arith_theorem_producer.h

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00001 /*****************************************************************************/
00002 /*!
00003  * \file arith_theorem_producer.h
00004  * \brief TRUSTED implementation of arithmetic proof rules
00005  * 
00006  * Author: Vijay Ganesh, Sergey Berezin
00007  * 
00008  * Created: Dec 13 02:09:04 GMT 2002
00009  *
00010  * <hr>
00011  *
00012  * License to use, copy, modify, sell and/or distribute this software
00013  * and its documentation for any purpose is hereby granted without
00014  * royalty, subject to the terms and conditions defined in the \ref
00015  * LICENSE file provided with this distribution.
00016  * 
00017  * <hr>
00018  * 
00019  */
00020 /*****************************************************************************/
00021 
00022 #ifndef _cvc3__arith_theorem_producer_h_
00023 #define _cvc3__arith_theorem_producer_h_
00024 
00025 #include "arith_proof_rules.h"
00026 #include "theorem_producer.h"
00027 #include "theory_arith_new.h"
00028 
00029 namespace CVC3 {
00030   class TheoryArithNew;
00031   
00032   class ArithTheoremProducer: public ArithProofRules, public TheoremProducer {
00033     TheoryArithNew* d_theoryArith;
00034   private:
00035     /*! \name Auxiliary functions for eqElimIntRule()
00036      * Methods that compute the subterms used in eqElimIntRule()
00037      *@{
00038      */
00039     //! Compute the special modulus: i - m*floor(i/m+1/2)
00040     Rational modEq(const Rational& i, const Rational& m);
00041     //! Create the term 't'.  See eqElimIntRule().
00042     Expr create_t(const Expr& eqn);
00043     //! Create the term 't2'.  See eqElimIntRule().
00044     Expr create_t2(const Expr& lhs, const Expr& rhs, const Expr& t);
00045     //! Create the term 't3'.  See eqElimIntRule().
00046     Expr create_t3(const Expr& lhs, const Expr& rhs, const Expr& t);
00047     /*! @brief Takes sum = a_0 + a_1*x_1+...+a_n*x_n and returns the
00048      * vector of similar monomials (in 'summands') with coefficients
00049      * mod(a_i, m).  If divide flag is true, the coefficients will be
00050      * mod(a_i,m)/m.
00051      */
00052     void sumModM(std::vector<Expr>& summands, const Expr& sum,
00053      const Rational& m, const Rational& divisor);
00054     Expr monomialModM(const Expr& e,
00055           const Rational& m, const Rational& divisor);
00056     void sumMulF(std::vector<Expr>& summands, const Expr& sum,
00057      const Rational& m, const Rational& divisor);
00058     Expr monomialMulF(const Expr& e,
00059           const Rational& m, const Rational& divisor);
00060     //! Compute floor(i/m+1/2) + mod(i,m)
00061     Rational f(const Rational& i, const Rational& m);
00062     Expr substitute(const Expr& term, ExprMap<Expr>& eMap);
00063 
00064     /*@}*/
00065   public:
00066     //! Constructor
00067     ArithTheoremProducer(TheoremManager* tm, TheoryArithNew* theoryArith):
00068       TheoremProducer(tm), d_theoryArith(theoryArith) { }
00069 
00070     //! Create Expr from Rational (for convenience)
00071     Expr rat(Rational r) { return d_em->newRatExpr(r); }
00072     Type realType() { return d_theoryArith->realType(); }
00073     Type intType() { return d_theoryArith->intType(); }
00074     //! Construct the dark shadow expression representing lhs <= rhs
00075     Expr darkShadow(const Expr& lhs, const Expr& rhs) {
00076       return d_theoryArith->darkShadow(lhs, rhs);
00077     }
00078     //! Construct the gray shadow expression representing c1 <= v - e <= c2
00079     /*! Alternatively, v = e + i for some i s.t. c1 <= i <= c2
00080      */
00081     Expr grayShadow(const Expr& v, const Expr& e,
00082         const Rational& c1, const Rational& c2) {
00083       return d_theoryArith->grayShadow(v, e, c1, c2);
00084     }
00085 
00086     ////////////////////////////////////////////////////////////////////
00087     // Canonization rules
00088     ////////////////////////////////////////////////////////////////////
00089 
00090     // ==> x = 1 * x
00091     virtual Theorem varToMult(const Expr& e);
00092 
00093     // ==> -(e) = (-1) * e
00094     virtual Theorem uMinusToMult(const Expr& e);
00095 
00096     // ==> x - y = x + (-1) * y
00097     virtual Theorem minusToPlus(const Expr& x, const Expr& y);
00098 
00099     // Rule for unary minus: it just converts it to division by -1,
00100     virtual Theorem canonUMinusToDivide(const Expr& e);
00101 
00102     // Rules for division by constant 'd'
00103     // (c) / d ==> (c/d), takes c and d
00104     virtual Theorem canonDivideConst(const Expr& c, const Expr& d);
00105     // (c * x) / d ==> (c/d) * x, takes (c*x) and d
00106     virtual Theorem canonDivideMult(const Expr& cx, const Expr& d);
00107     // (+ c ...)/d ==> push division to all the coefficients.
