RepresentingQuantifierFreeFormula - Maple Help

RegularChains[SemiAlgebraicSetTools]

 RepresentingQuantifierFreeFormula
 return the quantifier-free formula of a parametric box or a regular semi-algebraic system

 Calling Sequence RepresentingQuantifierFreeFormula(pbx) RepresentingQuantifierFreeFormula(rsas, R)

Parameters

 pbx - a parametric box rsas - a regular semi-algebraic system R - a polynomial ring

Description

 • The command RepresentingQuantifierFreeFormula(pbx) returns the representing quantifier-free formula of  the parametric box pbx.
 • The command RepresentingQuantifierFreeFormula(rsas, R) returns the representing quantifier-free formula of the regular semi-algebraic system rsas.
 • See the page SemiAlgebraicSetTools for the definition of a regular semi-algebraic system and that of a parametric box.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ParametricSystemTools}\right):$
 > $\mathrm{with}\left(\mathrm{SemiAlgebraicSetTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,b,a,c\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $F≔\left[a{x}^{2}+bx+c\right]$
 ${F}{≔}\left[{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}\right]$ (2)
 > $N≔\left[\right]$
 ${N}{≔}\left[\right]$ (3)
 > $P≔\left[x\right]$
 ${P}{≔}\left[{x}\right]$ (4)
 > $H≔\left[a\right]$
 ${H}{≔}\left[{a}\right]$ (5)
 > $\mathrm{rrc}≔\mathrm{RealRootClassification}\left(F,\left[\right],\left[x\right],\left[a\right],3,2,R\right)$
 ${\mathrm{rrc}}{≔}\left[\left[{\mathrm{regular_semi_algebraic_set}}\right]{,}{\mathrm{border_polynomial}}\right]$ (6)
 > $\mathrm{rsas}≔{{\mathrm{rrc}}_{1}}_{1}$
 ${\mathrm{rsas}}{≔}{\mathrm{regular_semi_algebraic_set}}$ (7)
 > $\mathrm{pbx}≔\mathrm{RepresentingBox}\left(\mathrm{rsas},R\right)$
 ${\mathrm{pbx}}{≔}{\mathrm{parametric_box}}$ (8)
 > $\mathrm{IsParametricBox}\left(\mathrm{pbx}\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{qff}≔\mathrm{RepresentingQuantifierFreeFormula}\left(\mathrm{pbx}\right)$
 ${\mathrm{qff}}{≔}{\mathrm{quantifier_free_formula}}$ (10)
 > $\mathrm{Info}\left(\mathrm{qff},R\right)$
 $\left[\left[{c}{,}{a}{,}{b}{,}{4}{}{a}{}{c}{-}{{b}}^{{2}}\right]{,}\left[\left[{-1}{,}{-1}{,}{1}{,}{-1}\right]{,}\left[{1}{,}{1}{,}{-1}{,}{-1}\right]\right]\right]$ (11)
 > $F≔\left[a{x}^{2}+bx+c=0,0
 ${F}{≔}\left[{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}{=}{0}{,}{0}{<}{x}{,}{a}{\ne }{0}\right]$ (12)
 > $R≔\mathrm{PolynomialRing}\left(\left[x,c,b,a\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (13)
 > $\mathrm{out}≔\mathrm{LazyRealTriangularize}\left(F,R,\mathrm{output}=\mathrm{list}\right)$
 ${\mathrm{out}}{≔}\left[{\mathrm{regular_semi_algebraic_system}}\right]$ (14)
 > $\mathrm{map}\left(\mathrm{Display},\mathrm{out},R\right)$
 $\left[\left\{\begin{array}{cc}{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}{=}{0}& {}\\ {x}{>}{0}& {}\\ \left\{\begin{array}{cc}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\ne }{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{<}{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\ne }{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{>}{0}& {}\end{array}\right\& {}\end{array}\right\\right]$ (15)
 > $P≔\mathrm{PositiveInequalities}\left({\mathrm{out}}_{1},R\right)$
 ${P}{≔}\left[{x}\right]$ (16)
 > $\mathrm{rc}≔\mathrm{RepresentingChain}\left({\mathrm{out}}_{1},R\right);$$\mathrm{Display}\left(\mathrm{rc},R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$
 $\left\{\begin{array}{cc}{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}{=}{0}& {}\\ {a}{\ne }{0}& {}\end{array}\right\$ (17)
 > $\mathrm{qff}≔\mathrm{RepresentingQuantifierFreeFormula}\left({\mathrm{out}}_{1}\right);$$\mathrm{Display}\left(\mathrm{qff},R\right)$
 ${\mathrm{qff}}{≔}{\mathrm{quantifier_free_formula}}$
 ${-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\ne }{0}$
 $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{<}{0}$
 $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\ne }{0}$
 $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{>}{0}$ (18)
 > $\mathrm{Display}\left({\mathrm{out}}_{1},R\right)$
 $\left\{\begin{array}{cc}{a}{}{{x}}^{{2}}{+}{b}{}{x}{+}{c}{=}{0}& {}\\ {x}{>}{0}& {}\\ \left\{\begin{array}{cc}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\ne }{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{<}{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{\ne }{0}& {}\\ \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{or}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{-}{4}{}{c}{}{a}{+}{{b}}^{{2}}{>}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{b}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{c}{<}{0}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathbf{and}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{a}{>}{0}& {}\end{array}\right\& {}\end{array}\right\$ (19)