The command MakeCaseDiscussion(lvc) returns a list lwd of value-under-constraints objects so that lwd is a case discussion and lwd refines lvc.

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

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

$\mathrm{vc1}≔\mathrm{ValueUnderConstraints}\left(\left[1\right]\,\left[a\,b\,c\,d\,e\,f\,g\,h\right]\,\left[\right]\,\left[a\,b\right]\,\left[c\,d\right]\,\left[\right]\,\left\{a\,b\,c\,d\,e\,f\,g\,h\right\}\right)$

$\mathrm{vc1}≔⟨\text{value}\left\{1\right\}\text{when}\left[0\le a\,0\le b\,0\le c-1\,0\le d-1\right]⟩$

$\mathrm{vc2}≔\mathrm{ValueUnderConstraints}\left(\left[2\right]\,\left[a\,b\,c\,d\,e\,f\,g\,h\right]\,\left[\right]\,\left[e\,f\right]\,\left[g\,h\right]\,\left[\right]\,\left\{a\,b\,c\,d\,e\,f\,g\,h\right\}\right)$

$\mathrm{vc2}≔⟨\text{value}\left\{2\right\}\text{when}\left[0\le e\,0\le f\,0\le g-1\,0\le h-1\right]⟩$

$\mathrm{PairCompare}\left(\mathrm{vc1}\,\mathrm{vc2}\right)$

$\left[⟨\text{value}\left\{1\right\}\text{when}\left[0\le a\,0\le b\,0\le c-1\,0\le d-1\,0\le -e-1\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[0\le a\,0\le b\,0\le e\,0\le c-1\,0\le d-1\,0\le -f-1\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[0\le a\,0\le b\,0\le e\,0\le f\,0\le -g\,0\le c-1\,0\le d-1\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[0\le a\,0\le b\,0\le e\,0\le f\,0\le -h\,0\le c-1\,0\le d-1\,0\le g-1\right]⟩\right],\left[⟨\text{values}\left\{1\,2\right\}\text{when}\left[0\le a\,0\le b\,0\le e\,0\le f\,0\le c-1\,0\le d-1\,0\le g-1\,0\le h-1\right]⟩\right],\left[⟨\text{value}\left\{2\right\}\text{when}\left[0\le e\,0\le f\,0\le -a-1\,0\le g-1\,0\le h-1\right]⟩\,⟨\text{value}\left\{2\right\}\text{when}\left[0\le a\,0\le e\,0\le f\,0\le -b-1\,0\le g-1\,0\le h-1\right]⟩\,⟨\text{value}\left\{2\right\}\text{when}\left[0\le a\,0\le b\,0\le e\,0\le f\,0\le -c\,0\le g-1\,0\le h-1\right]⟩\,⟨\text{value}\left\{2\right\}\text{when}\left[0\le a\,0\le b\,0\le e\,0\le f\,0\le -d\,0\le c-1\,0\le g-1\,0\le h-1\right]⟩\right]$

$\mathrm{MakeCaseDiscussion}\left(\left[\mathrm{vc1}\,\mathrm{vc2}\right]\right)$

$\left[⟨\text{value}\left\{1\right\}\text{when}\left[0\le a\,0\le b\,0\le c-1\,0\le d-1\,0\le -e-1\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[0\le a\,0\le b\,0\le e\,0\le c-1\,0\le d-1\,0\le -f-1\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[0\le a\,0\le b\,0\le e\,0\le f\,0\le -g\,0\le c-1\,0\le d-1\right]⟩\,⟨\text{value}\left\{1\right\}\text{when}\left[0\le a\,0\le b\,0\le e\,0\le f\,0\le -h\,0\le c-1\,0\le d-1\,0\le g-1\right]⟩\,⟨\text{values}\left\{1\,2\right\}\text{when}\left[0\le a\,0\le b\,0\le e\,0\le f\,0\le c-1\,0\le d-1\,0\le g-1\,0\le h-1\right]⟩\,⟨\text{value}\left\{2\right\}\text{when}\left[0\le e\,0\le f\,0\le -a-1\,0\le g-1\,0\le h-1\right]⟩\,⟨\text{value}\left\{2\right\}\text{when}\left[0\le a\,0\le e\,0\le f\,0\le -b-1\,0\le g-1\,0\le h-1\right]⟩\,⟨\text{value}\left\{2\right\}\text{when}\left[0\le a\,0\le b\,0\le e\,0\le f\,0\le -c\,0\le g-1\,0\le h-1\right]⟩\,⟨\text{value}\left\{2\right\}\text{when}\left[0\le a\,0\le b\,0\le e\,0\le f\,0\le -d\,0\le c-1\,0\le g-1\,0\le h-1\right]⟩\right]$