00108     // The result is not canonical, but canonizing the summands will
00109     // make it canonical.
00110     // Takes (+ c ...) and d
00111     virtual Theorem canonDividePlus(const Expr& e, const Expr& d);
00112     // x / d ==> (1/d) * x, takes x and d
00113     virtual Theorem canonDivideVar(const Expr& e1, const Expr& e2);
00114 
00115     // Canon Rules for multiplication
00116 
00117     // TODO Deepak:
00118     // t1 * t2 where t1 and t2 are canonized expressions, i.e. it can be a
00119     // 1) Rational constant
00120     // 2) Arithmetic Leaf (var or term from another theory)
00121     // 3) (POW rational leaf)
00122     // 4) (MULT rational mterm'_1 ...) where each mterm' is of type (2) or (3)
00123     // 5) (PLUS rational sterm_1 sterm_2 ...) where each sterm is of 
00124     //     type (2) or (3) or (4) 
00125 
00126     static bool greaterthan(const Expr &, const Expr &);
00127     virtual Expr simplifiedMultExpr(std::vector<Expr> & mulKids);
00128     virtual Expr canonMultConstMult(const Expr & c, const Expr & e);
00129     virtual Expr canonMultConstPlus(const Expr & e1, const Expr & e2);
00130     virtual Expr canonMultPowPow(const Expr & e1, const Expr & e2);
00131     virtual Expr canonMultPowLeaf(const Expr & e1, const Expr & e2);
00132     virtual Expr canonMultLeafLeaf(const Expr & e1, const Expr & e2);
00133     virtual Expr canonMultLeafOrPowMult(const Expr & e1, const Expr & e2);
00134     virtual Expr canonCombineLikeTerms(const std::vector<Expr> & sumExprs);
00135     virtual Expr 
00136   canonMultLeafOrPowOrMultPlus(const Expr & e1, const Expr & e2);
00137     virtual Expr canonMultPlusPlus(const Expr & e1, const Expr & e2);
00138     virtual Theorem canonMultMtermMterm(const Expr& e);
00139     virtual Theorem canonPlus(const Expr & e);
00140     virtual Theorem canonInvertConst(const Expr & e);
00141     virtual Theorem canonInvertLeaf(const Expr & e);
00142     virtual Theorem canonInvertPow(const Expr & e);
00143     virtual Theorem canonInvertMult(const Expr & e);
00144     virtual Theorem canonInvert(const Expr & e);
00145 
00146     /**
00147      * Transform e = (SUM r t1 ... tn) @ 0 into (SUM t1 ... tn) @ -r. The first 
00148      * sum term (r) must be a rational and t1 ... tn terms must be canonised.
00149      * 
00150      * @param e the expression to transform 
00151      * @return rewrite theorem representing the transformation  
00152      */
00153     virtual Theorem moveSumConstantRight(const Expr& e);
00154 
00155     /** e[0]/e[1] ==> e[0]*(e[1])^-1 */
00156     virtual Theorem canonDivide(const Expr & e);
00157     
00158     /** Multiply out the operands of the multiplication (each of them is expected to be canonised */
00159     virtual Theorem canonMult(const Expr & e);
00160 
00161     // t*c ==> c*t, takes constant c and term t
00162     virtual Theorem canonMultTermConst(const Expr& c, const Expr& t);
00163     // t1*t2 ==> Error, takes t1 and t2 where both are non-constants
00164     virtual Theorem canonMultTerm1Term2(const Expr& t1, const Expr& t2);
00165     // 0*t ==> 0, takes 0*t
00166     virtual Theorem canonMultZero(const Expr& e);
00167     // 1*t ==> t, takes 1*t
00168     virtual Theorem canonMultOne(const Expr& e);
00169     // c1*c2 ==> c', takes constant c1*c2 
00170     virtual Theorem canonMultConstConst(const Expr& c1, const Expr& c2);
00171     // c1*(c2*t) ==> c'*t, takes c1 and c2 and t
00172     virtual Theorem 
00173       canonMultConstTerm(const Expr& c1, const Expr& c2, const Expr&t);
00174     // c1*(+ c2 v1 ...) ==> (+ c' c1v1 ...), takes c1 and the sum
00175     virtual Theorem canonMultConstSum(const Expr& c1, const Expr& sum);
00176     // c^n = c' (compute the constant power expression)
00177     virtual Theorem canonPowConst(const Expr& pow);
00178 
00179     // Rules for addition
00180     // flattens the input. accepts a PLUS expr
00181     virtual Theorem canonFlattenSum(const Expr& e);
00182     
00183     // Rules for addition
00184     // combine like terms. accepts a flattened PLUS expr
00185     virtual Theorem canonComboLikeTerms(const Expr& e);
00186     
00187     // 0 = (* e1 e2 ...) <=> 0 = e1 OR 0 = e2 OR ...
