def test_ImageSet_simplification(): from sympy.abc import n, m assert imageset(Lambda(n, n), S.Integers) == S.Integers assert imageset(Lambda(n, sin(n)), imageset(Lambda(m, tan(m)), S.Integers)) == imageset( Lambda(m, sin(tan(m))), S.Integers )
def test_imageset_intersect_real(): from sympy import I from sympy.abc import n assert imageset(Lambda(n, n + (n - 1)*(n + 1)*I), S.Integers).intersect(S.Reals) == \ FiniteSet(-1, 1) s = ImageSet(Lambda(n, -I*(I*(2*pi*n - pi/4) + log(Abs(sqrt(-I))))), S.Integers) assert s.intersect(S.Reals) == imageset(Lambda(n, 2*n*pi - pi/4), S.Integers)
def test_Range_eval_imageset(): a, b, c = symbols('a b c') assert imageset(x, a*(x + b) + c, Range(3)) == \ imageset(x, a*x + a*b + c, Range(3)) eq = (x + 1)**2 assert imageset(x, eq, Range(3)).lamda.expr == eq eq = a*(x + b) + c r = Range(3, -3, -2) imset = imageset(x, eq, r) assert imset.lamda.expr != eq assert list(imset) == [eq.subs(x, i).expand() for i in list(r)]
def test_infinitely_indexed_set_2(): from sympy import exp from sympy.abc import n a = Symbol('a', integer=True) assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(n, n + a), S.Integers) assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(n, -n + a), S.Integers) assert imageset(Lambda(n, -6*n), S.Integers) == ImageSet(Lambda(n, 6*n), S.Integers) assert imageset(Lambda(n, 2*n + pi), S.Integers) == ImageSet(Lambda(n, 2*n + pi), S.Integers) assert imageset(Lambda(n, pi*n + pi), S.Integers) == ImageSet(Lambda(n, pi*n + pi), S.Integers) assert imageset(Lambda(n, exp(n)), S.Integers) != imageset(Lambda(n, n), S.Integers)
def _intersect(self, other): from sympy.solvers.diophantine import diophantine if self.base_set is S.Integers: g = None if isinstance(other, ImageSet) and other.base_set is S.Integers: g = other.lamda.expr m = other.lamda.variables[0] elif other is S.Integers: m = g = Dummy('x') if g is not None: f = self.lamda.expr n = self.lamda.variables[0] # Diophantine sorts the solutions according to the alphabetic # order of the variable names, since the result should not depend # on the variable name, they are replaced by the dummy variables # below a, b = Dummy('a'), Dummy('b') f, g = f.subs(n, a), g.subs(m, b) solns_set = diophantine(f - g) if solns_set == set(): return EmptySet() solns = list(diophantine(f - g)) if len(solns) != 1: return # since 'a' < 'b', select soln for n nsol = solns[0][0] t = nsol.free_symbols.pop() return imageset(Lambda(n, f.subs(a, nsol.subs(t, n))), S.Integers) if other == S.Reals: from sympy.solvers.solveset import solveset_real from sympy.core.function import expand_complex if len(self.lamda.variables) > 1: return None f = self.lamda.expr n = self.lamda.variables[0] n_ = Dummy(n.name, real=True) f_ = f.subs(n, n_) re, im = f_.as_real_imag() im = expand_complex(im) return imageset(Lambda(n_, re), self.base_set.intersect( solveset_real(im, n_)))
def test_infinitely_indexed_set_1(): from sympy.abc import n, m, t assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(m, m), S.Integers) assert imageset(Lambda(n, 2*n), S.Integers).intersect( imageset(Lambda(m, 2*m + 1), S.Integers)) is S.EmptySet assert imageset(Lambda(n, 2*n), S.Integers).intersect( imageset(Lambda(n, 2*n + 1), S.Integers)) is S.EmptySet assert imageset(Lambda(m, 2*m), S.Integers).intersect( imageset(Lambda(n, 3*n), S.Integers)) == \ ImageSet(Lambda(t, 6*t), S.Integers) assert imageset(x, x/2 + S(1)/3, S.Integers).intersect(S.Integers) is S.EmptySet assert imageset(x, x/2 + S.Half, S.Integers).intersect(S.Integers) is S.Integers
def _intersect(self, other): from sympy import Dummy from sympy.solvers.diophantine import diophantine from sympy.sets.sets import imageset if self.base_set is S.Integers: if isinstance(other, ImageSet) and other.base_set is S.Integers: f, g = self.lamda.expr, other.lamda.expr n, m = self.lamda.variables[0], other.lamda.variables[0] # Diophantine sorts the solutions according to the alphabetic # order of the variable names, since the result should not depend # on the variable name, they are replaced by the dummy variables # below a, b = Dummy('a'), Dummy('b') f, g = f.subs(n, a), g.subs(m, b) solns_set = diophantine(f - g) if solns_set == set(): return EmptySet() solns = list(diophantine(f - g)) if len(solns) == 1: t = list(solns[0][0].free_symbols)[0] else: return None # since 'a' < 'b' return imageset(Lambda(t, f.subs(a, solns[0][0])), S.Integers)
