Predicates *INTRO A predicate is a function that returns a boolean value, i.e. {True} or {False}. Predicates are often used in patterns, For instance, a rule that only holds for a positive integer would use a pattern such as {n_IsPositiveInteger}. *CMD != --- test for "not equal" *STD *CALL e1 != e2 Precedence: *EVAL OpPrecedence("!=") *PARMS {e1}, {e2} -- expressions to be compared *DESC Both expressions are evaluated and compared. If they turn out to be equal, the result is {False}. Otherwise, the result is {True}. The expression {e1 != e2} is equivalent to {Not(e1 = e2)}. *E.G. In> 1 != 2; Out> True; In> 1 != 1; Out> False; *SEE = *CMD = --- test for equality of expressions *STD *CALL e1 = e2 Precedence: *EVAL OpPrecedence("=") *PARMS {e1}, {e2} -- expressions to be compared *DESC Both expressions are evaluated and compared. If they turn out to be equal, the result is {True}. Otherwise, the result is {False}. The function {Equals} does the same. Note that the test is on syntactic equality, not mathematical equality. Hence even if the result is {False}, the expressions can still be mathematically equal; see the examples below. Put otherwise, this function tests whether the two expressions would be displayed in the same way if they were printed. *E.G. In> e1 := (x+1) * (x-1); Out> (x+1)*(x-1); In> e2 := x^2 - 1; Out> x^2-1; In> e1 = e2; Out> False; In> Expand(e1) = e2; Out> True; *SEE !=, Equals *CMD Not --- logical negation *CORE *CALL Not expr *PARMS {expr} -- a boolean expression *DESC Not returns the logical negation of the argument expr. If {expr} is {False} it returns {True}, and if {expr} is {True}, {Not expr} returns {False}. If the argument is neither {True} nor {False}, it returns the entire expression with evaluated arguments. *E.G. In> Not True Out> False; In> Not False Out> True; In> Not(a) Out> Not a; *SEE And, Or *CMD And --- logical conjunction *CORE *CALL a1 And a2 Precedence: *EVAL OpPrecedence("And") And(a1, a2, a3, ..., aN) *PARMS {a}1, ..., {a}N -- boolean values (may evaluate to {True} or {False}) *DESC This function returns {True} if all arguments are true. The {And} operation is "lazy", i.e. it returns {False} as soon as a {False} argument is found (from left to right). If an argument other than {True} or {False} is encountered a new {And} expression is returned with all arguments that didn't evaluate to {True} or {False} yet. *E.G. In> True And False Out> False; In> And(True,True) Out> True; In> False And a Out> False; In> True And a Out> And(a); In> And(True,a,True,b) Out> b And a; *SEE Or, Not *CMD Or --- logical disjunction *CORE *CALL a1 Or a2 Precedence: *EVAL OpPrecedence("Or") Or(a1, a2, a3, ..., aN) *PARMS {a}1, ..., {a}N -- boolean expressions (may evaluate to {True} or {False}) *DESC This function returns {True} if an argument is encountered that is true (scanning from left to right). The {Or} operation is "lazy", i.e. it returns {True} as soon as a {True} argument is found (from left to right). If an argument other than {True} or {False} is encountered, an unevaluated {Or} expression is returned with all arguments that didn't evaluate to {True} or {False} yet. *E.G. In> True Or False Out> True; In> False Or a Out> Or(a); In> Or(False,a,b,True) Out> True; *SEE And, Not *CMD IsFreeOf --- test whether expression depends on variable *STD *CALL IsFreeOf(var, expr) IsFreeOf({var, ...