$\mathrm{MakeCaseDiscussion}\left(\left[\mathrm{vc1}\,\mathrm{vc2}\right]\,\mathrm{output}=\mathrm{piecewise}\right)$

$\left\{\begin{array}{cc}\left\{1\right\}& 0\le a\wedge 0\le b\wedge 0\le c-1\wedge 0\le d-1\wedge 0\le -e-1\\ \left\{1\right\}& 0\le a\wedge 0\le b\wedge 0\le e\wedge 0\le c-1\wedge 0\le d-1\wedge 0\le -f-1\\ \left\{1\right\}& 0\le a\wedge 0\le b\wedge 0\le e\wedge 0\le f\wedge 0\le -g\wedge 0\le c-1\wedge 0\le d-1\\ \left\{1\right\}& 0\le a\wedge 0\le b\wedge 0\le e\wedge 0\le f\wedge 0\le -h\wedge 0\le c-1\wedge 0\le d-1\wedge 0\le g-1\\ \left\{1\,2\right\}& 0\le a\wedge 0\le b\wedge 0\le e\wedge 0\le f\wedge 0\le c-1\wedge 0\le d-1\wedge 0\le g-1\wedge 0\le h-1\\ \left\{2\right\}& 0\le e\wedge 0\le f\wedge 0\le -a-1\wedge 0\le g-1\wedge 0\le h-1\\ \left\{2\right\}& 0\le a\wedge 0\le e\wedge 0\le f\wedge 0\le -b-1\wedge 0\le g-1\wedge 0\le h-1\\ \left\{2\right\}& 0\le a\wedge 0\le b\wedge 0\le e\wedge 0\le f\wedge 0\le -c\wedge 0\le g-1\wedge 0\le h-1\\ \left\{2\right\}& 0\le a\wedge 0\le b\wedge 0\le e\wedge 0\le f\wedge 0\le -d\wedge 0\le c-1\wedge 0\le g-1\wedge 0\le h-1\end{array}\right.$

$\mathrm{vc5}≔\mathrm{ValueUnderConstraints}\left(\left[3\right]\,\left[a\,b\,c\,d\right]\,\left[a\right]\,\left[b\right]\,\left[c\right]\,\left[d\right]\,\left\{a\,b\,c\,d\right\}\right)$

$\mathrm{vc5}≔⟨\text{value}\left\{3\right\}\text{when}\left[a=0\,d\ne 0\,0\le b\,0\le c-1\right]⟩$

$\mathrm{vc6}≔\mathrm{ValueUnderConstraints}\left(\left[4\right]\,\left[a\,b\,c\,d\right]\,\left[b\right]\,\left[c\right]\,\left[d\right]\,\left[a\right]\,\left\{a\,b\,c\,d\right\}\right)$

$\mathrm{vc6}≔⟨\text{value}\left\{4\right\}\text{when}\left[b=0\,a\ne 0\,0\le c\,0\le d-1\right]⟩$

$\mathrm{vc7}≔\mathrm{ValueUnderConstraints}\left(\left[5\right]\,\left[a\,b\,c\,d\right]\,\left[c\right]\,\left[d\right]\,\left[a\right]\,\left[b\right]\,\left\{a\,b\,c\,d\right\}\right)$

$\mathrm{vc7}≔⟨\text{value}\left\{5\right\}\text{when}\left[c=0\,b\ne 0\,0\le d\,0\le a-1\right]⟩$

$\mathrm{vc8}≔\mathrm{ValueUnderConstraints}\left(\left[6\right]\,\left[a\,b\,c\,d\right]\,\left[d\right]\,\left[a\right]\,\left[b\right]\,\left[c\right]\,\left\{a\,b\,c\,d\right\}\right)$

$\mathrm{vc8}≔⟨\text{value}\left\{6\right\}\text{when}\left[d=0\,c\ne 0\,0\le a\,0\le b-1\right]⟩$

