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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

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

withRegularChains:

withConstructibleSetTools:

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

R:=PolynomialRingx,y,t

R:=polynomial_ring

(1)

p:=5t+5xy10t+7

p:=5t+5xy10t7

(2)

q:=5t5xt+2y7t+11

q:=5t5xt+2y7t+11

(3)

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

cs1:=GeneralConstructp,q,xt,R

cs1:=constructible_set

(4)

cs2:=GeneralConstructp,q,x+t,R

cs2:=constructible_set

(5)

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

IsContainedcs1,cs2,R

false

(6)

IsContainedcs2,cs1,R

false

(7)

IsContainedIntersectioncs2,cs1,R,cs2,R

true

(8)

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

emcs:=EmptyConstructibleSetR

emcs:=constructible_set

(9)

IsContainedemcs,cs2,R

true

(10)

IsContainedemcs,emcs,R

true

(11)

Semi-algebraic case:

lrsas1:=RealTriangularizep2+q2,,,xt,R

lrsas1:=regular_semi_algebraic_system

(12)

lrsas2:=RealTriangularizep,q,,,x+t,xt,R

lrsas2:=regular_semi_algebraic_system,regular_semi_algebraic_system

(13)

IsContainedlrsas1,lrsas2,R

false

(14)

IsContainedlrsas2,lrsas1,R

true

(15)

See Also

Complement, ConstructibleSet, ConstructibleSetTools, Difference, EmptyConstructibleSet, Intersection, RealTriangularize, RegularChains, Union

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.


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