theory_arith_new.h

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/*****************************************************************************/
/*!
 * \file theory_arith_new.h
 * 
 * Author: Dejan Jovanovic
 *
 * Created: Thu Jun 14 13:38:16 2007
 *
 * <hr>
 *
 * License to use, copy, modify, sell and/or distribute this software
 * and its documentation for any purpose is hereby granted without
 * royalty, subject to the terms and conditions defined in the \ref
 * LICENSE file provided with this distribution.
 * 
 * <hr>
 * 
 */
/*****************************************************************************/

#ifndef _cvc3__include__theory_arith_new_h_
#define _cvc3__include__theory_arith_new_h_

#include "theory_arith.h"

#include <hash_fun.h>
#include <hash_map.h>
#include <queryresult.h>
#include <map>

namespace CVC3 {


/**
 * This theory handles basic linear arithmetic.
 *
 * @author Clark Barrett
 * 
 * @since Sat Feb  8 14:44:32 2003
 */
class TheoryArithNew :public TheoryArith {
    
  /** For concrete model generation */
  CDList<Theorem> d_diseq;
  /** Index to the next unprocessed disequality */   
  CDO<size_t> d_diseqIdx; 
  ArithProofRules* d_rules;
  CDO<bool> d_inModelCreation;

  /** Data class for the strongest free constant in separation inqualities **/
  class FreeConst {
  private:
    Rational d_r;
    bool d_strict;
  public:
    FreeConst() { }
    FreeConst(const Rational& r, bool strict): d_r(r), d_strict(strict) { }
    const Rational& getConst() const { return d_r; }
    bool strict() const { return d_strict; }
  };
  //! Printing
  friend std::ostream& operator<<(std::ostream& os, const FreeConst& fc);
  //! Private class for an inequality in the Fourier-Motzkin database
  class Ineq {
  private:
    Theorem d_ineq; //!< The inequality
    bool d_rhs; //!< Var is isolated on the RHS
    const FreeConst* d_const; //!< The max/min const for subsumption check
    //! Default constructor is disabled
    Ineq() { }
  public:
    //! Initial constructor.  'r' is taken from the subsumption database.
    Ineq(const Theorem& ineq, bool varOnRHS, const FreeConst& c):
      d_ineq(ineq), d_rhs(varOnRHS), d_const(&c) { }
    //! Get the inequality
    const Theorem& ineq() const { return d_ineq; }
    //! Get the max/min constant
    const FreeConst& getConst() const { return *d_const; }
    //! Flag whether var is isolated on the RHS
    bool varOnRHS() const { return d_rhs; }
    //! Flag whether var is isolated on the LHS
    bool varOnLHS() const { return !d_rhs; }
    //! Auto-cast to Theorem
    operator Theorem() const { return d_ineq; }
  };
  //! Printing
  friend std::ostream& operator<<(std::ostream& os, const Ineq& ineq);
  //! Database of inequalities with a variable isolated on the right
  ExprMap<CDList<Ineq> *> d_inequalitiesRightDB;
  //! Database of inequalities with a variable isolated on the left
  ExprMap<CDList<Ineq> *> d_inequalitiesLeftDB;
  //! Mapping of inequalities to the largest/smallest free constant
  /*! The Expr is the original inequality with the free constant
   * removed and inequality converted to non-strict (for indexing
   * purposes).  I.e. ax<c+t becomes ax<=t.  This inequality is mapped
   * to a pair<c,strict>, the smallest (largest for c+t<ax) constant
   * among inequalities with the same 'a', 'x', and 't', and a boolean
   * flag indicating whether the strongest inequality is strict.
   */
  CDMap<Expr, FreeConst> d_freeConstDB;
  // Input buffer to store the incoming inequalities
  CDList<Theorem> d_buffer; //!< Buffer of input inequalities
  CDO<size_t> d_bufferIdx; //!< Buffer index of the next unprocessed inequality
  const int* d_bufferThres; //!< Threshold when the buffer must be processed
  // Statistics for the variables
  /*! @brief Mapping of a variable to the number of inequalities where
    the variable would be isolated on the right */
  CDMap<Expr, int> d_countRight;
  /*! @brief Mapping of a variable to the number of inequalities where
    the variable would be isolated on the left */
  CDMap<Expr, int> d_countLeft;
  //! Set of shared terms (for counterexample generation)
  CDMap<Expr, bool> d_sharedTerms;
  //! Set of shared integer variables (i-leaves)
  CDMap<Expr, bool> d_sharedVars;
  //Directed Acyclic Graph representing partial variable ordering for
  //variable projection over inequalities.
  class VarOrderGraph {
    ExprMap<std::vector<Expr> > d_edges;
    ExprMap<bool> d_cache;
    bool dfs(const Expr& e1, const Expr& e2);
  public:
    void addEdge(const Expr& e1, const Expr& e2);
    //returns true if e1 < e2, false otherwise.
    bool lessThan(const Expr& e1, const Expr& e2);
    //selects those variables which are largest and incomparable among
    //v1 and puts it into v2
    void selectLargest(const std::vector<Expr>& v1, std::vector<Expr>& v2);
    //selects those variables which are smallest and incomparable among
    //v1, removes them from v1 and  puts them into v2. 
    void selectSmallest( std::vector<Expr>& v1, std::vector<Expr>& v2);

