/// <summary>
/// Generates a slightly randomized expression.
/// </summary>
static BoolExpr MkRandomExpr(Context ctx, System.Random rng)
{
int limit = 1073741823;
Sort i = ctx.IntSort;
Sort b = ctx.BoolSort;
Symbol sr1 = ctx.MkSymbol(rng.Next(0, limit));
Symbol sr2 = ctx.MkSymbol(rng.Next(0, limit));
Symbol sr3 = ctx.MkSymbol(rng.Next(0, limit));
FuncDecl r1 = ctx.MkFuncDecl(sr1, i, b);
FuncDecl r2 = ctx.MkFuncDecl(sr2, i, b);
FuncDecl r3 = ctx.MkFuncDecl(sr3, i, b);
Symbol s = ctx.MkSymbol(rng.Next(0, limit));
Expr x = ctx.MkConst(s, i);
BoolExpr r1x = (BoolExpr)ctx.MkApp(r1, x);
BoolExpr r2x = (BoolExpr)ctx.MkApp(r2, x);
BoolExpr r3x = (BoolExpr)ctx.MkApp(r3, x);
Expr[] vars = { x };
BoolExpr rl1 = ctx.MkForall(vars, ctx.MkImplies(r1x, r2x));
BoolExpr rl2 = ctx.MkForall(vars, ctx.MkImplies(r2x, r1x));
BoolExpr rl3 = (BoolExpr)ctx.MkApp(r1, ctx.MkInt(3));
BoolExpr q = (BoolExpr)ctx.MkApp(r3, ctx.MkInt(2));
BoolExpr a1 = ctx.MkNot(q);
BoolExpr q1 = ctx.MkExists(vars, ctx.MkAnd(r3x, r2x));
BoolExpr q2 = ctx.MkExists(vars, ctx.MkAnd(r3x, r1x));
BoolExpr[] all = { a1, q1, q2 };
return ctx.MkAnd(all);
}
public static void ProveExample2(Context ctx)
{
Console.WriteLine("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((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(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);
Console.WriteLine("prove: not(g(g(x) - g(y)) = g(z)), x + z <= y <= x implies z < 0");
Prove(ctx, f, c1, c2, c3);
/* disprove z < -1 */
IntExpr minus_one = ctx.MkInt(-1);
f = ctx.MkLt(z, minus_one);
Console.WriteLine("disprove: not(g(g(x) - g(y)) = g(z)), x + z <= y <= x implies z < -1");
Disprove(ctx, f, c1, c2, c3);
}
//Constructor
public Z3Context()
{
//Initialize Config and Context
_config = new Config();
_config.SetParamValue("MODEL", "true"); // corresponds to /m switch
_config.SetParamValue("MACRO_FINDER", "true");
_context = new Context(_config);
//Setup custom conversion method BoolToInt (boolean -> integer)----------------------------------------------------------------
FuncDecl boolToInt = _context.MkFuncDecl("BoolToInt", _context.MkBoolSort(), _context.MkIntSort());
Term i = _context.MkConst("i", _context.MkBoolSort());
Term fDef = _context.MkIte(_context.MkEq(i, _context.MkTrue()), _context.MkIntNumeral(1), _context.MkIntNumeral(0)); // x == true => 1, x == false => 0
Term fStatement = _context.MkForall(0, new Term[] { i }, null, _context.MkEq(_context.MkApp(boolToInt, i), fDef));
_context.AssertCnstr(fStatement);
//
_functions.Add("BoolToInt", new Z3Function(boolToInt));
//-----------------------------------------------------------------------------------------------------------------------------
}
internal EnumSort(Context ctx, Symbol name, Symbol[] enumNames)
: base(ctx)
{
Contract.Requires(ctx != null);
Contract.Requires(name != null);
Contract.Requires(enumNames != null);
int n = enumNames.Length;
IntPtr[] n_constdecls = new IntPtr[n];
IntPtr[] n_testers = new IntPtr[n];
NativeObject = Native.Z3_mk_enumeration_sort(ctx.nCtx, name.NativeObject, (uint)n,
Symbol.ArrayToNative(enumNames), n_constdecls, n_testers);
_constdecls = new FuncDecl[n];
for (uint i = 0; i < n; i++)
_constdecls[i] = new FuncDecl(ctx, n_constdecls[i]);
_testerdecls = new FuncDecl[n];
for (uint i = 0; i < n; i++)
_testerdecls[i] = new FuncDecl(ctx, n_testers[i]);
_consts = new Expr[n];
for (uint i = 0; i < n; i++)
_consts[i] = ctx.MkApp(_constdecls[i]);
}
/// <summary>
/// Some basic tests.
