The IsContained command returns true if every integer point of zpoly is contained in one of the ZPolyhedralSet of the list list, and otherwise it returns false.

If zpoly is not bounded, an error is raised.

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

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

$\mathrm{ineqs}≔\left[3x-2y+z\le 7\,-2x+2y-z\le 12\,-4x+y+3z\le 15\,-y\le -25\right]$

$\mathrm{ineqs}≔\left[3x-2y+z\le 7\,-2x+2y-z\le 12\,-4x+y+3z\le 15\,-y\le \mathrm{-25}\right]$

$\mathrm{vars}≔\left[x\,y\,z\right]\:$

$\mathrm{zp}≔\mathrm{ZPolyhedralSet}\left(\mathrm{ineqs}\,\mathrm{vars}\right)$

$\mathrm{zp}≔\left\{\begin{array}{lll}\text{Relations}& \:& \left\{\begin{array}{:}-y\le \mathrm{-25}\\ -4x+y+3z\le 15\\ -2x+2y-z\le 12\\ 3x-2y+z\le 7\end{array}\right.\\ \text{Variables}& \:& \left[x\,y\,z\right]\\ \text{Parameters}& \:& \left[\right]\\ \text{ParameterConstraints}& \:& \left\{\begin{array}{}\end{array}\right.\\ \text{Lattice}& \:& \text{ZSpan}\left(\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\,\,\,\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\right)\end{array}\right.$

$\mathrm{theset}≔\mathrm{IntegerPointDecomposition}\left(\mathrm{zp}\right)$

$\mathrm{theset}≔\left[\left\{\begin{array}{lll}\text{Relations}& \:& \left\{\begin{array}{:}z\le 17\\ -y\le \mathrm{-25}\\ -z\le \mathrm{-2}\\ -5y+13z\le 67\\ 2y-z\le 48\\ -4x+y+3z\le 15\\ -2x+2y-z\le 12\\ 3x-2y+z\le 7\end{array}\right.\\ \text{Variables}& \:& \left[x\,y\,z\right]\\ \text{Parameters}& \:& \left[\right]\\ \text{ParameterConstraints}& \:& \left\{\begin{array}{}\end{array}\right.\\ \text{Lattice}& \:& \text{ZSpan}\left(\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\,\,\,\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\right)\end{array}\right.\,\left\{\begin{array}{lll}\text{Relations}& \:& \left\{\begin{array}{:}x=15\\ y=27\\ z=16\end{array}\right.\\ \text{Variables}& \:& \left[x\,y\,z\right]\\ \text{Parameters}& \:& \left[\right]\\ \text{ParameterConstraints}& \:& \left\{\begin{array}{}\end{array}\right.\\ \text{Lattice}& \:& \text{ZSpan}\left(\left[\begin{array}{c}7\\ 13\\ 5\end{array}\right]\,\,\,\left[\begin{array}{c}\mathrm{-6}\\ \mathrm{-12}\\ 1\end{array}\right]\right)\end{array}\right.\,\left\{\begin{array}{lll}\text{Relations}& \:& \left\{\begin{array}{:}x=14\\ y=25\\ z=15\end{array}\right.\\ \text{Variables}& \:& \left[x\,y\,z\right]\\ \text{Parameters}& \:& \left[\right]\\ \text{ParameterConstraints}& \:& \left\{\begin{array}{}\end{array}\right.\\ \text{Lattice}& \:& \text{ZSpan}\left(\left[\begin{array}{c}7\\ 13\\ 5\end{array}\right]\,\,\,\left[\begin{array}{c}\mathrm{-7}\\ \mathrm{-14}\\ 0\end{array}\right]\right)\end{array}\right.\,\left\{\begin{array}{lll}\text{Relations}& \:& \left\{\begin{array}{:}x=18\\ y=33\\ z=18\end{array}\right.\\ \text{Variables}& \:& \left[x\,y\,z\right]\\ \text{Parameters}& \:& \left[\right]\\ \text{ParameterConstraints}& \:& \left\{\begin{array}{}\end{array}\right.\\ \text{Lattice}& \:& \text{ZSpan}\left(\left[\begin{array}{c}7\\ 13\\ 5\end{array}\right]\,\,\,\left[\begin{array}{c}\mathrm{-3}\\ \mathrm{-6}\\ 3\end{array}\right]\right)\end{array}\right.\,\left\{\begin{array}{lll}\text{Relations}& \:& \left\{\begin{array}{:}x=19\\ y=\frac{z}{2}+25\\ z\le 18\\ -z\le 0\end{array}\right.\\ \text{Variables}& \:& \left[x\,y\,z\right]\\ \text{Parameters}& \:& \left[\right]\\ \text{ParameterConstraints}& \:& \left\{\begin{array}{}\end{array}\right.\\ \text{Lattice}& \:& \text{ZSpan}\left(\left[\begin{array}{c}0\\ 1\\ 2\end{array}\right]\,\,\,\left[\begin{array}{c}19\\ 25\\ 0\end{array}\right]\right)\end{array}\right.\right]$

