C# Context.MkImplies方法代码示例

    public void Run()
    {
        Dictionary<string, string> cfg = new Dictionary<string, string>() {
            { "AUTO_CONFIG", "true" } };

        using (Context ctx = new Context(cfg))
        {
            BoolExpr p1 = ctx.MkBoolConst("p1");
            BoolExpr p2 = ctx.MkBoolConst("p2");
            BoolExpr p3 = ctx.MkBoolConst("p3");

            IntExpr x = ctx.MkIntConst("x");
            IntExpr y = ctx.MkIntConst("y");

            Solver s = ctx.MkSolver();

            s.Assert(ctx.MkImplies(p1, ctx.MkGt(x, ctx.MkInt(10))),
                     ctx.MkImplies(p1, ctx.MkGt(y, x)),
                     ctx.MkImplies(p2, ctx.MkLt(y, ctx.MkInt(5))),
                     ctx.MkImplies(p3, ctx.MkGt(y, ctx.MkInt(0))));

            Console.WriteLine(s);
            Console.WriteLine(s.Check(p1, p2, p3));
            Console.WriteLine("Core: ");
            foreach (Expr e in s.UnsatCore)
                Console.WriteLine(e);

            Console.WriteLine(s.Check(p1, p3));
            Console.WriteLine(s.Model);
        }
    }

 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>
 /// 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 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());
        }
    }

        /// <summary>
        /// Prove <tt>store(a1, i1, v1) = store(a2, i2, v2) implies (i1 = i3 or i2 = i3 or select(a1, i3) = select(a2, i3))</tt>.
        /// </summary>
        /// <remarks>This example demonstrates how to use the array theory.</remarks>
        public static void ArrayExample2(Context ctx)
        {
            Console.WriteLine("ArrayExample2");

            Sort int_type = ctx.IntSort;
            Sort array_type = ctx.MkArraySort(int_type, int_type);

            ArrayExpr a1 = (ArrayExpr)ctx.MkConst("a1", array_type);
            ArrayExpr a2 = ctx.MkArrayConst("a2", int_type, int_type);
            Expr i1 = ctx.MkConst("i1", int_type);
            Expr i2 = ctx.MkConst("i2", int_type);
            Expr i3 = ctx.MkConst("i3", int_type);
            Expr v1 = ctx.MkConst("v1", int_type);
            Expr v2 = ctx.MkConst("v2", int_type);

            Expr st1 = ctx.MkStore(a1, i1, v1);
            Expr st2 = ctx.MkStore(a2, i2, v2);

            Expr sel1 = ctx.MkSelect(a1, i3);
            Expr sel2 = ctx.MkSelect(a2, i3);

            /* create antecedent */
            BoolExpr antecedent = ctx.MkEq(st1, st2);

            /* create consequent: i1 = i3 or  i2 = i3 or select(a1, i3) = select(a2, i3) */
            BoolExpr consequent = ctx.MkOr(new BoolExpr[] { ctx.MkEq(i1, i3), ctx.MkEq(i2, i3), ctx.MkEq(sel1, sel2) });

            /* prove store(a1, i1, v1) = store(a2, i2, v2) implies (i1 = i3 or i2 = i3 or select(a1, i3) = select(a2, i3)) */
            BoolExpr thm = ctx.MkImplies(antecedent, consequent);
            Console.WriteLine("prove: store(a1, i1, v1) = store(a2, i2, v2) implies (i1 = i3 or i2 = i3 or select(a1, i3) = select(a2, i3))");
            Console.WriteLine("{0}", (thm));
            Prove(ctx, thm);
        }

    public void Run()
    {
        Dictionary<string, string> cfg = new Dictionary<string, string>() {
            { "AUTO_CONFIG", "true" } };

        using (Context ctx = new Context(cfg))
        {
            Sort A = ctx.MkUninterpretedSort("A");
            Sort B = ctx.MkUninterpretedSort("B");

            FuncDecl f = ctx.MkFuncDecl("f", A, B);

            Expr a1 = ctx.MkConst("a1", A);
            Expr a2 = ctx.MkConst("a2", A);
            Expr b = ctx.MkConst("b", B);
            Expr x = ctx.MkConst("x", A);
            Expr y = ctx.MkConst("y", A);

