|
|
Model m = s.Model();
|
|
Expr[][] R = new Expr[9][9];
|
|
for (int i = 0; i < 9; i++)
|
|
for (int j = 0; j < 9; j++)
|
|
R[i][j] = m.Evaluate(X[i][j], false);
|
|
System.out.println("Sudoku solution:");
|
|
for (int i = 0; i < 9; i++)
|
|
{
|
|
for (int j = 0; j < 9; j++)
|
|
System.out.print(" " + R[i][j]);
|
|
System.out.println();
|
|
}
|
|
} else
|
|
{
|
|
System.out.println("Failed to solve sudoku");
|
|
throw new TestFailedException();
|
|
}
|
|
}
|
|
|
|
// / A basic example of how to use quantifiers.
|
|
|
|
void QuantifierExample1(Context ctx) throws Z3Exception
|
|
{
|
|
System.out.println("QuantifierExample");
|
|
|
|
Sort[] types = new Sort[3];
|
|
IntExpr[] xs = new IntExpr[3];
|
|
Symbol[] names = new Symbol[3];
|
|
IntExpr[] vars = new IntExpr[3];
|
|
|
|
for (int j = 0; j < 3; j++)
|
|
{
|
|
types[j] = ctx.IntSort();
|
|
names[j] = ctx.MkSymbol("x_" + Integer.toString(j));
|
|
xs[j] = (IntExpr) ctx.MkConst(names[j], types[j]);
|
|
vars[j] = (IntExpr) ctx.MkBound(2 - j, types[j]); // <-- vars
|
|
// reversed!
|
|
}
|
|
|
|
Expr body_vars = ctx
|
|
.MkAnd(new BoolExpr[] {
|
|
ctx.MkEq(
|
|
ctx.MkAdd(new ArithExpr[] { vars[0],
|
|
ctx.MkInt(1) }), ctx.MkInt(2)),
|
|
ctx.MkEq(
|
|
ctx.MkAdd(new ArithExpr[] { vars[1],
|
|
ctx.MkInt(2) }),
|
|
ctx.MkAdd(new ArithExpr[] { vars[2],
|
|
ctx.MkInt(3) })) });
|
|
|
|
Expr body_const = ctx.MkAnd(new BoolExpr[] {
|
|
ctx.MkEq(ctx.MkAdd(new ArithExpr[] { xs[0], ctx.MkInt(1) }),
|
|
ctx.MkInt(2)),
|
|
ctx.MkEq(ctx.MkAdd(new ArithExpr[] { xs[1], ctx.MkInt(2) }),
|
|
ctx.MkAdd(new ArithExpr[] { xs[2], ctx.MkInt(3) })) });
|
|
|
|
Expr x = ctx.MkForall(types, names, body_vars, 1, null, null,
|
|
ctx.MkSymbol("Q1"), ctx.MkSymbol("skid1"));
|
|
System.out.println("Quantifier X: " + x.toString());
|
|
|
|
Expr y = ctx.MkForall(xs, body_const, 1, null, null,
|
|
ctx.MkSymbol("Q2"), ctx.MkSymbol("skid2"));
|
|
System.out.println("Quantifier Y: " + y.toString());
|
|
}
|
|
|
|
void QuantifierExample2(Context ctx) throws Z3Exception
|
|
{
|
|
|
|
System.out.println("QuantifierExample2");
|
|
|
|
Expr q1, q2;
|
|
FuncDecl f = ctx.MkFuncDecl("f", ctx.IntSort(), ctx.IntSort());
|
|
FuncDecl g = ctx.MkFuncDecl("g", ctx.IntSort(), ctx.IntSort());
|
|
|
|
// Quantifier with Exprs as the bound variables.
|
|
{
|
|
Expr x = ctx.MkConst("x", ctx.IntSort());
|
|
Expr y = ctx.MkConst("y", ctx.IntSort());
|
|
Expr f_x = ctx.MkApp(f, new Expr[] { x });
|
|
Expr f_y = ctx.MkApp(f, new Expr[] { y });
|
|
Expr g_y = ctx.MkApp(g, new Expr[] { y });
|
|
Pattern[] pats = new Pattern[] { ctx.MkPattern(new Expr[] { f_x,
|
|
g_y }) };
|
|
Expr[] no_pats = new Expr[] { f_y };
|
|
Expr[] bound = new Expr[] { x, y };
|
|
Expr body = ctx.MkAnd(new BoolExpr[] { ctx.MkEq(f_x, f_y),
|
|
ctx.MkEq(f_y, g_y) });
|
|
|
|
q1 = ctx.MkForall(bound, body, 1, null, no_pats, ctx.MkSymbol("q"),
|
|
ctx.MkSymbol("sk"));
|
|
|
|
System.out.println(q1);
|
|
}
|
|
|
|
// Quantifier with de-Brujin indices.
|
|
{
|
|
Expr x = ctx.MkBound(1, ctx.IntSort());
|
|
Expr y = ctx.MkBound(0, ctx.IntSort());
|
|
Expr f_x = ctx.MkApp(f, new Expr[] { x });
|
|
Expr f_y = ctx.MkApp(f, new Expr[] { y });
|
|
Expr g_y = ctx.MkApp(g, new Expr[] { y });
|
|
Pattern[] pats = new Pattern[] { ctx.MkPattern(new Expr[] { f_x,
|
|
g_y }) };
|
|
Expr[] no_pats = new Expr[] { f_y };
|
|
Symbol[] names = new Symbol[] { ctx.MkSymbol("x"),
|
|
ctx.MkSymbol("y") };
|
|
Sort[] sorts = new Sort[] { ctx.IntSort(), ctx.IntSort() };
|
|
Expr body = ctx.MkAnd(new BoolExpr[] { ctx.MkEq(f_x, f_y),
|
|
ctx.MkEq(f_y, g_y) });
|
|
|
|
q2 = ctx.MkForall(sorts, names, body, 1, null, // pats,
|
|
no_pats, ctx.MkSymbol("q"), ctx.MkSymbol("sk"));
|
|
System.out.println(q2);
|
|
}
|
|
|
|
System.out.println(q1.Equals(q2));
|
|
}
|
|
|
|
// / Prove that <tt>f(x, y) = f(w, v) implies y = v</tt> when
|
|
// / <code>f</code> is injective in the second argument. <seealso
|
|
// cref="inj_axiom"/>
|
|
|
|
public void QuantifierExample3() throws Z3Exception, TestFailedException
|
|
{
|
|
System.out.println("QuantifierExample3");
|
|
|
|
HashMap<String, String> cfg = new HashMap<String, String>();
|
|
cfg.put("MBQI", "false");
|
|
cfg.put("PROOF_MODE", "2");
|
|
cfg.put("AUTO_CONFIG", "false");
|
|
|
|
/*
|
|
* If quantified formulas are asserted in a logical context, then the
|
|
* model produced by Z3 should be viewed as a potential model.
|
|
*/
|
|
|
|
{
|
|
Context ctx = new Context(cfg);
|
|
|
|
/* declare function f */
|
|
Sort I = ctx.IntSort();
|
|
FuncDecl f = ctx.MkFuncDecl("f", new Sort[] { I, I }, I);
|
|
|
|
/* f is injective in the second argument. */
|
|
BoolExpr inj = InjAxiom(ctx, f, 1);
|
|
|
|
/* create x, y, v, w, fxy, fwv */
|
|
Expr x = ctx.MkIntConst("x");
|
|
Expr y = ctx.MkIntConst("y");
|
|
Expr v = ctx.MkIntConst("v");
|
|
Expr w = ctx.MkIntConst("w");
|
|
Expr fxy = ctx.MkApp(f, new Expr[] { x, y });
|
|
Expr fwv = ctx.MkApp(f, new Expr[] { w, v });
|
|
|
|
/* f(x, y) = f(w, v) */
|
|
BoolExpr p1 = ctx.MkEq(fxy, fwv);
|
|
|
|
/* prove f(x, y) = f(w, v) implies y = v */
|
|
BoolExpr p2 = ctx.MkEq(y, v);
|
|
Prove(ctx, p2, new BoolExpr[] { inj, p1 });
|
|
|
|
/* disprove f(x, y) = f(w, v) implies x = w */
|
|
BoolExpr p3 = ctx.MkEq(x, w);
|
|
Disprove(ctx, p3, new BoolExpr[] { inj, p1 });
|
|
}
|
|
}
|
|
|
|
// / Prove that <tt>f(x, y) = f(w, v) implies y = v</tt> when
|
|
// / <code>f</code> is injective in the second argument. <seealso
|
|
// cref="inj_axiom"/>
|
|
|
|
public void QuantifierExample4() throws Z3Exception, TestFailedException
|
|
{
|
|
System.out.println("QuantifierExample4");
|
|
|
|
HashMap<String, String> cfg = new HashMap<String, String>();
|
|
cfg.put("MBQI", "false");
|
|
cfg.put("PROOF_MODE", "2");
|
|
cfg.put("AUTO_CONFIG", "false");
|
|
|
|
/*
|
|
* If quantified formulas are asserted in a logical context, then the
|
|
* model produced by Z3 should be viewed as a potential model.