$\mathrm{ineqs1}≔\left[x=19\,y=\frac{z}{2}+25\,-z\le 0\,z\le 18\right]$

$\mathrm{ineqs1}≔\left[x=19\,y=\frac{z}{2}+25\,-z\le 0\,z\le 18\right]$

$\mathrm{ineqs2}≔\left[x=18\,y=\frac{z}{2}+25\,-z\le 0\,z\le 18\right]$

$\mathrm{ineqs2}≔\left[x=18\,y=\frac{z}{2}+25\,-z\le 0\,z\le 18\right]$

$\mathrm{zp1}≔\mathrm{ZPolyhedralSet}\left(\mathrm{ineqs1}\,\mathrm{vars}\right)$

$\mathrm{zp1}≔\left\{\begin{array}{lll}\text{Relations}& \:& \left\{\begin{array}{:}x=19\\ y=\frac{z}{2}+25\\ z\le 18\\ -z\le 0\end{array}\right.\\ \text{Variables}& \:& \left[x\,y\,z\right]\\ \text{Parameters}& \:& \left[\right]\\ \text{ParameterConstraints}& \:& \left\{\begin{array}{}\end{array}\right.\\ \text{Lattice}& \:& \text{ZSpan}\left(\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\,\,\,\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\right)\end{array}\right.$

$\mathrm{zp2}≔\mathrm{ZPolyhedralSet}\left(\mathrm{ineqs2}\,\mathrm{vars}\right)$

$\mathrm{zp2}≔\left\{\begin{array}{lll}\text{Relations}& \:& \left\{\begin{array}{:}x=18\\ y=\frac{z}{2}+25\\ z\le 18\\ -z\le 0\end{array}\right.\\ \text{Variables}& \:& \left[x\,y\,z\right]\\ \text{Parameters}& \:& \left[\right]\\ \text{ParameterConstraints}& \:& \left\{\begin{array}{}\end{array}\right.\\ \text{Lattice}& \:& \text{ZSpan}\left(\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\,\,\,\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\right)\end{array}\right.$

