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# Interface to QEPCAD

The basic function of QEPCAD is to construct cylindrical algebraic decompositions (CADs) of \(\RR^k\), given a list of polynomials. Using this CAD, it is possible to perform quantifier elimination and formula simplification.

A CAD for a set \(A\) of \(k\)-variate polynomials decomposes \(\RR^j\) into disjoint cells, for each \(j\) in \(0 \leq j \leq k\). The sign of each polynomial in \(A\) is constant in each cell of \(\RR^k\), and for each cell in \(\RR^j\) (\(j > 1\)), the projection of that cell into \(\RR^{j-1}\) is a cell of \(\RR^{j-1}\). (This property makes the decomposition ‘cylindrical’.)

Given a formula \(\exists x. P(a,b,x) = 0\) (for a polynomial \(P\)), and a cylindrical algebraic decomposition for \(P\), we can eliminate the quantifier (find an equivalent formula in the two variables \(a\), \(b\) without the quantifier \(\exists\)) as follows. For each cell \(C\) in \(\RR^2\), find the cells of \(\RR^3\) which project to \(C\). (This collection is called the `stack` over \(C\).) Mark \(C\) as true if some member of the stack has sign \(= 0\); otherwise, mark \(C\) as false. Then, construct a polynomial formula in \(a\), \(b\) which specifies exactly the true cells (this is always possible). The same technique works if the body of the quantifier is any boolean combination of polynomial equalities and inequalities.

Formula simplification is a similar technique. Given a formula which describes a simple set of \(\RR^k\) in a complicated way as a boolean combination of polynomial equalities and inequalities, QEPCAD can construct a CAD for the polynomials and recover a simple equivalent formula.

Note that while the following documentation is tutorial in nature, and is written for people who may not be familiar with QEPCAD, it is documentation for the sage interface rather than for QEPCAD. As such, it does not cover several issues that are very important to use QEPCAD efficiently, such as variable ordering, the efficient use of the alternate quantifiers and `_root_` expressions, the `measure-zero-error` command, etc. For more information on QEPCAD, see the online documentation at url{http://www.cs.usna.edu/~qepcad/B/QEPCAD.html} and Chris Brown’s tutorial handout and slides from url{http://www.cs.usna.edu/~wcbrown/research/ISSAC04/Tutorial.html}. (Several of the examples in this documentation came from these sources.)

The examples below require that the optional qepcad package is installed.

QEPCAD can be run in a fully automatic fashion, or interactively. We first demonstrate the automatic use of QEPCAD.

Since sage has no built-in support for quantifiers, this interface provides `qepcad_formula` which helps construct quantified formulas in the syntax QEPCAD requires.

    sage: var('a,b,c,d,x,y,z')
    (a, b, c, d, x, y, z)
    sage: qf = qepcad_formula

We start with a simple example. Consider an arbitrarily-selected ellipse:

    sage: ellipse = 3*x^2 + 2*x*y + y^2 - x + y - 7

What is the projection onto the \(x\) axis of this ellipse? First we construct a formula asking this question.

    sage: F = qf.exists(y, ellipse == 0); F
    (E y)[3 x^2 + 2 x y + y^2 - x + y - 7 = 0]

Then we run qepcad to get the answer:

    sage: qepcad(F)                            # optional - qepcad
    8 x^2 - 8 x - 29 <= 0

How about the projection onto the \(y\) axis?

    sage: qepcad(qf.exists(x, ellipse == 0))   # optional - qepcad
    8 y^2 + 16 y - 85 <= 0

QEPCAD deals with more quantifiers than just ‘exists’, of course. Besides the standard ‘forall’, there are also ‘for infinitely many’, ‘for all but finitely many’, ‘for a connected subset’, and ‘for exactly k’. The `qepcad()` documentation has examples of all of these; here we will just give one example.

First we construct a circle:

    sage: circle = x^2 + y^2 - 3

For what values \(k\) does a vertical line \(x=k\) intersect the combined figure of the circle and ellipse exactly three times?

    sage: F = qf.exactly_k(3, y, circle * ellipse == 0); F  # optional - qepcad
    (X3 y)[(x^2 + y^2 - 3) (3 x^2 + 2 x y + y^2 - x + y - 7) = 0]
    sage: qepcad(F)                                         # optional - qepcad
    8 x^2 - 8 x - 29 <= 0 /\ x^2 - 3 <= 0 /\ 8 x^4 - 26 x^2 - 4 x + 13 >= 0 /\ [ 8 x^4 - 26 x^2 - 4 x + 13 = 0 \/ x^2 - 3 = 0 \/ 8 x^2 - 8 x - 29 = 0 ]

Here we see that the solutions are among the eight (\(4 + 2 + 2\)) roots of the three polynomials inside the brackets, but not all of these roots are solutions; the polynomial inequalities outside the brackets are needed to select those roots that are solutions.

QEPCAD also supports an extended formula language, where `_root_k` \(P(\bar x, y)\) refers to a particular zero of \(P(\bar x, y)\) (viewed as a polynomial in \(y\)). If there are \(n\) roots, then `_root_1` refers to the least root and `_root_n` refers to the greatest. Also, `_root_-n` refers to the least root and `_root_-1` refers to the greatest.

This extended language is available both on input and output; see the QEPCAD documentation for more information on how to use this syntax on input. We can request output that is intended to be easy to interpret geometrically; then QEPCAD will use the extended language to produce a solution formula without the selection polynomials.

    sage: qepcad(F, solution='geometric')                 # optional - qepcad
    x = _root_1 8 x^2 - 8 x - 29
    \/
    8 x^4 - 26 x^2 - 4 x + 13 = 0
    \/
    x = _root_-1 x^2 - 3

We then see that the 6 solutions correspond to the vertical tangent on the left side of the ellipse, the four intersections between the ellipse and the circle, and the vertical tangent on the right side of the circle.

Let us do some basic formula simplification and visualization. We will look at the region which is inside both the ellipse and the circle:

    sage: F = qf.and_(ellipse < 0, circle < 0); F         # optional - qepcad
    [3 x^2 + 2 x y + y^2 - x + y - 7 < 0 /\ x^2 + y^2 - 3 < 0]
    sage: qepcad(F)                                       # optional - qepcad
    y^2 + 2 x y + y + 3 x^2 - x - 7 < 0 /\ y^2 + x^2 - 3 < 0

We get back the same formula we put in. This is not surprising (we started with a pretty simple formula, after all), but it is not very enlightening either. Again, if we ask for a ‘geometric’ output, then we see an output that lets us understand something about the shape of the solution set.

    sage: qepcad(F, solution='geometric')                 # optional - qepcad
    [
      [
        x = _root_-2 8 x^4 - 26 x^2 - 4 x + 13
        \/
        x = _root_2 8 x^4 - 26 x^2 - 4 x + 13
        \/
        8 x^4 - 26 x^2 - 4 x + 13 < 0
      ]
      /\
      y^2 + 2 x y + y + 3 x^2 - x - 7 < 0
      /\
      y^2 + x^2 - 3 < 0
    ]
    \/
    [
      x > _root_2 8 x^4 - 26 x^2 - 4 x + 13
      /\
      x < _root_-2 8 x^4 - 26 x^2 - 4 x + 13
      /\
      y^2 + x^2 - 3 < 0
    ]