|
|
*/
|
|
|
|
{
|
|
Context ctx = new Context(cfg);
|
|
/* declare function f */
|
|
Sort I = ctx.IntSort();
|
|
FuncDecl f = ctx.MkFuncDecl("f", new Sort[] { I, I }, I);
|
|
|
|
/* f is injective in the second argument. */
|
|
BoolExpr inj = InjAxiomAbs(ctx, f, 1);
|
|
|
|
/* create x, y, v, w, fxy, fwv */
|
|
Expr x = ctx.MkIntConst("x");
|
|
Expr y = ctx.MkIntConst("y");
|
|
Expr v = ctx.MkIntConst("v");
|
|
Expr w = ctx.MkIntConst("w");
|
|
Expr fxy = ctx.MkApp(f, new Expr[] { x, y });
|
|
Expr fwv = ctx.MkApp(f, new Expr[] { w, v });
|
|
|
|
/* f(x, y) = f(w, v) */
|
|
BoolExpr p1 = ctx.MkEq(fxy, fwv);
|
|
|
|
/* prove f(x, y) = f(w, v) implies y = v */
|
|
BoolExpr p2 = ctx.MkEq(y, v);
|
|
Prove(ctx, p2, new BoolExpr[] { inj, p1 });
|
|
|
|
/* disprove f(x, y) = f(w, v) implies x = w */
|
|
BoolExpr p3 = ctx.MkEq(x, w);
|
|
Disprove(ctx, p3, new BoolExpr[] { inj, p1 });
|
|
}
|
|
}
|
|
|
|
// / Some basic tests.
|
|
|
|
void BasicTests(Context ctx) throws Z3Exception, TestFailedException
|
|
{
|
|
System.out.println("BasicTests");
|
|
|
|
Symbol fname = ctx.MkSymbol("f");
|
|
Symbol x = ctx.MkSymbol("x");
|
|
Symbol y = ctx.MkSymbol("y");
|
|
|
|
Sort bs = ctx.MkBoolSort();
|
|
|
|
Sort[] domain = { bs, bs };
|
|
FuncDecl f = ctx.MkFuncDecl(fname, domain, bs);
|
|
Expr fapp = ctx.MkApp(f,
|
|
new Expr[] { ctx.MkConst(x, bs), ctx.MkConst(y, bs) });
|
|
|
|
Expr[] fargs2 = { ctx.MkFreshConst("cp", bs) };
|
|
Sort[] domain2 = { bs };
|
|
Expr fapp2 = ctx.MkApp(ctx.MkFreshFuncDecl("fp", domain2, bs), fargs2);
|
|
|
|
BoolExpr trivial_eq = ctx.MkEq(fapp, fapp);
|
|
BoolExpr nontrivial_eq = ctx.MkEq(fapp, fapp2);
|
|
|
|
Goal g = ctx.MkGoal(true, false, true);
|
|
g.Assert(trivial_eq);
|
|
g.Assert(nontrivial_eq);
|
|
System.out.println("Goal: " + g);
|
|
|
|
Solver solver = ctx.MkSolver();
|
|
|
|
for (BoolExpr a : g.Formulas())
|
|
solver.Assert(a);
|
|
|
|
if (solver.Check() != Status.SATISFIABLE)
|
|
throw new TestFailedException();
|
|
|
|
ApplyResult ar = ApplyTactic(ctx, ctx.MkTactic("simplify"), g);
|
|
if (ar.NumSubgoals() == 1
|
|
&& (ar.Subgoals()[0].IsDecidedSat() || ar.Subgoals()[0]
|
|
.IsDecidedUnsat()))
|
|
throw new TestFailedException();
|
|
|
|
ar = ApplyTactic(ctx, ctx.MkTactic("smt"), g);
|
|
if (ar.NumSubgoals() != 1 || !ar.Subgoals()[0].IsDecidedSat())
|
|
throw new TestFailedException();
|
|
|
|
g.Assert(ctx.MkEq(ctx.MkNumeral(1, ctx.MkBitVecSort(32)),
|
|
ctx.MkNumeral(2, ctx.MkBitVecSort(32))));
|
|
ar = ApplyTactic(ctx, ctx.MkTactic("smt"), g);
|
|
if (ar.NumSubgoals() != 1 || !ar.Subgoals()[0].IsDecidedUnsat())
|
|
throw new TestFailedException();
|
|
|
|
Goal g2 = ctx.MkGoal(true, true, false);
|
|
ar = ApplyTactic(ctx, ctx.MkTactic("smt"), g2);
|
|
if (ar.NumSubgoals() != 1 || !ar.Subgoals()[0].IsDecidedSat())
|
|
throw new TestFailedException();
|
|
|
|
g2 = ctx.MkGoal(true, true, false);
|
|
g2.Assert(ctx.MkFalse());
|
|
ar = ApplyTactic(ctx, ctx.MkTactic("smt"), g2);
|
|
if (ar.NumSubgoals() != 1 || !ar.Subgoals()[0].IsDecidedUnsat())
|
|
throw new TestFailedException();
|
|
|
|
Goal g3 = ctx.MkGoal(true, true, false);
|
|
Expr xc = ctx.MkConst(ctx.MkSymbol("x"), ctx.IntSort());
|
|
Expr yc = ctx.MkConst(ctx.MkSymbol("y"), ctx.IntSort());
|
|
g3.Assert(ctx.MkEq(xc, ctx.MkNumeral(1, ctx.IntSort())));
|
|
g3.Assert(ctx.MkEq(yc, ctx.MkNumeral(2, ctx.IntSort())));
|
|
BoolExpr constr = ctx.MkEq(xc, yc);
|
|
g3.Assert(constr);
|
|
ar = ApplyTactic(ctx, ctx.MkTactic("smt"), g3);
|
|
if (ar.NumSubgoals() != 1 || !ar.Subgoals()[0].IsDecidedUnsat())
|
|
throw new TestFailedException();
|
|
|
|
ModelConverterTest(ctx);
|
|
|
|
// Real num/den test.
|
|
RatNum rn = ctx.MkReal(42, 43);
|
|
Expr inum = rn.Numerator();
|
|
Expr iden = rn.Denominator();
|
|
System.out.println("Numerator: " + inum + " Denominator: " + iden);
|
|
if (!inum.toString().equals("42") || !iden.toString().equals("43"))
|
|
throw new TestFailedException();
|
|
|
|
if (!rn.ToDecimalString(3).toString().equals("0.976?"))
|
|
throw new TestFailedException();
|
|
|
|
BigIntCheck(ctx, ctx.MkReal("-1231231232/234234333"));
|
|
BigIntCheck(ctx, ctx.MkReal("-123123234234234234231232/234234333"));
|
|
BigIntCheck(ctx, ctx.MkReal("-234234333"));
|
|
BigIntCheck(ctx, ctx.MkReal("234234333/2"));
|
|
|
|
String bn = "1234567890987654321";
|
|
|
|
if (!ctx.MkInt(bn).BigInteger().toString().equals(bn))
|
|
throw new TestFailedException();
|
|
|
|
if (!ctx.MkBV(bn, 128).BigInteger().toString().equals(bn))
|
|
throw new TestFailedException();
|
|
|
|
if (ctx.MkBV(bn, 32).BigInteger().toString().equals(bn))
|
|
throw new TestFailedException();
|
|
|
|
// Error handling test.
|
|
try
|
|
{
|
|
IntExpr i = ctx.MkInt("0.5");
|
|
throw new TestFailedException(); // unreachable
|
|
} catch (Z3Exception e)
|
|
{
|
|
System.out.println("GOT: " + e.getMessage());
|
|
}
|
|
}
|
|
|
|
// / Some basic expression casting tests.
|
|
|
|
void CastingTest(Context ctx) throws Z3Exception, TestFailedException
|
|
{
|
|
System.out.println("CastingTest");
|
|
|
|
Sort[] domain = { ctx.BoolSort(), ctx.BoolSort() };
|
|
FuncDecl f = ctx.MkFuncDecl("f", domain, ctx.BoolSort());
|
|
|
|
AST upcast = ctx.MkFuncDecl(ctx.MkSymbol("q"), domain, ctx.BoolSort());
|
|
|
|
try
|
|
{
|
|
FuncDecl downcast = (FuncDecl) f; // OK
|
|
} catch (ClassCastException e)
|
|
{
|
|
throw new TestFailedException();
|
|
}
|
|
|
|
try
|
|
{
|
|
Expr uc = (Expr) upcast;
|
|
throw new TestFailedException(); // should not be reachable!