def _intersect(self, other): from sympy import Dummy from sympy.solvers.diophantine import diophantine from sympy.sets.sets import imageset if self.base_set is S.Integers: if isinstance(other, ImageSet) and other.base_set is S.Integers: f, g = self.lamda.expr, other.lamda.expr n, m = self.lamda.variables[0], other.lamda.variables[0] # Diophantine sorts the solutions according to the alphabetic # order of the variable names, since the result should not depend # on the variable name, they are replaced by the dummy variables # below a, b = Dummy("a"), Dummy("b") f, g = f.subs(n, a), g.subs(m, b) solns_set = diophantine(f - g) if solns_set == set(): return EmptySet() solns = list(diophantine(f - g)) if len(solns) == 1: t = list(solns[0][0].free_symbols)[0] else: return None # since 'a' < 'b' return imageset(Lambda(t, f.subs(a, solns[0][0])), S.Integers) if other == S.Reals: from sympy.solvers.solveset import solveset_real from sympy.core.function import expand_complex if len(self.lamda.variables) > 1: return None f = self.lamda.expr n = self.lamda.variables[0] n_ = Dummy(n.name, real=True) f_ = f.subs(n, n_) re, im = f_.as_real_imag() im = expand_complex(im) return imageset(Lambda(n_, re), self.base_set.intersect(solveset_real(im, n_)))
def test_infinitely_indexed_set_1(): from sympy.abc import n, m, t assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(m, m), S.Integers) assert imageset(Lambda(n, 2 * n), S.Integers).intersect(imageset(Lambda(m, 2 * m + 1), S.Integers)) == EmptySet() assert imageset(Lambda(n, 2 * n), S.Integers).intersect(imageset(Lambda(n, 2 * n + 1), S.Integers)) == EmptySet() assert imageset(Lambda(m, 2 * m), S.Integers).intersect(imageset(Lambda(n, 3 * n), S.Integers)) == ImageSet( Lambda(t, 6 * t), S.Integers )
def test_infinitely_indexed_set_2(): from sympy.abc import n a = Symbol('a', integer=True) assert imageset(Lambda(n, n), S.Integers) == \ imageset(Lambda(n, n + a), S.Integers) assert imageset(Lambda(n, n + pi), S.Integers) == \ imageset(Lambda(n, n + a + pi), S.Integers) assert imageset(Lambda(n, n), S.Integers) == \ imageset(Lambda(n, -n + a), S.Integers) assert imageset(Lambda(n, -6*n), S.Integers) == \ ImageSet(Lambda(n, 6*n), S.Integers) assert imageset(Lambda(n, 2*n + pi), S.Integers) == \ ImageSet(Lambda(n, 2*n + pi - 2), S.Integers)
def test_infinitely_indexed_set_3(): from sympy.abc import n, m, t assert imageset(Lambda(m, 2*pi*m), S.Integers).intersect( imageset(Lambda(n, 3*pi*n), S.Integers)) == \ ImageSet(Lambda(t, 6*pi*t), S.Integers) assert imageset(Lambda(n, 2*n + 1), S.Integers) == \ imageset(Lambda(n, 2*n - 1), S.Integers) assert imageset(Lambda(n, 3*n + 2), S.Integers) == \ imageset(Lambda(n, 3*n - 1), S.Integers)
def test_infinitely_indexed_set_1(): from sympy.abc import n, m, t assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(m, m), S.Integers) assert (imageset(Lambda(n, 2 * n), S.Integers).intersect( imageset(Lambda(m, 2 * m + 1), S.Integers)) is S.EmptySet) assert (imageset(Lambda(n, 2 * n), S.Integers).intersect( imageset(Lambda(n, 2 * n + 1), S.Integers)) is S.EmptySet) assert imageset(Lambda(m, 2 * m), S.Integers).intersect( imageset(Lambda(n, 3 * n), S.Integers)) == ImageSet(Lambda(t, 6 * t), S.Integers) assert (imageset(x, x / 2 + Rational(1, 3), S.Integers).intersect( S.Integers) is S.EmptySet) assert imageset(x, x / 2 + S.Half, S.Integers).intersect( S.Integers) is S.Integers # https://github.com/sympy/sympy/issues/17355 S53 = ImageSet(Lambda(n, 5 * n + 3), S.Integers) assert S53.intersect(S.Integers) == S53
def test_Integers_eval_imageset(): ans = ImageSet(Lambda(x, 2*x + Rational(3, 7)), S.Integers) im = imageset(Lambda(x, -2*x + Rational(3, 7)), S.Integers) assert im == ans im = imageset(Lambda(x, -2*x - Rational(11, 7)), S.Integers) assert im == ans y = Symbol('y') L = imageset(x, 2*x + y, S.Integers) assert y + 4 in L a, b, c = 0.092, 0.433, 0.341 assert a in imageset(x, a + c*x, S.Integers) assert b in imageset(x, b + c*x, S.Integers) _x = symbols('x', negative=True) eq = _x**2 - _x + 1 assert imageset(_x, eq, S.Integers).lamda.expr == _x**2 + _x + 1 eq = 3*_x - 1 assert imageset(_x, eq, S.Integers).lamda.expr == 3*_x + 2 assert imageset(x, (x, 1/x), S.Integers) == \ ImageSet(Lambda(x, (x, 1/x)), S.Integers)
