CVCL::ArithProofRules Class Reference

#include <arith_proof_rules.h>

Inheritance diagram for CVCL::ArithProofRules:

[legend]List of all members.

Public Member Functions

virtual ~ArithProofRules ()
virtual Theorem uMinusToMult (const Expr &e)=0
==> -(e) = (-1) * e
virtual Theorem minusToPlus (const Expr &x, const Expr &y)=0
==> x - y = x + (-1) * y
virtual Theorem canonUMinusToDivide (const Expr &e)=0
-(e) ==> e / (-1); takes 'e'
virtual Theorem canonDivideConst (const Expr &c, const Expr &d)=0
(c) / d ==> (c/d), takes c and d
virtual Theorem canonDivideMult (const Expr &cx, const Expr &d)=0
(c * x) / d ==> (c/d) * x, takes (c*x) and d
virtual Theorem canonDividePlus (const Expr &e, const Expr &d)=0
(+ c ...)/d ==> push division to all the coefficients.
virtual Theorem canonDivideVar (const Expr &e, const Expr &e)=0
x / d ==> (1/d) * x, takes x and d
virtual Theorem canonMultMtermMterm (const Expr &e)=0
virtual Theorem canonPlus (const Expr &e)=0
Canonize (PLUS t1 ... tn).
virtual Theorem canonMult (const Expr &e)=0
Canonize (MULT t1 ... tn).
virtual Theorem canonInvert (const Expr &e)=0
==> 1/e = e' where e' is canonical; takes e.
virtual Theorem canonDivide (const Expr &e)=0
Canonize t1/t2.
virtual Theorem canonMultTermConst (const Expr &c, const Expr &t)=0
t*c ==> c*t, takes constant c and term t
virtual Theorem canonMultTerm1Term2 (const Expr &t1, const Expr &t2)=0
t1*t2 ==> Error, takes t1 and t2 where both are non-constants
virtual Theorem canonMultZero (const Expr &e)=0
0*t ==> 0, takes 0*t
virtual Theorem canonMultOne (const Expr &e)=0
1*t ==> t, takes 1*t
virtual Theorem canonMultConstConst (const Expr &c1, const Expr &c2)=0
c1*c2 ==> c', takes constant c1*c2
virtual Theorem canonMultConstTerm (const Expr &c1, const Expr &c2, const Expr &t)=0
c1*(c2*t) ==> c'*t, takes c1 and c2 and t
virtual Theorem canonMultConstSum (const Expr &c1, const Expr &sum)=0
c1*(+ c2 v1 ...) ==> (+ c' c1v1 ...), takes c1 and the sum
virtual Theorem canonPowConst (const Expr &pow)=0
c^n = c' (compute the constant power expression)
virtual Theorem canonFlattenSum (const Expr &e)=0
flattens the input. accepts a PLUS expr
virtual Theorem canonComboLikeTerms (const Expr &e)=0
combine like terms. accepts a flattened PLUS expr
virtual Theorem constPredicate (const Expr &e)=0
e0 @ e1 <==> true | false, where @ is {=,<,<=,>,>=}
virtual Theorem rightMinusLeft (const Expr &e)=0
e[0] @ e[1] <==> 0 @ e[1] - e[0], where @ is {=,<,<=,>,>=}
virtual Theorem plusPredicate (const Expr &x, const Expr &y, const Expr &z, int kind)=0
x @ y <==> x + z @ y + z, where @ is {=,<,<=,>,>=} (given as 'kind')
virtual Theorem multEqn (const Expr &x, const Expr &y, const Expr &z)=0
x = y <==> x * z = y * z, where z is a non-zero constant
virtual Theorem multIneqn (const Expr &e, const Expr &z)=0
Multiplying inequation by a non-zero constant.
virtual Theorem flipInequality (const Expr &e)=0
"op1 GE|GT op2" <==> op2 LE|LT op1
