The command IsContained(cs1, cs2, R) returns true if cs1 is contained in cs2; otherwise false. The polynomial ring may have characteristic zero or a prime characteristic. cs1 and cs2 must be defined over the same ring R.

The command IsContained('lrsas1', 'lrsas2', 'R') returns true if lrsas1 is contained in lrsas2; otherwise false. The polynomial ring must have characteristic zero. lrsas1 and lrsas2 must be defined over the same ring R.

A constructible set is encoded as an constructible_set object, see the type definition in ConstructibleSetTools.

A semi-algebraic set is encoded by a list of regular_semi_algebraic_system, see the type definition in RealTriangularize.

This command is available once either the RegularChains[ConstructibleSetTools] submodule or RegularChains[SemiAlgebraicSetTools] submodule has been loaded. It can also be accessed through the long form of the command by using RegularChains[ConstructibleSetTools][IsContained] or RegularChains[SemiAlgebraicSetTools][IsContained].

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

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

$R≔\mathrm{PolynomialRing}\left(\left[x\,y\,t\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$p≔\left(5t+5\right)x-y-\left(10t+7\right)$

$p≔\left(5t+5\right)x-y-10t-7$

$q≔\left(5t-5\right)x-\left(t+2\right)y+\left(-7t+11\right)$

$q≔\left(5t-5\right)x-\left(t+2\right)y-7t+11$

$\mathrm{cs1}≔\mathrm{GeneralConstruct}\left(\left[p\,q\right]\,\left[x-t\right]\,R\right)$

$\mathrm{cs1}≔\mathrm{constructible\_set}$

$\mathrm{cs2}≔\mathrm{GeneralConstruct}\left(\left[p\,q\right]\,\left[x+t\right]\,R\right)$

$\mathrm{cs2}≔\mathrm{constructible\_set}$

$\mathrm{IsContained}\left(\mathrm{cs1}\,\mathrm{cs2}\,R\right)$

$\mathrm{false}$

$\mathrm{IsContained}\left(\mathrm{cs2}\,\mathrm{cs1}\,R\right)$

$\mathrm{false}$

$\mathrm{IsContained}\left(\mathrm{Intersection}\left(\mathrm{cs2}\,\mathrm{cs1}\,R\right)\,\mathrm{cs2}\,R\right)$

$\mathrm{true}$

$\mathrm{emcs}≔\mathrm{EmptyConstructibleSet}\left(R\right)$

$\mathrm{emcs}≔\mathrm{constructible\_set}$

$\mathrm{IsContained}\left(\mathrm{emcs}\,\mathrm{cs2}\,R\right)$

$\mathrm{true}$

$\mathrm{IsContained}\left(\mathrm{emcs}\,\mathrm{emcs}\,R\right)$

$\mathrm{true}$

$\mathrm{lrsas1}≔\mathrm{RealTriangularize}\left(\left[p^2+q^2\right]\,\left[\right]\,\left[\right]\,\left[x-t\right]\,R\right)$

$\mathrm{lrsas1}≔\left[\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{lrsas2}≔\mathrm{RealTriangularize}\left(\left[p\,q\right]\,\left[\right]\,\left[\right]\,\left[x+t\,x-t\right]\,R\right)$

$\mathrm{lrsas2}≔\left[\mathrm{regular\_semi\_algebraic\_system}\,\mathrm{regular\_semi\_algebraic\_system}\right]$

$\mathrm{IsContained}\left(\mathrm{lrsas1}\,\mathrm{lrsas2}\,R\right)$

$\mathrm{false}$

$\mathrm{IsContained}\left(\mathrm{lrsas2}\,\mathrm{lrsas1}\,R\right)$

$\mathrm{true}$

Chen, C.; Golubitsky, O.; Lemaire, F.; Moreno Maza, M.; and Pan, W. "Comprehensive Triangular Decomposition". Proc. CASC 2007, LNCS, Vol. 4770: 73-101. Springer, 2007.

Chen, C.; Davenport, J.-D.; Moreno Maza, M.; Xia, B.; and Xiao, R. "Computing with semi-algebraic sets represented by triangular decomposition". Proceedings of 2011 International Symposium on Symbolic and Algebraic Computation (ISSAC 2011), ACM Press, pp. 75--82, 2011.

The RegularChains[SemiAlgebraicSetTools][IsContained] command was introduced in Maple 16.

The lrsas1 parameter was introduced in Maple 16.

For more information on Maple 16 changes, see Updates in Maple 16.