def test_Integers_eval_imageset(): ans = ImageSet(Lambda(x, 2*x + S(3)/7), S.Integers) im = imageset(Lambda(x, -2*x + S(3)/7), S.Integers) assert im == ans im = imageset(Lambda(x, -2*x - S(11)/7), S.Integers) assert im == ans y = Symbol('y') assert imageset(x, 2*x + y, S.Integers) == \ imageset(x, 2*x + y % 2, S.Integers) _x = symbols('x', negative=True) eq = _x**2 - _x + 1 assert imageset(_x, eq, S.Integers).lamda.expr == _x**2 + _x + 1 eq = 3*_x - 1 assert imageset(_x, eq, S.Integers).lamda.expr == 3*_x + 2 assert imageset(x, (x, 1/x), S.Integers) == \ ImageSet(Lambda(x, (x, 1/x)), S.Integers)
def test_Integers_eval_imageset(): ans = ImageSet(Lambda(x, 2 * x + S(3) / 7), S.Integers) im = imageset(Lambda(x, -2 * x + S(3) / 7), S.Integers) assert im == ans im = imageset(Lambda(x, -2 * x - S(11) / 7), S.Integers) assert im == ans y = Symbol('y') assert imageset(x, 2*x + y, S.Integers) == \ imageset(x, 2*x + y % 2, S.Integers) _x = symbols('x', negative=True) eq = _x**2 - _x + 1 assert imageset(_x, eq, S.Integers).lamda.expr == _x**2 + _x + 1 eq = 3 * _x - 1 assert imageset(_x, eq, S.Integers).lamda.expr == 3 * _x + 2 assert imageset(x, (x, 1/x), S.Integers) == \ ImageSet(Lambda(x, (x, 1/x)), S.Integers)
def test_ImageSet(): assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1) assert ImageSet(Lambda(x, y), S.Integers) == FiniteSet(y) squares = ImageSet(Lambda(x, x**2), S.Naturals) assert 4 in squares assert 5 not in squares assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9) assert 16 not in squares.intersect(Interval(0, 10)) si = iter(squares) a, b, c, d = next(si), next(si), next(si), next(si) assert (a, b, c, d) == (1, 4, 9, 16) harmonics = ImageSet(Lambda(x, 1 / x), S.Naturals) assert Rational(1, 5) in harmonics assert Rational(.25) in harmonics assert 0.25 not in harmonics assert Rational(.3) not in harmonics assert harmonics.is_iterable assert imageset(x, -x, Interval(0, 1)) == Interval(-1, 0) assert ImageSet(Lambda(x, x**2), Interval(0, 2)).doit() == Interval(0, 4) c = ComplexRegion(Interval(1, 3) * Interval(1, 3)) assert Tuple(2, 6) in ImageSet(Lambda((x, y), (x, 2 * y)), c) assert Tuple(2, S.Half) in ImageSet(Lambda((x, y), (x, 1 / y)), c) assert Tuple(2, -2) not in ImageSet(Lambda((x, y), (x, y**2)), c) assert Tuple(2, -2) in ImageSet(Lambda((x, y), (x, -2)), c) c3 = Interval(3, 7) * Interval(8, 11) * Interval(5, 9) assert Tuple(8, 3, 9) in ImageSet(Lambda((t, y, x), (y, t, x)), c3) assert Tuple(S(1) / 8, 3, 9) in ImageSet(Lambda((t, y, x), (1 / y, t, x)), c3) assert 2 / pi not in ImageSet(Lambda((x, y), 2 / x), c) assert 2 / S(100) not in ImageSet(Lambda((x, y), 2 / x), c) assert 2 / S(3) in ImageSet(Lambda((x, y), 2 / x), c)
def _eval_imageset(self, f): from sympy.core.function import expand_mul 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))
def _eval_imageset(self, f): from sympy.core.function import expand_mul 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))
def test_ImageSet(): assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1) assert ImageSet(Lambda(x, y), S.Integers) == FiniteSet(y) squares = ImageSet(Lambda(x, x**2), S.Naturals) assert 4 in squares assert 5 not in squares assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9) assert 16 not in squares.intersect(Interval(0, 10)) si = iter(squares) a, b, c, d = next(si), next(si), next(si), next(si) assert (a, b, c, d) == (1, 4, 9, 16) harmonics = ImageSet(Lambda(x, 1/x), S.Naturals) assert Rational(1, 5) in harmonics assert Rational(.25) in harmonics assert 0.25 not in harmonics assert Rational(.3) not in harmonics assert harmonics.is_iterable assert imageset(x, -x, Interval(0, 1)) == Interval(-1, 0) assert ImageSet(Lambda(x, x**2), Interval(0, 2)).doit() == Interval(0, 4) c = ComplexRegion(Interval(1, 3)*Interval(1, 3)) assert Tuple(2, 6) in ImageSet(Lambda((x, y), (x, 2*y)), c) assert Tuple(2, S.Half) in ImageSet(Lambda((x, y), (x, 1/y)), c) assert Tuple(2, -2) not in ImageSet(Lambda((x, y), (x, y**2)), c) assert Tuple(2, -2) in ImageSet(Lambda((x, y), (x, -2)), c) c3 = Interval(3, 7)*Interval(8, 11)*Interval(5, 9) assert Tuple(8, 3, 9) in ImageSet(Lambda((t, y, x), (y, t, x)), c3) assert Tuple(S(1)/8, 3, 9) in ImageSet(Lambda((t, y, x), (1/y, t, x)), c3) assert 2/pi not in ImageSet(Lambda((x, y), 2/x), c) assert 2/S(100) not in ImageSet(Lambda((x, y), 2/x), c) assert 2/S(3) in ImageSet(Lambda((x, y), 2/x), c)
