def test_Range_symbolic(): # symbolic Range sr = Range(x, y, t) i = Symbol('i', integer=True) ip = Symbol('i', integer=True, positive=True) ir = Range(i, i + 20, 2) inf = symbols('inf', infinite=True) # args assert sr.args == (x, y, t) assert ir.args == (i, i + 20, 2) # reversed raises(ValueError, lambda: sr.reversed) assert ir.reversed == Range(i + 18, i - 2, -2) # contains assert inf not in sr assert inf not in ir assert .1 not in sr assert .1 not in ir assert i + 1 not in ir assert i + 2 in ir raises(TypeError, lambda: 1 in sr) # XXX is this what contains is supposed to do? # iter raises(ValueError, lambda: next(iter(sr))) assert next(iter(ir)) == i assert sr.intersect(S.Integers) == sr assert sr.intersect(FiniteSet(x)) == Intersection({x}, sr) raises(ValueError, lambda: sr[:2]) raises(ValueError, lambda: sr[0]) raises(ValueError, lambda: sr.as_relational(x)) # len assert len(ir) == ir.size == 10 raises(ValueError, lambda: len(sr)) raises(ValueError, lambda: sr.size) # bool assert bool(ir) == bool(sr) == True # getitem raises(ValueError, lambda: sr[0]) raises(ValueError, lambda: sr[-1]) raises(ValueError, lambda: sr[:2]) assert ir[:2] == Range(i, i + 4, 2) assert ir[0] == i assert ir[-2] == i + 16 assert ir[-1] == i + 18 raises(ValueError, lambda: Range(i)[-1]) assert Range(ip)[-1] == ip - 1 assert ir.inf == i assert ir.sup == i + 18 assert Range(ip).inf == 0 assert Range(ip).sup == ip - 1 raises(ValueError, lambda: Range(i).inf) # as_relational raises(ValueError, lambda: sr.as_relational(x)) assert ir.as_relational(x) == (x >= i) & Eq(x, floor(x)) & (x <= i + 18) assert Range(i, i + 1).as_relational(x) == Eq(x, i) # contains() for symbolic values (issue #18146) e = Symbol('e', integer=True, even=True) o = Symbol('o', integer=True, odd=True) assert Range(5).contains(i) == And(i >= 0, i <= 4) assert Range(1).contains(i) == Eq(i, 0) assert Range(-oo, 5, 1).contains(i) == (i <= 4) assert Range(-oo, oo).contains(i) == True assert Range(0, 8, 2).contains(i) == Contains(i, Range(0, 8, 2)) assert Range(0, 8, 2).contains(e) == And(e >= 0, e <= 6) assert Range(0, 8, 2).contains(2 * i) == And(2 * i >= 0, 2 * i <= 6) assert Range(0, 8, 2).contains(o) == False assert Range(1, 9, 2).contains(e) == False assert Range(1, 9, 2).contains(o) == And(o >= 1, o <= 7) assert Range(8, 0, -2).contains(o) == False assert Range(9, 1, -2).contains(o) == And(o >= 3, o <= 9) assert Range(-oo, 8, 2).contains(i) == Contains(i, Range(-oo, 8, 2))
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 _(a, b): return ConditionSet(a.sym, a.condition, Intersection(a.base_set, b))
def continuous_domain(f, symbol, domain): """ Returns the intervals in the given domain for which the function is continuous. This method is limited by the ability to determine the various singularities and discontinuities of the given function. Examples ======== >>> from sympy import Symbol, S, tan, log, pi, sqrt >>> from sympy.sets import Interval >>> from sympy.calculus.util import continuous_domain >>> x = Symbol('x') >>> continuous_domain(1/x, x, S.Reals) Union(Interval.open(-oo, 0), Interval.open(0, oo)) >>> continuous_domain(tan(x), x, Interval(0, pi)) Union(Interval.Ropen(0, pi/2), Interval.Lopen(pi/2, pi)) >>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5)) Interval(2, 5) >>> continuous_domain(log(2*x - 1), x, S.Reals) Interval.open(1/2, oo) """ from sympy.solvers.inequalities import solve_univariate_inequality from sympy.solvers.solveset import solveset, _has_rational_power if domain.is_subset(S.Reals): constrained_interval = domain for atom in f.atoms(Pow): predicate, denomin = _has_rational_power(atom, symbol) constraint = S.EmptySet if predicate and denomin == 2: constraint = solve_univariate_inequality( atom.base >= 0, symbol).as_set() constrained_interval = Intersection(constraint, constrained_interval) for atom in f.atoms(log): constraint = solve_univariate_inequality(atom.args[0] > 0, symbol).as_set() constrained_interval = Intersection(constraint, constrained_interval) domain = constrained_interval try: sings = S.EmptySet if f.has(Abs): sings = solveset(1/f, symbol, domain) + \ solveset(denom(together(f)), symbol, domain) else: for atom in f.atoms(Pow): predicate, denomin = _has_rational_power(atom, symbol) if predicate and denomin == 2: sings = solveset(1/f, symbol, domain) +\ solveset(denom(together(f)), symbol, domain) break else: sings = Intersection(solveset(1/f, symbol), domain) + \ solveset(denom(together(f)), symbol, domain) except NotImplementedError: import sys raise (NotImplementedError( "Methods for determining the continuous domains" " of this function have not been developed."), None, sys.exc_info()[2]) return domain - sings