  };
  
  VarOrderGraph d_graph;

  // Private methods
  //! Check the term t for integrality.  
  /*! \return a theorem of IS_INTEGER(t) or Null. */
  Theorem isIntegerThm(const Expr& e);
  //! A helper method for isIntegerThm()
  /*! Check if IS_INTEGER(e) is easily derivable from the given 'thm' */
  Theorem isIntegerDerive(const Expr& isIntE, const Theorem& thm);
  //! Check if the kids of e are fully simplified and canonized (for debugging)
  bool kidsCanonical(const Expr& e);
  
  //! Canonize the expression e, assuming all children are canonical
  Theorem canon(const Expr& e);
  /*! @brief Canonize and reduce e w.r.t. union-find database; assume
   * all children are canonical */
  Theorem canonSimplify(const Expr& e);
  /*! @brief Composition of canonSimplify(const Expr&) by
   * transitivity: take e0 = e1, canonize and simplify e1 to e2,
   * return e0 = e2. */
  Theorem canonSimplify(const Theorem& thm) {
    return transitivityRule(thm, canonSimplify(thm.getRHS()));
  }
  //! Canonize predicate (x = y, x < y, etc.)
  Theorem canonPred(const Theorem& thm);
  //! Canonize predicate like canonPred except that the input theorem
  //! is an equivalent transformation.
  Theorem canonPredEquiv(const Theorem& thm);
  //! Solve an equation and return an equivalent Theorem in the solved form
  Theorem doSolve(const Theorem& e);
  //! takes in a conjunction equivalence Thm and canonizes it.
  Theorem canonConjunctionEquiv(const Theorem& thm);
  //! picks the monomial with the smallest abs(coeff) from the input
  //integer equation.
  Expr pickIntEqMonomial(const Expr& right);
  //! processes equalities with 1 or more vars of type REAL
  Theorem processRealEq(const Theorem& eqn);
  //! processes equalities whose vars are all of type INT
  Theorem processIntEq(const Theorem& eqn);
  //! One step of INT equality processing (aux. method for processIntEq())
  Theorem processSimpleIntEq(const Theorem& eqn);
  //! Take an inequality and isolate a variable
  Theorem isolateVariable(const Theorem& inputThm, bool& e1);
  //! Update the statistics counters for the variable with a coeff. c
  void updateStats(const Rational& c, const Expr& var);
  //! Update the statistics counters for the monomial
  void updateStats(const Expr& monomial);
  //! Add an inequality to the input buffer.  See also d_buffer
  void addToBuffer(const Theorem& thm);
  Expr pickMonomial(const Expr& right);
 public: // ArithTheoremProducer needs this function, so make it public
  //! Separate monomial e = c*p1*...*pn into c and 1*p1*...*pn
  void separateMonomial(const Expr& e, Expr& c, Expr& var);
  //! Check the term t for integrality (return bool)
  bool isInteger(const Expr& e) { return !(isIntegerThm(e).isNull()); }
 private:
  bool lessThanVar(const Expr& isolatedVar, const Expr& var2);
  //! Check if the term expression is "stale"
  bool isStale(const Expr& e);
  //! Check if the inequality is "stale" or subsumed
  bool isStale(const Ineq& ineq);
  void projectInequalities(const Theorem& theInequality,bool isolatedVarOnRHS);
  void assignVariables(std::vector<Expr>&v);
  void findRationalBound(const Expr& varSide, const Expr& ratSide, 
       const Expr& var,
       Rational &r);
  bool findBounds(const Expr& e, Rational& lub, Rational&  glb);

