venv/lib/python3.11/site-packages/sympy/sets/handlers/functions.py

from sympy.core.singleton import S
from sympy.sets.sets import Set
from sympy.calculus.singularities import singularities
from sympy.core import Expr, Add
from sympy.core.function import Lambda, FunctionClass, diff, expand_mul
from sympy.core.numbers import Float, oo
from sympy.core.symbol import Dummy, symbols, Wild
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.miscellaneous import Min, Max
from sympy.logic.boolalg import true
from sympy.multipledispatch import Dispatcher
from sympy.sets import (imageset, Interval, FiniteSet, Union, ImageSet,
    Intersection, Range, Complement)
from sympy.sets.sets import EmptySet, is_function_invertible_in_set
from sympy.sets.fancysets import Integers, Naturals, Reals
from sympy.functions.elementary.exponential import match_real_imag


_x, _y = symbols("x y")

FunctionUnion = (FunctionClass, Lambda)

_set_function = Dispatcher('_set_function')


@_set_function.register(FunctionClass, Set)
def _(f, x):
    return None

@_set_function.register(FunctionUnion, FiniteSet)
def _(f, x):
    return FiniteSet(*map(f, x))

@_set_function.register(Lambda, Interval)
def _(f, x):
    from sympy.solvers.solveset import solveset
    from sympy.series import limit
    # TODO: handle functions with infinitely many solutions (eg, sin, tan)
    # TODO: handle multivariate functions

    expr = f.expr
    if len(expr.free_symbols) > 1 or len(f.variables) != 1:
        return
    var = f.variables[0]
    if not var.is_real:
        if expr.subs(var, Dummy(real=True)).is_real is False:
            return

    if expr.is_Piecewise:
        result = S.EmptySet
        domain_set = x
        for (p_expr, p_cond) in expr.args:
            if p_cond is true:
                intrvl = domain_set
            else:
                intrvl = p_cond.as_set()
                intrvl = Intersection(domain_set, intrvl)

            if p_expr.is_Number:
                image = FiniteSet(p_expr)
            else:
                image = imageset(Lambda(var, p_expr), intrvl)
            result = Union(result, image)

            # remove the part which has been `imaged`
            domain_set = Complement(domain_set, intrvl)
            if domain_set is S.EmptySet:
                break
        return result

    if not x.start.is_comparable or not x.end.is_comparable:
        return

    try:
        from sympy.polys.polyutils import _nsort
        sing = list(singularities(expr, var, x))
        if len(sing) > 1:
            sing = _nsort(sing)
    except NotImplementedError:
        return

    if x.left_open:
        _start = limit(expr, var, x.start, dir="+")
    elif x.start not in sing:
        _start = f(x.start)
    if x.right_open:
        _end = limit(expr, var, x.end, dir="-")
    elif x.end not in sing:
        _end = f(x.end)

    if len(sing) == 0:
        soln_expr = solveset(diff(expr, var), var)
        if not (isinstance(soln_expr, FiniteSet)
                or soln_expr is S.EmptySet):
            return
        solns = list(soln_expr)

        extr = [_start, _end] + [f(i) for i in solns
                                 if i.is_real and i in x]
        start, end = Min(*extr), Max(*extr)

        left_open, right_open = False, False
        if _start <= _end:
            # the minimum or maximum value can occur simultaneously
            # on both the edge of the interval and in some interior
            # point
            if start == _start and start not in solns:
                left_open = x.left_open
            if end == _end and end not in solns:
                right_open = x.right_open
        else:
            if start == _end and start not in solns:
                left_open = x.right_open
            if end == _start and end not in solns:
                right_open = x.left_open

        return Interval(start, end, left_open, right_open)
    else:
        return imageset(f, Interval(x.start, sing[0],
                                    x.left_open, True)) + \
            Union(*[imageset(f, Interval(sing[i], sing[i + 1], True, True))
                    for i in range(0, len(sing) - 1)]) + \
            imageset(f, Interval(sing[-1], x.end, True, x.right_open))