            Solver s = ctx.MkSolver();

            s.Assert(ctx.MkNot(ctx.MkEq(a1, a2)));
            s.Assert(ctx.MkEq(f[a1], b));
            s.Assert(ctx.MkEq(f[a2], b));
            s.Assert(ctx.MkForall(new Expr[] { x, y }, ctx.MkImplies(ctx.MkEq(f[x], f[y]),
                                                                     ctx.MkEq(x, y)),
                                                       1,
                                                       new Pattern[] { ctx.MkPattern(f[x], f[y]) }));
            Console.WriteLine(s);
            Console.WriteLine(s.Check());
        }
    }

 public BoolExpr DependsOn(Context ctx, BoolExpr pack, BoolExpr[] deps)
 {
     BoolExpr[] q = new BoolExpr[deps.Length];
     for (uint i = 0; i < deps.Length; i++)
         q[i] = ctx.MkImplies(pack, deps[i]);
     return ctx.MkAnd(q);
 }

    public void Run()
    {
        using (Context ctx = new Context())
        {
            BoolExpr p = ctx.MkBoolConst("p");
            BoolExpr q = ctx.MkBoolConst("q");
            Console.WriteLine(ctx.MkAnd(p, q));
            Console.WriteLine(ctx.MkOr(p, q));
            Console.WriteLine(ctx.MkAnd(p, ctx.MkTrue()));
            Console.WriteLine(ctx.MkOr(p, ctx.MkFalse()));
            Console.WriteLine(ctx.MkNot(p));
            Console.WriteLine(ctx.MkImplies(p, q));
            Console.WriteLine(ctx.MkEq(p, q).Simplify());
            Console.WriteLine(ctx.MkEq(p, q));

            BoolExpr r = ctx.MkBoolConst("r");

            Console.WriteLine(ctx.MkNot(ctx.MkEq(p, ctx.MkNot(ctx.MkEq(q, r)))));
            Console.WriteLine(ctx.MkNot(ctx.MkEq(ctx.MkNot(ctx.MkEq(p, q)), r)));
            Console.WriteLine(ctx.MkEq(p, ctx.MkTrue()));
            Console.WriteLine(ctx.MkEq(p, ctx.MkFalse()));
            Console.WriteLine(ctx.MkEq(p, ctx.MkTrue()).Simplify());
            Console.WriteLine(ctx.MkEq(p, ctx.MkFalse()).Simplify());
            Console.WriteLine(ctx.MkEq(p, p).Simplify());
            Console.WriteLine(ctx.MkEq(p, q).Simplify());
            Console.WriteLine(ctx.MkAnd(p, q, r));
            Console.WriteLine(ctx.MkOr(p, q, r));

            IntExpr x = ctx.MkIntConst("x");

            Console.WriteLine(x is BoolExpr);
            Console.WriteLine(p is BoolExpr);
            Console.WriteLine(ctx.MkAnd(p, q) is BoolExpr);
            Console.WriteLine(p is BoolExpr);
            Console.WriteLine(ctx.MkAdd(x, ctx.MkInt(1)) is BoolExpr);
            Console.WriteLine(p.IsAnd);
            Console.WriteLine(ctx.MkOr(p, q).IsOr);
            Console.WriteLine(ctx.MkAnd(p, q).IsAnd);
            Console.WriteLine(x.IsNot);
            Console.WriteLine(p.IsNot);
            Console.WriteLine(ctx.MkNot(p));
            Console.WriteLine(ctx.MkNot(p).IsDistinct);
            Console.WriteLine(ctx.MkEq(p, q).IsDistinct);
            Console.WriteLine(ctx.MkDistinct(p, q).IsDistinct);
            Console.WriteLine(ctx.MkDistinct(x, ctx.MkAdd(x, ctx.MkInt(1)), ctx.MkAdd(x, ctx.MkInt(2))).IsDistinct);

            Console.WriteLine();

            Console.WriteLine(ctx.MkBool(true));
            Console.WriteLine(ctx.MkBool(false));
            Console.WriteLine(ctx.BoolSort);