$\mathrm{MakeCaseDiscussion}\left(\left[\mathrm{vc5}\,\mathrm{vc6}\,\mathrm{vc7}\,\mathrm{vc8}\right]\right)$

$\left[⟨\text{value}\left\{3\right\}\text{when}\left[a=0\,d\ne 0\,0\le b\,0\le c-1\right]⟩\,⟨\text{value}\left\{4\right\}\text{when}\left[b=0\,a\ne 0\,0\le c\,0\le d-1\right]⟩\,⟨\text{value}\left\{5\right\}\text{when}\left[c=0\,b\ne 0\,0\le d\,0\le a-1\right]⟩\,⟨\text{value}\left\{6\right\}\text{when}\left[d=0\,c\ne 0\,0\le a\,0\le b-1\right]⟩\right]$

$\mathrm{MakeCaseDiscussion}\left(\left[\mathrm{vc5}\,\mathrm{vc6}\,\mathrm{vc7}\,\mathrm{vc8}\right]\,\mathrm{output}=\mathrm{piecewise}\right)$

$\left\{\begin{array}{cc}\left\{3\right\}& a=0\wedge d\ne 0\wedge 0\le b\wedge 0\le c-1\\ \left\{4\right\}& b=0\wedge a\ne 0\wedge 0\le c\wedge 0\le d-1\\ \left\{5\right\}& c=0\wedge b\ne 0\wedge 0\le d\wedge 0\le a-1\\ \left\{6\right\}& d=0\wedge c\ne 0\wedge 0\le a\wedge 0\le b-1\end{array}\right.$

$\mathrm{vc14}≔\mathrm{ValueUnderConstraints}\left(\left[1\right]\,\left[a\,b\,c\,d\,e\,f\,g\,h\right]\,\left[a\right]\,\left[b\right]\,\left[c\,1-b\right]\,\left[d\right]\,\varnothing \right)$

$\mathrm{vc14}≔⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;d\ne 0\&comma;0<c\&comma;0<-b+1\&comma;0\le b\right]⟩$

$\mathrm{vc15}≔\mathrm{ValueUnderConstraints}\left(\left[2\right]\&comma;\left[a\&comma;b\&comma;c\&comma;d\&comma;e\&comma;f\&comma;g\&comma;h\right]\&comma;\left[e\right]\&comma;\left[f\right]\&comma;\left[g\&comma;1-f\right]\&comma;\left[h\right]\&comma;\varnothing \right)$

$\mathrm{vc15}≔⟨\text{value}\left\{2\right\}\text{when}\left[e=0\&comma;h\ne 0\&comma;0<g\&comma;0<1-f\&comma;0\le f\right]⟩$

$\mathrm{MakeCaseDiscussion}\left(\left[\mathrm{vc14}\&comma;\mathrm{vc15}\right]\&comma;\mathrm{output}=\mathrm{list}\right)$

$\left[⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;d\ne 0\&comma;0<c\&comma;0<-b+1\&comma;0\le b\&comma;0\le -g\right]⟩\&comma;⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;d\ne 0\&comma;0<c\&comma;0<g\&comma;0<-b+1\&comma;0\le b\&comma;0\le -1+f\right]⟩\&comma;⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;d\ne 0\&comma;0<c\&comma;0<g\&comma;0<-f\&comma;0<-b+1\&comma;0\le b\right]⟩\&comma;⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;d\ne 0\&comma;e\ne 0\&comma;0<c\&comma;0<g\&comma;0<1-f\&comma;0<-b+1\&comma;0\le b\&comma;0\le f\right]⟩\&comma;⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;e=0\&comma;h=0\&comma;d\ne 0\&comma;0<c\&comma;0<g\&comma;0<1-f\&comma;0<-b+1\&comma;0\le b\&comma;0\le f\right]⟩\&comma;⟨\text{values}\left\{1\&comma;2\right\}\text{when}\left[a=0\&comma;e=0\&comma;d\ne 0\&comma;h\ne 0\&comma;0<c\&comma;0<g\&comma;0<1-f\&comma;0<-b+1\&comma;0\le b\&comma;0\le f\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[e=0\&comma;h\ne 0\&comma;0<g\&comma;0<1-f\&comma;0\le f\&comma;0\le -c\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[e=0\&comma;h\ne 0\&comma;0<c\&comma;0<g\&comma;0<1-f\&comma;0\le f\&comma;0\le b-1\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[e=0\&comma;h\ne 0\&comma;0<c\&comma;0<g\&comma;0<-b\&comma;0<1-f\&comma;0\le f\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[e=0\&comma;a\ne 0\&comma;h\ne 0\&comma;0<c\&comma;0<g\&comma;0<1-f\&comma;0<-b+1\&comma;0\le b\&comma;0\le f\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[a=0\&comma;d=0\&comma;e=0\&comma;h\ne 0\&comma;0<c\&comma;0<g\&comma;0<1-f\&comma;0<-b+1\&comma;0\le b\&comma;0\le f\right]⟩\right]$