00188     virtual Theorem multEqZero(const Expr& expr);
00189 
00190     // 0 = (^ c x) <=> false if c <=0
00191     //             <=> 0 = x if c > 0
00192     virtual Theorem powEqZero(const Expr& expr);
00193 
00194     // x^n = y^n <=> x = y (if n is odd)
00195     // x^n = y^n <=> x = y OR x = -y (if n is even)
00196     virtual Theorem elimPower(const Expr& expr);
00197 
00198     // x^n = c <=> x = root (if n is odd and root^n = c)
00199     // x^n = c <=> x = root OR x = -root (if n is even and root^n = c)
00200     virtual Theorem elimPowerConst(const Expr& expr, const Rational& root);
00201 
00202     // x^n = c <=> false (if n is even and c is negative)
00203     virtual Theorem evenPowerEqNegConst(const Expr& expr);
00204 
00205     // x^n = c <=> false (if x is an integer and c is not a perfect n power)
00206     virtual Theorem intEqIrrational(const Expr& expr, const Theorem& isInt);
00207 
00208     // e[0] kind e[1] <==> true when e[0] kind e[1],
00209     // false when e[0] !kind e[1], for constants only
00210     virtual Theorem constPredicate(const Expr& e);
00211 
00212     // e[0] kind e[1] <==> 0 kind e[1] - e[0]
00213     virtual Theorem rightMinusLeft(const Expr& e);
00214 
00215     // e[0] kind e[1] <==> e[0] - e[1] kind 0
00216     virtual Theorem leftMinusRight(const Expr& e);
00217 
00218     // x kind y <==> x + z kind y + z
00219     virtual Theorem plusPredicate(const Expr& x, 
00220           const Expr& y, 
00221           const Expr& z, int kind);
00222 
00223     // x = y <==> x * z = y * z
00224     virtual Theorem multEqn(const Expr& x, const Expr& y, const Expr& z);
00225 
00226     // x = y <==> z=0 OR x * 1/z = y * 1/z
00227     virtual Theorem divideEqnNonConst(const Expr& x, const Expr& y, const Expr& z);
00228 
00229     // if z is +ve, return e[0] LT|LE|GT|GE e[1] <==> e[0]*z LT|LE|GT|GE e[1]*z
00230     // if z is -ve, return e[0] LT|LE|GT|GE e[1] <==> e[1]*z LT|LE|GT|GE e[0]*z
00231     virtual Theorem multIneqn(const Expr& e, const Expr& z);
00232 
00233     // x = y ==> x <= y and x >= y
00234     virtual Theorem eqToIneq(const Expr& e);
00235 
00236     // "op1 GE|GT op2" <==> op2 LE|LT op1
00237     virtual Theorem flipInequality(const Expr& e);
00238     
00239     // NOT (op1 LT op2)  <==> (op1 GE op2)
00240     // NOT (op1 LE op2)  <==> (op1 GT op2)
00241     // NOT (op1 GT op2)  <==> (op1 LE op2)
00242     // NOT (op1 GE op2)  <==> (op1 LT op2)
00243     Theorem negatedInequality(const Expr& e);
00244 
00245     Theorem realShadow(const Theorem& alphaLTt, const Theorem& tLTbeta);
00246     Theorem realShadowEq(const Theorem& alphaLEt, const Theorem& tLEalpha);
00247     Theorem finiteInterval(const Theorem& aLEt, const Theorem& tLEac,
00248          const Theorem& isInta, const Theorem& isIntt);
00249     
00250     Theorem darkGrayShadow2ab(const Theorem& betaLEbx, 
00251             const Theorem& axLEalpha,
00252             const Theorem& isIntAlpha,
00253             const Theorem& isIntBeta,
00254             const Theorem& isIntx);
00255       
00256     Theorem darkGrayShadow2ba(const Theorem& betaLEbx, 
00257             const Theorem& axLEalpha,
00258             const Theorem& isIntAlpha,
00259             const Theorem& isIntBeta,
00260             const Theorem& isIntx);
00261         
00262     Theorem expandDarkShadow(const Theorem& darkShadow);
00263     Theorem expandGrayShadow0(const Theorem& grayShadow);
00264     Theorem splitGrayShadow(const Theorem& grayShadow);