There is another reason to prefer output using _root_ expressions; not only does it sometimes give added insight into the geometric structure, it also can be more efficient to construct. Consider this formula for the projection of a particular semicircle onto the \(x\) axis:

    sage: F = qf.exists(y, qf.and_(circle == 0, x + y > 0)); F   # optional - qepcad
    (E y)[x^2 + y^2 - 3 = 0 /\ x + y > 0]
    sage: qepcad(F)                                  # optional - qepcad
    x^2 - 3 <= 0 /\ [ x > 0 \/ 2 x^2 - 3 < 0 ]

Here, the formula \(x > 0\) had to be introduced in order to get a solution formula; the original CAD of F did not include the polynomial \(x\). To avoid having QEPCAD do the extra work to come up with a solution formula, we can tell it to use the extended language; it is always possible to construct a solution formula in the extended language without introducing new polynomials.

    sage: qepcad(F, solution='extended')                 # optional - qepcad
    x^2 - 3 <= 0 /\ x > _root_1 2 x^2 - 3

Up to this point, all the output we have seen has basically been in the form of strings; there is no support (yet) for parsing these outputs back into sage polynomials (partly because sage does not yet have support for symbolic conjunctions and disjunctions). The function `qepcad()` supports three more output types that give numbers which can be manipulated in sage: any-point, all-points, and cell-points.

These output types give dictionaries mapping variable names to values. With any-point, `qepcad()` either produces a single dictionary specifying a point where the formula is true, or raises an exception if the formula is false everywhere. With all-points, `qepcad()` either produces a list of dictionaries for all points where the formula is true, or raises an exception if the formula is true on infinitely many points. With cell-points, `qepcad()` produces a list of dictionaries with one point for each cell where the formula is true. (This means you will have at least one point in each connected component of the solution, although you will often have many more points than that.)

Let us revisit some of the above examples and get some points to play with. We will start by finding a point on our ellipse.

    sage: p = qepcad(ellipse == 0, solution='any-point'); p  # optional - qepcad
    {'y': 0.9685019685029527?, 'x': -1.468501968502953?}

(Note that despite the decimal printing and the question marks, these are really exact numbers.)

We can verify that this point is a solution. To do so, we create a copy of ellipse as a polynomial over \(\QQ\) (instead of a symbolic expression).

    sage: pellipse = QQ['x,y'](ellipse)                # optional - qepcad
    sage: pellipse(**p) == 0                           # optional - qepcad
    True

For cell-points, let us look at points not on the ellipse.

    sage: pts = qepcad(ellipse != 0, solution='cell-points'); pts   # optional - qepcad
    [{'y': 0, 'x': 4}, {'y': 1, 'x': 2.468501968502953?}, {'y': -9, 'x': 2.468501968502953?}, {'y': 9, 'x': 1/2}, {'y': -1, 'x': 1/2}, {'y': -5, 'x': 1/2}, {'y': 3, 'x': -1.468501968502953?}, {'y': -1, 'x': -1.468501968502953?}, {'y': 0, 'x': -3}]

For the points here which are in full-dimensional cells, QEPCAD has the freedom to choose rational sample points, and it does so.

And, of course, all these points really are not on the ellipse.

    sage: [pellipse(**p) != 0 for p in pts]               # optional - qepcad
    [True, True, True, True, True, True, True, True, True]

Finally, for all-points, let us look again at finding vertical lines that intersect the union of the circle and the ellipse exactly three times.

    sage: F = qf.exactly_k(3, y, circle * ellipse == 0); F   # optional - qepcad
    (X3 y)[(x^2 + y^2 - 3) (3 x^2 + 2 x y + y^2 - x + y - 7) = 0]
    sage: pts = qepcad(F, solution='all-points'); pts       # optional - qepcad
    [{'x': 1.732050807568878?}, {'x': 1.731054913462534?}, {'x': 0.678911384208004?}, {'x': -0.9417727377417167?}, {'x': -1.468193559928821?}, {'x': -1.468501968502953?}]

Since \(y\) is bound by the quantifier, the solutions only refer to \(x\).

We can substitute one of these solutions into the original equation:

    sage: pt = pts[0]                             # optional - qepcad
    sage: pcombo = QQ['x,y'](circle * ellipse)     # optional - qepcad
    sage: intersections = pcombo(y=polygen(AA, 'y'), **pt); intersections       # optional - qepcad
    y^4 + 4.464101615137755?*y^3 + 0.2679491924311227?*y^2

and verify that it does have three roots:

    sage: intersections.roots()                                                 # optional - qepcad
    [(-4.403249005600958?, 1), (-0.06085260953679653?, 1), (0, 2)]

Let us check all six solutions.

    sage: [len(pcombo(y=polygen(AA, 'y'), **p).roots()) for p in pts]    # optional - qepcad
    [3, 3, 3, 3, 3, 3]

We said earlier that we can run QEPCAD either automatically or interactively. Now that we have discussed the automatic modes, let us turn to interactive uses.

If the `qepcad()` function is passed `interact=True`, then instead of returning a result, it returns an object of class `Qepcad` representing a running instance of QEPCAD that you can interact with. For example:

    sage: qe = qepcad(qf.forall(x, x^2 + b*x + c > 0), interact=True); qe     # optional - qepcad
    QEPCAD object in phase 'Before Normalization'

This object is a fairly thin wrapper over QEPCAD; most QEPCAD commands are available as methods on the `Qepcad` object. Given a `Qepcad` object `qe`, you can type `qe.[tab]` to see the available QEPCAD commands; to see the documentation for an individual QEPCAD command, for example `d_setting`, you can type `qe.d_setting?`. (In QEPCAD, this command is called `d-setting`. We systematically replace hyphens with underscores for this interface.)

The execution of QEPCAD is divided into four phases. Most commands are not available during all phases. We saw above that QEPCAD starts out in phase `'Before Normalization'`. We see that the `d_cell` command is not available in this phase:

    sage: qe.d_cell()                                # optional - qepcad
    Error GETCID: This command is not active here.

We will focus here on the fourth (and last) phase, `'Before Solution'`, because this interface has special support for some operations in this phase. Consult the QEPCAD documentation for information on the other phases.