virtual Theorem negatedInequality (const Expr &e)=0
Propagating negation over <,<=,>,>=.
virtual Theorem realShadow (const Theorem &alphaLTt, const Theorem &tLTbeta)=0
Real shadow: a <(=) t, t <(=) b ==> a <(=) b.
virtual Theorem realShadowEq (const Theorem &alphaLEt, const Theorem &tLEalpha)=0
Projecting a tight inequality: alpha <= t <= alpha ==> t = alpha.
virtual Theorem finiteInterval (const Theorem &aLEt, const Theorem &tLEac, const Theorem &isInta, const Theorem &isIntt)=0
Finite interval for integers: a <= t <= a + c ==> G(t, a, 0, c).
virtual Theorem darkGrayShadow2ab (const Theorem &betaLEbx, const Theorem &axLEalpha, const Theorem &isIntAlpha, const Theorem &isIntBeta, const Theorem &isIntx)=0
Dark & Gray shadows when a <= b.
virtual Theorem darkGrayShadow2ba (const Theorem &betaLEbx, const Theorem &axLEalpha, const Theorem &isIntAlpha, const Theorem &isIntBeta, const Theorem &isIntx)=0
Dark & Gray shadows when b <= a.
virtual Theorem expandDarkShadow (const Theorem &darkShadow)=0
DARK_SHADOW(t1, t2) ==> t1 <= t2.
virtual Theorem expandGrayShadow0 (const Theorem &g)=0
GRAY_SHADOW(v, e, c, c) ==> v=e+c.
virtual Theorem splitGrayShadow (const Theorem &g)=0
G(x, e, c1, c2) ==> (G1 or G2) and (!G1 or !G2).
virtual Theorem expandGrayShadow (const Theorem &g)=0
G(x, e, c1, c2) ==> e+c1 <= x AND x <= e+c2.
virtual Theorem expandGrayShadowConst (const Theorem &g)=0
Optimized rules: GRAY_SHADOW(a*x, c, c1, c2) ==> [expansion].
virtual Theorem grayShadowConst (const Theorem &g)=0
|- G(ax, c, c1, c2) ==> |- G(x, 0, ceil((c1+c)/a), floor((c2+c)/a))
virtual Theorem lessThanToLE (const Theorem &less, const Theorem &isIntLHS, const Theorem &isIntRHS, bool changeRight)=0
a,b: INT; a < b ==> a <= b-1 [or a+1 <= b]
virtual Theorem intVarEqnConst (const Expr &eqn, const Theorem &isIntx)=0
virtual Theorem eqElimIntRule (const Theorem &eqn, const Theorem &isIntx, const std::vector< Theorem > &isIntVars)=0
Equality elimination rule for integers:
$\frac{\mathsf{int}(a\cdot x)\quad \mathsf{int}(C+\sum_{i=1}^{n}a_{i}\cdot x_{i})} {a\cdot x=C+\sum_{i=1}^{n}a_{i}\cdot x_{i} \quad\equiv\quad x=t_{2}\wedge 0=t_{3}}$
See the detailed description for explanations.
$\frac{\Gamma\models ax=t\quad \Gamma'\models\mathsf{int}(x)\quad \{\Gamma_i\models\mathsf{int}(x_i) | x_i\mbox{ is var in }t\}} {\Gamma,\Gamma',\bigcup_i\Gamma_i\models \exists (y:\mathrm{int}).\ x=t_2(y)\wedge 0=t_3(y)}$
See detailed docs for more information.
virtual Theorem isIntConst (const Expr &e)=0
return e <=> TRUE/FALSE for e == IS_INTEGER(c), where c is a constant
virtual Theorem equalLeaves1 (const Theorem &thm)=0
virtual Theorem equalLeaves2 (const Theorem &thm)=0
virtual Theorem equalLeaves3 (const Theorem &thm)=0
virtual Theorem equalLeaves4 (const Theorem &thm)=0
virtual Theorem diseqToIneq (const Theorem &diseq)=0
x /= y ==> (x < y) OR (x > y)

Detailed Description

Definition at line 40 of file arith_proof_rules.h.

