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public MkBound ( uint index, Microsoft.Z3.Sort ty ) : |
||
| index | uint | The de-Bruijn index of the variable |
| ty | Microsoft.Z3.Sort | The sort of the variable |
| Résultat | ||
public void Run()
{
using (Context ctx = new Context()) {
ctx.UpdateParamValue("DL_ENGINE","1");
ctx.UpdateParamValue("DL_PDR_USE_FARKAS","true");
// ctx.UpdateParamValue("VERBOSE","2");
var s = ctx.MkFixedpoint();
BoolSort B = ctx.BoolSort;
IntSort I = ctx.IntSort;
FuncDecl mc = ctx.MkFuncDecl("mc", new Sort[]{I, I}, B);
ArithExpr x = (ArithExpr)ctx.MkBound(0,I);
ArithExpr y = (ArithExpr)ctx.MkBound(1,I);
ArithExpr z = (ArithExpr)ctx.MkBound(2,I);
s.RegisterRelation(mc);
BoolExpr gt = ctx.MkGt(x, ctx.MkInt(100));
s.AddRule(ctx.MkImplies(gt,(BoolExpr)mc[x,ctx.MkSub(x,ctx.MkInt(10))]));
s.AddRule(ctx.MkImplies(ctx.MkAnd(ctx.MkNot(gt),
(BoolExpr) mc[ctx.MkAdd(x,ctx.MkInt(11)),y],
(BoolExpr) mc[y,z]),
(BoolExpr) mc[x,z]));
Console.WriteLine(s.Query(ctx.MkAnd((BoolExpr)mc[x,y], ctx.MkGt(y,ctx.MkInt(100)))));
Console.WriteLine(s.GetAnswer());
Console.WriteLine(s.Query(ctx.MkAnd((BoolExpr)mc[x,y], ctx.MkLt(y,ctx.MkInt(91)))));
Console.WriteLine(s.GetAnswer());
}
}
public void Run()
{
using (Context ctx = new Context()) {
var s = ctx.MkFixedpoint();
BoolSort B = ctx.BoolSort;
Sort BV8 = ctx.MkBitVecSort(8);
FuncDecl edge = ctx.MkFuncDecl("edge", new Sort[]{BV8, BV8}, B);
FuncDecl path = ctx.MkFuncDecl("path", new Sort[]{BV8, BV8}, B);
BitVecExpr x = (BitVecExpr)ctx.MkBound(0,BV8);
BitVecExpr y = (BitVecExpr)ctx.MkBound(1,BV8);
BitVecExpr z = (BitVecExpr)ctx.MkBound(2,BV8);
s.RegisterRelation(edge);
s.RegisterRelation(path);
s.AddRule(ctx.MkImplies((BoolExpr)edge[x,y],(BoolExpr)path[x,y]));
s.AddRule(ctx.MkImplies(ctx.MkAnd((BoolExpr)path[x,y],(BoolExpr)path[y,z]),
(BoolExpr)path[x,z]));
for (uint i = 0; i < 128; ++i) {
s.AddFact(edge, i, i+1);
}
Console.WriteLine(s.Query((BoolExpr)path[ctx.MkBV(0,8),ctx.MkBV(129,8)]));
Console.WriteLine(s.GetAnswer());
Console.WriteLine(s.Query((BoolExpr)path[ctx.MkBV(0,8),ctx.MkBV(128,8)]));
Console.WriteLine(s.GetAnswer());
Console.WriteLine(s.Query((BoolExpr)path[x,ctx.MkBV(20,8)]));
Console.WriteLine(s.GetAnswer());
Console.WriteLine(s.Query(ctx.MkAnd((BoolExpr)path[x,y],
(BoolExpr)path[y,ctx.MkBV(20,8)])));
Console.WriteLine(s.GetAnswer());
}
}
/// <summary>
/// Create axiom: function f is injective in the i-th argument.
/// </summary>
/// <remarks>
/// The following axiom is produced:
/// <c>
/// forall (x_0, ..., x_n) finv(f(x_0, ..., x_i, ..., x_{n-1})) = x_i
/// </c>
/// Where, <code>finv</code>is a fresh function declaration.