We can tell QEPCAD to finish off the current phase and move to the next with its `go` command. (There is also the `step` command, which partially completes a phase for phases that have multiple steps, and the `finish` command, which runs QEPCAD to completion.)

    sage: qe.go()                                        # optional - qepcad
    QEPCAD object has moved to phase 'Before Projection (x)'
    sage: qe.go()                                        # optional - qepcad
    QEPCAD object has moved to phase 'Before Choice'
    sage: qe.go()                                        # optional - qepcad
    QEPCAD object has moved to phase 'Before Solution'

Note that the `Qepcad` object returns the new phase whenever the phase changes, as a convenience for interactive use; except that when the new phase is `'EXITED'`, the solution formula printed by QEPCAD is returned instead.

    sage: qe.go()                                        # optional - qepcad
    4 c - b^2 > 0
    sage: qe                                             # optional - qepcad
    QEPCAD object in phase 'EXITED'

Let us pick a nice, simple example, return to phase 4, and explore the resulting `qe` object.

    sage: qe = qepcad(circle == 0, interact=True); qe       # optional - qepcad
    QEPCAD object in phase 'Before Normalization'
    sage: qe.go(); qe.go(); qe.go()                         # optional - qepcad
    QEPCAD object has moved to phase 'Before Projection (y)'
    QEPCAD object has moved to phase 'Before Choice'
    QEPCAD object has moved to phase 'Before Solution'

We said before that QEPCAD creates ‘cylindrical algebraic decompositions’; since we have a bivariate polynomial, we get decompositions of \(\RR^0\), \(\RR^1\), and \(\RR^2\). In this case, where our example is a circle of radius \(\sqrt{3}\) centered on the origin, these decompositions are as follows:

The decomposition of \(\RR^0\) is trivial (of course). The decomposition of \(\RR^1\) has five cells: \(x < -\sqrt{3}\), \(x = -\sqrt{3}\), \(-\sqrt{3} < x < \sqrt{3}\), \(x = \sqrt{3}\), and \(x > \sqrt{3}\). These five cells comprise the `stack` over the single cell in the trivial decomposition of \(\RR^0\).

These five cells give rise to five stacks in \(\RR^2\). The first and fifth stack have just one cell apiece. The second and fourth stacks have three cells: \(y < 0\), \(y = 0\), and \(y > 0\). The third stack has five cells: below the circle, the lower semicircle, the interior of the circle, the upper semicircle, and above the circle.

QEPCAD (and this QEPCAD interface) number the cells in a stack starting with 1. Each cell has an `index`, which is a tuple of integers describing the path to the cell in the tree of all cells. For example, the cell ‘below the circle’ has index (3,1) (the first cell in the stack over the third cell of \(\RR^1\)) and the interior of the circle has index (3,3).

We can view these cells with the QEPCAD command `d_cell`. For instance, let us look at the cell for the upper semicircle:

    sage: qe.d_cell(3, 4)                              # optional - qepcad
    ---------- Information about the cell (3,4) ----------
    Level                       : 2
    Dimension                   : 1
    Number of children          : 0
    Truth value                 : T    by trial evaluation.
    Degrees after substitution  : Not known yet or No polynomial.
    Multiplicities              : ((1,1))
    Signs of Projection Factors
    Level 1  : (-)
    Level 2  : (0)
    ----------   Sample point  ----------
    The sample point is in a PRIMITIVE representation.
    <BLANKLINE>
    alpha = the unique root of x^2 - 3 between 0 and 4
          = 1.7320508076-
    <BLANKLINE>
    Coordinate 1 = 0
                 = 0.0000000000
    Coordinate 2 = alpha
                 = 1.7320508076-
    ----------------------------------------------------

We see that, the level of this cell is 2, meaning that it is part of the decomposition of \(\RR^2\). The dimension is 1, meaning that the cell is homeomorphic to a line (rather than a plane or a point). The sample point gives the coordinates of one point in the cell, both symbolically and numerically.

For programmatic access to cells, we have defined a sage wrapper class `QepcadCell`. These cells can be created with the `cell()` method; for example:

    sage: c = qe.cell(3, 4); c                       # optional - qepcad
    QEPCAD cell (3, 4)

A `QepcadCell` has accessor methods for the important state held within a cell. For instance:

    sage: c.level()                                  # optional - qepcad
    2
    sage: c.index()                                  # optional - qepcad
    (3, 4)
    sage: qe.cell(3).number_of_children()            # optional - qepcad
    5
    sage: len(qe.cell(3))                            # optional - qepcad
    5

One particularly useful thing we can get from a cell is its sample point, as sage algebraic real numbers.

    sage: c.sample_point()                           # optional - qepcad
    (0, 1.732050807568878?)
    sage: c.sample_point_dict()                      # optional - qepcad
    {'y': 1.732050807568878?, 'x': 0}

We have seen that we can get cells using the `cell()` method. There are several QEPCAD commands that print lists of cells; we can also get cells using the `make_cells()` method, passing it the output of one of these commands.

    sage: qe.make_cells(qe.d_true_cells())           # optional - qepcad
    [QEPCAD cell (4, 2), QEPCAD cell (3, 4), QEPCAD cell (3, 2),
    QEPCAD cell (2, 2)]

Also, the cells in the stack over a given cell can be accessed using array subscripting or iteration. (Remember that cells in a stack are numbered starting with one; we preserve this convention in the array-subscripting syntax.)

    sage: c = qe.cell(3)                             # optional - qepcad
    sage: c[1]                                       # optional - qepcad
    QEPCAD cell (3, 1)
    sage: [c2 for c2 in c]                           # optional - qepcad
    [QEPCAD cell (3, 1), QEPCAD cell (3, 2), QEPCAD cell (3, 3),
    QEPCAD cell (3, 4), QEPCAD cell (3, 5)]

We can do one more thing with a cell: we can set its truth value. Once the truth values of the cells have been set, we can get QEPCAD to produce a formula which is true in exactly the cells we have selected. This is useful if QEPCAD’s quantifier language is insufficient to express your problem.

For example, consider again our combined figure of the circle and the ellipse. Suppose you want to find all vertical lines that intersect the circle twice, and also intersect the ellipse twice. The vertical lines that intersect the circle twice can be found by simplifying:

    sage: F = qf.exactly_k(2, y, circle == 0); F         # optional - qepcad
    (X2 y)[x^2 + y^2 - 3 = 0]

and the vertical lines that intersect the ellipse twice are expressed by:

    sage: G = qf.exactly_k(2, y, ellipse == 0); G        # optional - qepcad
    (X2 y)[3 x^2 + 2 x y + y^2 - x + y - 7 = 0]

and the lines that intersect both figures would be:

    sage: qf.and_(F, G)                                  # optional - qepcad
    Traceback (most recent call last):
    ...
    ValueError: QEPCAD formulas must be in prenex (quantifiers outermost) form

...except that QEPCAD does not support formulas like this; in QEPCAD input, all logical connectives must be inside all quantifiers.

Instead, we can get QEPCAD to construct a CAD for our combined figure and set the truth values ourselves. (The exact formula we use doesn’t matter, since we’re going to replace the truth values in the cells; we just need to use a formula that uses both polynomials.)

    sage: qe = qepcad(qf.and_(ellipse == 0, circle == 0), interact=True)       # optional - qepcad
    sage: qe.go(); qe.go(); qe.go()                      # optional - qepcad
    QEPCAD object has moved to phase 'Before Projection (y)'
    QEPCAD object has moved to phase 'Before Choice'
    QEPCAD object has moved to phase 'Before Solution'

Now we want to find all cells \(c\) in the decomposition of \(\RR^1\) such that the stack over \(c\) contains exactly two cells on the ellipse, and also contains exactly two cells on the circle.