  Theorem normalizeProjectIneqs(const Theorem& ineqThm1,
        const Theorem& ineqThm2);
  //! Take a system of equations and turn it into a solved form
  Theorem solvedForm(const std::vector<Theorem>& solvedEqs);
  /*! @brief Substitute all vars in term 't' according to the
   * substitution 'subst' and canonize the result.
   */
  Theorem substAndCanonize(const Expr& t, ExprMap<Theorem>& subst);
  /*! @brief Substitute all vars in the RHS of the equation 'eq' of
   * the form (x = t) according to the substitution 'subst', and
   * canonize the result.
   */
  Theorem substAndCanonize(const Theorem& eq, ExprMap<Theorem>& subst);
  //! Traverse 'e' and push all the i-leaves into 'vars' vector
  void collectVars(const Expr& e, std::vector<Expr>& vars,
       std::set<Expr>& cache);
  /*! @brief Check if alpha <= ax & bx <= beta is a finite interval
   *  for integer var 'x', and assert the corresponding constraint
   */
  void processFiniteInterval(const Theorem& alphaLEax,
           const Theorem& bxLEbeta);
  //! For an integer var 'x', find and process all constraints A <= ax <= A+c 
  void processFiniteIntervals(const Expr& x);
  //! Recursive setup for isolated inequalities (and other new expressions)
  void setupRec(const Expr& e);

public:
  TheoryArithNew(TheoryCore* core);
  ~TheoryArithNew();

  // Trusted method that creates the proof rules class (used in constructor).
  // Implemented in arith_theorem_producer.cpp
  ArithProofRules* createProofRules();

  // Theory interface
  void addSharedTerm(const Expr& e);
  void assertFact(const Theorem& e);
  void refineCounterExample();
  void computeModelBasic(const std::vector<Expr>& v);
  void computeModel(const Expr& e, std::vector<Expr>& vars);
  void checkSat(bool fullEffort);
  Theorem rewrite(const Expr& e);
  void setup(const Expr& e);
  void update(const Theorem& e, const Expr& d);
  Theorem solve(const Theorem& e);
  void checkAssertEqInvariant(const Theorem& e);
  void checkType(const Expr& e);
  Cardinality finiteTypeInfo(Expr& e, Unsigned& n,
                             bool enumerate, bool computeSize);
  void computeType(const Expr& e);
  Type computeBaseType(const Type& t);
  void computeModelTerm(const Expr& e, std::vector<Expr>& v);
  Expr computeTypePred(const Type& t, const Expr& e);
  Expr computeTCC(const Expr& e);
  ExprStream& print(ExprStream& os, const Expr& e);
  virtual Expr parseExprOp(const Expr& e);

  // DDDDDDDDDDDDDDDDDDDDDDDDEEEEEEEEEEEEEEEEEEEEEEEJJJJJJJJJJJJJJJJJJJJJAAAAAAAAAAAAAAAAAAAAAAANNNNNNNNNNNNNNNNNNNNNNN
  
    public: 
    
      /**
       * EpsRational class ecapsulates the rationals with a symbolic small \f$\epsilon\f$ added. Each rational
       * number is presented as a pair \f$(q, k) = q + k\epsilon\f$, where \f$\epsilon\f$ is treated symbolically.
       * The operations on the new rationals are defined as
       * <ul>
       *  <li>\f$(q_1, k_1) + (q_2, k_2) \equiv (q_1 + q_2, k_1 + k_2)\f$
       *  <li>\f$a \times (q, k) \equiv (a \times q, a \times k)\f$
       *  <li>\f$(q_1, k_1) \leq (q_2, k_2) \equiv (q_1 < q_2) \vee (q_1 = q_2 \wedge k_1 \leq k_2)\f$
       * </ul>
       * 
       * Note that the operations on the infinite values are not defined, as they are never used currently. Infinities can
       * only be asigned or compared. 
       */
      class EpsRational {
        
        protected:
        
          /** Type of rationals, normal and the two infinities */
          typedef enum { FINITE, PLUS_INFINITY, MINUS_INFINITY } RationalType;
          
          /** The type of this rational */
          RationalType type;      

          /** The rational part */
          Rational q; 
          
          /** The epsilon multiplier */
          Rational k;       
        
          /**
           * Private constructor to construt infinities.
           */
          EpsRational(RationalType type) : type(type) {}
        
        public:
        
          /**
           * Returns if the number is a plain rational.
           * 
           * @return true if rational, false otherwise
           */
          inline bool isRational() const { return k == 0; }
          
          /**
           * Returns if the number is a plain integer.
           * 
           * @return true if rational, false otherwise
           */
          inline bool isInteger() const { return k == 0 && q.isInteger(); }
          
          /** 
           * Returns the floor of the number \f$x = q + k \epsilon\f$ using the following formula
           * \f[
           * \lfloor \beta(x) \rfloor = \left\{
           * \begin{tabular}{ll}
           * $\lfloor q \rfloor$ & $\mathrm{if\ } q \notin Z$\\
           * $q$ & $\mathrm{if\ } q \in Z \mathrm{\ and\ } k \geq 0$\\
           * $q - 1$ & $\mathrm{if\ } q \in Z \mathrm{\ and\ } k < 0$
           * \end{tabular}\right.
           * \f]
           */
          inline Rational getFloor() const { 
            if (q.isInteger()) {
              if (k >= 0) return q;
              else return q - 1;
            } else 
              // If not an integer, just floor it
              return floor(q);
          } 
          

