"Evolutionary" method offering incrementality within QuantifierElimination.

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

InsertFormula is a routine offering generic incrementality including insertion of formulae at generic atomic positions. This means that arbitrary formulae can be added within a targeted subformula of a formula originally used for elimination.

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 inserting formula in light of the previously used formula. This means QE of similar formulae can better be explored without traversing the entire elimination process from scratch.

Atomic positions start their indexing from 1 and not 0.

If using InsertFormula starting from CADData, then if newformula introduces new free variables to the formula, the CAD will be relifted regardless of the value of r, which may be slow. One should begin with a core formula that covers all free variables, if possible.

$\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:-InsertFormula command was introduced in Maple 2023.

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