$\mathrm{IsContained}\left(\mathrm{zp1}\,\mathrm{theset}\right)$

$\mathrm{true}$

$\mathrm{IsContained}\left(\mathrm{zp2}\,\mathrm{theset}\right)$

$\mathrm{false}$

$\mathrm{EnumerateIntegerPoints}\left(\mathrm{zp}\right)$

$\left[\left[x=19\,y=25\,z=0\right]\,\left[x=18\,y=25\,z=2\right]\,\left[x=19\,y=26\,z=2\right]\,\left[x=18\,y=25\,z=3\right]\,\left[x=17\,y=25\,z=4\right]\,\left[x=18\,y=26\,z=4\right]\,\left[x=19\,y=27\,z=4\right]\,\left[x=17\,y=25\,z=5\right]\,\left[x=18\,y=26\,z=5\right]\,\left[x=16\,y=25\,z=6\right]\,\left[x=17\,y=25\,z=6\right]\,\left[x=17\,y=26\,z=6\right]\,\left[x=18\,y=27\,z=6\right]\,\left[x=19\,y=28\,z=6\right]\,\left[x=16\,y=25\,z=7\right]\,\left[x=17\,y=26\,z=7\right]\,\left[x=18\,y=27\,z=7\right]\,\left[x=15\,y=25\,z=8\right]\,\left[x=16\,y=25\,z=8\right]\,\left[x=16\,y=26\,z=8\right]\,\left[x=17\,y=26\,z=8\right]\,\left[x=17\,y=27\,z=8\right]\,\left[x=18\,y=28\,z=8\right]\,\left[x=19\,y=29\,z=8\right]\,\left[x=15\,y=25\,z=9\right]\,\left[x=16\,y=25\,z=9\right]\,\left[x=16\,y=26\,z=9\right]\,\left[x=17\,y=27\,z=9\right]\,\left[x=18\,y=28\,z=9\right]\,\left[x=14\,y=25\,z=10\right]\,\left[x=15\,y=25\,z=10\right]\,\left[x=15\,y=26\,z=10\right]\,\left[x=16\,y=26\,z=10\right]\,\left[x=16\,y=27\,z=10\right]\,\left[x=17\,y=27\,z=10\right]\,\left[x=17\,y=28\,z=10\right]\,\left[x=18\,y=29\,z=10\right]\,\left[x=19\,y=30\,z=10\right]\,\left[x=14\,y=25\,z=11\right]\,\left[x=15\,y=25\,z=11\right]\,\left[x=15\,y=26\,z=11\right]\,\left[x=16\,y=26\,z=11\right]\,\left[x=16\,y=27\,z=11\right]\,\left[x=17\,y=28\,z=11\right]\,\left[x=18\,y=29\,z=11\right]\,\left[x=13\,y=25\,z=12\right]\,\left[x=14\,y=25\,z=12\right]\,\left[x=15\,y=25\,z=12\right]\,\left[x=14\,y=26\,z=12\right]\,\left[x=15\,y=26\,z=12\right]\,\left[x=15\,y=27\,z=12\right]\,\left[x=16\,y=27\,z=12\right]\,\left[x=16\,y=28\,z=12\right]\,\left[x=17\,y=28\,z=12\right]\,\left[x=17\,y=29\,z=12\right]\,\left[x=18\,y=30\,z=12\right]\,\left[x=19\,y=31\,z=12\right]\,\left[x=13\,y=25\,z=13\right]\,\left[x=14\,y=25\,z=13\right]\,\left[x=14\,y=26\,z=13\right]\,\left[x=15\,y=26\,z=13\right]\,\left[x=15\,y=27\,z=13\right]\,\left[x=16\,y=27\,z=13\right]\,\left[x=16\,y=28\,z=13\right]\,\left[x=17\,y=29\,z=13\right]\,\left[x=18\,y=30\,z=13\right]\,\left[x=13\,y=25\,z=14\right]\,\left[x=14\,y=25\,z=14\right]\,\left[x=14\,y=26\,z=14\right]\,\left[x=15\,y=26\,z=14\right]\,\left[x=14\,y=27\,z=14\right]\,\left[x=15\,y=27\,z=14\right]\,\left[x=15\,y=28\,z=14\right]\,\left[x=16\,y=28\,z=14\right]\,\left[x=16\,y=29\,z=14\right]\,\left[x=17\,y=29\,z=14\right]\,\left[x=17\,y=30\,z=14\right]\,\left[x=18\,y=31\,z=14\right]\,\left[x=19\,y=32\,z=14\right]\,\left[x=14\,y=25\,z=15\right]\,\left[x=14\,y=26\,z=15\right]\,\left[x=15\,y=27\,z=15\right]\,\left[x=15\,y=28\,z=15\right]\,\left[x=16\,y=28\,z=15\right]\,\left[x=16\,y=29\,z=15\right]\,\left[x=17\,y=30\,z=15\right]\,\left[x=18\,y=31\,z=15\right]\,\left[x=15\,y=27\,z=16\right]\,\left[x=16\,y=29\,z=16\right]\,\left[x=16\,y=30\,z=16\right]\,\left[x=17\,y=30\,z=16\right]\,\left[x=17\,y=31\,z=16\right]\,\left[x=18\,y=32\,z=16\right]\,\left[x=19\,y=33\,z=16\right]\,\left[x=17\,y=31\,z=17\right]\,\left[x=18\,y=32\,z=17\right]\,\left[x=18\,y=33\,z=18\right]\,\left[x=19\,y=34\,z=18\right]\right]$

$\mathrm{EnumerateIntegerPoints}\left(\mathrm{zp1}\right)$

$\left[\left[x=19\,y=25\,z=0\right]\,\left[x=19\,y=26\,z=2\right]\,\left[x=19\,y=27\,z=4\right]\,\left[x=19\,y=28\,z=6\right]\,\left[x=19\,y=29\,z=8\right]\,\left[x=19\,y=30\,z=10\right]\,\left[x=19\,y=31\,z=12\right]\,\left[x=19\,y=32\,z=14\right]\,\left[x=19\,y=33\,z=16\right]\,\left[x=19\,y=34\,z=18\right]\right]$

$\mathrm{EnumerateIntegerPoints}\left(\mathrm{zp2}\right)$

$\left[\left[x=18\,y=25\,z=0\right]\,\left[x=18\,y=26\,z=2\right]\,\left[x=18\,y=27\,z=4\right]\,\left[x=18\,y=28\,z=6\right]\,\left[x=18\,y=29\,z=8\right]\,\left[x=18\,y=30\,z=10\right]\,\left[x=18\,y=31\,z=12\right]\,\left[x=18\,y=32\,z=14\right]\,\left[x=18\,y=33\,z=16\right]\,\left[x=18\,y=34\,z=18\right]\right]$

Rachid Seghir, Vincent Loechner, and Benoı̂t Meister. "Integer affine transformations of parametric Z-polytopes and applications to loop nest optimization." Proceedings of TACO, Vol. 9(2):8:1–8:27, 2012.

Rui-Juan Jing and Marc Moreno Maza. "Computing the Integer Points of a Polyhedron, I: Algorithm." Proceedings of CASC 2017: 225-241, Springer.

Rui-Juan Jing and Marc Moreno Maza. "Computing the Integer Points of a Polyhedron, II: Complexity Estimates." Proceedings of CASC 2017: 242-256, Springer.

The PolyhedralSets:-ZPolyhedralSets:-IsContained command was introduced in Maple 2023.

For more information on Maple 2023 changes, see Updates in Maple 2023.