def test_Integers_eval_imageset(): ans = ImageSet(Lambda(x, 2 * x + Rational(3, 7)), S.Integers) im = imageset(Lambda(x, -2 * x + Rational(3, 7)), S.Integers) assert im == ans im = imageset(Lambda(x, -2 * x - Rational(11, 7)), S.Integers) assert im == ans y = Symbol("y") L = imageset(x, 2 * x + y, S.Integers) assert y + 4 in L _x = symbols("x", negative=True) eq = _x**2 - _x + 1 assert imageset(_x, eq, S.Integers).lamda.expr == _x**2 + _x + 1 eq = 3 * _x - 1 assert imageset(_x, eq, S.Integers).lamda.expr == 3 * _x + 2 assert imageset(x, (x, 1 / x), S.Integers) == ImageSet(Lambda(x, (x, 1 / x)), S.Integers)
def test_infinitely_indexed_failed_diophantine(): from sympy.abc import n, m, t assert imageset(Lambda(m, 2*pi*m), S.Integers).intersect(imageset(Lambda(n, 3*pi*n), S.Integers)) == \ ImageSet(Lambda(t, -6*pi*t), S.Integers)
def test_image_is_ImageSet(): assert isinstance(imageset(x, sqrt(sin(x)), Range(5)), ImageSet)
def intersection_sets(self, other): from sympy.solvers.diophantine import diophantine # Only handle the straight-forward univariate case if (len(self.lamda.variables) > 1 or self.lamda.signature != self.lamda.variables): return None base_set = self.base_sets[0] # Intersection between ImageSets with Integers as base set # For {f(n) : n in Integers} & {g(m) : m in Integers} we solve the # diophantine equations f(n)=g(m). # If the solutions for n are {h(t) : t in Integers} then we return # {f(h(t)) : t in integers}. if base_set is S.Integers: gm = None if isinstance(other, ImageSet) and other.base_sets == (S.Integers, ): gm = other.lamda.expr m = other.lamda.variables[0] elif other is S.Integers: m = gm = Dummy('x') if gm is not None: fn = self.lamda.expr n = self.lamda.variables[0] solns = list(diophantine(fn - gm, syms=(n, m))) if len(solns) == 0: return EmptySet elif len(solns) != 1: return else: soln, solm = solns[0] (t, ) = soln.free_symbols expr = fn.subs(n, soln.subs(t, n)) return imageset(Lambda(n, expr), S.Integers) if other == S.Reals: from sympy.solvers.solveset import solveset_real from sympy.core.function import expand_complex f = self.lamda.expr n = self.lamda.variables[0] n_ = Dummy(n.name, real=True) f_ = f.subs(n, n_) re, im = f_.as_real_imag() im = expand_complex(im) re = re.subs(n_, n) im = im.subs(n_, n) ifree = im.free_symbols lam = Lambda(n, re) if not im: # allow re-evaluation # of self in this case to make # the result canonical pass elif im.is_zero is False: return S.EmptySet elif ifree != {n}: return None else: # univarite imaginary part in same variable base_set = base_set.intersect(solveset_real(im, n)) return imageset(lam, base_set) elif isinstance(other, Interval): from sympy.solvers.solveset import (invert_real, invert_complex, solveset) f = self.lamda.expr n = self.lamda.variables[0] new_inf, new_sup = None, None new_lopen, new_ropen = other.left_open, other.right_open if f.is_real: inverter = invert_real else: inverter = invert_complex g1, h1 = inverter(f, other.inf, n) g2, h2 = inverter(f, other.sup, n) if all(isinstance(i, FiniteSet) for i in (h1, h2)): if g1 == n: if len(h1) == 1: new_inf = h1.args[0] if g2 == n: if len(h2) == 1: new_sup = h2.args[0] # TODO: Design a technique to handle multiple-inverse # functions # Any of the new boundary values cannot be determined if any(i is None for i in (new_sup, new_inf)): return range_set = S.EmptySet if all(i.is_real for i in (new_sup, new_inf)): # this assumes continuity of underlying function # however fixes the case when it is decreasing if new_inf > new_sup: new_inf, new_sup = new_sup, new_inf new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen) range_set = base_set.intersect(new_interval) else: if other.is_subset(S.Reals): solutions = solveset(f, n, S.Reals) if not isinstance(range_set, (ImageSet, ConditionSet)): range_set = solutions.intersect(other) else: return if range_set is S.EmptySet: return S.EmptySet elif isinstance(range_set, Range) and range_set.size is not S.Infinity: range_set = FiniteSet(*list(range_set)) if range_set is not None: return imageset(Lambda(n, f), range_set) return else: return