Constructor & Destructor Documentation

virtual CVCL::ArithProofRules::~ArithProofRules ( ) [inline, virtual]

Definition at line 43 of file arith_proof_rules.h.

Member Function Documentation

virtual Theorem CVCL::ArithProofRules::uMinusToMult ( const Expr & e ) [pure virtual]

==> -(e) = (-1) * e

Implemented in CVCL::ArithTheoremProducer, and CVCL::RefinedArithTheoremProducer.
Referenced by CVCL::TheoryArith::canon().

virtual Theorem CVCL::ArithProofRules::minusToPlus ( const Expr & x,

const Expr & y

) [pure virtual]

==> x - y = x + (-1) * y

Implemented in CVCL::ArithTheoremProducer, and CVCL::RefinedArithTheoremProducer.
Referenced by CVCL::TheoryArith::canon().

virtual Theorem CVCL::ArithProofRules::canonUMinusToDivide ( const Expr & e ) [pure virtual]

-(e) ==> e / (-1); takes 'e'
Canon Rule for unary minus: it just converts it to division by -1, the result is not yet canonical:
Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonDivideConst ( const Expr & c,

const Expr & d

) [pure virtual]

(c) / d ==> (c/d), takes c and d
Canon Rules for division by constant 'd'
Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonDivideMult ( const Expr & cx,

const Expr & d

) [pure virtual]

(c * x) / d ==> (c/d) * x, takes (c*x) and d

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonDividePlus ( const Expr & e,

const Expr & d

) [pure virtual]

(+ c ...)/d ==> push division to all the coefficients.
The result is not canonical, but canonizing the summands will make it canonical. Takes (+ c ...) and d
Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonDivideVar ( const Expr & e,

const Expr & e

) [pure virtual]

x / d ==> (1/d) * x, takes x and d

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonMultMtermMterm ( const Expr & e ) [pure virtual]

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonPlus ( const Expr & e ) [pure virtual]

Canonize (PLUS t1 ... tn).

Implemented in CVCL::ArithTheoremProducer, and CVCL::RefinedArithTheoremProducer.
Referenced by CVCL::TheoryArith::canon().

virtual Theorem CVCL::ArithProofRules::canonMult ( const Expr & e ) [pure virtual]

Canonize (MULT t1 ... tn).

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::canon().

virtual Theorem CVCL::ArithProofRules::canonInvert ( const Expr & e ) [pure virtual]

==> 1/e = e' where e' is canonical; takes e.

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonDivide ( const Expr & e ) [pure virtual]

Canonize t1/t2.

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::canon().

virtual Theorem CVCL::ArithProofRules::canonMultTermConst ( const Expr & c,

const Expr & t

) [pure virtual]

t*c ==> c*t, takes constant c and term t

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonMultTerm1Term2 ( const Expr & t1,

const Expr & t2

) [pure virtual]

t1*t2 ==> Error, takes t1 and t2 where both are non-constants

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonMultZero ( const Expr & e ) [pure virtual]

0*t ==> 0, takes 0*t

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonMultOne ( const Expr & e ) [pure virtual]

1*t ==> t, takes 1*t

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonMultConstConst ( const Expr & c1,

const Expr & c2

) [pure virtual]

c1*c2 ==> c', takes constant c1*c2

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonMultConstTerm ( const Expr & c1,

const Expr & c2,

const Expr & t

) [pure virtual]

c1*(c2*t) ==> c'*t, takes c1 and c2 and t

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonMultConstSum ( const Expr & c1,

const Expr & sum

) [pure virtual]

c1*(+ c2 v1 ...) ==> (+ c' c1v1 ...), takes c1 and the sum

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonPowConst ( const Expr & pow ) [pure virtual]

c^n = c' (compute the constant power expression)

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::canon().