        /** 
         * Returns the rational part of the number
         * 
         * @return the rational 
         */
        inline Rational getRational() const { return q; }

        /** The infinity constant */
        static const EpsRational PlusInfinity;
        /** The negative infinity constant */
        static const EpsRational MinusInfinity;
        /** The zero constant */
        static const EpsRational Zero;
        

        /** The blank constructor */
        EpsRational() : type(FINITE), q(0), k(0) {}

        /** Copy constructor */
        EpsRational(const EpsRational& r) : type(r.type), q(r.q), k(r.k) {} 
        
          /**
           * Constructor from a rational, constructs a new pair (q, 0).
           * 
           * @param q the rational
           */
        EpsRational(const Rational q) : type(FINITE), q(q), k(0) {} 
      
          /**
           * Constructor from a rational and a given epsilon multiplier, constructs a 
           * new pair (q, k).
           * 
           * @param q the rational
           * @param k the epsilon multiplier
           */
        EpsRational(const Rational q, const Rational k) : type(FINITE), q(q), k(k) {} 
      
          /**
           * Addition operator for two EpsRational numbers.
           * 
           * @param r the number to be added
           * @return the sum as defined in the class 
           */
          inline EpsRational operator + (const EpsRational& r) const {
            DebugAssert(type == FINITE, "EpsRational::operator +, adding to infinite number");
            DebugAssert(r.type == FINITE, "EpsRational::operator +, adding an infinite number");
            return EpsRational(q + r.q, k + r.k);
          }
          
          /**
           * Addition operator for two EpsRational numbers.
           * 
           * @param r the number to be added
           * @return the sum as defined in the class 
           */
          inline EpsRational& operator += (const EpsRational& r) {
            DebugAssert(type == FINITE, "EpsRational::operator +, adding to infinite number");
            q = q + r.q;
            k = k + r.k;
            return *this;
          }
          
          /**
           * Subtraction operator for two EpsRational numbers.
           * 
           * @param r the number to be added
           * @return the sum as defined in the class 
           */
          inline EpsRational operator - (const EpsRational& r) const {
          DebugAssert(type == FINITE, "EpsRational::operator -, subtracting from infinite number");
            DebugAssert(r.type == FINITE, "EpsRational::operator -, subtracting an infinite number");
            return EpsRational(q - r.q, k - r.k);
          }
          
          /**
           * Multiplication operator EpsRational number and a rational number.
           * 
           * @param a the number to be multiplied
           * @return the product as defined in the class 
           */
          inline EpsRational operator * (const Rational& a) const {
          DebugAssert(type == FINITE, "EpsRational::operator *, multiplying an infinite number");
            return EpsRational(a * q, a * k);
          }
          
          /**
           * Division operator EpsRational number and a rational number.
           * 
           * @param a the number to be multiplied
           * @return the product as defined in the class 
           */
          inline EpsRational operator / (const Rational& a) const {
          DebugAssert(type == FINITE, "EpsRational::operator *, dividing an infinite number");
            return EpsRational(q / a, k / a);
          }
          
          /**
           * Equality comparison operator.
           */
          inline bool operator == (const EpsRational& r) const { return (q == r.q && k == r.k); }
          
          /**
           * Less than or equal comparison operator.
           */
          inline bool operator <= (const EpsRational& r) const { 
            switch (r.type) {
              case FINITE:
                if (type == FINITE) 
                  // Normal comparison
                  return (q < r.q || (q == r.q && k <= r.k));
                else
                  // Finite number is bigger only of the negative infinity 
                  return type == MINUS_INFINITY;
              case PLUS_INFINITY:
                // Everything is less then or equal than +inf
                return true;
              case MINUS_INFINITY:
                // Only -inf is less then or equal than -inf
                return (type == MINUS_INFINITY);
              default:
                // Ohohohohohoooooo, whats up
                FatalAssert(false, "EpsRational::operator <=, what kind of number is this????");    
            }
            return false;
          }
          
          /**
           * Less than comparison operator.
           */
          inline bool operator < (const EpsRational& r) const { return !(r <= *this); }
          
          /**
           * Greater than comparison operator.
           */
          inline bool operator > (const EpsRational& r) const { return !(*this <= r); }
          