}, expr) *PARMS {expr} -- expression to test {var} -- variable to look for in "expr" *DESC This function checks whether the expression "expr" (after being evaluated) depends on the variable "var". It returns {False} if this is the case and {True} otherwise. The second form test whether the expression depends on any of the variables named in the list. The result is {True} if none of the variables appear in the expression and {False} otherwise. *E.G. In> IsFreeOf(x, Sin(x)); Out> False; In> IsFreeOf(y, Sin(x)); Out> True; In> IsFreeOf(x, D(x) a*x+b); Out> True; In> IsFreeOf({x,y}, Sin(x)); Out> False; The third command returns {True} because the expression {D(x) a*x+b} evaluates to {a}, which does not depend on {x}. *SEE Contains *CMD IsZeroVector --- test whether list contains only zeroes *STD *CALL IsZeroVector(list) *PARMS {list} -- list to compare against the zero vector *DESC The only argument given to {IsZeroVector} should be a list. The result is {True} if the list contains only zeroes and {False} otherwise. *E.G. In> IsZeroVector({0, x, 0}); Out> False; In> IsZeroVector({x-x, 1 - D(x) x}); Out> True; *SEE IsList, ZeroVector *CMD IsNonObject --- test whether argument is not an {Object()} *STD *CALL IsNonObject(expr) *PARMS {expr} -- the expression to examine *DESC This function returns {True} if "expr" is not of the form {Object(...)} and {False} otherwise. *HEAD Bugs In fact, the result is always {True}. *SEE Object *CMD IsEven --- test for an even integer *STD *CALL IsEven(n) *PARMS {n} -- integer to test *DESC This function tests whether the integer "n" is even. An integer is even if it is divisible by two. Hence the even numbers are 0, 2, 4, 6, 8, 10, etc., and -2, -4, -6, -8, -10, etc. *E.G. In> IsEven(4); Out> True; In> IsEven(-1); Out> False; *SEE IsOdd, IsInteger *CMD IsOdd --- test for an odd integer *STD *CALL IsOdd(n) *PARMS {n} -- integer to test *DESC This function tests whether the integer "n" is odd. An integer is odd if it is not divisible by two. Hence the odd numbers are 1, 3, 5, 7, 9, etc., and -1, -3, -5, -7, -9, etc. *E.G. In> IsOdd(4); Out> False; In> IsOdd(-1); Out> True; *SEE IsEven, IsInteger *CMD IsEvenFunction --- Return true if function is an even function, False otherwise *CMD IsOddFunction --- Return true if function is an odd function, False otherwise *STD *CALL IsEvenFunction(expression,variable) IsOddFunction(expression,variable) *PARMS {expression} -- mathematical expression {variable} -- variable *DESC These functions return True if Yacas can determine that the function is even or odd respectively. Even functions are defined to be functions that have the property: \$\$ f(x) = f(-x) \$\$ And odd functions have the property: \$\$ f(x) = -f(-x) \$\$ {Sin(x)} is an example of an odd function, and {Cos(x)} is an example of an even function. As a side note, one can decompose a function into an even and an odd part: \$\$ f(x) = f_even(x) + f_odd(x) \$\$ Where \$\$ f_even(x) = (f(x)+f(-x))/2 \$\$ and \$\$ f_odd(x) = (f(x)-f(-x))/2 \$\$ *EG In> IsEvenFunction(Cos(b*x),x) Out> True In> IsOddFunction(Cos(b*x),x) Out> False In> IsOddFunction(Sin(b*x),x) Out> True In> IsEvenFunction(Sin(b*x),x) Out> False In> IsEvenFunction(1/x^2,x) Out> True In> IsEvenFunction(1/x,x) Out> False In> IsOddFunction(1/x,x) Out> True In> IsOddFunction(1/x^2,x) Out> False *SEE Sin, Cos *CMD IsFunction --- test for a composite object *CORE *CALL IsFunction(expr) *PARMS {expr} -- expression to test *DESC This function tests whether "expr" is a composite object, i.e. not an atom. This includes not only obvious functions such as {f(x)}, but also expressions such as {x+5} and lists. *E.G. In> IsFunction(x+5); Out> True; In> IsFunction(x); Out> False; *SEE IsAtom, IsList, Type *CMD IsAtom --- test for an atom *CORE *CALL IsAtom(expr) *PARMS {expr} -- expression to test *DESC This function tests whether "expr" is an atom. Numbers, strings, and variables are all