@_set_function.register(FunctionClass, Interval)
def _(f, x):
    if f == exp:
        return Interval(exp(x.start), exp(x.end), x.left_open, x.right_open)
    elif f == log:
        return Interval(log(x.start), log(x.end), x.left_open, x.right_open)
    return ImageSet(Lambda(_x, f(_x)), x)

@_set_function.register(FunctionUnion, Union)
def _(f, x):
    return Union(*(imageset(f, arg) for arg in x.args))

@_set_function.register(FunctionUnion, Intersection)
def _(f, x):
    # If the function is invertible, intersect the maps of the sets.
    if is_function_invertible_in_set(f, x):
        return Intersection(*(imageset(f, arg) for arg in x.args))
    else:
        return ImageSet(Lambda(_x, f(_x)), x)

@_set_function.register(FunctionUnion, EmptySet)
def _(f, x):
    return x

@_set_function.register(FunctionUnion, Set)
def _(f, x):
    return ImageSet(Lambda(_x, f(_x)), x)

@_set_function.register(FunctionUnion, Range)
def _(f, self):
    if not self:
        return S.EmptySet
    if not isinstance(f.expr, Expr):
        return
    if self.size == 1:
        return FiniteSet(f(self[0]))
    if f is S.IdentityFunction:
        return self

    x = f.variables[0]
    expr = f.expr
    # handle f that is linear in f's variable
    if x not in expr.free_symbols or x in expr.diff(x).free_symbols:
        return
    if self.start.is_finite:
        F = f(self.step*x + self.start)  # for i in range(len(self))
    else:
        F = f(-self.step*x + self[-1])
    F = expand_mul(F)
    if F != expr:
        return imageset(x, F, Range(self.size))

@_set_function.register(FunctionUnion, Integers)
def _(f, self):
    expr = f.expr
    if not isinstance(expr, Expr):
        return

    n = f.variables[0]
    if expr == abs(n):
        return S.Naturals0

    # f(x) + c and f(-x) + c cover the same integers
    # so choose the form that has the fewest negatives
    c = f(0)
    fx = f(n) - c
    f_x = f(-n) - c
    neg_count = lambda e: sum(_.could_extract_minus_sign()
        for _ in Add.make_args(e))
    if neg_count(f_x) < neg_count(fx):
        expr = f_x + c

    a = Wild('a', exclude=[n])
    b = Wild('b', exclude=[n])
    match = expr.match(a*n + b)
    if match and match[a] and (
            not match[a].atoms(Float) and
            not match[b].atoms(Float)):
        # canonical shift
        a, b = match[a], match[b]
        if a in [1, -1]:
            # drop integer addends in b
            nonint = []
            for bi in Add.make_args(b):
                if not bi.is_integer:
                    nonint.append(bi)
            b = Add(*nonint)
        if b.is_number and a.is_real:
            # avoid Mod for complex numbers, #11391
            br, bi = match_real_imag(b)
            if br and br.is_comparable and a.is_comparable:
                br %= a
                b = br + S.ImaginaryUnit*bi
        elif b.is_number and a.is_imaginary:
            br, bi = match_real_imag(b)
            ai = a/S.ImaginaryUnit
            if bi and bi.is_comparable and ai.is_comparable:
                bi %= ai
                b = br + S.ImaginaryUnit*bi
        expr = a*n + b

    if expr != f.expr:
        return ImageSet(Lambda(n, expr), S.Integers)


@_set_function.register(FunctionUnion, Naturals)
def _(f, self):
    expr = f.expr
    if not isinstance(expr, Expr):
        return

    x = f.variables[0]
    if not expr.free_symbols - {x}:
        if expr == abs(x):
            if self is S.Naturals:
                return self
            return S.Naturals0
        step = expr.coeff(x)
        c = expr.subs(x, 0)
        if c.is_Integer and step.is_Integer and expr == step*x + c:
            if self is S.Naturals:
                c += step
            if step > 0:
                if step == 1:
                    if c == 0:
                        return S.Naturals0
                    elif c == 1:
                        return S.Naturals
                return Range(c, oo, step)
            return Range(c, -oo, step)


@_set_function.register(FunctionUnion, Reals)
def _(f, self):
    expr = f.expr
    if not isinstance(expr, Expr):
        return
    return _set_function(f, Interval(-oo, oo))