          /**
           * Returns the string representation of the number.
           * 
           * @return the string representation of the number
           */
          std::string toString() const { 
            switch (type) {
              case FINITE:
                return "(" + q.toString() + ", " + k.toString() + ")";
              case PLUS_INFINITY:
                return "+inf";
              case MINUS_INFINITY:
                return "-inf";
              default:
                FatalAssert(false, "EpsRational::toString, what kind of number is this????");
            } 
            return "hm, what am I?";
          }
      };
  
      /**
       * Registers the atom given from the core. This atoms are stored so that they can be used
       * for theory propagation.
       * 
       * @param e the expression (atom) that is part of the input formula
       */ 
    void registerAtom(const Expr& e);
 
    private:    
        
        /** A set of all integer variables */
        std::set<Expr> intVariables;
        
        /**
         * Return a Gomory cut plane derivation of the variable $x_i$. Mixed integer
         * Gomory cut can be constructed if 
         * <ul>
         * <li>\f$x_i\f$ is a integer basic variable with a non-integer value
         * <li>all non-basic variables in the row of \f$x_i\f$ are assigned to their
         *     upper or lower bounds
         * <li>all the values on the right side of the row have rational values (non
         *     eps-rational values)
         * </ul>
         */ 
        Theorem deriveGomoryCut(const Expr& x_i);
        
        /**
         * Tries to rafine the integer constraint of the theorem. For example,
         * x < 1 is rewritten as x <= 0, and x <(=) 1.5 is rewritten as x <= 1.
         * The constraint should be in the normal form.
         * 
         * @param thm the derivation of the constraint 
         */
        Theorem rafineIntegerConstraints(const Theorem& thm);
        
        /** Are we consistent or not */
        CDO<QueryResult> consistent;
        
        /** The theorem explaining the inconsistency */
        Theorem explanation;
        
        /** 
         * The structure necessaty to hold the bounds.
         */
        struct BoundInfo {
          /** The bound itself */
          EpsRational bound;
          /** The assertion theoreem of the bound */
          Theorem theorem;
          
          /** Constructor */
          BoundInfo(const EpsRational& bound, const Theorem& thm): bound(bound), theorem(thm) {}
          
          /** The empty constructor for the map */
          BoundInfo(): bound(0), theorem() {}
          
          /** 
           * The comparator, just if we need it. Compares first by expressions then by bounds 
           */
          bool operator < (const BoundInfo& bI) const {
            // Get tje expressoins
            const Expr& expr1 = (theorem.isRewrite() ? theorem.getRHS() : theorem.getExpr());
            const Expr& expr2 = (bI.theorem.isRewrite() ? bI.theorem.getRHS() : bI.theorem.getExpr());
            
            std::cout << expr1 << " @ " << expr2 << std::endl;
            
            // Compare first by the expressions (right sides of expressions)
            if (expr1[1] == expr2[1])
              // If the same, just return the bound comparison (plus a trick to order equal bounds, different relations) 
              if (bound == bI.bound && expr1.getKind() != expr2.getKind())
                return expr1.getKind() == LE; // LE before GE -- only case that can happen                              
              else
                return bound < bI.bound;
            else
              // Return the expression comparison
              return expr1[1] < expr2[1];
          }
        };


        /** 
         * The structure necessaty to hold the bounds on expressions (for theory propagation).
         */
        struct ExprBoundInfo {
          /** The bound itself */
          EpsRational bound;
          /** The assertion theoreem of the bound */
          Expr e;
          
          /** Constructor */
          ExprBoundInfo(const EpsRational& bound, const Expr& e): bound(bound), e(e) {}
          
          /** The empty constructor for the map */
          ExprBoundInfo(): bound(0) {}
          
          /** 
           * The comparator, just if we need it. Compares first by expressions then by bounds 
           */
          bool operator < (const ExprBoundInfo& bI) const {
            
            // Compare first by the expressions (right sides of expressions)
            if (e[1] == bI.e[1])
              // If the same, just return the bound comparison (plus a trick to order equal bounds, different relations) 
              if (bound == bI.bound && e.getKind() != bI.e.getKind())
                return e.getKind() == LE; // LE before GE -- only case that can happen                              
              else
                return bound < bI.bound;
            else
              // Return the expression comparison
              return e[1] < bI.e[1];
          }
        };
        
        /** The map from variables to lower bounds (these must be backtrackable) */  
      CDMap<Expr, BoundInfo> lowerBound;
      /** The map from variables to upper bounds (these must be backtrackable) */ 
      CDMap<Expr, BoundInfo> upperBound;
      /** The current real valuation of the variables (these must be backtrackable for the last succesefull checkSAT!!!) */
      CDMap<Expr, EpsRational> beta;
          
      typedef Hash::hash_map<Expr, Theorem> TebleauxMap;
      //typedef google::sparse_hash_map<Expr, Theorem, Hash::hash<Expr> > TebleauxMap;
      //typedef std::map<Expr, Theorem> TebleauxMap;
      
      typedef std::set<Expr> SetOfVariables;
      typedef Hash::hash_map<Expr, SetOfVariables> DependenciesMap;
      