$\mathrm{MakeCaseDiscussion}\left(\left[\mathrm{vc14}\&comma;\mathrm{vc15}\right]\&comma;\mathrm{output}=\mathrm{piecewise}\right)$

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

$\mathrm{vc14}≔\mathrm{ValueUnderConstraints}\left(\left[1\right]\&comma;\left[a\&comma;b\&comma;c\&comma;d\&comma;e\&comma;f\&comma;g\&comma;h\right]\&comma;\left[a\right]\&comma;\left[b\right]\&comma;\left[c\&comma;1-b\right]\&comma;\left[d\right]\&comma;\left\{b\&comma;f\right\}\right)$

$\mathrm{vc14}≔⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;b=0\&comma;d\ne 0\&comma;0<c\right]⟩$

$\mathrm{vc15}≔\mathrm{ValueUnderConstraints}\left(\left[2\right]\&comma;\left[a\&comma;b\&comma;c\&comma;d\&comma;e\&comma;f\&comma;g\&comma;h\right]\&comma;\left[e\right]\&comma;\left[f\right]\&comma;\left[g\&comma;1-f\right]\&comma;\left[h\right]\&comma;\left\{b\&comma;f\right\}\right)$

$\mathrm{vc15}≔⟨\text{value}\left\{2\right\}\text{when}\left[e=0\&comma;f=0\&comma;h\ne 0\&comma;0<g\right]⟩$

$\mathrm{MakeCaseDiscussion}\left(\left[\mathrm{vc14}\&comma;\mathrm{vc15}\right]\right)$

$\left[⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;b=0\&comma;d\ne 0\&comma;0<c\&comma;0\le -g\right]⟩\&comma;⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;b=0\&comma;d\ne 0\&comma;0<c\&comma;0<g\&comma;0\le -f-1\right]⟩\&comma;⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;b=0\&comma;d\ne 0\&comma;0<c\&comma;0<g\&comma;0\le -1+f\right]⟩\&comma;⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;b=0\&comma;f=0\&comma;d\ne 0\&comma;e\ne 0\&comma;0<c\&comma;0<g\right]⟩\&comma;⟨\text{value}\left\{1\right\}\text{when}\left[a=0\&comma;e=0\&comma;h=0\&comma;b=0\&comma;f=0\&comma;d\ne 0\&comma;0<c\&comma;0<g\right]⟩\&comma;⟨\text{values}\left\{1\&comma;2\right\}\text{when}\left[a=0\&comma;e=0\&comma;b=0\&comma;f=0\&comma;d\ne 0\&comma;h\ne 0\&comma;0<c\&comma;0<g\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[e=0\&comma;f=0\&comma;h\ne 0\&comma;0<g\&comma;0\le -c\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[e=0\&comma;f=0\&comma;h\ne 0\&comma;0<c\&comma;0<g\&comma;0\le -b-1\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[e=0\&comma;f=0\&comma;h\ne 0\&comma;0<c\&comma;0<g\&comma;0\le b-1\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[e=0\&comma;b=0\&comma;f=0\&comma;a\ne 0\&comma;h\ne 0\&comma;0<c\&comma;0<g\right]⟩\&comma;⟨\text{value}\left\{2\right\}\text{when}\left[a=0\&comma;d=0\&comma;e=0\&comma;b=0\&comma;f=0\&comma;h\ne 0\&comma;0<c\&comma;0<g\right]⟩\right]$

$\mathrm{MakeCaseDiscussion}\left(\left[\mathrm{vc14}\&comma;\mathrm{vc15}\right]\&comma;\mathrm{output}=\mathrm{piecewise}\right)$

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

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[MakeCaseDiscussion] command was introduced in Maple 2025.

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