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RegularChains[ConstructibleSetTools][IsContained] - check whether or not a constructible set is a subset of another one
RegularChains[SemiAlgebraicSetTools][IsContained] - check whether or not a semi-algebraic set is a subset of another one
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Calling Sequence
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IsContained(cs1, cs2, R)
IsContained(lrsas1, lrsas2, R)
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Parameters
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cs1, cs2
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constructible sets
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lrsas1, lrsas2
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lists of regular semi-algebraic systems
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R
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polynomial ring
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Description
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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.
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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.
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A constructible set is encoded as an constructible_set object, see the type definition in ConstructibleSetTools.
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A semi-algebraic set is encoded by a list of regular_semi_algebraic_system, see the type definition in RealTriangularize.
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This command is available once RegularChains[ConstructibleSetTools] submodule or RegularChains[SemiAlgebraicSetTools] submodule have been loaded. be accessed through the long form of the command by using RegularChains[ConstructibleSetTools][IsContained] or RegularChains[SemiAlgebraicSetTools][IsContained].
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Compatibility
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The RegularChains[SemiAlgebraicSetTools][IsContained] command was introduced in Maple 16.
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The lrsas1 parameter was introduced in Maple 16.
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Examples
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First, define the polynomial ring  and two polynomials of  .
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![R := PolynomialRing([x, y, t])](/support/helpjp/helpview.aspx?si=6390/file06474/math166.png)
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Using the GeneralConstruct function and adding one inequality, you can build a constructible set. By  and  , two constructible sets cs1 and cs2 are different.
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![cs1 := GeneralConstruct([p, q], [x-t], R)](/support/helpjp/helpview.aspx?si=6390/file06474/math199.png)
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![cs2 := GeneralConstruct([p, q], [x+t], R)](/support/helpjp/helpview.aspx?si=6390/file06474/math206.png)
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Use the IsContained function to check if one is contained in another.
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The empty constructible set is contained in any other constructible set.
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Semi-algebraic case:
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![lrsas1 := RealTriangularize([p^2+q^2], [], [], [x-t], R)](/support/helpjp/helpview.aspx?si=6390/file06474/math263.png)
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![lrsas1 := [regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6390/file06474/math266.png)
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![lrsas2 := RealTriangularize([p, q], [], [], [x+t, x-t], R)](/support/helpjp/helpview.aspx?si=6390/file06474/math270.png)
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![lrsas2 := [regular_semi_algebraic_system, regular_semi_algebraic_system]](/support/helpjp/helpview.aspx?si=6390/file06474/math273.png)
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References
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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.
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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.
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