atoms. *E.G. In> IsAtom(x+5); Out> False; In> IsAtom(5); Out> True; *SEE IsFunction, IsNumber, IsString *CMD IsString --- test for an string *CORE *CALL IsString(expr) *PARMS {expr} -- expression to test *DESC This function tests whether "expr" is a string. A string is a text within quotes, e.g. {"duh"}. *E.G. In> IsString("duh"); Out> True; In> IsString(duh); Out> False; *SEE IsAtom, IsNumber *CMD IsNumber --- test for a number *CORE *CALL IsNumber(expr) *PARMS {expr} -- expression to test *DESC This function tests whether "expr" is a number. There are two kinds of numbers, integers (e.g. 6) and reals (e.g. -2.75 or 6.0). Note that a complex number is represented by the {Complex} function, so {IsNumber} will return {False}. *E.G. In> IsNumber(6); Out> True; In> IsNumber(3.25); Out> True; In> IsNumber(I); Out> False; In> IsNumber("duh"); Out> False; *SEE IsAtom, IsString, IsInteger, IsPositiveNumber, IsNegativeNumber, Complex *CMD IsList --- test for a list *CORE *CALL IsList(expr) *PARMS {expr} -- expression to test *DESC This function tests whether "expr" is a list. A list is a sequence between curly braces, e.g. {{2, 3, 5}}. *E.G. In> IsList({2,3,5}); Out> True; In> IsList(2+3+5); Out> False; *SEE IsFunction *CMD IsNumericList --- test for a list of numbers *STD *CALL IsNumericList({list}) *PARMS {{list}} -- a list *DESC Returns {True} when called on a list of numbers or expressions that evaluate to numbers using {N()}. Returns {False} otherwise. *SEE N, IsNumber *CMD IsBound --- test for a bound variable *CORE *CALL IsBound(var) *PARMS {var} -- variable to test *DESC This function tests whether the variable "var" is bound, i.e. whether it has been assigned a value. The argument "var" is not evaluated. *E.G. In> IsBound(x); Out> False; In> x := 5; Out> 5; In> IsBound(x); Out> True; *SEE IsAtom *CMD IsBoolean --- test for a Boolean value *STD *CALL IsBoolean(expression) *PARMS {expression} -- an expression *DESC IsBoolean returns True if the argument is of a boolean type. This means it has to be either True, False, or an expression involving functions that return a boolean result, e.g. {=}, {>}, {<}, {>=}, {<=}, {!=}, {And}, {Not}, {Or}. *E.G. In> IsBoolean(a) Out> False; In> IsBoolean(True) Out> True; In> IsBoolean(a And b) Out> True; *SEE True, False *CMD IsNegativeNumber --- test for a negative number *STD *CALL IsNegativeNumber(n) *PARMS {n} -- number to test *DESC {IsNegativeNumber(n)} evaluates to {True} if \$n\$ is (strictly) negative, i.e. if \$n<0\$. If {n} is not a number, the functions return {False}. *E.G. In> IsNegativeNumber(6); Out> False; In> IsNegativeNumber(-2.5); Out> True; *SEE IsNumber, IsPositiveNumber, IsNotZero, IsNegativeInteger, IsNegativeReal *CMD IsNegativeInteger --- test for a negative integer *STD *CALL IsNegativeInteger(n) *PARMS {n} -- integer to test *DESC This function tests whether the integer {n} is (strictly) negative. The negative integers are -1, -2, -3, -4, -5, etc. If {n} is not a integer, the function returns {False}. *E.G. In> IsNegativeInteger(31); Out> False; In> IsNegativeInteger(-2); Out> True; *SEE IsPositiveInteger, IsNonZeroInteger, IsNegativeNumber *CMD IsPositiveNumber --- test for a positive number *STD *CALL IsPositiveNumber(n) *PARMS {n} -- number to test *DESC {IsPositiveNumber(n)} evaluates to {True} if \$n\$ is (strictly) positive, i.e. if \$n>0\$. If {n} is not a number the function returns {False}. *E.G. In> IsPositiveNumber(6); Out> True; In> IsPositiveNumber(-2.5); Out> False; *SEE IsNumber, IsNegativeNumber, IsNotZero, IsPositiveInteger, IsPositiveReal *CMD IsPositiveInteger --- test for a positive integer *STD *CALL IsPositiveInteger(n) *PARMS {n} -- integer to