/// </summary>
public static BoolExpr InjAxiom(Context ctx, FuncDecl f, int i)
{
Sort[] domain = f.Domain;
uint sz = f.DomainSize;
if (i >= sz)
{
Console.WriteLine("failed to create inj axiom");
return null;
}
/* declare the i-th inverse of f: finv */
Sort finv_domain = f.Range;
Sort finv_range = domain[i];
FuncDecl finv = ctx.MkFuncDecl("f_fresh", finv_domain, finv_range);
/* allocate temporary arrays */
Expr[] xs = new Expr[sz];
Symbol[] names = new Symbol[sz];
Sort[] types = new Sort[sz];
/* fill types, names and xs */
for (uint j = 0; j < sz; j++)
{
types[j] = domain[j];
names[j] = ctx.MkSymbol(String.Format("x_{0}", j));
xs[j] = ctx.MkBound(j, types[j]);
}
Expr x_i = xs[i];
/* create f(x_0, ..., x_i, ..., x_{n-1}) */
Expr fxs = f[xs];
/* create f_inv(f(x_0, ..., x_i, ..., x_{n-1})) */
Expr finv_fxs = finv[fxs];
/* create finv(f(x_0, ..., x_i, ..., x_{n-1})) = x_i */
Expr eq = ctx.MkEq(finv_fxs, x_i);
/* use f(x_0, ..., x_i, ..., x_{n-1}) as the pattern for the quantifier */
Pattern p = ctx.MkPattern(new Expr[] { fxs });
/* create & assert quantifier */
BoolExpr q = ctx.MkForall(
types, /* types of quantified variables */
names, /* names of quantified variables */
eq,
1,
new Pattern[] { p } /* patterns */);
return q;
}
public void Run()
{
Dictionary<string, string> cfg = new Dictionary<string, string>() {
{ "AUTO_CONFIG", "true" } };
using (Context ctx = new Context(cfg))
{
FuncDecl f = ctx.MkFuncDecl("f", new Sort[] { ctx.IntSort, ctx.IntSort }, ctx.IntSort);
IntExpr x = ctx.MkIntConst("x");
IntExpr y = ctx.MkIntConst("y");
Quantifier qf = ctx.MkForall(new Expr[] { x, y }, ctx.MkEq(f[x, y], ctx.MkInt(0)));
Console.WriteLine(qf.Body);
Expr v1 = qf.Body.Args[0].Args[0];
Console.WriteLine(v1);
Console.WriteLine(v1 == ctx.MkBound(1, ctx.IntSort));
}
}
public void Run()
{
using (Context ctx = new Context()) {
var s = ctx.MkFixedpoint();
BoolSort B = ctx.BoolSort;
Sort BV8 = ctx.MkBitVecSort(8);
FuncDecl f = ctx.MkFuncDecl("f", BV8, B);
FuncDecl g = ctx.MkFuncDecl("g", BV8, B);
BitVecExpr b0 = (BitVecExpr)ctx.MkBound(0,BV8);
s.RegisterRelation(f);
s.RegisterRelation(g);
s.AddRule((BoolExpr)f[b0]);
BitVecExpr mask0 = ctx.MkBV(0xFE,8);
BoolExpr even = ctx.MkEq(b0,ctx.MkBVAND(b0,mask0));
s.AddRule(ctx.MkImplies(ctx.MkAnd((BoolExpr)f[b0],even), (BoolExpr)g[b0]));
Console.WriteLine(s.Query((BoolExpr)g[b0]));
Console.WriteLine(s.GetAnswer());
}
}
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 basic example of how to use quantifiers.
/// </summary>
static void QuantifierExample1(Context ctx)
{
Console.WriteLine("QuantifierExample");
Sort[] types = new Sort[3];
IntExpr[] xs = new IntExpr[3];
Symbol[] names = new Symbol[3];
IntExpr[] vars = new IntExpr[3];
for (uint j = 0; j < 3; j++)
{
types[j] = ctx.IntSort;
names[j] = ctx.MkSymbol(String.Format("x_{0}", 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(ctx.MkEq(ctx.MkAdd(vars[0], ctx.MkInt(1)), ctx.MkInt(2)),
ctx.MkEq(ctx.MkAdd(vars[1], ctx.MkInt(2)),
ctx.MkAdd(vars[2], ctx.MkInt(3))));
Expr body_const = ctx.MkAnd(ctx.MkEq(ctx.MkAdd(xs[0], ctx.MkInt(1)), ctx.MkInt(2)),
ctx.MkEq(ctx.MkAdd(xs[1], ctx.MkInt(2)),
ctx.MkAdd(xs[2], ctx.MkInt(3))));
Expr x = ctx.MkForall(types, names, body_vars, 1, null, null, ctx.MkSymbol("Q1"), ctx.MkSymbol("skid1"));
Console.WriteLine("Quantifier X: " + x.ToString());
Expr y = ctx.MkForall(xs, body_const, 1, null, null, ctx.MkSymbol("Q2"), ctx.MkSymbol("skid2"));
Console.WriteLine("Quantifier Y: " + y.ToString());
}