Our input polynomials are ‘level-2 projection factors’, we see:

    sage: qe.d_proj_factors()                           # optional - qepcad
    P_1,1  = fac(J_1,1) = fac(dis(A_2,1))
           = 8 x^2 - 8 x - 29
    P_1,2  = fac(J_1,2) = fac(dis(A_2,2))
           = x^2 - 3
    P_1,3  = fac(J_1,3) = fac(res(A_2,1|A_2,2))
           = 8 x^4 - 26 x^2 - 4 x + 13
    A_2,1  = input
           = y^2 + 2 x y + y + 3 x^2 - x - 7
    A_2,2  = input
           = y^2 + x^2 - 3

so we can test whether a cell is on the ellipse by checking that the sign of the corresponding projection factor is 0 in our cell. For instance, the cell (12,2) is on the ellipse:

    sage: qe.cell(12,2).signs()[1][0]                  # optional - qepcad
    0

So we can update the truth values as desired like this:

    sage: for c in qe.cell():                                        # optional - qepcad
    ....:     count_ellipse = 0
    ....:     count_circle = 0
    ....:     for c2 in c:
    ....:         count_ellipse += (c2.signs()[1][0] == 0)
    ....:         count_circle += (c2.signs()[1][1] == 0)
    ....:     c.set_truth(count_ellipse == 2 and count_circle == 2)

and then we can get our desired solution formula. (The `'G'` stands for `'geometric'`, and gives solutions using the same rules as `solution='geometric'` described above.)

    sage: qe.solution_extension('G')     # optional - qepcad
    8 x^2 - 8 x - 29 < 0
    /\
    x^2 - 3 < 0

TESTS:

Check the qepcad configuration file:

    sage: from sage.misc.misc import SAGE_LOCAL
    sage: open('%s/default.qepcadrc'%SAGE_LOCAL).readlines()[-1]
    'SINGULAR .../local/bin\n'

AUTHORS:

- Carl Witty (2008-03): initial version

class `sage.interfaces.qepcad.``Qepcad`(formula, vars=None, logfile=None, verbose=False, memcells=None, server=None)

  The wrapper for QEPCAD.

  `answer`()

    For a QEPCAD instance which is finished, return the simplified quantifier-free formula that it printed just before exiting.

    EXAMPLES:

        sage: qe = qepcad(x^3 - x == 0, interact=True) # optional - qepcad
        sage: qe.finish() # optional - qepcad
        x - 1 <= 0 /\ x + 1 >= 0 /\ [ x = 0 \/ x - 1 = 0 \/ x + 1 = 0 ]
        sage: qe.answer() # optional - qepcad
        x - 1 <= 0 /\ x + 1 >= 0 /\ [ x = 0 \/ x - 1 = 0 \/ x + 1 = 0 ]

  `assume`(assume)

    The following documentation is from `qepcad.help`.

    Add an assumption to the problem. These will not be included in the solution formula.

    For example, with input (E x)[ a x^2 + b x + c = 0], if we issue the command

    > assume [ a /= 0 ]

    we will get the solution formula b^2 - 4 a c >= 0. Without the assumption we’d get something like [a = 0 /b /= 0] / [a /= 0 /4 a c - b^2 <= 0] / [a = 0 /b = 0 /c = 0].

    EXAMPLES:

        sage: var('a,b,c,x')
        (a, b, c, x)
        sage: qf = qepcad_formula
        sage: qe = qepcad(qf.exists(x, a*x^2 + b*x + c == 0), interact=True) # optional - qepcad
        sage: qe.assume(a != 0) # optional - qepcad
        sage: qe.finish() # optional - qepcad
        4 a c - b^2 <= 0

  `cell`(*index)

    Given a cell index, returns a `QepcadCell` wrapper for that cell. Uses a cache for efficiency.

    EXAMPLES:

        sage: qe = qepcad(x + 3 == 42, interact=True) # optional - qepcad
        sage: qe.go(); qe.go(); qe.go() # optional - qepcad
        QEPCAD object has moved to phase 'At the end of projection phase'
        QEPCAD object has moved to phase 'Before Choice'
        QEPCAD object has moved to phase 'Before Solution'
        sage: qe.cell(2) # optional - qepcad
        QEPCAD cell (2)
        sage: qe.cell(2) is qe.cell(2) # optional - qepcad
        True

  `final_stats`()

    For a QEPCAD instance which is finished, return the statistics that it printed just before exiting.

    EXAMPLES:

        sage: qe = qepcad(x == 0, interact=True) # optional
        sage: qe.finish() # optional
        x = 0
        sage: qe.final_stats() # random, optional
        -----------------------------------------------------------------------------
        0 Garbage collections, 0 Cells and 0 Arrays reclaimed, in 0 milliseconds.
        492840 Cells in AVAIL, 500000 Cells in SPACE.
        System time: 8 milliseconds.
        System time after the initialization: 4 milliseconds.
        -----------------------------------------------------------------------------

  `make_cells`(text)

    Given the result of some QEPCAD command that returns cells (such as `d_cell()`, `d_witness_list()`, etc.), return a list of cell objects.

    EXAMPLES:

        sage: var('x,y')
        (x, y)
        sage: qe = qepcad(x^2 + y^2 == 1, interact=True) # optional - qepcad
        sage: qe.go(); qe.go(); qe.go() # optional - qepcad
        QEPCAD object has moved to phase 'Before Projection (y)'
        QEPCAD object has moved to phase 'Before Choice'
        QEPCAD object has moved to phase 'Before Solution'
        sage: qe.make_cells(qe.d_false_cells()) # optional - qepcad
        [QEPCAD cell (5, 1), QEPCAD cell (4, 3), QEPCAD cell (4, 1), QEPCAD cell (3, 5), QEPCAD cell (3, 3), QEPCAD cell (3, 1), QEPCAD cell (2, 3), QEPCAD cell (2, 1), QEPCAD cell (1, 1)]

  `phase`()

    Return the current phase of the QEPCAD program.

    EXAMPLES:

        sage: qe = qepcad(x > 2/3, interact=True) # optional - qepcad
        sage: qe.phase() # optional - qepcad
        'Before Normalization'
        sage: qe.go() # optional
        QEPCAD object has moved to phase 'At the end of projection phase'
        sage: qe.phase() # optional  - qepcad
        'At the end of projection phase'
        sage: qe.go() # optional
        QEPCAD object has moved to phase 'Before Choice'
        sage: qe.phase() # optional - qepcad
        'Before Choice'
        sage: qe.go() # optional
        QEPCAD object has moved to phase 'Before Solution'
        sage: qe.phase() # optional - qepcad
        'Before Solution'
        sage: qe.go() # optional - qepcad
        3 x - 2 > 0
        sage: qe.phase() # optional - qepcad
        'EXITED'

  `set_truth_value`(index, nv)

    Given a cell index (or a cell) and an integer, set the truth value of the cell to that integer.

    Valid integers are 0 (false), 1 (true), and 2 (undetermined).

