Denote by V1 and D1 the value and domain of vc. Denote by D2 the union of the domains of the value-under-constraints objects in lwd.

The command RefineCaseDiscussion(vc,lwd) returns L1, L12, L2 where L1, L12 and L2 are three lists of value-under-constraints so that:

any value-under-constraint object in L1 has value V1 and its domain is contained in the set theoretic difference of D1 and D2;

the domain of any value-under-constraint object in L12 is contained in the intersection of D1 and D2;

the domain of any value-under-constraint object in L2 is contained in the set theoretic difference of D2 and D1;

each of L1, L12 and L2 is a case discussion;

the list [op(L1), op(L12), op(L2)] refines the list [vc, op(lwd)].

By including the option output = piecewise, you instruct the command to return the result as a single piecewise expression instead. You can explicitly select the default output of three lists by passing the option output = lists.

$\mathrm{with}\left(\mathrm{ValuesUnderConstraints}\right)\:$

$\mathrm{vc1}≔\mathrm{ValueUnderConstraints}\left(\left[1\right]\&comma;\left[a=0\&comma;0\le b\&comma;c<0\&comma;d\ne 0\right]\right)$

$\mathrm{vc1}≔⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;d\ne 0\&comma;0<-c\&comma;0\le b\right]⟩$

$\mathrm{vc2}≔\mathrm{ValueUnderConstraints}\left(\left[2\right]\&comma;\left[a\ne 0\&comma;0\le b\&comma;c<0\&comma;d=0\right]\right)$

$\mathrm{vc2}≔⟨\text{value}\left\{2\right\}\text{when}\left[d=0\&comma;a\ne 0\&comma;0<-c\&comma;0\le b\right]⟩$

$\mathrm{vc3}≔\mathrm{ValueUnderConstraints}\left(\left[3\right]\&comma;\left[b=0\&comma;0\le a\&comma;d\ne 0\&comma;c\ne 0\right]\right)$

$\mathrm{vc3}≔⟨\text{value}\left\{3\right\}\text{when}\left[b=0\&comma;c\ne 0\&comma;d\ne 0\&comma;0\le a\right]⟩$

$\mathrm{vc4}≔\mathrm{ValueUnderConstraints}\left(\left[4\right]\&comma;\left[b\ne 0\&comma;0\le a\&comma;d\ne 0\&comma;c=0\right]\right)$

$\mathrm{vc4}≔⟨\text{value}\left\{4\right\}\text{when}\left[c=0\&comma;b\ne 0\&comma;d\ne 0\&comma;0\le a\right]⟩$

$\mathrm{lvc}≔\mathrm{MakeCaseDiscussion}\left(\left[\mathrm{vc1}\&comma;\mathrm{vc2}\&comma;\mathrm{vc3}\&comma;\mathrm{vc4}\right]\right)$

$\mathrm{lvc}≔\left[⟨\text{value}\left\{4\right\}\text{when}\left[c=0\&comma;b\ne 0\&comma;d\ne 0\&comma;0\le a\right]⟩\&comma;⟨\text{value}\left\{3\right\}\text{when}\left[b=0\&comma;d\ne 0\&comma;0<a\&comma;0<-c\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[d=0\&comma;a\ne 0\&comma;0<-c\&comma;0\le b\right]⟩\&comma;⟨\text{values}\left\{1\&comma;3\right\}\text{when}\left[a=0\&comma;b=0\&comma;d\ne 0\&comma;0<-c\right]⟩\&comma;⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;d\ne 0\&comma;0<b\&comma;0<-c\right]⟩\&comma;⟨\text{value}\left\{3\right\}\text{when}\left[b=0\&comma;d\ne 0\&comma;0<c\&comma;0\le a\right]⟩\right]$

$\mathrm{MergeTwoCaseDiscussions}\left(\left[\mathrm{vc1}\&comma;\mathrm{vc2}\right]\&comma;\left[\mathrm{vc3}\&comma;\mathrm{vc4}\right]\right)$