            Context ctx1 = new Context();
            Console.WriteLine(ctx1.MkBool(true));
            Console.WriteLine(ctx1.BoolSort);
            Console.WriteLine(ctx1.MkBool(true).Sort == ctx1.BoolSort);
            Console.WriteLine(ctx1.MkBool(true).Sort == ctx.BoolSort);
            Console.WriteLine(ctx1.MkBool(true).Sort != ctx.BoolSort);
        }
    }

 public void Run()
 {
     using (Context ctx = new Context()) {
         var s = ctx.MkFixedpoint();
         BoolExpr a = ctx.MkBoolConst("a");
         BoolExpr b = ctx.MkBoolConst("b");
         BoolExpr c = ctx.MkBoolConst("c");
         s.RegisterRelation(a.FuncDecl);
         s.RegisterRelation(b.FuncDecl);
         s.RegisterRelation(c.FuncDecl);
         s.AddRule(ctx.MkImplies(a, b));
         s.AddRule(ctx.MkImplies(b, c));
         Console.WriteLine(s.Query(c));
         s.AddRule(a);
         Console.WriteLine(s.Query(c));
     }
 }

    public void Run()
    {
        Dictionary<string, string> cfg = new Dictionary<string, string>() {
            { "AUTO_CONFIG", "true" } };

        using (Context ctx = new Context(cfg))
        {
            Sort T = ctx.MkUninterpretedSort("Type");
            FuncDecl subtype = ctx.MkFuncDecl("subtype", new Sort[] { T, T }, ctx.BoolSort);
            FuncDecl array_of = ctx.MkFuncDecl("array_of", T, T);
            Expr root = ctx.MkConst("root", T);

            Expr x = ctx.MkConst("x", T);
            Expr y = ctx.MkConst("y", T);
            Expr z = ctx.MkConst("z", T);

            BoolExpr[] axioms = new BoolExpr[] {
                ctx.MkForall(new Expr[] { x }, subtype[x, x]),
                ctx.MkForall(new Expr[] { x, y , z }, ctx.MkImplies(ctx.MkAnd((BoolExpr)subtype[x,y], (BoolExpr)subtype[y,z]), (BoolExpr)subtype[x,z])),
                ctx.MkForall(new Expr[] { x, y }, ctx.MkImplies(ctx.MkAnd((BoolExpr)subtype[x, y], (BoolExpr)subtype[y,x]), ctx.MkEq(x, y))),
                ctx.MkForall(new Expr[] { x, y, z }, ctx.MkImplies(ctx.MkAnd((BoolExpr)subtype[x,y],(BoolExpr)subtype[x,z]),
                                                                   ctx.MkOr((BoolExpr)subtype[y,z], (BoolExpr)subtype[z,y]))),
                ctx.MkForall(new Expr[] { x, y }, ctx.MkImplies((BoolExpr)subtype[x,y], (BoolExpr)subtype[array_of[x], array_of[y]])),
                ctx.MkForall(new Expr[] { x }, (BoolExpr) subtype[root, x])
            };

            Solver s = ctx.MkSolver();
            s.Assert(axioms);
            Console.WriteLine(s);
            Console.WriteLine(s.Check());
            Expr[] universe = s.Model.SortUniverse(T);
            foreach (var e in universe)
                Console.WriteLine(e);
            Console.WriteLine(s.Model);
        }
    }

 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());
     }
 }

//.........这里部分代码省略.........
            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>
        /// Prove <tt>x = y implies g(x) = g(y)</tt>, and
        /// disprove <tt>x = y implies g(g(x)) = g(y)</tt>.
        /// </summary>
        /// <remarks>This function demonstrates how to create uninterpreted
        /// types and functions.</remarks>
        public static void ProveExample1(Context ctx)
        {
            Console.WriteLine("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[x];
            Expr gy = g[y];

            /* assert x = y */
            BoolExpr eq = ctx.MkEq(x, y);

            /* prove g(x) = g(y) */
            BoolExpr f = ctx.MkEq(gx, gy);
            Console.WriteLine("prove: x = y implies g(x) = g(y)");
            Prove(ctx, ctx.MkImplies(eq, f));

            /* create g(g(x)) */
            Expr ggx = g[gx];

            /* disprove g(g(x)) = g(y) */
            f = ctx.MkEq(ggx, gy);
            Console.WriteLine("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);
            Console.WriteLine(m);
        }