    EXAMPLES:

        sage: qe = qepcad(x == 1, interact=True) # optional - qepcad
        sage: qe.go(); qe.go(); qe.go() # optional - qepcad
        QEPCAD object has moved to phase 'At the end of projection phase'
        QEPCAD object has moved to phase 'Before Choice'
        QEPCAD object has moved to phase 'Before Solution'
        sage: qe.set_truth_value(1, 1) # optional - qepcad

  `solution_extension`(kind)

    The following documentation is modified from `qepcad.help`:

    solution-extension x

    Use an alternative solution formula construction method. The parameter x is allowed to be T,E, or G. If x is T, then a formula in the usual language of Tarski formulas is produced. If x is E, a formula in the language of Extended Tarski formulas is produced. If x is G, then a geometry-based formula is produced.

    EXAMPLES:

        sage: var('x,y')
        (x, y)
        sage: qf = qepcad_formula
        sage: qe = qepcad(qf.and_(x^2 + y^2 - 3 == 0, x + y > 0), interact=True) # optional
        sage: qe.go(); qe.go(); qe.go() # optional - qepcad
        QEPCAD object has moved to phase 'Before Projection (y)'
        QEPCAD object has moved to phase 'Before Choice'
        QEPCAD object has moved to phase 'Before Solution'
        sage: qe.solution_extension('E') # optional - qepcad
        x > _root_1 2 x^2 - 3 /\ y^2 + x^2 - 3 = 0 /\ [ 2 x^2 - 3 > 0 \/ y = _root_-1 y^2 + x^2 - 3 ]
        sage: qe.solution_extension('G') # optional - qepcad
        [
          [
            2 x^2 - 3 < 0
            \/
            x = _root_-1 2 x^2 - 3
          ]
          /\
          y = _root_-1 y^2 + x^2 - 3
        ]
        \/
        [
          x^2 - 3 <= 0
          /\
          x > _root_-1 2 x^2 - 3
          /\
          y^2 + x^2 - 3 = 0
        ]
        sage: qe.solution_extension('T') # optional - qepcad
        y + x > 0 /\ y^2 + x^2 - 3 = 0

  `trait_names`()

    Return a list of the QEPCAD commands which are available as extra methods on a `Qepcad` object.

    EXAMPLES:

        sage: qe = qepcad(x^2 < 0, interact=True) # optional - qepcad
        sage: len(qe.trait_names()) # random, optional - qepcad
        97
        sage: 'd_cell' in qe.trait_names() # optional - qepcad
        True

class `sage.interfaces.qepcad.``QepcadCell`(parent, lines)

  A wrapper for a QEPCAD cell.

  `index`()

    Give the index of a QEPCAD cell.

    EXAMPLES:

        sage: var('x,y')
        (x, y)
        sage: qe = qepcad(x^2 + y^2 == 1, interact=True) # optional - qepcad
        sage: qe.go(); qe.go(); qe.go() # optional - qepcad
        QEPCAD object has moved to phase 'Before Projection (y)'
        QEPCAD object has moved to phase 'Before Choice'
        QEPCAD object has moved to phase 'Before Solution'
        sage: qe.cell().index() # optional - qepcad
        ()
        sage: qe.cell(1).index() # optional - qepcad
        (1,)
        sage: qe.cell(2, 2).index() # optional - qepcad
        (2, 2)

  `level`()

    Return the level of a QEPCAD cell.

    EXAMPLES:

        sage: var('x,y')
        (x, y)
        sage: qe = qepcad(x^2 + y^2 == 1, interact=True) # optional - qepcad
        sage: qe.go(); qe.go(); qe.go() # optional - qepcad
        QEPCAD object has moved to phase 'Before Projection (y)'
        QEPCAD object has moved to phase 'Before Choice'
        QEPCAD object has moved to phase 'Before Solution'
        sage: qe.cell().level() # optional - qepcad
        0
        sage: qe.cell(1).level() # optional - qepcad
        1
        sage: qe.cell(2, 2).level() # optional - qepcad
        2

  `number_of_children`()

    Return the number of elements in the stack over a QEPCAD cell. (This is always an odd number, if the stack has been constructed.)

    EXAMPLES:

        sage: var('x,y')
        (x, y)
        sage: qe = qepcad(x^2 + y^2 == 1, interact=True) # optional - qepcad
        sage: qe.go(); qe.go(); qe.go() # optional - qepcad
        QEPCAD object has moved to phase 'Before Projection (y)'
        QEPCAD object has moved to phase 'Before Choice'
        QEPCAD object has moved to phase 'Before Solution'
        sage: qe.cell().number_of_children() # optional - qepcad
        5
        sage: [c.number_of_children() for c in qe.cell()] # optional - qepcad
        [1, 3, 5, 3, 1]

  `sample_point`()

    Return the coordinates of a point in the cell, as a tuple of sage algebraic reals.

    EXAMPLES:

        sage: qe = qepcad(x^2 - x - 1 == 0, interact=True) # optional - qepcad
        sage: qe.go(); qe.go(); qe.go() # optional - qepcad
        QEPCAD object has moved to phase 'At the end of projection phase'
        QEPCAD object has moved to phase 'Before Choice'
        QEPCAD object has moved to phase 'Before Solution'
        sage: v1 = qe.cell(2).sample_point()[0]; v1 # optional - qepcad
        -0.618033988749895?
        sage: v2 = qe.cell(4).sample_point()[0]; v2 # optional - qepcad
        1.618033988749895?
        sage: v1 + v2 == 1 # optional - qepcad
        True

  `sample_point_dict`()

    Return the coordinates of a point in the cell, as a dictionary mapping variable names (as strings) to sage algebraic reals.

    EXAMPLES:

        sage: qe = qepcad(x^2 - x - 1 == 0, interact=True) # optional - qepcad
        sage: qe.go(); qe.go(); qe.go() # optional - qepcad
        QEPCAD object has moved to phase 'At the end of projection phase'
        QEPCAD object has moved to phase 'Before Choice'
        QEPCAD object has moved to phase 'Before Solution'
        sage: qe.cell(4).sample_point_dict() # optional - qepcad
        {'x': 1.618033988749895?}

  `set_truth`(v)

    Set the truth value of this cell, as used by QEPCAD for solution formula construction.

    The argument `v` should be either a boolean or `None` (which will set the truth value to `'undetermined'`).

    EXAMPLES:

        sage: var('x,y')
        (x, y)
        sage: qe = qepcad(x^2 + y^2 == 1, interact=True) # optional - qepcad
        sage: qe.go(); qe.go(); qe.go() # optional - qepcad
        QEPCAD object has moved to phase 'Before Projection (y)'
        QEPCAD object has moved to phase 'Before Choice'
        QEPCAD object has moved to phase 'Before Solution'
        sage: qe.solution_extension('T') # optional - qepcad
        y^2 + x^2 - 1 = 0
        sage: qe.cell(3, 3).set_truth(True) # optional - qepcad
        sage: qe.solution_extension('T') # optional - qepcad
        y^2 + x^2 - 1 <= 0

  `signs`()

    Return the sign vector of a QEPCAD cell.

    This is a list of lists. The outer list contains one element for each level of the cell; the inner list contains one element for each projection factor at that level. These elements are either -1, 0, or 1.