      /** Maps variables to sets of variables that depend on it in the tableaux */
      DependenciesMap dependenciesMap;  
          
        /** The tableaux, a map from basic variables to the expressions over the non-basic ones (theorems that explain them how we got there) */ 
      TebleauxMap tableaux;

    /** Additional tableaux map from expressions asserted to the corresponding theorems explaining the introduction of the new variables */
    TebleauxMap freshVariables;
    
    /** A set containing all the unsatisfied basic variables */
    std::set<Expr> unsatBasicVariables;
    
    /** The vector to keep the assignments from fresh variables to expressions they represent */  
    std::vector<Expr> assertedExpr;
    /** The backtrackable number of fresh variables asserted so far */
    CDO<unsigned int> assertedExprCount;
    
    /** A set of BoundInfo objects */
    typedef std::set<ExprBoundInfo> BoundInfoSet;
    
    /** Internal variable to see wheather to propagate or not */
    bool propagate;
    
    /** 
     * Propagate all that is possible from given assertion and its bound
     */
    void propagateTheory(const Expr& assertExpr, const EpsRational& bound, const EpsRational& oldBound);    
    
    /**
     * Store all the atoms from the formula in this set. It is searchable by an expression
     * and the bound. To get all the implied atoms, one just has to go up (down) and check if the 
     * atom or it's negation are implied.
     */
    BoundInfoSet allBounds;
    
    /**
     * Adds var to the dependencies sets of all the variables in the sum.
     * 
     * @param var the variable on the left side 
     * @param sum the sum that defines the variable
     */
    void updateDependenciesAdd(const Expr& var, const Expr& sum);
    
    /**
     * Remove var from the dependencies sets of all the variables in the sum.
     * 
     * @param var the variable on the left side 
     * @param sum the sum that defines the variable
     */
    void updateDependenciesRemove(const Expr& var, const Expr& sum);
  
    /**
     * Updates the dependencies if a right side of an expression in the tableaux is changed. For example,
     * if oldExpr is x + y and newExpr is y + z, var will be added to the dependency list of z, and removed
     * from the dependency list of x.
     * 
     * @param oldExpr the old right side of the tableaux 
     * @param newExpr the new right side of the tableaux
     * @param var the variable that is defined by these two expressions
     * @param skipVar a variable to skip when going through the expressions
     */
    void updateDependencies(const Expr& oldExpr, const Expr& newExpr, const Expr& var, const Expr& skipVar);
    
    /**
     * Update the values of variables that have appeared in the tableaux due to backtracking.
     */
    void updateFreshVariables();
    
    /** 
     * Updates the value of variable var by computing the value of expression e.
     * 
     * @param var the variable to update
     * @param e the expression to compute
     */
    void updateValue(const Expr& var, const Expr& e);
    
    /**
     * Returns a string representation of the tableaux.
     * 
     * @return tableaux as string
     */
    std::string tableauxAsString() const;
    
    /**
     * Returns a string representation of the unsat variables.
     * 
     * @return unsat as string
     */
    std::string unsatAsString() const;
    
    /**
     * Returns a string representation of the current bounds.
     * 
     * @return tableaux as string
     */
    std::string boundsAsString();

    /**
     * Gets the equality of the fresh variable tableaux variable corresponding to this expression. If the expression has already been
     * asserted, the coresponding variable is returned, othervise a fresh variable is created and the theorem is returned.
     * 
     * @param leftSide the left side of the asserted constraint
     * @return the equality theorem s = leftSide
     */
    Theorem getVariableIntroThm(const Expr& leftSide);

    /**
     * Find the coefficient standing by the variable var in the expression expr. Expresion is expected to be
     * in canonical form, i.e either a rational constant, an arithmetic leaf (i.e. variable or term from some 
     * other theory), (MULT rat leaf) where rat is a non-zero rational constant, leaf is an arithmetic leaf or
     * (PLUS \f$const term_0 term_1 ... term_n\f$) where each \f$term_i\f$ is either a leaf or (MULT \f$rat leaf\f$) 
     * and each leaf in \f$term_i\f$ must be strictly greater than the leaf in \f$term_{i+1}\f$.               
     * 
     * @param var the variable
     * @param expr the expression to search in
     */     
      const Rational& findCoefficient(const Expr& var, const Expr& expr); 
    