def interval(self): """Sequence is defined on the intersection of all the intervals of respective sequences """ return Intersection(*(a.interval for a in self.args))
def _intersect(self, other): if other.is_Interval: return Intersection(S.Integers, other, Interval(self._inf, S.Infinity)) return None
def continuous_domain(f, symbol, domain): """ Returns the intervals in the given domain for which the function is continuous. This method is limited by the ability to determine the various singularities and discontinuities of the given function. Parameters ========== f : :py:class:`~.Expr` The concerned function. symbol : :py:class:`~.Symbol` The variable for which the intervals are to be determined. domain : :py:class:`~.Interval` The domain over which the continuity of the symbol has to be checked. Examples ======== >>> from sympy import Interval, Symbol, S, tan, log, pi, sqrt >>> from sympy.calculus.util import continuous_domain >>> x = Symbol('x') >>> continuous_domain(1/x, x, S.Reals) Union(Interval.open(-oo, 0), Interval.open(0, oo)) >>> continuous_domain(tan(x), x, Interval(0, pi)) Union(Interval.Ropen(0, pi/2), Interval.Lopen(pi/2, pi)) >>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5)) Interval(2, 5) >>> continuous_domain(log(2*x - 1), x, S.Reals) Interval.open(1/2, oo) Returns ======= :py:class:`~.Interval` Union of all intervals where the function is continuous. Raises ====== NotImplementedError If the method to determine continuity of such a function has not yet been developed. """ from sympy.solvers.inequalities import solve_univariate_inequality if domain.is_subset(S.Reals): constrained_interval = domain for atom in f.atoms(Pow): den = atom.exp.as_numer_denom()[1] if den.is_even and den.is_nonzero: constraint = solve_univariate_inequality( atom.base >= 0, symbol).as_set() constrained_interval = Intersection(constraint, constrained_interval) for atom in f.atoms(log): constraint = solve_univariate_inequality(atom.args[0] > 0, symbol).as_set() constrained_interval = Intersection(constraint, constrained_interval) return constrained_interval - singularities(f, symbol, domain)
def set(self): if self.is_symbolic: return Intersection(S.Naturals0, Interval(0, self.n)) return set(self.dict.keys())
def test_Range_symbolic(): # symbolic Range xr = Range(x, x + 4, 5) sr = Range(x, y, t) i = Symbol('i', integer=True) ip = Symbol('i', integer=True, positive=True) ipr = Range(ip) inr = Range(0, -ip, -1) ir = Range(i, i + 19, 2) ir2 = Range(i, i * 8, 3 * i) i = Symbol('i', integer=True) inf = symbols('inf', infinite=True) raises(ValueError, lambda: Range(inf)) raises(ValueError, lambda: Range(inf, 0, -1)) raises(ValueError, lambda: Range(inf, inf, 1)) raises(ValueError, lambda: Range(1, 1, inf)) # args assert xr.args == (x, x + 5, 5) assert sr.args == (x, y, t) assert ir.args == (i, i + 20, 2) assert ir2.args == (i, 10 * i, 3 * i) # reversed raises(ValueError, lambda: xr.reversed) raises(ValueError, lambda: sr.reversed) assert ipr.reversed.args == (ip - 1, -1, -1) assert inr.reversed.args == (-ip + 1, 1, 1) assert ir.reversed.args == (i + 18, i - 2, -2) assert ir2.reversed.args == (7 * i, -2 * i, -3 * i) # contains assert inf not in sr assert inf not in ir assert 0 in ipr assert 0 in inr raises(TypeError, lambda: 1 in ipr) raises(TypeError, lambda: -1 in inr) assert .1 not in sr assert .1 not in ir assert i + 1 not in ir assert i + 2 in ir raises(TypeError, lambda: x in xr) # XXX is this what contains is supposed to do? raises(TypeError, lambda: 1 in sr) # XXX is this what contains is supposed to do? # iter raises(ValueError, lambda: next(iter(xr))) raises(ValueError, lambda: next(iter(sr))) assert next(iter(ir)) == i assert