def test_ImageSet_simplification(): from sympy.abc import n, m assert imageset(Lambda(n, n), S.Integers) == S.Integers assert imageset(Lambda(n, sin(n)), imageset(Lambda(m, tan(m)), S.Integers)) == \ imageset(Lambda(m, sin(tan(m))), S.Integers)
def test_ImageSet_contains(): from sympy.abc import x assert (2, S.Half) in imageset(x, (x, 1 / x), S.Integers)
def _intersect(self, other): from sympy.solvers.diophantine import diophantine if self.base_set is S.Integers: g = None if isinstance(other, ImageSet) and other.base_set is S.Integers: g = other.lamda.expr m = other.lamda.variables[0] elif other is S.Integers: m = g = Dummy('x') if g is not None: f = self.lamda.expr n = self.lamda.variables[0] # Diophantine sorts the solutions according to the alphabetic # order of the variable names, since the result should not depend # on the variable name, they are replaced by the dummy variables # below a, b = Dummy('a'), Dummy('b') f, g = f.subs(n, a), g.subs(m, b) solns_set = diophantine(f - g) if solns_set == set(): return EmptySet() solns = list(diophantine(f - g)) if len(solns) != 1: return # since 'a' < 'b', select soln for n nsol = solns[0][0] t = nsol.free_symbols.pop() return imageset(Lambda(n, f.subs(a, nsol.subs(t, n))), S.Integers) if other == S.Reals: from sympy.solvers.solveset import solveset_real from sympy.core.function import expand_complex if len(self.lamda.variables) > 1: return None f = self.lamda.expr n = self.lamda.variables[0] n_ = Dummy(n.name, real=True) f_ = f.subs(n, n_) re, im = f_.as_real_imag() im = expand_complex(im) return imageset(Lambda(n_, re), self.base_set.intersect( solveset_real(im, n_))) elif isinstance(other, Interval): from sympy.solvers.solveset import (invert_real, invert_complex, solveset) f = self.lamda.expr n = self.lamda.variables[0] base_set = self.base_set new_inf, new_sup = None, None new_lopen, new_ropen = other.left_open, other.right_open if f.is_real: inverter = invert_real else: inverter = invert_complex g1, h1 = inverter(f, other.inf, n) g2, h2 = inverter(f, other.sup, n) if all(isinstance(i, FiniteSet) for i in (h1, h2)): if g1 == n: if len(h1) == 1: new_inf = h1.args[0] if g2 == n: if len(h2) == 1: new_sup = h2.args[0] # TODO: Design a technique to handle multiple-inverse # functions # Any of the new boundary values cannot be determined if any(i is None for i in (new_sup, new_inf)): return range_set = S.EmptySet if all(i.is_real for i in (new_sup, new_inf)): new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen) range_set = base_set._intersect(new_interval) else: if other.is_subset(S.Reals): solutions = solveset(f, n, S.Reals) if not isinstance(range_set, (ImageSet, ConditionSet)): range_set = solutions._intersect(other) else: return if range_set is S.EmptySet: return S.EmptySet elif isinstance(range_set, Range) and range_set.size is not S.Infinity: range_set = FiniteSet(*list(range_set)) if range_set is not None: return imageset(Lambda(n, f), range_set) return else: return
def test_ImageSet(): raises(ValueError, lambda: ImageSet(x, S.Integers)) assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1) assert ImageSet(Lambda(x, y), S.Integers) == {y} assert ImageSet(Lambda(x, 1), S.EmptySet) == S.EmptySet empty = Intersection(FiniteSet(log(2) / pi), S.Integers) assert unchanged(ImageSet, Lambda(x, 1), empty) # issue #17471 squares = ImageSet(Lambda(x, x**2), S.Naturals) assert 4 in squares assert 5 not in squares assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9) assert 16 not in squares.intersect(Interval(0, 10)) si = iter(squares) a, b, c, d = next(si), next(si), next(si), next(si) assert (a, b, c, d) == (1, 4, 9, 16) harmonics = ImageSet(Lambda(x, 1 / x), S.Naturals) assert Rational(1, 5) in harmonics assert Rational(.25) in harmonics assert 0.25 not in harmonics assert Rational(.3) not in harmonics assert (1, 2) not in harmonics assert harmonics.is_iterable assert imageset(x, -x, Interval(0, 1)) == Interval(-1, 0) assert ImageSet(Lambda(x, x**2), Interval(0, 2)).doit() == Interval(0, 4) assert ImageSet(Lambda((x, y), 2 * x), {4}, {3}).doit() == FiniteSet(8) assert (ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {10, 20, 30}).doit() == FiniteSet(11, 12, 13, 21, 22, 23, 31, 32, 33)) c = Interval(1, 3) * Interval(1, 3) assert Tuple(2, 6) in ImageSet(Lambda(((x, y), ), (x, 2 * y)), c) assert Tuple(2, S.Half) in ImageSet(Lambda(((x, y), ), (x, 1 / y)), c) assert Tuple(2, -2) not in ImageSet(Lambda(((x, y), ), (x, y**2)), c) assert Tuple(2, -2) in ImageSet(Lambda(((x, y), ), (x, -2)), c) c3 = ProductSet(Interval(3, 7), Interval(8, 11), Interval(5, 9)) assert Tuple(8, 3, 9) in ImageSet(Lambda(((t, y, x), ), (y, t, x)), c3) assert Tuple(Rational(1, 8), 3, 9) in ImageSet(Lambda(((t, y, x), ), (1 / y, t, x)), c3) assert 2 / pi not in ImageSet(Lambda(((x, y), ), 2 / x), c) assert 2 / S(100) not in ImageSet(Lambda(((x, y), ), 2 / x), c) assert Rational(2, 3) in ImageSet(Lambda(((x, y), ), 2 / x), c) S1 = imageset(lambda x, y: x + y, S.Integers, S.Naturals) assert S1.base_pset == ProductSet(S.Integers, S.Naturals) assert S1.base_sets == (S.Integers, S.Naturals) # Passing a set instead of a FiniteSet shouldn't raise assert unchanged(ImageSet, Lambda(x, x**2), {1, 2, 3}) S2 = ImageSet(Lambda(((x, y), ), x + y), {(1, 2), (3, 4)}) assert 3 in S2.doit() # FIXME: This doesn't yet work: #assert 3 in S2 assert S2._contains(3) is None raises(TypeError, lambda: ImageSet(Lambda(x, x**2), 1))
def _intersect(self, other): from sympy.solvers.diophantine import diophantine if self.base_set is S.Integers: g = None if isinstance(other, ImageSet) and other.base_set is S.Integers: g = other.lamda.expr m = other.lamda.variables[0] elif other is S.Integers: m = g = Dummy('x') if g is not None: f = self.lamda.expr n = self.lamda.variables[0] # Diophantine sorts the solutions according to the alphabetic # order of the variable names, since the result should not depend # on the variable name, they are replaced by the dummy variables # below a, b = Dummy('a'), Dummy('b') f, g = f.subs(n, a), g.subs(m, b) solns_set = diophantine(f - g) if solns_set == set(): return EmptySet() solns = list(diophantine(f - g)) if len(solns) != 1: return # since 'a' < 'b', select soln for n nsol = solns[0][0] t = nsol.free_symbols.pop() return imageset(Lambda(n, f.subs(a, nsol.subs(t, n))), S.Integers) if other == S.Reals: from sympy.solvers.solveset import solveset_real from sympy.core.function import expand_complex if len(self.lamda.variables) > 1: return None f = self.lamda.expr n = self.lamda.variables[0] n_ = Dummy(n.name, real=True) f_ = f.subs(n, n_) re, im = f_.as_real_imag() im = expand_complex(im) return imageset(Lambda(n_, re), self.base_set.intersect(solveset_real(im, n_))) elif isinstance(other, Interval): from sympy.solvers.solveset import (invert_real, invert_complex, solveset) f = self.lamda.expr n = self.lamda.variables[0] base_set = self.base_set new_inf, new_sup = None, None new_lopen, new_ropen = other.left_open, other.right_open if f.is_real: inverter = invert_real else: inverter = invert_complex g1, h1 = inverter(f, other.inf, n) g2, h2 = inverter(f, other.sup, n) if all(isinstance(i, FiniteSet) for i in (h1, h2)): if g1 == n: if len(h1) == 1: new_inf = h1.args[0] if g2 == n: if len(h2) == 1: new_sup = h2.args[0] # TODO: Design a technique to handle multiple-inverse # functions # Any of the new boundary values cannot be determined if any(i is None for i in (new_sup, new_inf)): return range_set = S.EmptySet if all(i.is_real for i in (new_sup, new_inf)): # this assumes continuity of underlying function # however fixes the case when it is decreasing if new_inf > new_sup: new_inf, new_sup = new_sup, new_inf new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen) range_set = base_set._intersect(new_interval) else: if other.is_subset(S.Reals): solutions = solveset(f, n, S.Reals) if not isinstance(range_set, (ImageSet, ConditionSet)): range_set = solutions._intersect(other) else: return if range_set is S.EmptySet: return S.EmptySet elif isinstance(range_set, Range) and range_set.size is not S.Infinity: range_set = FiniteSet(*list(range_set)) if range_set is not None: return imageset(Lambda(n, f), range_set) return else: return
