IsContained - Maple Help

RegularChains

 ConstructibleSetTools[IsContained]
 check whether or not a constructible set is a subset of another one
 SemiAlgebraicSetTools[IsContained]
 check whether or not a semi-algebraic set is a subset of another one

 Calling Sequence IsContained(cs1, cs2, R) IsContained(lrsas1, lrsas2, R)

Parameters

 cs1, cs2 - constructible sets lrsas1, lrsas2 - lists of regular semi-algebraic systems R - polynomial ring

Description

 • 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].

Examples

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

First, define the polynomial ring $R$ and two polynomials of $R$.

 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,t\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $p≔\left(5t+5\right)x-y-\left(10t+7\right)$
 ${p}{≔}\left({5}{}{t}{+}{5}\right){}{x}{-}{y}{-}{10}{}{t}{-}{7}$ (2)
 > $q≔\left(5t-5\right)x-\left(t+2\right)y+\left(-7t+11\right)$
 ${q}{≔}\left({5}{}{t}{-}{5}\right){}{x}{-}\left({t}{+}{2}\right){}{y}{-}{7}{}{t}{+}{11}$ (3)

Using the GeneralConstruct function and adding one inequality, you can build a constructible set. By $x-t$ and $x+t$, two constructible sets cs1 and cs2 are different.

 > $\mathrm{cs1}≔\mathrm{GeneralConstruct}\left(\left[p,q\right],\left[x-t\right],R\right)$
 ${\mathrm{cs1}}{≔}{\mathrm{constructible_set}}$ (4)
 > $\mathrm{cs2}≔\mathrm{GeneralConstruct}\left(\left[p,q\right],\left[x+t\right],R\right)$
 ${\mathrm{cs2}}{≔}{\mathrm{constructible_set}}$ (5)

Use the IsContained function to check if one is contained in another.

 > $\mathrm{IsContained}\left(\mathrm{cs1},\mathrm{cs2},R\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{IsContained}\left(\mathrm{cs2},\mathrm{cs1},R\right)$
 ${\mathrm{false}}$ (7)
 > $\mathrm{IsContained}\left(\mathrm{Intersection}\left(\mathrm{cs2},\mathrm{cs1},R\right),\mathrm{cs2},R\right)$
 ${\mathrm{true}}$ (8)

The empty constructible set is contained in any other constructible set.

 > $\mathrm{emcs}≔\mathrm{EmptyConstructibleSet}\left(R\right)$
 ${\mathrm{emcs}}{≔}{\mathrm{constructible_set}}$ (9)
 > $\mathrm{IsContained}\left(\mathrm{emcs},\mathrm{cs2},R\right)$
 ${\mathrm{true}}$ (10)
 > $\mathrm{IsContained}\left(\mathrm{emcs},\mathrm{emcs},R\right)$
 ${\mathrm{true}}$ (11)

Semi-algebraic case:

 > $\mathrm{lrsas1}≔\mathrm{RealTriangularize}\left(\left[{p}^{2}+{q}^{2}\right],\left[\right],\left[\right],\left[x-t\right],R\right)$
 ${\mathrm{lrsas1}}{≔}\left[{\mathrm{regular_semi_algebraic_system}}\right]$ (12)
 > $\mathrm{lrsas2}≔\mathrm{RealTriangularize}\left(\left[p,q\right],\left[\right],\left[\right],\left[x+t,x-t\right],R\right)$
 ${\mathrm{lrsas2}}{≔}\left[{\mathrm{regular_semi_algebraic_system}}{,}{\mathrm{regular_semi_algebraic_system}}\right]$ (13)
 > $\mathrm{IsContained}\left(\mathrm{lrsas1},\mathrm{lrsas2},R\right)$
 ${\mathrm{false}}$ (14)
 > $\mathrm{IsContained}\left(\mathrm{lrsas2},\mathrm{lrsas1},R\right)$
 ${\mathrm{true}}$ (15)

References

 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.

Compatibility

 • The RegularChains[SemiAlgebraicSetTools][IsContained] command was introduced in Maple 16.
 • The lrsas1 parameter was introduced in Maple 16.