The assume routine sets variable properties and relationships between variables. Similar functionality is provided by the assuming command.

A common use of the assume function is assume(a>0). This states that the symbol a is assumed to be a positive real constant.  Having made such an assumption, Maple routines are able to use this information to simplify expressions, for example, sqrt(a^2*b) to a*sqrt(b), and evaluate inequalities, for example, is(a+1>0) returns true. Note, both sides of an inequality assumption are implicitly assumed real (and finite), unless they are identical to infinity or -infinity.

When you make assumptions about a variable, thereafter it prints with an appended tilde (~) to indicate that the variable carries assumptions.  This behavior can be changed, if desired, by updating the showassumed interface setting (see interface).

The assume function takes arguments in the form of multiple pairs or multiple relations. When given multiple arguments, all the assumptions are taken to be true simultaneously.  For example, to define the ordering a<b<c, one can enter assume(a<b, b<c).  This specifies a<b, b<c, and a<c.

When the assume function is used to make an assumption about an expression x, all previous assumptions on x are removed. For example, if you enter assume(x>0) then assume(x<0), there is no contradiction.  Therefore, assume(0<x) followed by assume(x<1) is not equivalent to assume(0<x, x<1).

The additionally function adds additional assumptions without removing previous assumptions. It has the same syntax as assume.  Its functionality differs from assume only in that additionally adds the given properties to the set of preexisting properties of the object(s) included. If you want to assume various properties in several steps, the first call should be to assume and further calls to additionally.  For example, to assume $0<x<1$, the user can use assume(x>0) then additionally(x<1).

Assumptions made on names can be removed by unassigning names. For example, having made assumption(s) on x, x := 'x' clears them.

The is routine determines whether x1 satisfies the property prop1. It returns true, false, or FAIL. The is routine returns true if all possible values of x1 satisfy the property prop1. The is routine returns false if any possible value of x1 does not satisfy the property prop1. The is function returns FAIL if it cannot determine whether the property is always satisfied. This is a result of insufficient information or an inability to compute the logical derivation.

Important: If the is command is used with a Maple type typename, it returns true if the corresponding type(..., typename) command returns true. This may lead to unexpected results.

There are convenient tests that can be used when programming with variables that carry assumptions. To test if the expression is real, use is(Im(x)=0). To test whether x is real and positive, use is(signum(x)=1). To test whether x is an integer, use is(Im(x)=0) and is(frac(x)=0).

The coulditbe routine determines whether there is a value of x1 such that prop1 is satisfied. It returns true, false, or FAIL. The coulditbe routine returns true if there is a possible value of x1 that satisfies prop1. The coulditbe routine returns false if all possible values of x1 do not satisfy the property prop1. The coulditbe routine returns FAIL if it cannot determine whether the property is true or false.  This is a result of insufficient information or an ability to compute the logical derivation.

The environment variable _EnvTry can be used to specify the intensity of the testing by the is and coulditbe routines. Currently _EnvTry can be set to normal (the default) or hard. If _EnvTry is set to hard, is and coulditbe calls can take exponential time.

The about(x1) function returns assumption and property information for x1.

The hasassumptions(x1) function returns true if something has been assumed about x1 and false otherwise.

The getassumptions(x1) function returns the set of all relevant assumptions in the form expression::property.

If the extra option relation=relationtype is provided, getassumptions will decide which assumptions are considered relevant based on the value of relationtype, according to the following table. A name v is assumable if type(v, assumable_name) returns true. Almost all names are assumable; the main exceptions are constants, names of type last_name_eval, and names reserved for internal use by the Maple system. The default value of relationtype is default.

The addproperty(prop1, parents, children) function installs a new property in the property tables.  The function must be called with the new property name and a set of the parent properties and children properties that must already be defined in the property lattice.  The empty set can be given to specify no parent or children properties. For example, suppose you have defined properties complex and integer and wish to define the new property real. Use addproperty(real, {complex}, {integer}) to specify that all reals are complex and all integers are real.

A property can be a:

For more information on parametric properties for the assume facility, see assume/parametric.

$\mathrm{sqrt}\left(u^{2}c\right)$

$\sqrt{u^{2}c}$

$\mathrm{int}\left(\exp \left(-ux\right)x^{\frac{1}{3}}\&comma;x=0..\mathrm{\infty }\right)$

${\int }_0^{\mathrm{\infty }}{\&ExponentialE;}^{-u\mathrm{x~}}{\mathrm{x~}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\&DifferentialD;\mathrm{x~}$

$\mathrm{assume}\left(0<u\right)$

$\mathrm{sqrt}\left(u^{2}c\right)$

$\mathrm{u~}\sqrt{c}$

$\mathrm{int}\left(\exp \left(-ux\right)x^{\frac{1}{3}}\&comma;x=0..\mathrm{\infty }\right)$