/// </summary>
static void BasicTests(Context ctx)
{
Console.WriteLine("BasicTests");
Symbol qi = ctx.MkSymbol(1);
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, 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);
g.Assert(trivial_eq);
g.Assert(nontrivial_eq);
Console.WriteLine("Goal: " + g);
Solver solver = ctx.MkSolver();
foreach (BoolExpr a in 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);
ar = ApplyTactic(ctx, ctx.MkTactic("smt"), g2);
if (ar.NumSubgoals != 1 || !ar.Subgoals[0].IsDecidedSat)
throw new TestFailedException();
g2 = ctx.MkGoal(true, true);
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);
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;
Console.WriteLine("Numerator: " + inum + " Denominator: " + iden);
if (inum.ToString() != "42" || iden.ToString() != "43")
throw new TestFailedException();
if (rn.ToDecimalString(3) != "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() != bn)
throw new TestFailedException();
if (ctx.MkBV(bn, 128).BigInteger.ToString() != bn)
throw new TestFailedException();
if (ctx.MkBV(bn, 32).BigInteger.ToString() == bn)
throw new TestFailedException();
// Error handling test.
try
{
IntExpr i = ctx.MkInt("1/2");
throw new TestFailedException(); // unreachable
}
catch (Z3Exception)
{
}
}
/// <summary>
/// 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"/>
/// </summary>
public static void QuantifierExample4(Context ctx)
{
Console.WriteLine("QuantifierExample4");
/* If quantified formulas are asserted in a logical context, then
the model produced by Z3 should be viewed as a potential model. */
/* 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, x, y);
Expr fwv = ctx.MkApp(f, 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, false, inj, p1);
/* disprove f(x, y) = f(w, v) implies x = w */
BoolExpr p3 = ctx.MkEq(x, w);
Disprove(ctx, p3, false, inj, p1);
}
static void QuantifierExample2(Context ctx)
{
Console.WriteLine("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, x);
Expr f_y = ctx.MkApp(f, y);
Expr g_y = ctx.MkApp(g, y);
Pattern[] pats = new Pattern[] { ctx.MkPattern(new Expr[] { f_x, g_y }) };
Expr[] no_pats = new Expr[] { f_y };
Expr[] bound = new Expr[2] { x, y };
Expr body = ctx.MkAnd(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"));
Console.WriteLine("{0}", 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, x);
Expr f_y = ctx.MkApp(f, y);
Expr g_y = ctx.MkApp(g, 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(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")
);
Console.WriteLine("{0}", q2);
}
Console.WriteLine("{0}", (q1.Equals(q2)));
}
/// <summary>
/// A simple array example.