    EXAMPLES:

        sage: var('x,y')
        (x, y)
        sage: qe = qepcad(x^2 + y^2 == 1, interact=True) # optional - qepcad
        sage: qe.go(); qe.go(); qe.go() # optional - qepcad
        QEPCAD object has moved to phase 'Before Projection (y)'
        QEPCAD object has moved to phase 'Before Choice'
        QEPCAD object has moved to phase 'Before Solution'
        sage: from sage.interfaces.qepcad import QepcadCell
        sage: all_cells = flatten(qe.cell(), ltypes=QepcadCell, max_level=1) # optional - qepcad
        sage: [(c, c.signs()[1][0]) for c in all_cells] # optional - qepcad
        [(QEPCAD cell (1, 1), 1), (QEPCAD cell (2, 1), 1), (QEPCAD cell (2, 2), 0), (QEPCAD cell (2, 3), 1), (QEPCAD cell (3, 1), 1), (QEPCAD cell (3, 2), 0), (QEPCAD cell (3, 3), -1), (QEPCAD cell (3, 4), 0), (QEPCAD cell (3, 5), 1), (QEPCAD cell (4, 1), 1), (QEPCAD cell (4, 2), 0), (QEPCAD cell (4, 3), 1), (QEPCAD cell (5, 1), 1)]

class `sage.interfaces.qepcad.``QepcadFunction`(parent, name)

  Bases: `sage.interfaces.expect.ExpectFunction`

  A wrapper for a QEPCAD command.

class `sage.interfaces.qepcad.``Qepcad_expect`(memcells=None, maxread=100000, logfile=None, server=None)

  Bases: `sage.interfaces.expect.Expect`

  The low-level wrapper for QEPCAD.

`sage.interfaces.qepcad.``qepcad`(formula, assume=None, interact=False, solution=None, vars=None, **kwargs)

  Quantifier elimination and formula simplification using QEPCAD B.

  If `assume` is specified, then the given formula is `'assumed'`, which is taken into account during final solution formula construction.

  If `interact=True` is given, then a `Qepcad` object is returned which can be interacted with either at the command line or programmatically.

  The type of solution returned can be adjusted with `solution`. The options are `'geometric'`, which tries to construct a solution formula with geometric meaning; `'extended'`, which gives a solution formula in an extended language that may be more efficient to construct; `'any-point'`, which returns any point where the formula is true; `'all-points'`, which returns a list of all points where the formula is true (or raises an exception if there are infinitely many); and `'cell-points'`, which returns one point in each cell where the formula is true.

  All other keyword arguments are passed through to the `Qepcad` constructor.

  For much more documentation and many more examples, see the module docstring for this module (type `sage.interfaces.qepcad?` to read this docstring from the sage command line).

  The examples below require that the optional qepcad package is installed.

  EXAMPLES:

      sage: qf = qepcad_formula

      sage: var('a,b,c,d,x,y,z,long_with_underscore_314159')
      (a, b, c, d, x, y, z, long_with_underscore_314159)
      sage: K.<q,r> = QQ[]

      sage: qepcad('(E x)[a x + b > 0]', vars='(a,b,x)')      # optional - qepcad
      a /= 0 \/ b > 0

      sage: qepcad(a > b)                            # optional - qepcad
      b - a < 0

      sage: qepcad(qf.exists(x, a*x^2 + b*x + c == 0))        # optional - qepcad
      4 a c - b^2 <= 0 /\ [ c = 0 \/ a /= 0 \/ 4 a c - b^2 < 0 ]

      sage: qepcad(qf.exists(x, a*x^2 + b*x + c == 0), assume=(a != 0))    # optional - qepcad
      4 a c - b^2 <= 0

  For which values of \(a\), \(b\), \(c\) does \(a x^2 + b x + c\) have 2 real zeroes?

      sage: exact2 = qepcad(qf.exactly_k(2, x, a*x^2 + b*x + c == 0)); exact2   # optional - qepcad
      a /= 0 /\ 4 a c - b^2 < 0

  one real zero?

      sage: exact1 = qepcad(qf.exactly_k(1, x, a*x^2 + b*x + c == 0)); exact1   # optional - qepcad
      [ a > 0 /\ 4 a c - b^2 = 0 ] \/ [ a < 0 /\ 4 a c - b^2 = 0 ] \/ [ a = 0 /\ 4 a c - b^2 < 0 ]

  No real zeroes?

      sage: exact0 = qepcad(qf.forall(x, a*x^2 + b*x + c != 0)); exact0     # optional - qepcad
      4 a c - b^2 >= 0 /\ c /= 0 /\ [ b = 0 \/ 4 a c - b^2 > 0 ]

  \(3^{75}\) real zeroes?

      sage: qepcad(qf.exactly_k(3^75, x, a*x^2 + b*x + c == 0))    # optional - qepcad
      FALSE

  We can check that the results don’t overlap:

      sage: qepcad(r'[[%s] /\ [%s]]' % (exact0, exact1), vars='a,b,c')      # optional - qepcad
      FALSE
      sage: qepcad(r'[[%s] /\ [%s]]' % (exact0, exact2), vars='a,b,c')      # optional - qepcad
      FALSE
      sage: qepcad(r'[[%s] /\ [%s]]' % (exact1, exact2), vars='a,b,c')      # optional - qepcad
      FALSE

  and that the union of the results is as expected:

      sage: qepcad(r'[[%s] \/ [%s] \/ [%s]]' % (exact0, exact1, exact2), vars=(a,b,c))  # optional - qepcad
      b /= 0 \/ a /= 0 \/ c /= 0

  So we have finitely many zeroes if \(a\), \(b\), or \(c\) is nonzero; which means we should have infinitely many zeroes if they are all zero.

      sage: qepcad(qf.infinitely_many(x, a*x^2 + b*x + c == 0))          # optional - qepcad
      a = 0 /\ b = 0 /\ c = 0

  The polynomial is nonzero almost everywhere iff it is not identically zero.

      sage: qepcad(qf.all_but_finitely_many(x, a*x^2 + b*x + c != 0))    # optional - qepcad
      b /= 0 \/ a /= 0 \/ c /= 0

  The non-zeroes are continuous iff there are no zeroes or if the polynomial is zero.

      sage: qepcad(qf.connected_subset(x, a*x^2 + b*x + c != 0))         # optional - qepcad
      4 a c - b^2 >= 0 /\ [ a = 0 \/ 4 a c - b^2 > 0 ]

  The zeroes are continuous iff there are no or one zeroes, or if the polynomial is zero:

      sage: qepcad(qf.connected_subset(x, a*x^2 + b*x + c == 0))         # optional - qepcad
      a = 0 \/ 4 a c - b^2 >= 0
      sage: qepcad(r'[[%s] \/ [%s] \/ [a = 0 /\ b = 0 /\ c = 0]]' % (exact0, exact1), vars='a,b,c')   # optional - qepcad
      a = 0 \/ 4 a c - b^2 >= 0

  Since polynomials are continuous and \(y > 0\) is an open set, they are positive infinitely often iff they are positive at least once.