virtual Theorem CVCL::ArithProofRules::canonFlattenSum ( const Expr & e ) [pure virtual]

flattens the input. accepts a PLUS expr

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::canonComboLikeTerms ( const Expr & e ) [pure virtual]

combine like terms. accepts a flattened PLUS expr

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::constPredicate ( const Expr & e ) [pure virtual]

e0 @ e1 <==> true | false, where @ is {=,<,<=,>,>=}

Parameters:

e takes e is (e0@e1) where e0 and e1 are constants

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::doSolve(), CVCL::TheoryArith::isolateVariable(), CVCL::TheoryArith::normalizeProjectIneqs(), and CVCL::TheoryArith::rewrite().

virtual Theorem CVCL::ArithProofRules::rightMinusLeft ( const Expr & e ) [pure virtual]

e[0] @ e[1] <==> 0 @ e[1] - e[0], where @ is {=,<,<=,>,>=}

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::addToBuffer(), CVCL::TheoryArith::doSolve(), CVCL::TheoryArith::normalizeProjectIneqs(), and CVCL::TheoryArith::rewrite().

virtual Theorem CVCL::ArithProofRules::plusPredicate ( const Expr & x,

const Expr & y,

const Expr & z,

int kind

) [pure virtual]

x @ y <==> x + z @ y + z, where @ is {=,<,<=,>,>=} (given as 'kind')

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::isolateVariable(), CVCL::TheoryArith::processRealEq(), and CVCL::TheoryArith::processSimpleIntEq().

virtual Theorem CVCL::ArithProofRules::multEqn ( const Expr & x,

const Expr & y,

const Expr & z

) [pure virtual]

x = y <==> x * z = y * z, where z is a non-zero constant

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::normalize(), CVCL::TheoryArith::processRealEq(), and CVCL::TheoryArith::processSimpleIntEq().

virtual Theorem CVCL::ArithProofRules::multIneqn ( const Expr & e,

const Expr & z

) [pure virtual]

Multiplying inequation by a non-zero constant.
z>0 ==> e[0] @ e[1] <==> e[0]*z @ e[1]*z
z<0 ==> e[0] @ e[1] <==> e[1]*z @ e[0]*z
for @ in {<,<=,>,>=}:
Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::isolateVariable(), CVCL::TheoryArith::normalize(), CVCL::TheoryArith::normalizeProjectIneqs(), and CVCL::TheoryArith::processFiniteInterval().

virtual Theorem CVCL::ArithProofRules::flipInequality ( const Expr & e ) [pure virtual]

"op1 GE|GT op2" <==> op2 LE|LT op1

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::rewrite().

virtual Theorem CVCL::ArithProofRules::negatedInequality ( const Expr & e ) [pure virtual]

Propagating negation over <,<=,>,>=.
NOT (op1 < op2) <==> (op1 >= op2)
NOT (op1 <= op2) <==> (op1 > op2)
NOT (op1 > op2) <==> (op1 <= op2)
NOT (op1 >= op2) <==> (op1 < op2)
Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::rewrite().

virtual Theorem CVCL::ArithProofRules::realShadow ( const Theorem & alphaLTt,

const Theorem & tLTbeta

) [pure virtual]

Real shadow: a <(=) t, t <(=) b ==> a <(=) b.

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::normalizeProjectIneqs().

virtual Theorem CVCL::ArithProofRules::realShadowEq ( const Theorem & alphaLEt,

const Theorem & tLEalpha

) [pure virtual]

Projecting a tight inequality: alpha <= t <= alpha ==> t = alpha.

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::normalizeProjectIneqs().

virtual Theorem CVCL::ArithProofRules::finiteInterval ( const Theorem & aLEt,

const Theorem & tLEac,

const Theorem & isInta,

const Theorem & isIntt

) [pure virtual]

Finite interval for integers: a <= t <= a + c ==> G(t, a, 0, c).