test *DESC This function tests whether the integer {n} is (strictly) positive. The positive integers are 1, 2, 3, 4, 5, etc. If {n} is not a integer, the function returns {False}. *E.G. In> IsPositiveInteger(31); Out> True; In> IsPositiveInteger(-2); Out> False; *SEE IsNegativeInteger, IsNonZeroInteger, IsPositiveNumber *CMD IsNotZero --- test for a nonzero number *STD *CALL IsNotZero(n) *PARMS {n} -- number to test *DESC {IsNotZero(n)} evaluates to {True} if {n} is not zero. In case {n} is not a number, the function returns {False}. *E.G. In> IsNotZero(3.25); Out> True; In> IsNotZero(0); Out> False; *SEE IsNumber, IsPositiveNumber, IsNegativeNumber, IsNonZeroInteger *CMD IsNonZeroInteger --- test for a nonzero integer *STD *CALL IsNonZeroInteger(n) *PARMS {n} -- integer to test *DESC This function tests whether the integer {n} is not zero. If {n} is not an integer, the result is {False}. *E.G. In> IsNonZeroInteger(0) Out> False; In> IsNonZeroInteger(-2) Out> True; *SEE IsPositiveInteger, IsNegativeInteger, IsNotZero *CMD IsInfinity --- test for an infinity *STD *CALL IsInfinity(expr) *PARMS {expr} -- expression to test *DESC This function tests whether {expr} is an infinity. This is only the case if {expr} is either {Infinity} or {-Infinity}. *E.G. In> IsInfinity(10^1000); Out> False; In> IsInfinity(-Infinity); Out> True; *SEE Integer *CMD IsPositiveReal --- test for a numerically positive value *STD *CALL IsPositiveReal(expr) *PARMS {expr} -- expression to test *DESC This function tries to approximate "expr" numerically. It returns {True} if this approximation is positive. In case no approximation can be found, the function returns {False}. Note that round-off errors may cause incorrect results. *E.G. In> IsPositiveReal(Sin(1)-3/4); Out> True; In> IsPositiveReal(Sin(1)-6/7); Out> False; In> IsPositiveReal(Exp(x)); Out> False; The last result is because {Exp(x)} cannot be numerically approximated if {x} is not known. Hence Yacas can not determine the sign of this expression. *SEE IsNegativeReal, IsPositiveNumber, N *CMD IsNegativeReal --- test for a numerically negative value *STD *CALL IsNegativeReal(expr) *PARMS {expr} -- expression to test *DESC This function tries to approximate {expr} numerically. It returns {True} if this approximation is negative. In case no approximation can be found, the function returns {False}. Note that round-off errors may cause incorrect results. *E.G. In> IsNegativeReal(Sin(1)-3/4); Out> False; In> IsNegativeReal(Sin(1)-6/7); Out> True; In> IsNegativeReal(Exp(x)); Out> False; The last result is because {Exp(x)} cannot be numerically approximated if {x} is not known. Hence Yacas can not determine the sign of this expression. *SEE IsPositiveReal, IsNegativeNumber, N *CMD IsConstant --- test for a constant *STD *CALL IsConstant(expr) *PARMS {expr} -- some expression *DESC {IsConstant} returns {True} if the expression is some constant or a function with constant arguments. It does this by checking that no variables are referenced in the expression. {Pi} is considered a constant. *E.G. In> IsConstant(Cos(x)) Out> False; In> IsConstant(Cos(2)) Out> True; In> IsConstant(Cos(2+x)) Out> False; *SEE IsNumber, IsInteger, VarList *CMD IsGaussianInteger --- test for a Gaussian integer *STD *CALL IsGaussianInteger(z) *PARMS {z} -- a complex or real number *DESC This function returns {True} if the argument is a Gaussian integer and {False} otherwise. A Gaussian integer is a generalization of integers into the complex plane. A complex number \$a+b*I\$ is a Gaussian integer if and only if \$a\$ and \$b\$ are integers. *E.G. In> IsGaussianInteger(5) Out> True; In> IsGaussianInteger(5+6*I) Out> True; In> IsGaussianInteger(1+2.5*I) Out> False; *SEE