      sage: qepcad(qf.infinitely_many(x, a*x^2 + b*x + c > 0))        # optional - qepcad
      c > 0 \/ a > 0 \/ 4 a c - b^2 < 0
      sage: qepcad(qf.exists(x, a*x^2 + b*x + c > 0))        # optional - qepcad
      c > 0 \/ a > 0 \/ 4 a c - b^2 < 0

  However, since \(y >= 0\) is not open, the equivalence does not hold if you replace ‘positive’ with ‘nonnegative’. (We assume \(a \neq 0\) to get simpler formulas.)

      sage: qepcad(qf.infinitely_many(x, a*x^2 + b*x + c >= 0), assume=(a != 0))    # optional - qepcad
      a > 0 \/ 4 a c - b^2 < 0
      sage: qepcad(qf.exists(x, a*x^2 + b*x + c >= 0), assume=(a != 0))             # optional - qepcad
      a > 0 \/ 4 a c - b^2 <= 0

  TESTS:

  We verify that long variable names work. (Note that QEPCAD does not support underscores, so they are stripped from the formula.)

      sage: qepcad(qf.exists(a, a*long_with_underscore_314159 == 1))                # optional - qepcad
      longwithunderscore314159 /= 0

`sage.interfaces.qepcad.``qepcad_banner`()

  Return the QEPCAD startup banner.

  EXAMPLES:

      sage: from sage.interfaces.qepcad import qepcad_banner
      sage: qepcad_banner() # optional - qepcad
      =======================================================
                      Quantifier Elimination
                                in
                  Elementary Algebra and Geometry
                                by
            Partial Cylindrical Algebraic Decomposition
      ...
                                by
                             Hoon Hong
                       (hhong@math.ncsu.edu)
      With contributions by: Christopher W. Brown, George E.
      Collins, Mark J. Encarnacion, Jeremy R. Johnson
      Werner Krandick, Richard Liska, Scott McCallum,
      Nicolas Robidoux, and Stanly Steinberg
      =======================================================

`sage.interfaces.qepcad.``qepcad_console`(memcells=None)

  Run QEPCAD directly. To exit early, press Control-C.

  EXAMPLES:

      sage: qepcad_console() # not tested
      ...
      Enter an informal description  between '[' and ']':

class `sage.interfaces.qepcad.``qepcad_formula_factory`

  Contains routines to help construct formulas in QEPCAD syntax.

  `A`(v, formula)

    Given a variable (or list of variables) and a formula, returns the universal quantification of the formula over the variables.

    This method is available both as `forall()` and `A()` (the QEPCAD name for this quantifier).

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLE:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.forall(a, a^2 + b > b^2 + a)
        (A a)[a^2 + b > b^2 + a]
        sage: qf.forall((a, b), a^2 + b^2 > 0)
        (A a)(A b)[a^2 + b^2 > 0]
        sage: qf.A(b, b^2 != a)
        (A b)[b^2 /= a]

  `C`(v, formula, allow_multi=False)

    Given a variable and a formula, returns a new formula which is true iff the set of values for the variable at which the original formula was true is connected (including cases where this set is empty or is a single point).

    This method is available both as `connected_subset()` and `C()` (the QEPCAD name for this quantifier).

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLE:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.connected_subset(a, a^2 + b > b^2 + a)
        (C a)[a^2 + b > b^2 + a]
        sage: qf.C(b, b^2 != a)
        (C b)[b^2 /= a]

  `E`(v, formula)

    Given a variable (or list of variables) and a formula, returns the existential quantification of the formula over the variables.

    This method is available both as `exists()` and `E()` (the QEPCAD name for this quantifier).

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLE:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.exists(a, a^2 + b > b^2 + a)
        (E a)[a^2 + b > b^2 + a]
        sage: qf.exists((a, b), a^2 + b^2 < 0)
        (E a)(E b)[a^2 + b^2 < 0]
        sage: qf.E(b, b^2 == a)
        (E b)[b^2 = a]

  `F`(v, formula)

    Given a variable and a formula, returns a new formula which is true iff the original formula was true for infinitely many values of the variable.

    This method is available both as `infinitely_many()` and `F()` (the QEPCAD name for this quantifier).

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLE:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.infinitely_many(a, a^2 + b > b^2 + a)
        (F a)[a^2 + b > b^2 + a]
        sage: qf.F(b, b^2 != a)
        (F b)[b^2 /= a]

  `G`(v, formula)

    Given a variable and a formula, returns a new formula which is true iff the original formula was true for all but finitely many values of the variable.

    This method is available both as `all_but_finitely_many()` and `G()` (the QEPCAD name for this quantifier).

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLE:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.all_but_finitely_many(a, a^2 + b > b^2 + a)
        (G a)[a^2 + b > b^2 + a]
        sage: qf.G(b, b^2 != a)
        (G b)[b^2 /= a]

  `X`(k, v, formula, allow_multi=False)

    Given a nonnegative integer \(k\), a variable, and a formula, returns a new formula which is true iff the original formula is true for exactly \(k\) values of the variable.

    This method is available both as `exactly_k()` and `X()` (the QEPCAD name for this quantifier).

    (Note that QEPCAD does not support \(k=0\) with this syntax, so if \(k=0\) is requested we implement it with `forall()` and `not_()`.)

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLE:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.exactly_k(1, x, x^2 + a*x + b == 0)
        (X1 x)[a x + x^2 + b = 0]
        sage: qf.exactly_k(0, b, a*b == 1)
        (A b)[~a b = 1]

  `all_but_finitely_many`(v, formula)

    Given a variable and a formula, returns a new formula which is true iff the original formula was true for all but finitely many values of the variable.

    This method is available both as `all_but_finitely_many()` and `G()` (the QEPCAD name for this quantifier).

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLE:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.all_but_finitely_many(a, a^2 + b > b^2 + a)
        (G a)[a^2 + b > b^2 + a]
        sage: qf.G(b, b^2 != a)
        (G b)[b^2 /= a]

  `and_`(*formulas)

    Return the conjunction of its input formulas.

    (This method would be named ‘and’ if that were not a Python keyword.)

    Each input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLES:

        sage: var('a,b,c,x')
        (a, b, c, x)
        sage: qf = qepcad_formula
        sage: qf.and_(a*b, a*c, b*c != 0)
        [a b = 0 /\ a c = 0 /\ b c /= 0]
        sage: qf.and_(a*x^2 == 3, qf.or_(a > b, b > c))
        [a x^2 = 3 /\ [a > b \/ b > c]]

  `atomic`(lhs, op='=', rhs=0)

    Construct a QEPCAD formula from the given inputs.

    INPUT:

    - `lhs` – a polynomial, or a symbolic equality or inequality
    - `op` – a relational operator, default ‘=’
    - `rhs` – a polynomial, default 0

    If `lhs` is a symbolic equality or inequality, then `op` and `rhs` are ignored.

    This method works by printing the given polynomials, so we do not care what ring they are in (as long as they print with integral or rational coefficients).