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::processFiniteInterval().

virtual Theorem CVCL::ArithProofRules::darkGrayShadow2ab ( const Theorem & betaLEbx,

const Theorem & axLEalpha,

const Theorem & isIntAlpha,

const Theorem & isIntBeta,

const Theorem & isIntx

) [pure virtual]

Dark & Gray shadows when a <= b.
takes two integer ineqs (i.e. all vars are ints) "|- beta <= b.x" and "|- a.x <= alpha" and checks if "1 <= a <= b" and returns (D or G) and (!D or !G), where D = Dark_Shadow(ab-1, b.alpha - a.beta), G = Gray_Shadow(a.x, alpha, -(a-1), 0).
Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::normalizeProjectIneqs().

virtual Theorem CVCL::ArithProofRules::darkGrayShadow2ba ( const Theorem & betaLEbx,

const Theorem & axLEalpha,

const Theorem & isIntAlpha,

const Theorem & isIntBeta,

const Theorem & isIntx

) [pure virtual]

Dark & Gray shadows when b <= a.
takes two integer ineqs (i.e. all vars are ints) "|- beta <= b.x" and "|- a.x <= alpha" and checks if "1 <= b < a" and returns (D or G) and (!D or !G), where D = Dark_Shadow(ab-1, b.alpha - a.beta), G = Gray_Shadow(b.x, beta, 0, b-1).
Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::normalizeProjectIneqs().

virtual Theorem CVCL::ArithProofRules::expandDarkShadow ( const Theorem & darkShadow ) [pure virtual]

DARK_SHADOW(t1, t2) ==> t1 <= t2.

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::assertFact().

virtual Theorem CVCL::ArithProofRules::expandGrayShadow0 ( const Theorem & g ) [pure virtual]

GRAY_SHADOW(v, e, c, c) ==> v=e+c.

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::assertFact().

virtual Theorem CVCL::ArithProofRules::splitGrayShadow ( const Theorem & g ) [pure virtual]

G(x, e, c1, c2) ==> (G1 or G2) and (!G1 or !G2).
Here G1 = G(x,e,c1,c), G2 = G(x,e,c+1,c2), and c = floor((c1+c2)/2).
Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::assertFact().

virtual Theorem CVCL::ArithProofRules::expandGrayShadow ( const Theorem & g ) [pure virtual]

G(x, e, c1, c2) ==> e+c1 <= x AND x <= e+c2.

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::assertFact().

virtual Theorem CVCL::ArithProofRules::expandGrayShadowConst ( const Theorem & g ) [pure virtual]

Optimized rules: GRAY_SHADOW(a*x, c, c1, c2) ==> [expansion].
Implements three versions of the rule:

$\frac{\mathrm{GRAY\_SHADOW}(ax,c,c1,c2)} {ax = c + i - \mathrm{sign}(i)\cdot j(c,i,a) \lor GRAY\_SHADOW(ax, c, i-\mathrm{sign}(i)\cdot (a+j(c,i,a)))}$

where the conclusion may become FALSE or without the GRAY_SHADOW part, depending on the values of a, c and i.
Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::grayShadowConst ( const Theorem & g ) [pure virtual]

|- G(ax, c, c1, c2) ==> |- G(x, 0, ceil((c1+c)/a), floor((c2+c)/a))
In the case the new c1 > c2, return |- FALSE
Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::assertFact().

virtual Theorem CVCL::ArithProofRules::lessThanToLE ( const Theorem & less,

const Theorem & isIntLHS,

const Theorem & isIntRHS,

bool changeRight

) [pure virtual]

a,b: INT; a < b ==> a <= b-1 [or a+1 <= b]
Takes a Theorem(\alpha < \beta) and returns Theorem(\alpha < \beta <==> \alpha <= \beta -1) or Theorem(\alpha < \beta <==> \alpha + 1 <= \beta), depending on which side must be changed (changeRight flag)
Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::projectInequalities().

virtual Theorem CVCL::ArithProofRules::intVarEqnConst ( const Expr & eqn,

const Theorem & isIntx

) [pure virtual]