|
|
} catch (ClassCastException e)
|
|
{
|
|
}
|
|
|
|
Symbol s = ctx.MkSymbol(42);
|
|
IntSymbol si = (s.getClass() == IntSymbol.class) ? (IntSymbol) s : null;
|
|
if (si == null)
|
|
throw new TestFailedException();
|
|
try
|
|
{
|
|
IntSymbol si2 = (IntSymbol) s;
|
|
} catch (ClassCastException e)
|
|
{
|
|
throw new TestFailedException();
|
|
}
|
|
|
|
s = ctx.MkSymbol("abc");
|
|
StringSymbol ss = (s.getClass() == StringSymbol.class) ? (StringSymbol) s
|
|
: null;
|
|
if (ss == null)
|
|
throw new TestFailedException();
|
|
try
|
|
{
|
|
StringSymbol ss2 = (StringSymbol) s;
|
|
} catch (ClassCastException e)
|
|
{
|
|
throw new TestFailedException();
|
|
}
|
|
try
|
|
{
|
|
IntSymbol si2 = (IntSymbol) s;
|
|
throw new TestFailedException(); // unreachable
|
|
} catch (Exception e)
|
|
{
|
|
}
|
|
|
|
Sort srt = ctx.MkBitVecSort(32);
|
|
BitVecSort bvs = null;
|
|
try
|
|
{
|
|
bvs = (BitVecSort) srt;
|
|
} catch (ClassCastException e)
|
|
{
|
|
throw new TestFailedException();
|
|
}
|
|
|
|
if (bvs.Size() != 32)
|
|
throw new TestFailedException();
|
|
|
|
Expr q = ctx.MkAdd(new ArithExpr[] { ctx.MkInt(1), ctx.MkInt(2) });
|
|
Expr q2 = q.Args()[1];
|
|
Sort qs = q2.Sort();
|
|
if (qs.getClass() != IntSort.class)
|
|
throw new TestFailedException();
|
|
try
|
|
{
|
|
IntSort isrt = (IntSort) qs;
|
|
} catch (ClassCastException e)
|
|
{
|
|
throw new TestFailedException();
|
|
}
|
|
|
|
AST a = ctx.MkInt(42);
|
|
Expr ae = (a.getClass() == Expr.class) ? ((Expr) a) : null;
|
|
if (ae == null)
|
|
throw new TestFailedException();
|
|
ArithExpr aae = (a.getClass() == ArithExpr.class) ? ((ArithExpr) a)
|
|
: null;
|
|
if (aae == null)
|
|
throw new TestFailedException();
|
|
IntExpr aie = (a.getClass() == IntExpr.class) ? ((IntExpr) a) : null;
|
|
if (aie == null)
|
|
throw new TestFailedException();
|
|
IntNum ain = (a.getClass() == IntNum.class) ? ((IntNum) a) : null;
|
|
if (ain == null)
|
|
throw new TestFailedException();
|
|
|
|
Expr[][] earr = new Expr[2][];
|
|
earr[0] = new Expr[2];
|
|
earr[1] = new Expr[2];
|
|
earr[0][0] = ctx.MkTrue();
|
|
earr[0][1] = ctx.MkTrue();
|
|
earr[1][0] = ctx.MkFalse();
|
|
earr[1][1] = ctx.MkFalse();
|
|
for (Expr[] ea : earr)
|
|
for (Expr e : ea)
|
|
{
|
|
try
|
|
{
|
|
Expr ns = ctx.MkNot((BoolExpr) e);
|
|
BoolExpr ens = (BoolExpr) ns;
|
|
} catch (ClassCastException ex)
|
|
{
|
|
throw new TestFailedException();
|
|
}
|
|
}
|
|
}
|
|
|
|
// / Shows how to read an SMT1 file.
|
|
|
|
void SMT1FileTest(String filename) throws Z3Exception
|
|
{
|
|
System.out.print("SMT File test ");
|
|
|
|
{
|
|
HashMap<String, String> cfg = new HashMap<String, String>();
|
|
Context ctx = new Context(cfg);
|
|
ctx.ParseSMTLIBFile(filename, null, null, null, null);
|
|
|
|
BoolExpr a = ctx.MkAnd(ctx.SMTLIBFormulas());
|
|
System.out.println("read formula: " + a);
|
|
}
|
|
}
|
|
|
|
// / Shows how to read an SMT2 file.
|
|
|
|
void SMT2FileTest(String filename) throws Z3Exception
|
|
{
|
|
Date before = new Date();
|
|
|
|
System.out.println("SMT2 File test ");
|
|
System.gc();
|
|
|
|
{
|
|
HashMap<String, String> cfg = new HashMap<String, String>();
|
|
cfg.put("MODEL", "true");
|
|
Context ctx = new Context(cfg);
|
|
Expr a = ctx.ParseSMTLIB2File(filename, null, null, null, null);
|
|
|
|
long t_diff = ((new Date()).getTime() - before.getTime()) / 1000;
|
|
|
|
System.out.println("SMT2 file read time: " + t_diff + " sec");
|
|
|
|
// Iterate over the formula.
|
|
|
|
LinkedList<Expr> q = new LinkedList<Expr>();
|
|
q.add(a);
|
|
int cnt = 0;
|
|
while (q.size() > 0)
|
|
{
|
|
AST cur = (AST) q.removeFirst();
|
|
cnt++;
|
|
|
|
if (cur.getClass() == Expr.class)
|
|
if (!(cur.IsVar()))
|
|
for (Expr c : ((Expr) cur).Args())
|
|
q.add(c);
|
|
}
|
|
System.out.println(cnt + " ASTs");
|
|
}
|
|
|
|
long t_diff = ((new Date()).getTime() - before.getTime()) / 1000;
|
|
System.out.println("SMT2 file test took " + t_diff + " sec");
|
|
}
|
|
|
|
// / Shows how to use Solver(logic)
|
|
|
|
// / <param name="ctx"></param>
|
|
void LogicExample(Context ctx) throws Z3Exception, TestFailedException
|
|
{
|
|
System.out.println("LogicTest");
|
|
|
|
Context.ToggleWarningMessages(true);
|
|
|
|
BitVecSort bvs = ctx.MkBitVecSort(32);
|
|
Expr x = ctx.MkConst("x", bvs);
|
|
Expr y = ctx.MkConst("y", bvs);
|
|
BoolExpr eq = ctx.MkEq(x, y);
|
|
|
|
// Use a solver for QF_BV
|
|
Solver s = ctx.MkSolver("QF_BV");
|
|
s.Assert(eq);
|
|
Status res = s.Check();
|
|
System.out.println("solver result: " + res);
|
|
|
|
// Or perhaps a tactic for QF_BV
|
|
Goal g = ctx.MkGoal(true, false, true);
|
|
g.Assert(eq);
|
|
|
|
Tactic t = ctx.MkTactic("qfbv");
|
|
ApplyResult ar = t.Apply(g);
|
|
System.out.println("tactic result: " + ar);
|
|
|
|
if (ar.NumSubgoals() != 1 || !ar.Subgoals()[0].IsDecidedSat())
|
|
throw new TestFailedException();
|
|
}
|
|
|
|
// / Demonstrates how to use the ParOr tactic.
|
|
|
|
void ParOrExample(Context ctx) throws Z3Exception, TestFailedException
|
|
{
|
|
System.out.println("ParOrExample");
|
|
|
|
BitVecSort bvs = ctx.MkBitVecSort(32);
|
|
Expr x = ctx.MkConst("x", bvs);
|
|
Expr y = ctx.MkConst("y", bvs);
|
|
BoolExpr q = ctx.MkEq(x, y);
|
|
|
|
Goal g = ctx.MkGoal(true, false, true);
|
|
g.Assert(q);
|
|
|
|
Tactic t1 = ctx.MkTactic("qfbv");
|
|
Tactic t2 = ctx.MkTactic("qfbv");
|
|
Tactic p = ctx.ParOr(new Tactic[] { t1, t2 });
|
|
|
|
ApplyResult ar = p.Apply(g);
|
|
|
|
if (ar.NumSubgoals() != 1 || !ar.Subgoals()[0].IsDecidedSat())
|
|
throw new TestFailedException();
|
|
}
|
|
|
|
void BigIntCheck(Context ctx, RatNum r) throws Z3Exception
|
|
{
|
|
System.out.println("Num: " + r.BigIntNumerator());
|
|
System.out.println("Den: " + r.BigIntDenominator());
|
|
}
|
|
|
|
// / Find a model for <code>x xor y</code>.
|
|
|
|
public void FindModelExample1(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("FindModelExample1");
|
|
|
|
BoolExpr x = ctx.MkBoolConst("x");
|
|
BoolExpr y = ctx.MkBoolConst("y");
|
|
BoolExpr x_xor_y = ctx.MkXor(x, y);
|
|
|
|
Model model = Check(ctx, x_xor_y, Status.SATISFIABLE);
|
|
System.out.println("x = " + model.Evaluate(x, false) + ", y = "
|
|
+ model.Evaluate(y, false));
|
|
}
|
|
|
|
// / Find a model for <tt>x < y + 1, x > 2</tt>.
|
|
// / Then, assert <tt>not(x = y)</tt>, and find another model.
|
|
|
|
public void FindModelExample2(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("find_model_example2");
|
|
|
|
IntExpr x = ctx.MkIntConst("x");
|
|
IntExpr y = ctx.MkIntConst("y");
|
|
IntExpr one = ctx.MkInt(1);
|
|
IntExpr two = ctx.MkInt(2);
|
|
|
|
ArithExpr y_plus_one = ctx.MkAdd(new ArithExpr[] { y, one });
|
|
|
|
BoolExpr c1 = ctx.MkLt(x, y_plus_one);
|
|
BoolExpr c2 = ctx.MkGt(x, two);
|
|
|
|
BoolExpr q = ctx.MkAnd(new BoolExpr[] { c1, c2 });
|
|
|
|
System.out.println("model for: x < y + 1, x > 2");
|
|
Model model = Check(ctx, q, Status.SATISFIABLE);
|
|
System.out.println("x = " + model.Evaluate(x, false) + ", y ="
|
|
+ model.Evaluate(y, false));
|
|
|
|
/* assert not(x = y) */
|
|
BoolExpr x_eq_y = ctx.MkEq(x, y);
|
|
BoolExpr c3 = ctx.MkNot(x_eq_y);
|
|
|
|
q = ctx.MkAnd(new BoolExpr[] { q, c3 });
|
|
|
|
System.out.println("model for: x < y + 1, x > 2, not(x = y)");
|
|
model = Check(ctx, q, Status.SATISFIABLE);
|
|
System.out.println("x = " + model.Evaluate(x, false) + ", y = "
|
|
+ model.Evaluate(y, false));
|
|
}
|
|
|
|
// / Prove <tt>x = y implies g(x) = g(y)</tt>, and
|
|
// / disprove <tt>x = y implies g(g(x)) = g(y)</tt>.