$\frac{2\mathrm{\pi }\sqrt{3}}{9\mathrm{\Gamma }\left(\frac{2}{3}\right){\mathrm{u~}}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

$\mathrm{assume}\left(u<0\right)$

$\mathrm{sqrt}\left(u^{2}c\right)$

$-\mathrm{u~}\sqrt{c}$

$\mathrm{int}\left(\exp \left(-ux\right)x^{\frac{1}{3}}\&comma;x=0..\mathrm{\infty }\right)$

$\mathrm{\infty }$

$\mathrm{is}\left(5\&comma;\mathrm{RealRange}\left(5\&comma;\mathrm{\infty }\right)\right)$

$\mathrm{true}$

$\mathrm{is}\left(5\&comma;\mathrm{RealRange}\left(\mathrm{Open}\left(5\right)\&comma;\mathrm{\infty }\right)\right)$

$\mathrm{false}$

$\mathrm{assume}\left(a\&comma;'\mathrm{SquareMatrix}'\right)$

$\mathrm{assume}\left(n\&comma;'\mathrm{integer}'\right)$

$\mathrm{is}\left(a^n\&comma;'\mathrm{SquareMatrix}'\right)$

$\mathrm{true}$

$\mathrm{coulditbe}\left(a^n\&comma;'\mathrm{tridiagonal}'\right)$

$\mathrm{true}$

$\mathrm{about}\left('\mathrm{SquareMatrix}'\right)$

SquareMatrix:
  a known property having {matrix} as immediate parents
  and {NonSingular, NonSymmetric, antisymmetric, idempotent, nilpotent, singular, symmetric, triangular, tridiagonal, ElementaryMatrix, PositiveSemidefinite} as immediate children.
  mutually exclusive with {0, 1, Catalan, Pi, complex, integer, list, prime, real, set, table, GaussianInteger, GaussianPrime, NumeralNonZero, composite, constant, fraction, irrational, rational, RectangularMatrix, RealRange(0,infinity), RealRange(1,infinity), RealRange(2,infinity), RealRange(-infinity,Open(0)), RealRange(Open(0),infinity)}

$\mathrm{addproperty}\left('\mathrm{orthonormal}'\&comma;\left\{'\mathrm{vector}'\right\}\&comma;\varnothing \right)$

$\mathrm{assume}\left(u\&comma;'\mathrm{orthonormal}'\right)$

$\mathrm{is}\left(u\&comma;'\mathrm{orthonormal}'\right)$

$\mathrm{true}$

$\mathrm{is}\left(u\&comma;'\mathrm{vector}'\right)$

$\mathrm{true}$

$\mathrm{addproperty}\left('\mathrm{orthogonal}'\&comma;\left\{'\mathrm{vector}'\right\}\&comma;\left\{'\mathrm{orthonormal}'\right\}\right)$

$\mathrm{assume}\left(v\&comma;'\mathrm{orthogonal}'\right)$

$\mathrm{is}\left(v\&comma;'\mathrm{orthonormal}'\right)$

$\mathrm{false}$

$\mathrm{coulditbe}\left(v\&comma;'\mathrm{orthonormal}'\right)$

$\mathrm{true}$

$\mathrm{assume}\left(a\&comma;'\mathrm{integer}'\right)$

$\mathrm{is}\left(a\mathbf{in}'\mathrm{SetOf}'\left('\mathrm{real}'\right)\right)$

$\mathrm{true}$

$\mathrm{is}\left(a\mathbf{in}'\mathrm{SetOf}'\left('\mathrm{integer}'\right)\right)$

$\mathrm{true}$

$\mathrm{coulditbe}\left(a\mathbf{in}'\mathrm{SetOf}'\left('\mathrm{integer}'\right)\right)$

$\mathrm{true}$

$\mathrm{assume}\left(a\mathbf{in}'\mathrm{SetOf}'\left('\mathrm{prime}'\right)\mathbf{minus}\left\{2\right\}\right)$

$\mathrm{is}\left(a\mathbf{in}'\mathrm{SetOf}'\left('\mathrm{integer}'\right)\right)$

$\mathrm{true}$

$\mathrm{assume}\left(x<\mathrm{I}\&comma;y<-\mathrm{\infty }\right)$

Error, (in assume) the assumed property or properties cannot be satisfied

The assume facility for Maple is based on the algorithms described in the following references:

Weibel, Trudy, and Gonnet, Gaston. "An Algebra of Properties." Proceedings of the ISSAC-91 Conference, pp. 352-359. Bonn, July 1991.

Weibel, Trudy and Gonnet, Gaston. "An assume facility for CAS with a sample implementation for Maple." Conference Presentation at DISCO '92, Bath, England, April 14, 1992.

A general discussion of how to use the assume facility, its design and limitations is given in:

Corless, Robert, and Monagan, Michael. "Simplification and the Assume Facility." Maple Technical Newsletter, Vol. 1 No. 1. Birkhauser, 1994.