      /**
       * Return true iof the given variable is basic in the tableaux, i.e. it is on the left side, expressed
       * in terms of the non-basic variables.
       * 
       * @param x the variable to be checked
       * @return true if the variable is basic, false if the variable is non-basic
       */
      bool isBasic(const Expr& x) const;
      
      /**
       * Returns the coefficient at a_ij in the current tableaux, i.e. the coefficient 
       * at x_j in the row of x_i.
       * 
       * @param x_i a basic variable
       * @param x_j a non-basic variable
       * @return the reational coefficient 
       */
      Rational getTableauxEntry(const Expr& x_i, const Expr& x_j); 
      
      /**
       * Swaps a basic variable \f$x_r\f$ and a non-basic variable \f$x_s\f$ such
       * that ars \f$a_{rs} \neq 0\f$. After pivoting, \f$x_s\f$ becomes basic and \f$x_r\f$ becomes non-basic. 
       * The tableau is updated by replacing equation \f[x_r = \sum_{x_j \in N}{a_{rj}xj}\f] with
       * \f[x_s = \frac{x_r}{a_{rs}} - \sum_{x_j \in N }{\frac{a_{rj}}{a_rs}x_j}\f] and this equation 
       * is used to eliminate \f$x_s\f$ from the rest of the tableau by substitution.
       * 
       * @param x_r a basic variable
       * @param x_s a non-basic variable
       */
      void pivot(const Expr& x_r, const Expr& x_s);
    
      /**
       * Sets the value of a non-basic variable \f$x_i\f$ to \f$v\f$ and adjusts the value of all 
       * the basic variables so that all equations remain satisfied.
       * 
       * @param x_i a non-basic variable
       * @param v the value to set the variable \f$x_i\f$ to 
       */
      void update(const Expr& x_i, const EpsRational& v);
      
      /**
       * Pivots the basic variable \f$x_i\f$ and the non-basic variable \f$x_j\f$. It also sets \f$x_i\f$ to \f$v\f$ and adjusts all basic 
       * variables to keep the equations satisfied.
       * 
       * @param x_i a basic variable
       * @param x_j a non-basic variable
       * @param v the valie to assign to x_i
       */ 
      void pivotAndUpdate(const Expr& x_i, const Expr& x_j, const EpsRational& v);
      
      /**
       * Asserts a new upper bound constraint on a variable and performs a simple check for consistency (not complete).
       * 
       * @param x_i the variable to assert the bound on
       * @param c the bound to assert
     */
      QueryResult assertUpper(const Expr& x_i, const EpsRational& c, const Theorem& thm);

      /**
       * Asserts a new lower bound constraint on a variable and performs a simple check for consistency (not complete).
       * 
       * @param x_i the variable to assert the bound on
       * @param c the bound to assert
     */
      QueryResult assertLower(const Expr& x_i, const EpsRational& c, const Theorem& thm);
      
      /**
       * Asserts a new equality constraint on a variable by asserting both upper and lower bounds.
       * 
       * @param x_i the variable to assert the bound on
       * @param c the bound to assert
     */
      QueryResult assertEqual(const Expr& x_i, const EpsRational& c, const Theorem& thm);     

    /**
       * Type of noramlization GCD = 1 or just first coefficient 1
       */
      typedef enum { NORMALIZE_GCD, NORMALIZE_UNIT } NormalizationType;
      
      /**
       * Given a canonized term, compute a factor to make all coefficients integer and relatively prime 
       */
      Expr computeNormalFactor(const Expr& rhs, NormalizationType type = NORMALIZE_GCD);
  
      /**
       * Normalize an equation (make all coefficients rel. prime integers)
       */
      Theorem normalize(const Expr& e, NormalizationType type = NORMALIZE_GCD);
      
      /**
       * Normalize an equation (make all coefficients rel. prime integers) accepts a rewrite theorem over 
       * eqn|ineqn and normalizes it and returns a theorem to that effect.
       */
      Theorem normalize(const Theorem& thm, NormalizationType type = NORMALIZE_GCD);
    
    /**
     * Canonise the equation using the tebleaux equations, i.e. replace all the tableaux right sides
     * with the corresponding left sides and canonise the result. 
     * 
     * @param eq the equation to canonise
     * @return the explaining theorem
     */   
      Theorem substAndCanonizeModTableaux(const Theorem& eq);
      
      /**
     * Canonise the sum using the tebleaux equations, i.e. replace all the tableaux right sides
     * with the corresponding left sides and canonise the result. 
     * 
     * @param sum the canonised sum to canonise
     * @return the explaining theorem
     */   
      Theorem substAndCanonizeModTableaux(const Expr& sum);
      