|
|
|
|
// / <remarks>This function demonstrates how to create uninterpreted
|
|
// / types and functions.</remarks>
|
|
public void ProveExample1(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("ProveExample1");
|
|
|
|
/* create uninterpreted type. */
|
|
Sort U = ctx.MkUninterpretedSort(ctx.MkSymbol("U"));
|
|
|
|
/* declare function g */
|
|
FuncDecl g = ctx.MkFuncDecl("g", U, U);
|
|
|
|
/* create x and y */
|
|
Expr x = ctx.MkConst("x", U);
|
|
Expr y = ctx.MkConst("y", U);
|
|
/* create g(x), g(y) */
|
|
Expr gx = g.Apply(x);
|
|
Expr gy = g.Apply(y);
|
|
|
|
/* assert x = y */
|
|
BoolExpr eq = ctx.MkEq(x, y);
|
|
|
|
/* prove g(x) = g(y) */
|
|
BoolExpr f = ctx.MkEq(gx, gy);
|
|
System.out.println("prove: x = y implies g(x) = g(y)");
|
|
Prove(ctx, ctx.MkImplies(eq, f));
|
|
|
|
/* create g(g(x)) */
|
|
Expr ggx = g.Apply(gx);
|
|
|
|
/* disprove g(g(x)) = g(y) */
|
|
f = ctx.MkEq(ggx, gy);
|
|
System.out.println("disprove: x = y implies g(g(x)) = g(y)");
|
|
Disprove(ctx, ctx.MkImplies(eq, f));
|
|
|
|
/* Print the model using the custom model printer */
|
|
Model m = Check(ctx, ctx.MkNot(f), Status.SATISFIABLE);
|
|
System.out.println(m);
|
|
}
|
|
|
|
// / Prove <tt>not(g(g(x) - g(y)) = g(z)), x + z <= y <= x implies z < 0
|
|
// </tt>.
|
|
// / Then, show that <tt>z < -1</tt> is not implied.
|
|
|
|
// / <remarks>This example demonstrates how to combine uninterpreted
|
|
// functions
|
|
// / and arithmetic.</remarks>
|
|
public void ProveExample2(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("ProveExample2");
|
|
|
|
/* declare function g */
|
|
Sort I = ctx.IntSort();
|
|
|
|
FuncDecl g = ctx.MkFuncDecl("g", I, I);
|
|
|
|
/* create x, y, and z */
|
|
IntExpr x = ctx.MkIntConst("x");
|
|
IntExpr y = ctx.MkIntConst("y");
|
|
IntExpr z = ctx.MkIntConst("z");
|
|
|
|
/* create gx, gy, gz */
|
|
Expr gx = ctx.MkApp(g, x);
|
|
Expr gy = ctx.MkApp(g, y);
|
|
Expr gz = ctx.MkApp(g, z);
|
|
|
|
/* create zero */
|
|
IntExpr zero = ctx.MkInt(0);
|
|
|
|
/* assert not(g(g(x) - g(y)) = g(z)) */
|
|
ArithExpr gx_gy = ctx.MkSub(new ArithExpr[] { (IntExpr) gx,
|
|
(IntExpr) gy });
|
|
Expr ggx_gy = ctx.MkApp(g, gx_gy);
|
|
BoolExpr eq = ctx.MkEq(ggx_gy, gz);
|
|
BoolExpr c1 = ctx.MkNot(eq);
|
|
|
|
/* assert x + z <= y */
|
|
ArithExpr x_plus_z = ctx.MkAdd(new ArithExpr[] { x, z });
|
|
BoolExpr c2 = ctx.MkLe(x_plus_z, y);
|
|
|
|
/* assert y <= x */
|
|
BoolExpr c3 = ctx.MkLe(y, x);
|
|
|
|
/* prove z < 0 */
|
|
BoolExpr f = ctx.MkLt(z, zero);
|
|
System.out
|
|
.println("prove: not(g(g(x) - g(y)) = g(z)), x + z <= y <= x implies z < 0");
|
|
Prove(ctx, f, new BoolExpr[] { c1, c2, c3 });
|
|
|
|
/* disprove z < -1 */
|
|
IntExpr minus_one = ctx.MkInt(-1);
|
|
f = ctx.MkLt(z, minus_one);
|
|
System.out
|
|
.println("disprove: not(g(g(x) - g(y)) = g(z)), x + z <= y <= x implies z < -1");
|
|
Disprove(ctx, f, new BoolExpr[] { c1, c2, c3 });
|
|
}
|
|
|
|
// / Show how push & pop can be used to create "backtracking" points.
|
|
|
|
// / <remarks>This example also demonstrates how big numbers can be
|
|
// / created in ctx.</remarks>
|
|
public void PushPopExample1(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("PushPopExample1");
|
|
|
|
/* create a big number */
|
|
IntSort int_type = ctx.IntSort();
|
|
IntExpr big_number = ctx
|
|
.MkInt("1000000000000000000000000000000000000000000000000000000");
|
|
|
|
/* create number 3 */
|
|
IntExpr three = (IntExpr) ctx.MkNumeral("3", int_type);
|
|
|
|
/* create x */
|
|
IntExpr x = ctx.MkIntConst("x");
|
|
|
|
Solver solver = ctx.MkSolver();
|
|
|
|
/* assert x >= "big number" */
|
|
BoolExpr c1 = ctx.MkGe(x, big_number);
|
|
System.out.println("assert: x >= 'big number'");
|
|
solver.Assert(c1);
|
|
|
|
/* create a backtracking point */
|
|
System.out.println("push");
|
|
solver.Push();
|
|
|
|
/* assert x <= 3 */
|
|
BoolExpr c2 = ctx.MkLe(x, three);
|
|
System.out.println("assert: x <= 3");
|
|
solver.Assert(c2);
|
|
|
|
/* context is inconsistent at this point */
|
|
if (solver.Check() != Status.UNSATISFIABLE)
|
|
throw new TestFailedException();
|
|
|
|
/*
|
|
* backtrack: the constraint x <= 3 will be removed, since it was
|
|
* asserted after the last ctx.Push.
|
|
*/
|
|
System.out.println("pop");
|
|
solver.Pop(1);
|
|
|
|
/* the context is consistent again. */
|
|
if (solver.Check() != Status.SATISFIABLE)
|
|
throw new TestFailedException();
|
|
|
|
/* new constraints can be asserted... */
|
|
|
|
/* create y */
|
|
IntExpr y = ctx.MkIntConst("y");
|
|
|
|
/* assert y > x */
|
|
BoolExpr c3 = ctx.MkGt(y, x);
|
|
System.out.println("assert: y > x");
|
|
solver.Assert(c3);
|
|
|
|
/* the context is still consistent. */
|
|
if (solver.Check() != Status.SATISFIABLE)
|
|
throw new TestFailedException();
|
|
}
|
|
|
|
// / Tuples.
|
|
|
|
// / <remarks>Check that the projection of a tuple
|
|
// / returns the corresponding element.</remarks>
|
|
public void TupleExample(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("TupleExample");
|
|
|
|
Sort int_type = ctx.IntSort();
|
|
TupleSort tuple = ctx.MkTupleSort(ctx.MkSymbol("mk_tuple"), // name of
|
|
// tuple
|
|
// constructor
|
|
new Symbol[] { ctx.MkSymbol("first"), ctx.MkSymbol("second") }, // names
|
|
// of
|
|
// projection
|
|
// operators
|
|
new Sort[] { int_type, int_type } // types of projection
|
|
// operators
|
|
);
|
|
FuncDecl first = tuple.FieldDecls()[0]; // declarations are for
|
|
// projections
|
|
FuncDecl second = tuple.FieldDecls()[1];
|
|
Expr x = ctx.MkConst("x", int_type);
|
|
Expr y = ctx.MkConst("y", int_type);
|
|
Expr n1 = tuple.MkDecl().Apply(new Expr[] { x, y });
|
|
Expr n2 = first.Apply(n1);
|
|
BoolExpr n3 = ctx.MkEq(x, n2);
|
|
System.out.println("Tuple example: " + n3);
|
|
Prove(ctx, n3);
|
|
}
|
|
|
|
// / Simple bit-vector example.