/// </summary>
/// <param name="ctx"></param>
static void ArrayExample1(Context ctx)
{
Console.WriteLine("ArrayExample1");
Goal g = ctx.MkGoal(true);
ArraySort asort = ctx.MkArraySort(ctx.IntSort, ctx.MkBitVecSort(32));
ArrayExpr aex = (ArrayExpr)ctx.MkConst(ctx.MkSymbol("MyArray"), asort);
Expr sel = ctx.MkSelect(aex, ctx.MkInt(0));
g.Assert(ctx.MkEq(sel, ctx.MkBV(42, 32)));
Symbol xs = ctx.MkSymbol("x");
IntExpr xc = (IntExpr)ctx.MkConst(xs, ctx.IntSort);
Symbol fname = ctx.MkSymbol("f");
Sort[] domain = { ctx.IntSort };
FuncDecl fd = ctx.MkFuncDecl(fname, domain, ctx.IntSort);
Expr[] fargs = { ctx.MkConst(xs, ctx.IntSort) };
IntExpr fapp = (IntExpr)ctx.MkApp(fd, fargs);
g.Assert(ctx.MkEq(ctx.MkAdd(xc, fapp), ctx.MkInt(123)));
Solver s = ctx.MkSolver();
foreach (BoolExpr a in g.Formulas)
s.Assert(a);
Console.WriteLine("Solver: " + s);
Status q = s.Check();
Console.WriteLine("Status: " + q);
if (q != Status.SATISFIABLE)
throw new TestFailedException();
Console.WriteLine("Model = " + s.Model);
Console.WriteLine("Interpretation of MyArray:\n" + s.Model.FuncInterp(aex.FuncDecl));
Console.WriteLine("Interpretation of x:\n" + s.Model.ConstInterp(xc));
Console.WriteLine("Interpretation of f:\n" + s.Model.FuncInterp(fd));
Console.WriteLine("Interpretation of MyArray as Term:\n" + s.Model.FuncInterp(aex.FuncDecl));
}
/// <summary>
/// Demonstrate how to use #Eval on tuples.
/// </summary>
public static void EvalExample2(Context ctx)
{
Console.WriteLine("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;
Console.WriteLine("{0}", model);
Console.WriteLine("evaluating tup1 {0}", (model.Evaluate(tup1)));
Console.WriteLine("evaluating tup2 {0}", (model.Evaluate(tup2)));
Console.WriteLine("evaluating second(tup2) {0}",
(model.Evaluate(ctx.MkApp(second, tup2))));
}
else
{
Console.WriteLine("BUG, the constraints are satisfiable.");
}
}
/// <summary>
/// Create a forest of trees.
/// </summary>
/// <remarks>
/// forest ::= nil | cons(tree, forest)
/// tree ::= nil | cons(forest, forest)
/// </remarks>
public static void ForestExample(Context ctx)
{
Console.WriteLine("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 };
uint[] sort1_refs = new uint[] { 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 };
uint[] sort2_refs = new uint[] { 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, nil2, nil1);
t1 = ctx.MkApp(cons2_decl, nil1, nil1);
t2 = ctx.MkApp(cons2_decl, f1, nil1);
t3 = ctx.MkApp(cons2_decl, f1, f1);
t4 = ctx.MkApp(cons2_decl, nil1, f1);
f2 = ctx.MkApp(cons1_decl, t1, nil1);
f3 = ctx.MkApp(cons1_decl, 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, x, u);
l2 = ctx.MkApp(cons1_decl, 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((BoolExpr)ctx.MkApp(is_nil1_decl, u),
(BoolExpr)ctx.MkApp(is_cons1_decl, u)));
/* occurs check u != cons(x,u) */
Prove(ctx, ctx.MkNot(ctx.MkEq(u, l1)));
}
/// <summary>
/// Create a binary tree datatype.
/// </summary>
public static void TreeExample(Context ctx)
{
Console.WriteLine("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 };
uint[] sort_refs = new uint[] { 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, nil, nil);
l2 = ctx.MkApp(cons_decl, 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, x, u);
l2 = ctx.MkApp(cons_decl, 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((BoolExpr)ctx.MkApp(is_nil_decl, u), (BoolExpr)ctx.MkApp(is_cons_decl, 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, ctx.MkApp(car_decl, u), ctx.MkApp(cdr_decl, u)));
fml = ctx.MkImplies((BoolExpr)ctx.MkApp(is_cons_decl, u), fml1);
Console.WriteLine("Formula {0}", fml);
Prove(ctx, fml);
Disprove(ctx, fml1);
}
/// <summary>
/// Create a list datatype.
