"Evolutionary" method offering decrementality within QuantifierElimination.

Performs QE on the input expression associated with data, but modified via arguments described above. Minimizes recomputation by using existing data from data.

QEData and CADData objects can be obtained by requesting data via the output keyword option for all QuantifierElimination routines offering quantifier elimination. CADData is returned when the quantifier elimination was provided by PartialCylindricalAlgebraicDecompose.

DeleteFormula is a routine offering generic decrementality including deletion of formulae at generic atomic positions for a previously used formula for quantifier elimination. This means that arbitrary subformulae can be deleted within a targeted subformula of a formula originally used for elimination. This means QE of similar formulae can better be explored without traversing the entire elimination process from scratch.

The atomic position atpos provided must be a positive integer, or a list of such which describes the path through the unquantified part of the prenex formula previously used for elimination which is associated to data. The atomic position provided must be a viable atomic position for the formula to delete in light of the previously used formula.

Atomic positions start their indexing from 1 and not 0.

Relifting the CAD tree completely may be a performance detriment, but may be useful to prune the CAD tree such that it is of fewer cells.

$\mathrm{with}\left(\mathrm{QuantifierElimination}\right)\:$$\mathrm{with}\left(\mathrm{QuantifierTools}\right)\:$

$\mathrm{expr}≔\mathrm{forall}\left(x\&comma;0<x\Rightarrow 0<x^2\right)$

$\mathrm{expr}≔\forall \left(x\&comma;0<x\Rightarrow 0<x^2\right)$

$\mathrm{expr}≔\mathrm{ConvertToPrenexForm}\left(\mathrm{expr}\right)$

$\mathrm{expr}≔\forall \left(x\&comma;0\le -x\vee -x^2<0\right)$

$\mathrm{qf},w,\mathrm{data}≔\mathrm{QuantifierEliminate}\left(\mathrm{expr}\&comma;'\mathrm{output}=\left[\mathrm{qf}\&comma;\mathrm{witnesses}\&comma;\mathrm{data}\right]'\right)$

$\mathrm{qf},w,\mathrm{data}≔\mathrm{true},\left[\right],\mathrm{QEData\; for}\forall \left(x\&comma;x\le 0\vee -x^2<0\right)$

$\mathrm{qf},w,\mathrm{data}≔\mathrm{DeleteFormula}\left(\mathrm{data}\&comma;1\&comma;'\mathrm{output}=\left[\mathrm{qf}\&comma;\mathrm{witnesses}\&comma;\mathrm{data}\right]'\right)$

$\mathrm{qf},w,\mathrm{data}≔\mathrm{false},\left[\left[\mathrm{false}\&comma;x=0\right]\right],\mathrm{QEData\; for}\forall \left(x\&comma;-x^2<0\right)$

$q,w,\mathrm{data}≔\mathrm{InsertFormula}\left(\mathrm{data}\&comma;'\mathrm{Or}'\&comma;1\&comma;x<0\&comma;'\mathrm{output}=\left[\mathrm{qf}\&comma;\mathrm{witnesses}\&comma;\mathrm{data}\right]'\right)$

$q,w,\mathrm{data}≔\mathrm{false},\left[\left[\mathrm{false}\&comma;x=0\right]\right],\mathrm{QEData\; for}\forall \left(x\&comma;x<0\vee -x^2<0\right)$

The QuantifierElimination:-DeleteFormula command was introduced in Maple 2023.

For more information on Maple 2023 changes, see Updates in Maple 2023.