|
|
|
|
// / <remarks>
|
|
// / This example disproves that x - 10 <= 0 IFF x <= 10 for (32-bit)
|
|
// machine integers
|
|
// / </remarks>
|
|
public void BitvectorExample1(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("BitvectorExample1");
|
|
|
|
Sort bv_type = ctx.MkBitVecSort(32);
|
|
BitVecExpr x = (BitVecExpr) ctx.MkConst("x", bv_type);
|
|
BitVecNum zero = (BitVecNum) ctx.MkNumeral("0", bv_type);
|
|
BitVecNum ten = ctx.MkBV(10, 32);
|
|
BitVecExpr x_minus_ten = ctx.MkBVSub(x, ten);
|
|
/* bvsle is signed less than or equal to */
|
|
BoolExpr c1 = ctx.MkBVSLE(x, ten);
|
|
BoolExpr c2 = ctx.MkBVSLE(x_minus_ten, zero);
|
|
BoolExpr thm = ctx.MkIff(c1, c2);
|
|
System.out
|
|
.println("disprove: x - 10 <= 0 IFF x <= 10 for (32-bit) machine integers");
|
|
Disprove(ctx, thm);
|
|
}
|
|
|
|
// / Find x and y such that: x ^ y - 103 == x * y
|
|
|
|
public void BitvectorExample2(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("BitvectorExample2");
|
|
|
|
/* construct x ^ y - 103 == x * y */
|
|
Sort bv_type = ctx.MkBitVecSort(32);
|
|
BitVecExpr x = ctx.MkBVConst("x", 32);
|
|
BitVecExpr y = ctx.MkBVConst("y", 32);
|
|
BitVecExpr x_xor_y = ctx.MkBVXOR(x, y);
|
|
BitVecExpr c103 = (BitVecNum) ctx.MkNumeral("103", bv_type);
|
|
BitVecExpr lhs = ctx.MkBVSub(x_xor_y, c103);
|
|
BitVecExpr rhs = ctx.MkBVMul(x, y);
|
|
BoolExpr ctr = ctx.MkEq(lhs, rhs);
|
|
|
|
System.out
|
|
.println("find values of x and y, such that x ^ y - 103 == x * y");
|
|
|
|
/* find a model (i.e., values for x an y that satisfy the constraint */
|
|
Model m = Check(ctx, ctr, Status.SATISFIABLE);
|
|
System.out.println(m);
|
|
}
|
|
|
|
// / Demonstrates how to use the SMTLIB parser.
|
|
|
|
public void ParserExample1(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("ParserExample1");
|
|
|
|
ctx.ParseSMTLIBString(
|
|
"(benchmark tst :extrafuns ((x Int) (y Int)) :formula (> x y) :formula (> x 0))",
|
|
null, null, null, null);
|
|
for (BoolExpr f : ctx.SMTLIBFormulas())
|
|
System.out.println("formula " + f);
|
|
|
|
Model m = Check(ctx, ctx.MkAnd(ctx.SMTLIBFormulas()),
|
|
Status.SATISFIABLE);
|
|
}
|
|
|
|
// / Demonstrates how to initialize the parser symbol table.
|
|
|
|
public void ParserExample2(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("ParserExample2");
|
|
|
|
Symbol[] declNames = { ctx.MkSymbol("a"), ctx.MkSymbol("b") };
|
|
FuncDecl a = ctx.MkConstDecl(declNames[0], ctx.MkIntSort());
|
|
FuncDecl b = ctx.MkConstDecl(declNames[1], ctx.MkIntSort());
|
|
FuncDecl[] decls = new FuncDecl[] { a, b };
|
|
|
|
ctx.ParseSMTLIBString("(benchmark tst :formula (> a b))", null, null,
|
|
declNames, decls);
|
|
BoolExpr f = ctx.SMTLIBFormulas()[0];
|
|
System.out.println("formula: " + f);
|
|
Check(ctx, f, Status.SATISFIABLE);
|
|
}
|
|
|
|
// / Demonstrates how to initialize the parser symbol table.
|
|
|
|
public void ParserExample3(Context ctx) throws Exception
|
|
{
|
|
System.out.println("ParserExample3");
|
|
|
|
/* declare function g */
|
|
Sort I = ctx.MkIntSort();
|
|
FuncDecl g = ctx.MkFuncDecl("g", new Sort[] { I, I }, I);
|
|
|
|
BoolExpr ca = CommAxiom(ctx, g);
|
|
|
|
ctx.ParseSMTLIBString(
|
|
"(benchmark tst :formula (forall (x Int) (y Int) (implies (= x y) (= (gg x 0) (gg 0 y)))))",
|
|
null, null, new Symbol[] { ctx.MkSymbol("gg") },
|
|
new FuncDecl[] { g });
|
|
|
|
BoolExpr thm = ctx.SMTLIBFormulas()[0];
|
|
System.out.println("formula: " + thm);
|
|
Prove(ctx, thm, ca);
|
|
}
|
|
|
|
// / Display the declarations, assumptions and formulas in a SMT-LIB string.
|
|
|
|
public void ParserExample4(Context ctx) throws Z3Exception
|
|
{
|
|
System.out.println("ParserExample4");
|
|
|
|
ctx.ParseSMTLIBString(
|
|
"(benchmark tst :extrafuns ((x Int) (y Int)) :assumption (= x 20) :formula (> x y) :formula (> x 0))",
|
|
null, null, null, null);
|
|
for (FuncDecl decl : ctx.SMTLIBDecls())
|
|
{
|
|
System.out.println("Declaration: " + decl);
|
|
}
|
|
for (BoolExpr f : ctx.SMTLIBAssumptions())
|
|
{
|
|
System.out.println("Assumption: " + f);
|
|
}
|
|
for (BoolExpr f : ctx.SMTLIBFormulas())
|
|
{
|
|
System.out.println("Formula: " + f);
|
|
}
|
|
}
|
|
|
|
// / Demonstrates how to handle parser errors using Z3 error handling
|
|
// support.
|
|
|
|
// / <remarks></remarks>
|
|
public void ParserExample5(Context ctx)
|
|
{
|
|
System.out.println("ParserExample5");
|
|
|
|
try
|
|
{
|
|
ctx.ParseSMTLIBString(
|
|
/*
|
|
* the following string has a parsing error: missing
|
|
* parenthesis
|
|
*/
|
|
"(benchmark tst :extrafuns ((x Int (y Int)) :formula (> x y) :formula (> x 0))",
|
|
null, null, null, null);
|
|
} catch (Z3Exception e)
|
|
{
|
|
System.out.println("Z3 error: " + e);
|
|
}
|
|
}
|
|
|
|
// / Create an ite-Expr (if-then-else Exprs).
|
|
|
|
public void ITEExample(Context ctx) throws Z3Exception
|
|
{
|
|
System.out.println("ITEExample");
|
|
|
|
BoolExpr f = ctx.MkFalse();
|
|
Expr one = ctx.MkInt(1);
|
|
Expr zero = ctx.MkInt(0);
|
|
Expr ite = ctx.MkITE(f, one, zero);
|
|
|
|
System.out.println("Expr: " + ite);
|
|
}
|
|
|
|
// / Create an enumeration data type.
|
|
|
|
public void EnumExample(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("EnumExample");
|
|
|
|
Symbol name = ctx.MkSymbol("fruit");
|
|
|
|
EnumSort fruit = ctx.MkEnumSort(ctx.MkSymbol("fruit"),
|
|
new Symbol[] { ctx.MkSymbol("apple"), ctx.MkSymbol("banana"),
|
|
ctx.MkSymbol("orange") });
|
|
|
|
System.out.println((fruit.Consts()[0]));
|
|
System.out.println((fruit.Consts()[1]));
|
|
System.out.println((fruit.Consts()[2]));
|
|
|
|
System.out.println((fruit.TesterDecls()[0]));
|
|
System.out.println((fruit.TesterDecls()[1]));
|
|
System.out.println((fruit.TesterDecls()[2]));
|
|
|
|
Expr apple = fruit.Consts()[0];
|
|
Expr banana = fruit.Consts()[1];
|
|
Expr orange = fruit.Consts()[2];
|
|
|
|
/* Apples are different from oranges */
|
|
Prove(ctx, ctx.MkNot(ctx.MkEq(apple, orange)));
|
|
|
|
/* Apples pass the apple test */
|
|
Prove(ctx, (BoolExpr) ctx.MkApp(fruit.TesterDecls()[0], apple));
|
|
|
|
/* Oranges fail the apple test */
|
|
Disprove(ctx, (BoolExpr) ctx.MkApp(fruit.TesterDecls()[0], orange));
|
|
Prove(ctx, (BoolExpr) ctx.MkNot((BoolExpr) ctx.MkApp(
|
|
fruit.TesterDecls()[0], orange)));
|
|
|
|
Expr fruity = ctx.MkConst("fruity", fruit);
|
|
|
|
/* If something is fruity, then it is an apple, banana, or orange */
|
|
|
|
Prove(ctx,
|
|
ctx.MkOr(new BoolExpr[] { ctx.MkEq(fruity, apple),
|
|
ctx.MkEq(fruity, banana), ctx.MkEq(fruity, orange) }));
|
|
}
|
|
|
|
// / Create a list datatype.