      /**
       * Sustitute the given equation everywhere in the tableaux.
       * 
       * @param eq the equation to use for substitution
       */
      void substAndCanonizeTableaux(const Theorem& eq);
      
      /**
       * Given an equality eq: \f$\sum a_i x_i = y\f$ and a variable $var$ that appears in
       * on the left hand side, pivots $y$ and $var$ so that $y$ comes to the right-hand 
       * side.
       * 
       * @param eq the proof of the equality
       * @param var the variable to move to the right-hand side
       */
      Theorem pivotRule(const Theorem& eq, const Expr& var);
      
      /**
       * Knowing that the tableaux row for \f$x_i\f$ is the problematic one, generate the
       * explanation clause. The variables in the row of \f$x_i = \sum_{x_j \in \mathcal{N}}{a_ij x_j}\f$ are separated to
       * <ul>
       * <li>\f$\mathcal{N}^+ = \left\lbrace x_j \in \mathcal{N} \; | \; a_{ij} > 0 \right\rbrace\f$
     * <li>\f$\mathcal{N}^- = \left\lbrace  x_j \in \mathcal{N} \; | \; a_{ij} < 0\right\rbrace\f$ 
       * </ul>
       * Then, the explanation clause to be returned is 
       * \f[\Gamma = \left\lbrace x_j \leq u_j \; | \; x_j \in \mathcal{N}^+\right\rbrace \cup \left\lbrace l_j \leq x_j \; | \; 
       * x_j \in \mathcal{N}^-\right\rbrace \cup \left\lbrace l_i \leq x_i  \right\rbrace\f]
       * 
       * @param var_it the variable that caused the clash
       * @return the theorem explainang the clash
       */ 
      Theorem getLowerBoundExplanation(const TebleauxMap::iterator& var_it);

      /**
       * Knowing that the tableaux row for \f$x_i\f$ is the problematic one, generate the
       * explanation clause. The variables in the row of \f$x_i = \sum_{x_j \in \mathcal{N}}{a_ij x_j}\f$ are separated to
       * <ul>
       * <li>\f$\mathcal{N}^+ = \left\lbrace x_j \in \mathcal{N} \; | \; a_{ij} > 0 \right\rbrace\f$
     * <li>\f$\mathcal{N}^- = \left\lbrace  x_j \in \mathcal{N} \; | \; a_{ij} < 0\right\rbrace\f$ 
       * </ul>
       * Then, the explanation clause to be returned is 
       * \f[\Gamma = \left\lbrace x_j \leq u_j \; | \; x_j \in \mathcal{N}^-\right\rbrace \cup \left\lbrace l_j \leq x_j \; | \; 
       * x_j \in \mathcal{N}^+\right\rbrace \cup \left\lbrace x_i \leq u_i \right\rbrace\f]
       * 
       * @param var_it the variable that caused the clash
       * @return the theorem explainang the clash
       */ 
    Theorem getUpperBoundExplanation(const TebleauxMap::iterator& var_it);
      
      Theorem addInequalities(const Theorem& le_1, const Theorem& le_2);
        
      /**
       * Check the satisfiability 
       */
      QueryResult checkSatSimplex();
      
      /**
       * Check the satisfiability of integer constraints
       */
      QueryResult checkSatInteger();
      
      /** 
       * The last lemma that we asserted to check the integer satisfiability. We don't do checksat until
       * the lemma split has been asserted.
       */
      CDO<Theorem> integer_lemma;
      
  public:
    
      /**
       * Gets the current lower bound on variable x.
       * 
       * @param x the variable
       * @return the current lower bound on x
       */
      EpsRational getLowerBound(const Expr& x) const;
      
      /**
       * Get the current upper bound on variable x.
       * 
       * @param x the variable
       * @return the current upper bound on x
       */
      EpsRational getUpperBound(const Expr& x) const;

    /**
       * Gets the theorem of the current lower bound on variable x.
       * 
       * @param x the variable
       * @return the current lower bound on x
       */
      Theorem getLowerBoundThm(const Expr& x) const;
      
      /**
       * Get the theorem of the current upper bound on variable x.
       * 
       * @param x the variable
       * @return the current upper bound on x
       */
      Theorem getUpperBoundThm(const Expr& x) const;
      
      /**
       * Gets the current valuation of variable x (beta).
       * 
       * @param x the variable
       * @return the current value of variable x
       */
      EpsRational getBeta(const Expr& x);
          
  // DDDDDDDDDDDDDDDDDDDDDDDDEEEEEEEEEEEEEEEEEEEEEEEJJJJJJJJJJJJJJJJJJJJJAAAAAAAAAAAAAAAAAAAAAAANNNNNNNNNNNNNNNNNNNNNNN

};

}

#endif

---