The ValuesUnderConstraints package is a collection of commands for manipulating binary relations with the main objective of supporting the implementation of algorithms operating on algebraic objects depending on parameters.

Typically, such an algorithm decomposes the space of parameters into regions and associates each region with zero, one or more functions defined on that region. Each association of a parameter region with the functions defined on it can be regarded as a binary relation with that region as source, and the union of the images of these functions as target.

Some algorithms operating on algebraic objects depending on parameters rely on recursive calls and/or other algorithms operating on algebraic objects depending on parameters. As a result, during the execution of such an algorithm, one may obtain several partitions of the parameter space and one may need to refine these different partitions into a single partition. This challenge occurs in polynomial algebra with the computations of comprehensive Groebner bases or comprehensive triangular decompositions. It occurs also in polyhedral geometry when counting the number of integer points of a parametric polyhedral set.

The command MakeCaseDiscussion of the ValuesUnderConstraints package solves that challenge when the parameters take integer values or real values. In its current version, the package is optimized for the former case, while more optimization opportunities remain to be implemented for the latter case.

The commands MergeTwoCaseDiscussions and RefineCaseDiscussion solve the same challenge, but under some assumptions which allow those commands to consume less resources (time and space).

The command ValueUnderConstraints creates a value-under-constraints object, see the terminology section below.

The commands Value and Constraints inspect such as object, while the commands Equations, NonNegativeInequalities, PositiveInequalities, and Inequations separate the constraints of such an object between equations, non-strict inequalities, strict inequalities and inequations.

The commands Eval, Simplifier and Simplify deal with the simplification of the value of a value-under-constraints object w.r.t. its constraints.

The command Symbols (resp. IntegerSymbols) returns the indeterminates (resp. indeterminates taking integer values) of such an object.

The command HasInconsistentConstraints checks whether such an object is consistent or not.

The command AreEqual checks whether two such objects are equal.

The command ToPiecewise converts a list of such objects into a piecewise function.

Rui-Juan Jing, Yuzhuo Lei, Christopher F. S. Maligec, Marc Moreno Maza: "Counting the Integer Points of Parametric Polytopes: A Maple Implementation." Proceedings of Computer Algebra in Scientific Computing - 26th International Workshop (CASC) 2024: 140-160, Lecture Notes in Computer Science, vol. 14938, Springer.

The ValuesUnderConstraints package was introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.