|
|
|
|
public void ListExample(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("ListExample");
|
|
|
|
Sort int_ty;
|
|
ListSort int_list;
|
|
Expr nil, l1, l2, x, y, u, v;
|
|
BoolExpr fml, fml1;
|
|
|
|
int_ty = ctx.MkIntSort();
|
|
|
|
int_list = ctx.MkListSort(ctx.MkSymbol("int_list"), int_ty);
|
|
|
|
nil = ctx.MkConst(int_list.NilDecl());
|
|
l1 = ctx.MkApp(int_list.ConsDecl(), new Expr[] { ctx.MkInt(1), nil });
|
|
l2 = ctx.MkApp(int_list.ConsDecl(), new Expr[] { ctx.MkInt(2), nil });
|
|
|
|
/* nil != cons(1, nil) */
|
|
Prove(ctx, ctx.MkNot(ctx.MkEq(nil, l1)));
|
|
|
|
/* cons(2,nil) != cons(1, nil) */
|
|
Prove(ctx, ctx.MkNot(ctx.MkEq(l1, l2)));
|
|
|
|
/* cons(x,nil) = cons(y, nil) => x = y */
|
|
x = ctx.MkConst("x", int_ty);
|
|
y = ctx.MkConst("y", int_ty);
|
|
l1 = ctx.MkApp(int_list.ConsDecl(), new Expr[] { x, nil });
|
|
l2 = ctx.MkApp(int_list.ConsDecl(), new Expr[] { y, nil });
|
|
Prove(ctx, ctx.MkImplies(ctx.MkEq(l1, l2), ctx.MkEq(x, y)));
|
|
|
|
/* cons(x,u) = cons(x, v) => u = v */
|
|
u = ctx.MkConst("u", int_list);
|
|
v = ctx.MkConst("v", int_list);
|
|
l1 = ctx.MkApp(int_list.ConsDecl(), new Expr[] { x, u });
|
|
l2 = ctx.MkApp(int_list.ConsDecl(), new Expr[] { y, v });
|
|
Prove(ctx, ctx.MkImplies(ctx.MkEq(l1, l2), ctx.MkEq(u, v)));
|
|
Prove(ctx, ctx.MkImplies(ctx.MkEq(l1, l2), ctx.MkEq(x, y)));
|
|
|
|
/* is_nil(u) or is_cons(u) */
|
|
Prove(ctx,
|
|
ctx.MkOr(new BoolExpr[] {
|
|
(BoolExpr) ctx.MkApp(int_list.IsNilDecl(),
|
|
new Expr[] { u }),
|
|
(BoolExpr) ctx.MkApp(int_list.IsConsDecl(),
|
|
new Expr[] { u }) }));
|
|
|
|
/* occurs check u != cons(x,u) */
|
|
Prove(ctx, ctx.MkNot(ctx.MkEq(u, l1)));
|
|
|
|
/* destructors: is_cons(u) => u = cons(head(u),tail(u)) */
|
|
fml1 = ctx.MkEq(u, ctx.MkApp(int_list.ConsDecl(),
|
|
new Expr[] { ctx.MkApp(int_list.HeadDecl(), new Expr[] { u }),
|
|
ctx.MkApp(int_list.TailDecl(), new Expr[] { u }) }));
|
|
fml = ctx.MkImplies(
|
|
(BoolExpr) ctx.MkApp(int_list.IsConsDecl(), new Expr[] { u }),
|
|
fml1);
|
|
System.out.println("Formula " + fml);
|
|
|
|
Prove(ctx, fml);
|
|
|
|
Disprove(ctx, fml1);
|
|
}
|
|
|
|
// / Create a binary tree datatype.
|
|
|
|
public void TreeExample(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("TreeExample");
|
|
|
|
Sort cell;
|
|
FuncDecl nil_decl, is_nil_decl, cons_decl, is_cons_decl, car_decl, cdr_decl;
|
|
Expr nil, l1, l2, x, y, u, v;
|
|
BoolExpr fml, fml1;
|
|
String[] head_tail = new String[] { "car", "cdr" };
|
|
Sort[] sorts = new Sort[] { null, null };
|
|
int[] sort_refs = new int[] { 0, 0 };
|
|
Constructor nil_con, cons_con;
|
|
|
|
nil_con = ctx.MkConstructor("nil", "is_nil", null, null, null);
|
|
cons_con = ctx.MkConstructor("cons", "is_cons", head_tail, sorts,
|
|
sort_refs);
|
|
Constructor[] constructors = new Constructor[] { nil_con, cons_con };
|
|
|
|
cell = ctx.MkDatatypeSort("cell", constructors);
|
|
|
|
nil_decl = nil_con.ConstructorDecl();
|
|
is_nil_decl = nil_con.TesterDecl();
|
|
cons_decl = cons_con.ConstructorDecl();
|
|
is_cons_decl = cons_con.TesterDecl();
|
|
FuncDecl[] cons_accessors = cons_con.AccessorDecls();
|
|
car_decl = cons_accessors[0];
|
|
cdr_decl = cons_accessors[1];
|
|
|
|
nil = ctx.MkConst(nil_decl);
|
|
l1 = ctx.MkApp(cons_decl, new Expr[] { nil, nil });
|
|
l2 = ctx.MkApp(cons_decl, new Expr[] { l1, nil });
|
|
|
|
/* nil != cons(nil, nil) */
|
|
Prove(ctx, ctx.MkNot(ctx.MkEq(nil, l1)));
|
|
|
|
/* cons(x,u) = cons(x, v) => u = v */
|
|
u = ctx.MkConst("u", cell);
|
|
v = ctx.MkConst("v", cell);
|
|
x = ctx.MkConst("x", cell);
|
|
y = ctx.MkConst("y", cell);
|
|
l1 = ctx.MkApp(cons_decl, new Expr[] { x, u });
|
|
l2 = ctx.MkApp(cons_decl, new Expr[] { y, v });
|
|
Prove(ctx, ctx.MkImplies(ctx.MkEq(l1, l2), ctx.MkEq(u, v)));
|
|
Prove(ctx, ctx.MkImplies(ctx.MkEq(l1, l2), ctx.MkEq(x, y)));
|
|
|
|
/* is_nil(u) or is_cons(u) */
|
|
Prove(ctx,
|
|
ctx.MkOr(new BoolExpr[] {
|
|
(BoolExpr) ctx.MkApp(is_nil_decl, new Expr[] { u }),
|
|
(BoolExpr) ctx.MkApp(is_cons_decl, new Expr[] { u }) }));
|
|
|
|
/* occurs check u != cons(x,u) */
|
|
Prove(ctx, ctx.MkNot(ctx.MkEq(u, l1)));
|
|
|
|
/* destructors: is_cons(u) => u = cons(car(u),cdr(u)) */
|
|
fml1 = ctx.MkEq(
|
|
u,
|
|
ctx.MkApp(
|
|
cons_decl,
|
|
new Expr[] { ctx.MkApp(car_decl, u),
|
|
ctx.MkApp(cdr_decl, u) }));
|
|
fml = ctx.MkImplies((BoolExpr) ctx.MkApp(is_cons_decl, u), fml1);
|
|
System.out.println("Formula " + fml);
|
|
Prove(ctx, fml);
|
|
|
|
Disprove(ctx, fml1);
|
|
}
|
|
|
|
// / Create a forest of trees.
|
|
|
|
// / <remarks>
|
|
// / forest ::= nil | cons(tree, forest)
|
|
// / tree ::= nil | cons(forest, forest)
|
|
// / </remarks>
|
|
public void ForestExample(Context ctx) throws Z3Exception,
|
|
TestFailedException
|
|
{
|
|
System.out.println("ForestExample");
|
|
|
|
Sort tree, forest;
|
|
FuncDecl nil1_decl, is_nil1_decl, cons1_decl, is_cons1_decl, car1_decl, cdr1_decl;
|
|
FuncDecl nil2_decl, is_nil2_decl, cons2_decl, is_cons2_decl, car2_decl, cdr2_decl;
|
|
Expr nil1, nil2, t1, t2, t3, t4, f1, f2, f3, l1, l2, x, y, u, v;
|
|
|
|
//
|
|
// Declare the names of the accessors for cons.
|
|
// Then declare the sorts of the accessors.
|
|
// For this example, all sorts refer to the new types 'forest' and
|
|
// 'tree'
|
|
// being declared, so we pass in null for both sorts1 and sorts2.
|
|
// On the other hand, the sort_refs arrays contain the indices of the
|
|
// two new sorts being declared. The first element in sort1_refs
|
|
// points to 'tree', which has index 1, the second element in sort1_refs
|
|
// array
|
|
// points to 'forest', which has index 0.
|
|
//
|
|
Symbol[] head_tail1 = new Symbol[] { ctx.MkSymbol("head"),
|
|
ctx.MkSymbol("tail") };
|
|
Sort[] sorts1 = new Sort[] { null, null };
|
|
int[] sort1_refs = new int[] { 1, 0 }; // the first item points to a
|
|
// tree, the second to a forest
|
|
|
|
Symbol[] head_tail2 = new Symbol[] { ctx.MkSymbol("car"),
|
|
ctx.MkSymbol("cdr") };
|
|
Sort[] sorts2 = new Sort[] { null, null };
|
|
int[] sort2_refs = new int[] { 0, 0 }; // both items point to the forest
|
|
// datatype.