def intersection_sets(self, other): from sympy.solvers.diophantine import diophantine # Only handle the straight-forward univariate case if (len(self.lamda.variables) > 1 or self.lamda.signature != self.lamda.variables): return None base_set = self.base_sets[0] if base_set is S.Integers: g = None if isinstance(other, ImageSet) and other.base_sets == (S.Integers, ): g = other.lamda.expr m = other.lamda.variables[0] elif other is S.Integers: m = g = Dummy('x') if g is not None: f = self.lamda.expr n = self.lamda.variables[0] # Diophantine sorts the solutions according to the alphabetic # order of the variable names, since the result should not depend # on the variable name, they are replaced by the dummy variables # below a, b = Dummy('a'), Dummy('b') fa, ga = f.subs(n, a), g.subs(m, b) solns = list(diophantine(fa - ga)) if not solns: return EmptySet() if len(solns) != 1: return nsol = solns[0][0] # since 'a' < 'b', nsol is first t = nsol.free_symbols.pop() # diophantine supplied symbol nsol = nsol.subs(t, n) if nsol != n: # if nsol == n and we know were are working with # a base_set of Integers then this was an unevaluated # ImageSet representation of Integers, otherwise # it is a new ImageSet intersection with a subset # of integers nsol = f.subs(n, nsol) return imageset(Lambda(n, nsol), S.Integers) if other == S.Reals: from sympy.solvers.solveset import solveset_real from sympy.core.function import expand_complex f = self.lamda.expr n = self.lamda.variables[0] n_ = Dummy(n.name, real=True) f_ = f.subs(n, n_) re, im = f_.as_real_imag() im = expand_complex(im) re = re.subs(n_, n) im = im.subs(n_, n) ifree = im.free_symbols lam = Lambda(n, re) if not im: # allow re-evaluation # of self in this case to make # the result canonical pass elif im.is_zero is False: return S.EmptySet elif ifree != {n}: return None else: # univarite imaginary part in same variable base_set = base_set.intersect(solveset_real(im, n)) return imageset(lam, base_set) elif isinstance(other, Interval): from sympy.solvers.solveset import (invert_real, invert_complex, solveset) f = self.lamda.expr n = self.lamda.variables[0] new_inf, new_sup = None, None new_lopen, new_ropen = other.left_open, other.right_open if f.is_real: inverter = invert_real else: inverter = invert_complex g1, h1 = inverter(f, other.inf, n) g2, h2 = inverter(f, other.sup, n) if all(isinstance(i, FiniteSet) for i in (h1, h2)): if g1 == n: if len(h1) == 1: new_inf = h1.args[0] if g2 == n: if len(h2) == 1: new_sup = h2.args[0] # TODO: Design a technique to handle multiple-inverse # functions # Any of the new boundary values cannot be determined if any(i is None for i in (new_sup, new_inf)): return range_set = S.EmptySet if all(i.is_real for i in (new_sup, new_inf)): # this assumes continuity of underlying function # however fixes the case when it is decreasing if new_inf > new_sup: new_inf, new_sup = new_sup, new_inf new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen) range_set = base_set.intersect(new_interval) else: if other.is_subset(S.Reals): solutions = solveset(f, n, S.Reals) if not isinstance(range_set, (ImageSet, ConditionSet)): range_set = solutions.intersect(other) else: return if range_set is S.EmptySet: return S.EmptySet elif isinstance(range_set, Range) and range_set.size is not S.Infinity: range_set = FiniteSet(*list(range_set)) if range_set is not None: return imageset(Lambda(n, f), range_set) return else: return
def test_infinitely_indexed_set_3(): from sympy.abc import n assert imageset(Lambda(n, 2*n + 1), S.Integers) == imageset(Lambda(n, 2*n - 1), S.Integers) assert imageset(Lambda(n, 3*n + 2), S.Integers) == imageset(Lambda(n, 3*n - 1), S.Integers)