    EXAMPLES:

        sage: qf = qepcad_formula
        sage: var('a,b,c')
        (a, b, c)
        sage: K.<x,y> = QQ[]
        sage: def test_qf(qf):
        ....:     return qf, qf.vars
        sage: test_qf(qf.atomic(a^2 + 17))
        (a^2 + 17 = 0, frozenset({'a'}))
        sage: test_qf(qf.atomic(a*b*c <= c^3))
        (a b c <= c^3, frozenset({'a', 'b', 'c'}))
        sage: test_qf(qf.atomic(x+y^2, '!=', a+b))
        (y^2 + x /= a + b, frozenset({'a', 'b', 'x', 'y'}))
        sage: test_qf(qf.atomic(x, operator.lt))
        (x < 0, frozenset({'x'}))

  `connected_subset`(v, formula, allow_multi=False)

    Given a variable and a formula, returns a new formula which is true iff the set of values for the variable at which the original formula was true is connected (including cases where this set is empty or is a single point).

    This method is available both as `connected_subset()` and `C()` (the QEPCAD name for this quantifier).

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLE:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.connected_subset(a, a^2 + b > b^2 + a)
        (C a)[a^2 + b > b^2 + a]
        sage: qf.C(b, b^2 != a)
        (C b)[b^2 /= a]

  `exactly_k`(k, v, formula, allow_multi=False)

    Given a nonnegative integer \(k\), a variable, and a formula, returns a new formula which is true iff the original formula is true for exactly \(k\) values of the variable.

    This method is available both as `exactly_k()` and `X()` (the QEPCAD name for this quantifier).

    (Note that QEPCAD does not support \(k=0\) with this syntax, so if \(k=0\) is requested we implement it with `forall()` and `not_()`.)

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLE:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.exactly_k(1, x, x^2 + a*x + b == 0)
        (X1 x)[a x + x^2 + b = 0]
        sage: qf.exactly_k(0, b, a*b == 1)
        (A b)[~a b = 1]

  `exists`(v, formula)

    Given a variable (or list of variables) and a formula, returns the existential quantification of the formula over the variables.

    This method is available both as `exists()` and `E()` (the QEPCAD name for this quantifier).

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLE:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.exists(a, a^2 + b > b^2 + a)
        (E a)[a^2 + b > b^2 + a]
        sage: qf.exists((a, b), a^2 + b^2 < 0)
        (E a)(E b)[a^2 + b^2 < 0]
        sage: qf.E(b, b^2 == a)
        (E b)[b^2 = a]

  `forall`(v, formula)

    Given a variable (or list of variables) and a formula, returns the universal quantification of the formula over the variables.

    This method is available both as `forall()` and `A()` (the QEPCAD name for this quantifier).

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLE:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.forall(a, a^2 + b > b^2 + a)
        (A a)[a^2 + b > b^2 + a]
        sage: qf.forall((a, b), a^2 + b^2 > 0)
        (A a)(A b)[a^2 + b^2 > 0]
        sage: qf.A(b, b^2 != a)
        (A b)[b^2 /= a]

  `formula`(formula)

    Constructs a QEPCAD formula from the given input.

    INPUTS:

    - `formula` – a polynomial, a symbolic equality or inequality, or a list of polynomials, equalities, or inequalities

    A polynomial \(p\) is interpreted as the equation \(p = 0\). A list is interpreted as the conjunction (‘and’) of the elements.

    EXAMPLES:

        sage: var('a,b,c,x')
        (a, b, c, x)
        sage: qf = qepcad_formula
        sage: qf.formula(a*x + b)
        a x + b = 0
        sage: qf.formula((a*x^2 + b*x + c, a != 0))
        [a x^2 + b x + c = 0 /\ a /= 0]

  `iff`(f1, f2)

    Return the equivalence of its input formulas (that is, given formulas \(P\) and \(Q\), returns ‘\(P\) iff \(Q\)‘).

    The input formulas may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLES:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.iff(a, b)
        [a = 0 <==> b = 0]
        sage: qf.iff(a^2 < b, b^2 < a)
        [a^2 < b <==> b^2 < a]

  `implies`(f1, f2)

    Return the implication of its input formulas (that is, given formulas \(P\) and \(Q\), returns ‘\(P\) implies \(Q\)‘).

    The input formulas may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLES:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.implies(a, b)
        [a = 0 ==> b = 0]
        sage: qf.implies(a^2 < b, b^2 < a)
        [a^2 < b ==> b^2 < a]

  `infinitely_many`(v, formula)

    Given a variable and a formula, returns a new formula which is true iff the original formula was true for infinitely many values of the variable.

    This method is available both as `infinitely_many()` and `F()` (the QEPCAD name for this quantifier).

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLE:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.infinitely_many(a, a^2 + b > b^2 + a)
        (F a)[a^2 + b > b^2 + a]
        sage: qf.F(b, b^2 != a)
        (F b)[b^2 /= a]

  `not_`(formula)

    Return the negation of its input formula.

    (This method would be named ‘not’ if that were not a Python keyword.)

    The input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLES:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.not_(a > b)
        [~a > b]
        sage: qf.not_(a^2 + b^2)
        [~a^2 + b^2 = 0]
        sage: qf.not_(qf.and_(a > 0, b < 0))
        [~[a > 0 /\ b < 0]]

  `or_`(*formulas)

    Return the disjunction of its input formulas.

    (This method would be named ‘or’ if that were not a Python keyword.)

    Each input formula may be a `qformula` as returned by the methods of `qepcad_formula`, a symbolic equality or inequality, or a polynomial \(p\) (meaning \(p = 0\)).

    EXAMPLES:

        sage: var('a,b,c,x')
        (a, b, c, x)
        sage: qf = qepcad_formula
        sage: qf.or_(a*b, a*c, b*c != 0)
        [a b = 0 \/ a c = 0 \/ b c /= 0]
        sage: qf.or_(a*x^2 == 3, qf.and_(a > b, b > c))
        [a x^2 = 3 \/ [a > b /\ b > c]]

  `quantifier`(kind, v, formula, allow_multi=True)

    A helper method for building quantified QEPCAD formulas; not expected to be called directly.

    Takes the quantifier kind (the string label of this quantifier), a variable or list of variables, and a formula, and returns the quantified formula.

    EXAMPLES:

        sage: var('a,b')
        (a, b)
        sage: qf = qepcad_formula
        sage: qf.quantifier('NOT_A_REAL_QEPCAD_QUANTIFIER', a, a*b==0)
        (NOT_A_REAL_QEPCAD_QUANTIFIER a)[a b = 0]
        sage: qf.quantifier('FOO', (a, b), a*b)
        (FOO a)(FOO b)[a b = 0]

`sage.interfaces.qepcad.``qepcad_version`()

  Return a string containing the current QEPCAD version number.

  EXAMPLES:

      sage: qepcad_version() # random, optional - qepcad
      'Version B 1.48, 25 Oct 2007'

  TESTS:

      sage: qepcad_version() # optional - qepcad
      'Version B ..., ...'

class `sage.interfaces.qepcad.``qformula`(formula, vars, qvars=[])

  A qformula holds a string describing a formula in QEPCAD’s syntax, and a set of variables used.

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