00265     Theorem splitGrayShadowSmall(const Theorem& grayShadow);
00266     Theorem expandGrayShadow(const Theorem& grayShadow);
00267     Theorem expandGrayShadowConst(const Theorem& grayShadow);
00268     Theorem grayShadowConst(const Theorem& g);
00269 
00270     //! Implements j(c,b,a)
00271     /*! accepts 3 integers a,b,c and returns
00272      *  (b > 0)? (c+b) mod a :  (a - (c+b)) mod a
00273      */
00274     Rational constRHSGrayShadow(const Rational& c,
00275         const Rational& b,
00276         const Rational& a);
00277 
00278     Theorem lessThanToLE(const Theorem& less, const Theorem& isIntLHS,
00279        const Theorem& isIntRHS, bool changeRight);
00280     
00281     Theorem lessThanToLERewrite(const Expr& ineq, const Theorem& isIntLHS,
00282            const Theorem& isIntRHS, bool changeRight);
00283     
00284     Theorem intVarEqnConst(const Expr& eqn, const Theorem& isIntx);
00285 
00286     Theorem IsIntegerElim(const Theorem& isIntx);
00287     Theorem eqElimIntRule(const Theorem& eqn, const Theorem& isIntx,
00288         const std::vector<Theorem>& isIntVars);
00289 
00290     Theorem isIntConst(const Expr& e);
00291 
00292     Theorem equalLeaves1(const Theorem& e);
00293     Theorem equalLeaves2(const Theorem& e);
00294     Theorem equalLeaves3(const Theorem& e);
00295     Theorem equalLeaves4(const Theorem& e);
00296 
00297     Theorem diseqToIneq(const Theorem& diseq);
00298     
00299     Theorem dummyTheorem(const Expr& e);
00300     
00301     Theorem oneElimination(const Expr& x);
00302     
00303     Theorem clashingBounds(const Theorem& lowerBound, const Theorem& upperBound);
00304     
00305     Theorem addInequalities(const Theorem& thm1, const Theorem& thm2);
00306     Theorem addInequalities(const std::vector<Theorem>& thms);
00307     
00308     Theorem implyWeakerInequality(const Expr& expr1, const Expr& expr2);
00309     
00310     Theorem implyNegatedInequality(const Expr& expr1, const Expr& expr2);
00311 
00312     Theorem integerSplit(const Expr& intVar, const Rational& intPoint);
00313     
00314     Theorem trustedRewrite(const Expr& expr1, const Expr& expr2);
00315   
00316     Theorem rafineStrictInteger(const Theorem& isIntConstrThm, const Expr& constr);
00317     
00318     Theorem simpleIneqInt(const Expr& ineq, const Theorem& isIntRHS);
00319     
00320     Theorem intEqualityRationalConstant(const Theorem& isIntConstrThm, const Expr& constr);
00321     
00322     Theorem cycleConflict(const std::vector<Theorem>& inequalitites);
00323     
00324     Theorem implyEqualities(const std::vector<Theorem>& inequalities);
00325     
00326     Theorem implyWeakerInequalityDiffLogic(const std::vector<Theorem>& antecedentThms, const Expr& implied);
00327     
00328     Theorem implyNegatedInequalityDiffLogic(const std::vector<Theorem>& antecedentThms, const Expr& implied);
00329     
00330     Theorem expandGrayShadowRewrite(const Expr& theShadow);
00331     
00332     Theorem compactNonLinearTerm(const Expr& nonLinear);
00333     
00334     Theorem nonLinearIneqSignSplit(const Theorem& ineqThm);
00335     
00336     Theorem implyDiffLogicBothBounds(const Expr& x, std::vector<Theorem>& c1_le_x, Rational c1, 
00337                          std::vector<Theorem>& x_le_c2, Rational c2);
00338     
00339     Theorem powerOfOne(const Expr& e);
00340 
00341   }; // end of class ArithTheoremProducer
00342 
00343 } // end of namespace CVC3
00344 
00345 #endif

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