def test_ImageSet_contains(): from sympy.abc import x assert (2, S.Half) in imageset(x, (x, 1/x), S.Integers)
def intersection_sets(self, other): # noqa:F811 from sympy.solvers.diophantine import diophantine # Only handle the straight-forward univariate case if (len(self.lamda.variables) > 1 or self.lamda.signature != self.lamda.variables): return None base_set = self.base_sets[0] # Intersection between ImageSets with Integers as base set # For {f(n) : n in Integers} & {g(m) : m in Integers} we solve the # diophantine equations f(n)=g(m). # If the solutions for n are {h(t) : t in Integers} then we return # {f(h(t)) : t in integers}. # If the solutions for n are {n_1, n_2, ..., n_k} then we return # {f(n_i) : 1 <= i <= k}. if base_set is S.Integers: gm = None if isinstance(other, ImageSet) and other.base_sets == (S.Integers, ): gm = other.lamda.expr var = other.lamda.variables[0] # Symbol of second ImageSet lambda must be distinct from first m = Dummy('m') gm = gm.subs(var, m) elif other is S.Integers: m = gm = Dummy('m') if gm is not None: fn = self.lamda.expr n = self.lamda.variables[0] try: solns = list(diophantine(fn - gm, syms=(n, m), permute=True)) except (TypeError, NotImplementedError): # TypeError if equation not polynomial with rational coeff. # NotImplementedError if correct format but no solver. return # 3 cases are possible for solns: # - empty set, # - one or more parametric (infinite) solutions, # - a finite number of (non-parametric) solution couples. # Among those, there is one type of solution set that is # not helpful here: multiple parametric solutions. if len(solns) == 0: return EmptySet elif any(not isinstance(s, int) and s.free_symbols for tupl in solns for s in tupl): if len(solns) == 1: soln, solm = solns[0] (t, ) = soln.free_symbols expr = fn.subs(n, soln.subs(t, n)).expand() return imageset(Lambda(n, expr), S.Integers) else: return else: return FiniteSet(*(fn.subs(n, s[0]) for s in solns)) if other == S.Reals: from sympy.solvers.solveset import solveset_real from sympy.core.function import expand_complex f = self.lamda.expr n = self.lamda.variables[0] n_ = Dummy(n.name, real=True) f_ = f.subs(n, n_) re, im = f_.as_real_imag() im = expand_complex(im) re = re.subs(n_, n) im = im.subs(n_, n) ifree = im.free_symbols lam = Lambda(n, re) if not im: # allow re-evaluation # of self in this case to make # the result canonical pass elif im.is_zero is False: return S.EmptySet elif ifree != {n}: return None else: # univarite imaginary part in same variable base_set = base_set.intersect(solveset_real(im, n)) return imageset(lam, base_set) elif isinstance(other, Interval): from sympy.solvers.solveset import (invert_real, invert_complex, solveset) f = self.lamda.expr n = self.lamda.variables[0] new_inf, new_sup = None, None new_lopen, new_ropen = other.left_open, other.right_open if f.is_real: inverter = invert_real else: inverter = invert_complex g1, h1 = inverter(f, other.inf, n) g2, h2 = inverter(f, other.sup, n) if all(isinstance(i, FiniteSet) for i in (h1, h2)): if g1 == n: if len(h1) == 1: new_inf = h1.args[0] if g2 == n: if len(h2) == 1: new_sup = h2.args[0] # TODO: Design a technique to handle multiple-inverse # functions # Any of the new boundary values cannot be determined if any(i is None for i in (new_sup, new_inf)): return range_set = S.EmptySet if all(i.is_real for i in (new_sup, new_inf)): # this assumes continuity of underlying function # however fixes the case when it is decreasing if new_inf > new_sup: new_inf, new_sup = new_sup, new_inf new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen) range_set = base_set.intersect(new_interval) else: if other.is_subset(S.Reals): solutions = solveset(f, n, S.Reals) if not isinstance(range_set, (ImageSet, ConditionSet)): range_set = solutions.intersect(other) else: return if range_set is S.EmptySet: return S.EmptySet elif isinstance(range_set, Range) and range_set.size is not S.Infinity: range_set = FiniteSet(*list(range_set)) if range_set is not None: return imageset(Lambda(n, f), range_set) return else: return
def test_no_mod_on_imaginary(): assert imageset(Lambda(x, 2*x + 3*I), S.Integers ) == ImageSet(Lambda(x, 2*x + I), S.Integers)