IsGaussianUnit, IsGaussianPrime *CMD MatchLinear --- match an expression to a polynomial of degree one in a variable *STD *CALL MatchLinear(x,expr) *PARMS {x} -- variable to express the univariate polynomial in {expr} -- expression to match *DESC {MatchLinear} tries to match an expression to a linear (degree less than two) polynomial. The function returns {True} if it could match, and it stores the resulting coefficients in the variables "{a}" and "{b}" as a side effect. The function calling this predicate should declare local variables "{a}" and "{b}" for this purpose. {MatchLinear} tries to match to constant coefficients which don't depend on the variable passed in, trying to find a form "{a*x+b}" with "{a}" and "{b}" not depending on {x} if {x} is given as the variable. *E.G. In> MatchLinear(x,(R+1)*x+(T-1)) Out> True; In> {a,b}; Out> {R+1,T-1}; In> MatchLinear(x,Sin(x)*x+(T-1)) Out> False; *SEE Integrate *CMD HasExpr --- check for expression containing a subexpression *CMD HasExprArith --- check for expression containing a subexpression *CMD HasExprSome --- check for expression containing a subexpression *STD *CALL HasExpr(expr, x) HasExprArith(expr, x) HasExprSome(expr, x, list) *PARMS {expr} -- an expression {x} -- a subexpression to be found {list} -- list of function atoms to be considered "transparent" *DESC The command {HasExpr} returns {True} if the expression {expr} contains a literal subexpression {x}. The expression is recursively traversed. The command {HasExprSome} does the same, except it only looks at arguments of a given {list} of functions. All other functions become "opaque" (as if they do not contain anything). {HasExprArith} is defined through {HasExprSome} to look only at arithmetic operations {+}, {-}, {*}, {/}. Note that since the operators "{+}" and "{-}" are prefix as well as infix operators, it is currently required to use {Atom("+")} to obtain the unevaluated atom "{+}". *E.G. In> HasExpr(x+y*Cos(Ln(z)/z), z) Out> True; In> HasExpr(x+y*Cos(Ln(z)/z), Ln(z)) Out> True; In> HasExpr(x+y*Cos(Ln(z)/z), z/Ln(z)) Out> False; In> HasExprArith(x+y*Cos(Ln(x)/x), z) Out> False; In> HasExprSome({a+b*2,c/d},c/d,{List}) Out> True; In> HasExprSome({a+b*2,c/d},c,{List}) Out> False; *SEE FuncList, VarList, HasFunc *CMD HasFunc --- check for expression containing a function *CMD HasFuncArith --- check for expression containing a function *CMD HasFuncSome --- check for expression containing a function *STD *CALL HasFunc(expr, func) HasFuncArith(expr, func) HasFuncSome(expr, func, list) *PARMS {expr} -- an expression {func} -- a function atom to be found {list} -- list of function atoms to be considered "transparent" *DESC The command {HasFunc} returns {True} if the expression {expr} contains a function {func}. The expression is recursively traversed. The command {HasFuncSome} does the same, except it only looks at arguments of a given {list} of functions. Arguments of all other functions become "opaque" (as if they do not contain anything). {HasFuncArith} is defined through {HasFuncSome} to look only at arithmetic operations {+}, {-}, {*}, {/}. Note that since the operators "{+}" and "{-}" are prefix as well as infix operators, it is currently required to use {Atom("+")} to obtain the unevaluated atom "{+}". *E.G. In> HasFunc(x+y*Cos(Ln(z)/z), Ln) Out> True; In> HasFunc(x+y*Cos(Ln(z)/z), Sin) Out> False; In> HasFuncArith(x+y*Cos(Ln(x)/x), Cos) Out> True; In> HasFuncArith(x+y*Cos(Ln(x)/x), Ln) Out> False; In> HasFuncSome({a+b*2,c/d},/,{List}) Out> True; In> HasFuncSome({a+b*2,c/d},*,{List}) Out> False; *SEE FuncList, VarList, HasExpr