|
|
Constructor nil1_con, cons1_con, nil2_con, cons2_con;
|
|
Constructor[] constructors1 = new Constructor[2], constructors2 = new Constructor[2];
|
|
Symbol[] sort_names = { ctx.MkSymbol("forest"), ctx.MkSymbol("tree") };
|
|
|
|
/* build a forest */
|
|
nil1_con = ctx.MkConstructor(ctx.MkSymbol("nil"),
|
|
ctx.MkSymbol("is_nil"), null, null, null);
|
|
cons1_con = ctx.MkConstructor(ctx.MkSymbol("cons1"),
|
|
ctx.MkSymbol("is_cons1"), head_tail1, sorts1, sort1_refs);
|
|
constructors1[0] = nil1_con;
|
|
constructors1[1] = cons1_con;
|
|
|
|
/* build a tree */
|
|
nil2_con = ctx.MkConstructor(ctx.MkSymbol("nil2"),
|
|
ctx.MkSymbol("is_nil2"), null, null, null);
|
|
cons2_con = ctx.MkConstructor(ctx.MkSymbol("cons2"),
|
|
ctx.MkSymbol("is_cons2"), head_tail2, sorts2, sort2_refs);
|
|
constructors2[0] = nil2_con;
|
|
constructors2[1] = cons2_con;
|
|
|
|
Constructor[][] clists = new Constructor[][] { constructors1,
|
|
constructors2 };
|
|
|
|
Sort[] sorts = ctx.MkDatatypeSorts(sort_names, clists);
|
|
forest = sorts[0];
|
|
tree = sorts[1];
|
|
|
|
//
|
|
// Now that the datatype has been created.
|
|
// Query the constructors for the constructor
|
|
// functions, testers, and field accessors.
|
|
//
|
|
nil1_decl = nil1_con.ConstructorDecl();
|
|
is_nil1_decl = nil1_con.TesterDecl();
|
|
cons1_decl = cons1_con.ConstructorDecl();
|
|
is_cons1_decl = cons1_con.TesterDecl();
|
|
FuncDecl[] cons1_accessors = cons1_con.AccessorDecls();
|
|
car1_decl = cons1_accessors[0];
|
|
cdr1_decl = cons1_accessors[1];
|
|
|
|
nil2_decl = nil2_con.ConstructorDecl();
|
|
is_nil2_decl = nil2_con.TesterDecl();
|
|
cons2_decl = cons2_con.ConstructorDecl();
|
|
is_cons2_decl = cons2_con.TesterDecl();
|
|
FuncDecl[] cons2_accessors = cons2_con.AccessorDecls();
|
|
car2_decl = cons2_accessors[0];
|
|
cdr2_decl = cons2_accessors[1];
|
|
|
|
nil1 = ctx.MkConst(nil1_decl);
|
|
nil2 = ctx.MkConst(nil2_decl);
|
|
f1 = ctx.MkApp(cons1_decl, new Expr[] { nil2, nil1 });
|
|
t1 = ctx.MkApp(cons2_decl, new Expr[] { nil1, nil1 });
|
|
t2 = ctx.MkApp(cons2_decl, new Expr[] { f1, nil1 });
|
|
t3 = ctx.MkApp(cons2_decl, new Expr[] { f1, f1 });
|
|
t4 = ctx.MkApp(cons2_decl, new Expr[] { nil1, f1 });
|
|
f2 = ctx.MkApp(cons1_decl, new Expr[] { t1, nil1 });
|
|
f3 = ctx.MkApp(cons1_decl, new Expr[] { t1, f1 });
|
|
|
|
/* nil != cons(nil,nil) */
|
|
Prove(ctx, ctx.MkNot(ctx.MkEq(nil1, f1)));
|
|
Prove(ctx, ctx.MkNot(ctx.MkEq(nil2, t1)));
|
|
|
|
/* cons(x,u) = cons(x, v) => u = v */
|
|
u = ctx.MkConst("u", forest);
|
|
v = ctx.MkConst("v", forest);
|
|
x = ctx.MkConst("x", tree);
|
|
y = ctx.MkConst("y", tree);
|
|
l1 = ctx.MkApp(cons1_decl, new Expr[] { x, u });
|
|
l2 = ctx.MkApp(cons1_decl, new Expr[] { y, v });
|
|
Prove(ctx, ctx.MkImplies(ctx.MkEq(l1, l2), ctx.MkEq(u, v)));
|
|
Prove(ctx, ctx.MkImplies(ctx.MkEq(l1, l2), ctx.MkEq(x, y)));
|
|
|
|
/* is_nil(u) or is_cons(u) */
|
|
Prove(ctx, ctx.MkOr(new BoolExpr[] {
|
|
(BoolExpr) ctx.MkApp(is_nil1_decl, new Expr[] { u }),
|
|
(BoolExpr) ctx.MkApp(is_cons1_decl, new Expr[] { u }) }));
|
|
|
|
/* occurs check u != cons(x,u) */
|
|
Prove(ctx, ctx.MkNot(ctx.MkEq(u, l1)));
|
|
}
|
|
|
|
// / Demonstrate how to use #Eval.
|
|
|
|
public void EvalExample1(Context ctx) throws Z3Exception
|
|
{
|
|
System.out.println("EvalExample1");
|
|
|
|
IntExpr x = ctx.MkIntConst("x");
|
|
IntExpr y = ctx.MkIntConst("y");
|
|
IntExpr two = ctx.MkInt(2);
|
|
|
|
Solver solver = ctx.MkSolver();
|
|
|
|
/* assert x < y */
|
|
solver.Assert(ctx.MkLt(x, y));
|
|
|
|
/* assert x > 2 */
|
|
solver.Assert(ctx.MkGt(x, two));
|
|
|
|
/* find model for the constraints above */
|
|
Model model = null;
|
|
if (Status.SATISFIABLE == solver.Check())
|
|
{
|
|
model = solver.Model();
|
|
System.out.println(model);
|
|
System.out.println("\nevaluating x+y");
|
|
Expr v = model.Evaluate(ctx.MkAdd(new ArithExpr[] { x, y }), false);
|
|
if (v != null)
|
|
{
|
|
System.out.println("result = " + (v));
|
|
} else
|
|
{
|
|
System.out.println("Failed to evaluate: x+y");
|
|
}
|
|
} else
|
|
{
|
|
System.out.println("BUG, the constraints are satisfiable.");
|
|
}
|
|
}
|
|
|
|
// / Demonstrate how to use #Eval on tuples.
|
|
|
|
public void EvalExample2(Context ctx) throws Z3Exception
|
|
{
|
|
System.out.println("EvalExample2");
|
|
|
|
Sort int_type = ctx.IntSort();
|
|
TupleSort tuple = ctx.MkTupleSort(ctx.MkSymbol("mk_tuple"), // name of
|
|
// tuple
|
|
// constructor
|
|
new Symbol[] { ctx.MkSymbol("first"), ctx.MkSymbol("second") }, // names
|
|
// of
|
|
// projection
|
|
// operators
|
|
new Sort[] { int_type, int_type } // types of projection
|
|
// operators
|
|
);
|
|
FuncDecl first = tuple.FieldDecls()[0]; // declarations are for
|
|
// projections
|
|
FuncDecl second = tuple.FieldDecls()[1];
|
|
Expr tup1 = ctx.MkConst("t1", tuple);
|
|
Expr tup2 = ctx.MkConst("t2", tuple);
|
|
|
|
Solver solver = ctx.MkSolver();
|
|
|
|
/* assert tup1 != tup2 */
|
|
solver.Assert(ctx.MkNot(ctx.MkEq(tup1, tup2)));
|
|
/* assert first tup1 = first tup2 */
|
|
solver.Assert(ctx.MkEq(ctx.MkApp(first, tup1), ctx.MkApp(first, tup2)));
|
|
|
|
/* find model for the constraints above */
|
|
Model model = null;
|
|
if (Status.SATISFIABLE == solver.Check())
|
|
{
|
|
model = solver.Model();
|
|
System.out.println(model);
|
|
System.out.println("evaluating tup1 "
|
|
+ (model.Evaluate(tup1, false)));
|
|
System.out.println("evaluating tup2 "
|
|
+ (model.Evaluate(tup2, false)));
|
|
System.out.println("evaluating second(tup2) "
|
|
+ (model.Evaluate(ctx.MkApp(second, tup2), false)));
|
|
} else
|
|
{
|
|
System.out.println("BUG, the constraints are satisfiable.");
|
|
}
|
|
}
|
|
|
|
// / Demonstrate how to use <code>Push</code>and <code>Pop</code>to
|
|
// / control the size of models.
|
|
|
|
// / <remarks>Note: this test is specialized to 32-bit bitvectors.</remarks>
|
|
public void CheckSmall(Context ctx, Solver solver, BitVecExpr[] to_minimize)
|
|
throws Z3Exception
|
|
{
|
|
int num_Exprs = to_minimize.length;
|
|
int[] upper = new int[num_Exprs];
|
|
int[] lower = new int[num_Exprs];
|
|
for (int i = 0; i < upper.length; ++i)
|
|
{
|
|
upper[i] = Integer.MAX_VALUE;
|
|
lower[i] = 0;
|
|
}
|
|
boolean some_work = true;
|
|
int last_index = -1;
|
|
int last_upper = 0;
|
|
while (some_work)
|
|
{
|
|
solver.Push();
|
|
|
|
boolean check_is_sat = true;
|
|
while (check_is_sat && some_work)
|
|
{
|
|
// Assert all feasible bounds.
|
|
for (int i = 0; i < num_Exprs; ++i)
|
|
{
|
|
solver.Assert(ctx.MkBVULE(to_minimize[i],
|
|
ctx.MkBV(upper[i], 32)));
|
|
}
|
|
|
|
check_is_sat = Status.SATISFIABLE == solver.Check();
|
|
if (!check_is_sat)
|
|
{
|
|
if (last_index != -1)
|
|
{
|
|
lower[last_index] = last_upper + 1;
|
|
}
|
|
break;
|
|
}
|
|
System.out.println(solver.Model());
|
|
|
|
// narrow the bounds based on the current model.