/// </summary>
public static void ListExample(Context ctx)
{
Console.WriteLine("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, ctx.MkInt(1), nil);
l2 = ctx.MkApp(int_list.ConsDecl, 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, x, nil);
l2 = ctx.MkApp(int_list.ConsDecl, 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, x, u);
l2 = ctx.MkApp(int_list.ConsDecl, 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((BoolExpr)ctx.MkApp(int_list.IsNilDecl, u),
(BoolExpr)ctx.MkApp(int_list.IsConsDecl, 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, ctx.MkApp(int_list.HeadDecl, u),
ctx.MkApp(int_list.TailDecl, u)));
fml = ctx.MkImplies((BoolExpr)ctx.MkApp(int_list.IsConsDecl, u), fml1);
Console.WriteLine("Formula {0}", fml);
Prove(ctx, fml);
Disprove(ctx, fml1);
}
/// <summary>
/// Create an enumeration data type.
/// </summary>
public static void EnumExample(Context ctx)
{
Console.WriteLine("EnumExample");
Symbol name = ctx.MkSymbol("fruit");
EnumSort fruit = ctx.MkEnumSort(ctx.MkSymbol("fruit"), new Symbol[] { ctx.MkSymbol("apple"), ctx.MkSymbol("banana"), ctx.MkSymbol("orange") });
Console.WriteLine("{0}", (fruit.Consts[0]));
Console.WriteLine("{0}", (fruit.Consts[1]));
Console.WriteLine("{0}", (fruit.Consts[2]));
Console.WriteLine("{0}", (fruit.TesterDecls[0]));
Console.WriteLine("{0}", (fruit.TesterDecls[1]));
Console.WriteLine("{0}", (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) }));
}
/// <summary>
/// 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"/>
/// </summary>
public static void QuantifierExample3()
{
Console.WriteLine("QuantifierExample3");
Dictionary<string, string> cfg = new Dictionary<string, string>() {
{ "MBQI", "false" },
{ "PROOF_MODE", "2" },
{ "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. */
using (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, x, y);
Expr fwv = ctx.MkApp(f, 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, inj, p1);
/* disprove f(x, y) = f(w, v) implies x = w */
BoolExpr p3 = ctx.MkEq(x, w);
Disprove(ctx, p3, inj, p1);
}
}
// CMW: get_implied_equalities is deprecated.
///*!
// \brief Extract implied equalities.
//*/
public void get_implied_equalities_example()
{
if (this.z3 != null)
{
this.z3.Dispose();
}
Config p = new Config();
p.SetParam("ARITH_EQ_BOUNDS","true");
this.z3 = new Context(p);
Sort int_sort = z3.MkIntSort();
Expr a = mk_int_var("a");
Expr b = mk_int_var("b");
Expr c = mk_int_var("c");
Expr d = mk_int_var("d");
FuncDecl f = z3.MkFuncDecl("f", int_sort, int_sort);
Expr fa = z3.MkApp(f,a);
Expr fb = z3.MkApp(f,b);
Expr fc = z3.MkApp(f,c);
Expr[] Exprs = new Expr[]{ a, b, c, d, fa, fb, fc };
uint[] class_ids;
solver.Assert(z3.MkEq(a, b));
solver.Assert(z3.MkEq(b, c));
solver.Assert(z3.MkLe((ArithExpr)fc, (ArithExpr)a));
solver.Assert(z3.MkLe((ArithExpr)b, (ArithExpr)fb));
int num_Exprs = Exprs.Length;
z3.GetImpliedEqualities(Exprs, out class_ids);
for (int i = 0; i < num_Exprs; ++i) {
Console.WriteLine("Class {0} |-> {1}", Exprs[i], class_ids[i]);
}
}