$\left[⟨\text{value}\left\{4\right\}\text{when}\left[c=0\&comma;b\ne 0\&comma;d\ne 0\&comma;0\le a\right]⟩\&comma;⟨\text{value}\left\{3\right\}\text{when}\left[b=0\&comma;d\ne 0\&comma;0<a\&comma;0<-c\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[d=0\&comma;a\ne 0\&comma;0<-c\&comma;0\le b\right]⟩\right],\left[⟨\text{values}\left\{1\&comma;3\right\}\text{when}\left[a=0\&comma;b=0\&comma;d\ne 0\&comma;0<-c\right]⟩\right],\left[⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;d\ne 0\&comma;0<b\&comma;0<-c\right]⟩\&comma;⟨\text{value}\left\{3\right\}\text{when}\left[b=0\&comma;d\ne 0\&comma;0<c\&comma;0\le a\right]⟩\right]$

$A,B,C≔\mathrm{RefineCaseDiscussion}\left(\mathrm{vc3}\&comma;\left[\mathrm{vc1}\&comma;\mathrm{vc2}\right]\right)$

$A,B,C≔\left[⟨\text{value}\left\{3\right\}\text{when}\left[b=0\&comma;d\ne 0\&comma;0<c\&comma;0\le a\right]⟩\&comma;⟨\text{value}\left\{3\right\}\text{when}\left[b=0\&comma;d\ne 0\&comma;0<a\&comma;0<-c\right]⟩\right],\left[⟨\text{values}\left\{1\&comma;3\right\}\text{when}\left[a=0\&comma;b=0\&comma;d\ne 0\&comma;0<-c\right]⟩\right],\left[⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;d\ne 0\&comma;0<b\&comma;0<-c\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[d=0\&comma;a\ne 0\&comma;0<-c\&comma;0\le b\right]⟩\right]$

$\mathrm{lvc}≔\left[\mathrm{op}\left(A\right)\&comma;\mathrm{op}\left(B\right)\&comma;\mathrm{op}\left(C\right)\&comma;\mathrm{vc4}\right]$

$\mathrm{lvc}≔\left[⟨\text{value}\left\{3\right\}\text{when}\left[b=0\&comma;d\ne 0\&comma;0<c\&comma;0\le a\right]⟩\&comma;⟨\text{value}\left\{3\right\}\text{when}\left[b=0\&comma;d\ne 0\&comma;0<a\&comma;0<-c\right]⟩\&comma;⟨\text{values}\left\{1\&comma;3\right\}\text{when}\left[a=0\&comma;b=0\&comma;d\ne 0\&comma;0<-c\right]⟩\&comma;⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;d\ne 0\&comma;0<b\&comma;0<-c\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[d=0\&comma;a\ne 0\&comma;0<-c\&comma;0\le b\right]⟩\&comma;⟨\text{value}\left\{4\right\}\text{when}\left[c=0\&comma;b\ne 0\&comma;d\ne 0\&comma;0\le a\right]⟩\right]$

$\mathrm{RefineCaseDiscussion}\left(\mathrm{vc3}\&comma;\left[\mathrm{vc1}\&comma;\mathrm{vc2}\right]\&comma;\mathrm{output}=\mathrm{piecewise}\right)$

$\left\{\begin{array}{cc}\left\{3\right\}& b=0\wedge d\ne 0\wedge 0<c\wedge 0\le a\\ \left\{3\right\}& b=0\wedge d\ne 0\wedge 0<a\wedge 0<-c\\ \left\{1\&comma;3\right\}& a=0\wedge b=0\wedge d\ne 0\wedge 0<-c\\ \left\{1\right\}& a=0\wedge d\ne 0\wedge 0<b\wedge 0<-c\\ \left\{2\right\}& d=0\wedge a\ne 0\wedge 0<-c\wedge 0\le b\end{array}\right.$

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[RefineCaseDiscussion] command was introduced in Maple 2025.

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