|
|
for (int i = 0; i < num_Exprs; ++i)
|
|
{
|
|
Expr v = solver.Model().Evaluate(to_minimize[i], false);
|
|
int ui = ((BitVecNum) v).Int();
|
|
if (ui < upper[i])
|
|
{
|
|
upper[i] = (int) ui;
|
|
}
|
|
System.out.println(i + " " + lower[i] + " " + upper[i]);
|
|
}
|
|
|
|
// find a new bound to add
|
|
some_work = false;
|
|
last_index = 0;
|
|
for (int i = 0; i < num_Exprs; ++i)
|
|
{
|
|
if (lower[i] < upper[i])
|
|
{
|
|
last_upper = (upper[i] + lower[i]) / 2;
|
|
last_index = i;
|
|
solver.Assert(ctx.MkBVULE(to_minimize[i],
|
|
ctx.MkBV(last_upper, 32)));
|
|
some_work = true;
|
|
break;
|
|
}
|
|
}
|
|
}
|
|
solver.Pop();
|
|
}
|
|
}
|
|
|
|
// / Reduced-size model generation example.
|
|
|
|
public void FindSmallModelExample(Context ctx) throws Z3Exception
|
|
{
|
|
System.out.println("FindSmallModelExample");
|
|
|
|
BitVecExpr x = ctx.MkBVConst("x", 32);
|
|
BitVecExpr y = ctx.MkBVConst("y", 32);
|
|
BitVecExpr z = ctx.MkBVConst("z", 32);
|
|
|
|
Solver solver = ctx.MkSolver();
|
|
|
|
solver.Assert(ctx.MkBVULE(x, ctx.MkBVAdd(y, z)));
|
|
CheckSmall(ctx, solver, new BitVecExpr[] { x, y, z });
|
|
}
|
|
|
|
// / Simplifier example.
|
|
|
|
public void SimplifierExample(Context ctx) throws Z3Exception
|
|
{
|
|
System.out.println("SimplifierExample");
|
|
|
|
IntExpr x = ctx.MkIntConst("x");
|
|
IntExpr y = ctx.MkIntConst("y");
|
|
IntExpr z = ctx.MkIntConst("z");
|
|
IntExpr u = ctx.MkIntConst("u");
|
|
|
|
Expr t1 = ctx.MkAdd(new ArithExpr[] {
|
|
x,
|
|
ctx.MkSub(new ArithExpr[] { y,
|
|
ctx.MkAdd(new ArithExpr[] { x, z }) }) });
|
|
Expr t2 = t1.Simplify();
|
|
System.out.println((t1) + " -> " + (t2));
|
|
}
|
|
|
|
// / Extract unsatisfiable core example
|
|
|
|
public void UnsatCoreAndProofExample() throws Z3Exception
|
|
{
|
|
System.out.println("UnsatCoreAndProofExample");
|
|
|
|
HashMap<String, String> cfg = new HashMap<String, String>();
|
|
cfg.put("PROOF_MODE", "2");
|
|
|
|
{
|
|
Context ctx = new Context(cfg);
|
|
Solver solver = ctx.MkSolver();
|
|
|
|
BoolExpr pa = ctx.MkBoolConst("PredA");
|
|
BoolExpr pb = ctx.MkBoolConst("PredB");
|
|
BoolExpr pc = ctx.MkBoolConst("PredC");
|
|
BoolExpr pd = ctx.MkBoolConst("PredD");
|
|
BoolExpr p1 = ctx.MkBoolConst("P1");
|
|
BoolExpr p2 = ctx.MkBoolConst("P2");
|
|
BoolExpr p3 = ctx.MkBoolConst("P3");
|
|
BoolExpr p4 = ctx.MkBoolConst("P4");
|
|
BoolExpr[] assumptions = new BoolExpr[] { ctx.MkNot(p1),
|
|
ctx.MkNot(p2), ctx.MkNot(p3), ctx.MkNot(p4) };
|
|
BoolExpr f1 = ctx.MkAnd(new BoolExpr[] { pa, pb, pc });
|
|
BoolExpr f2 = ctx.MkAnd(new BoolExpr[] { pa, ctx.MkNot(pb), pc });
|
|
BoolExpr f3 = ctx.MkOr(new BoolExpr[] { ctx.MkNot(pa),
|
|
ctx.MkNot(pc) });
|
|
BoolExpr f4 = pd;
|
|
|
|
solver.Assert(ctx.MkOr(new BoolExpr[] { f1, p1 }));
|
|
solver.Assert(ctx.MkOr(new BoolExpr[] { f2, p2 }));
|
|
solver.Assert(ctx.MkOr(new BoolExpr[] { f3, p3 }));
|
|
solver.Assert(ctx.MkOr(new BoolExpr[] { f4, p4 }));
|
|
Status result = solver.Check(assumptions);
|
|
|
|
if (result == Status.UNSATISFIABLE)
|
|
{
|
|
System.out.println("unsat");
|
|
System.out.println("proof: " + solver.Proof());
|
|
System.out.println("core: ");
|
|
for (Expr c : solver.UnsatCore())
|
|
{
|
|
System.out.println(c);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
public void FiniteDomainExample(Context ctx) throws Z3Exception
|
|
{
|
|
System.out.println("FiniteDomainExample");
|
|
|
|
FiniteDomainSort s = ctx.MkFiniteDomainSort("S", 10);
|
|
FiniteDomainSort t = ctx.MkFiniteDomainSort("T", 10);
|
|
Expr s1 = ctx.MkNumeral(1, s);
|
|
Expr t1 = ctx.MkNumeral(1, t);
|
|
System.out.println(s);
|
|
System.out.println(t);
|
|
System.out.println(s1);
|
|
System.out.println(ctx.MkNumeral(2, s));
|
|
System.out.println(t1);
|
|
// But you cannot mix numerals of different sorts
|
|
// even if the size of their domains are the same:
|
|
// System.out.println(ctx.MkEq(s1, t1));
|
|
}
|
|
|
|
public static void main(String[] args)
|
|
{
|
|
JavaExample p = new JavaExample();
|
|
try
|
|
{
|
|
Context.ToggleWarningMessages(true);
|
|
Log.Open("test.log");
|
|
|
|
System.out.print("Z3 Major Version: ");
|
|
System.out.println(Version.Major());
|
|
System.out.print("Z3 Full Version: ");
|
|
System.out.println(Version.getString());
|
|
|
|
p.SimpleExample();
|
|
|
|
{
|
|
HashMap<String, String> cfg = new HashMap<>();
|
|
cfg.put("MODEL", "true");
|
|
cfg.put("PROOF_MODE", "2");
|
|
Context ctx = new Context(cfg);
|
|
p.BasicTests(ctx);
|
|
p.CastingTest(ctx);
|
|
p.SudokuExample(ctx);
|
|
p.QuantifierExample1(ctx);
|
|
p.QuantifierExample2(ctx);
|
|
p.LogicExample(ctx);
|
|
p.ParOrExample(ctx);
|
|
p.FindModelExample1(ctx);
|
|
p.FindModelExample2(ctx);
|
|
p.ProveExample1(ctx);
|
|
p.ProveExample2(ctx);
|
|
p.PushPopExample1(ctx);
|
|
p.ArrayExample1(ctx);
|
|
p.ArrayExample2(ctx);
|
|
p.ArrayExample3(ctx);
|
|
p.TupleExample(ctx);
|
|
p.BitvectorExample1(ctx);
|
|
p.BitvectorExample2(ctx);
|
|
p.ParserExample1(ctx);
|
|
p.ParserExample2(ctx);
|
|
p.ParserExample3(ctx);
|
|
p.ParserExample4(ctx);
|
|
p.ParserExample5(ctx);
|
|
p.ITEExample(ctx);
|
|
p.EnumExample(ctx);
|
|
p.ListExample(ctx);
|
|
p.TreeExample(ctx);
|
|
p.ForestExample(ctx);
|
|
p.EvalExample1(ctx);
|
|
p.EvalExample2(ctx);
|
|
p.FindSmallModelExample(ctx);
|
|
p.SimplifierExample(ctx);
|
|
p.FiniteDomainExample(ctx);
|
|
}
|
|
|
|
p.QuantifierExample3();
|
|
p.QuantifierExample4();
|
|
p.UnsatCoreAndProofExample();
|
|
|
|
Log.Close();
|
|
if (Log.isOpen())
|
|
System.out.println("Log is still open!");
|
|
} catch (Z3Exception ex)
|
|
{
|
|
System.out.println("Z3 Managed Exception: " + ex.getMessage());
|
|
System.out.println("Stack trace: ");
|
|
ex.printStackTrace(System.out);
|
|
} catch (TestFailedException ex)
|
|
{
|
|
System.out.println("TEST CASE FAILED: " + ex.getMessage());
|
|
System.out.println("Stack trace: ");
|
|
ex.printStackTrace(System.out);
|
|
} catch (Exception ex)
|
|
{
|
|
System.out.println("Unknown Exception: " + ex.getMessage());
|
|
System.out.println("Stack trace: ");
|
|
ex.printStackTrace(System.out);
|
|
}
|
|
}
|
|
}
|