Parameters:

eqn is an equation 0 = a.x or 0 = c + a.x, where x is integer

isIntx is a proof of IS_INTEGER(x)

Returns:
the theorem 0 = c + a.x <==> x=-c/a if -c/a is int else return the theorem 0 = c + a.x <==> false.
It also handles the special case of 0 = a.x <==> x = 0
Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::processSimpleIntEq().

virtual Theorem CVCL::ArithProofRules::eqElimIntRule ( const Theorem & eqn,

const Theorem & isIntx,

const std::vector< Theorem > & isIntVars

) [pure virtual]

Equality elimination rule for integers:
$\frac{\mathsf{int}(a\cdot x)\quad \mathsf{int}(C+\sum_{i=1}^{n}a_{i}\cdot x_{i})} {a\cdot x=C+\sum_{i=1}^{n}a_{i}\cdot x_{i} \quad\equiv\quad x=t_{2}\wedge 0=t_{3}}$
See the detailed description for explanations.
$\frac{\Gamma\models ax=t\quad \Gamma'\models\mathsf{int}(x)\quad \{\Gamma_i\models\mathsf{int}(x_i) | x_i\mbox{ is var in }t\}} {\Gamma,\Gamma',\bigcup_i\Gamma_i\models \exists (y:\mathrm{int}).\ x=t_2(y)\wedge 0=t_3(y)}$
See detailed docs for more information.
This rule implements a step in the iterative algorithm for eliminating integer equality. The terms in the rule are defined as follows:

$\begin{array}{rcl} t & = & C+\sum_{i=1}^na_{i}\cdot x_{i}\\ t_{2}(y) & = & -(C\ \mathbf{mod}\ m+\sum_{i=1}^{n}(a_{i}\ \mathbf{mod}\ m) \cdot x_{i}-m\cdot y)\\ & & \\ t_{3}(y) & = & \mathbf{f}(C,m)+\sum_{i=1}^{n}\mathbf{f}(a_{i},m)\cdot x_{i} -a\cdot y\\ & & \\ m & = & a+1\\ & & \\ \mathbf{f}(i,m) & = & \left\lfloor \frac{i}{m} +\frac{1}{2}\right\rfloor +i\ \mathbf{mod}\ m\\ & & \\ i\ \mathbf{mod}\ m & = & i-m\left\lfloor\frac{i}{m} +\frac{1}{2}\right\rfloor \end{array}$

All the variables and coefficients are integer, and a >= 2.

Parameters:

eqn is the equation ax=t:
$a\cdot x = C + \sum_{i=1}^n a_i\cdot x_i.$

isIntx is a Theorem deriving the integrality of the LHS variable: IS_INTEGER(x)

isIntVars is a vector of Theorems deriving the integrality of all variables on the RHS

Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::processSimpleIntEq().

virtual Theorem CVCL::ArithProofRules::isIntConst ( const Expr & e ) [pure virtual]

return e <=> TRUE/FALSE for e == IS_INTEGER(c), where c is a constant

Parameters:

e is a predicate IS_INTEGER(c) where c is a rational constant

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::equalLeaves1 ( const Theorem & thm ) [pure virtual]

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::equalLeaves2 ( const Theorem & thm ) [pure virtual]

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::equalLeaves3 ( const Theorem & thm ) [pure virtual]

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::equalLeaves4 ( const Theorem & thm ) [pure virtual]

Implemented in CVCL::ArithTheoremProducer.

virtual Theorem CVCL::ArithProofRules::diseqToIneq ( const Theorem & diseq ) [pure virtual]

x /= y ==> (x < y) OR (x > y)
Used in concrete model generation
Implemented in CVCL::ArithTheoremProducer.
Referenced by CVCL::TheoryArith::checkSat().

The documentation for this class was generated from the following file:

arith_proof_rules.h

Generated on Thu Apr 13 16:57:46 2006 for CVC Lite by

1.4.4