next(iter(ir2)) == i assert sr.intersect(S.Integers) == sr assert sr.intersect(FiniteSet(x)) == Intersection({x}, sr) raises(ValueError, lambda: sr[:2]) raises(ValueError, lambda: xr[0]) raises(ValueError, lambda: sr[0]) # len assert len(ir) == ir.size == 10 assert len(ir2) == ir2.size == 3 raises(ValueError, lambda: len(xr)) raises(ValueError, lambda: xr.size) raises(ValueError, lambda: len(sr)) raises(ValueError, lambda: sr.size) # bool assert bool(Range(0)) == False assert bool(xr) assert bool(ir) assert bool(ipr) assert bool(inr) raises(ValueError, lambda: bool(sr)) raises(ValueError, lambda: bool(ir2)) # inf raises(ValueError, lambda: xr.inf) raises(ValueError, lambda: sr.inf) assert ipr.inf == 0 assert inr.inf == -ip + 1 assert ir.inf == i raises(ValueError, lambda: ir2.inf) # sup raises(ValueError, lambda: xr.sup) raises(ValueError, lambda: sr.sup) assert ipr.sup == ip - 1 assert inr.sup == 0 assert ir.inf == i raises(ValueError, lambda: ir2.sup) # getitem raises(ValueError, lambda: xr[0]) raises(ValueError, lambda: sr[0]) raises(ValueError, lambda: sr[-1]) raises(ValueError, lambda: sr[:2]) assert ir[:2] == Range(i, i + 4, 2) assert ir[0] == i assert ir[-2] == i + 16 assert ir[-1] == i + 18 assert ir2[:2] == Range(i, 7 * i, 3 * i) assert ir2[0] == i assert ir2[-2] == 4 * i assert ir2[-1] == 7 * i raises(ValueError, lambda: Range(i)[-1]) assert ipr[0] == ipr.inf == 0 assert ipr[-1] == ipr.sup == ip - 1 assert inr[0] == inr.sup == 0 assert inr[-1] == inr.inf == -ip + 1 raises(ValueError, lambda: ipr[-2]) assert ir.inf == i assert ir.sup == i + 18 raises(ValueError, lambda: Range(i).inf) # as_relational assert ir.as_relational(x) == ((x >= i) & (x <= i + 18) & Eq(Mod(-i + x, 2), 0)) assert ir2.as_relational(x) == Eq(Mod(-i + x, 3 * i), 0) & (( (x >= i) & (x <= 7 * i) & (3 * i >= 1)) | ((x <= i) & (x >= 7 * i) & (3 * i <= -1))) assert Range(i, i + 1).as_relational(x) == Eq(x, i) assert sr.as_relational(z) == Eq(Mod(t, 1), 0) & Eq(Mod(x, 1), 0) & Eq( Mod(-x + z, t), 0) & (((z >= x) & (z <= -t + y) & (t >= 1)) | ((z <= x) & (z >= -t + y) & (t <= -1))) assert xr.as_relational(z) == Eq(z, x) & Eq(Mod(x, 1), 0) # symbols can clash if user wants (but it must be integer) assert xr.as_relational(x) == Eq(Mod(x, 1), 0) # contains() for symbolic values (issue #18146) e = Symbol('e', integer=True, even=True) o = Symbol('o', integer=True, odd=True) assert Range(5).contains(i) == And(i >= 0, i <= 4) assert Range(1).contains(i) == Eq(i, 0) assert Range(-oo, 5, 1).contains(i) == (i <= 4) assert Range(-oo, oo).contains(i) == True assert Range(0, 8, 2).contains(i) == Contains(i, Range(0, 8, 2)) assert Range(0, 8, 2).contains(e) == And(e >= 0, e <= 6) assert Range(0, 8, 2).contains(2 * i) == And(2 * i >= 0, 2 * i <= 6) assert Range(0, 8, 2).contains(o) == False assert Range(1, 9, 2).contains(e) == False assert Range(1, 9, 2).contains(o) == And(o >= 1, o <= 7) assert Range(8, 0, -2).contains(o) == False assert Range(9, 1, -2).contains(o) == And(o >= 3, o <= 9) assert Range(-oo, 8, 2).contains(i) == Contains(i, Range(-oo, 8, 2))
def set(self): return Intersection(S.Naturals0, Interval(0, self.n))**len(self.p)
def test_Range_set(): empty = Range(0) assert Range(5) == Range(0, 5) == Range(0, 5, 1) r = Range(10, 20, 2) assert 12 in r assert 8 not in r assert 11 not in r assert 30 not in r assert list(Range(0, 5)) == list(range(5)) assert list(Range(5, 0, -1)) == list(range(5, 0, -1)) assert Range(5, 15).sup == 14 assert Range(5, 15).inf == 5 assert Range(15, 5, -1).sup == 15 assert Range(15, 5, -1).inf == 6 assert Range(10, 67, 10).sup == 60 assert Range(60, 7, -10).inf == 10 assert len(Range(10, 38, 10)) == 3 assert Range(0, 0, 5) == empty assert Range(oo, oo, 1) == empty assert Range(oo, 1, 1) == empty assert Range(-oo, 1, -1) == empty assert Range(1, oo, -1) == empty assert Range(1, -oo, 1) == empty assert Range(1, -4, oo) == empty assert Range(1, -4, -oo) == Range(1, 2) assert Range(1, 4, oo) == Range(1, 2) assert Range(-oo, oo).size == oo assert Range(oo, -oo, -1).size == oo raises(ValueError, lambda: Range(-oo, oo, 2)) raises(ValueError, lambda: Range(x, pi, y)) raises(ValueError, lambda: Range(x, y, 0)) assert 5 in Range(0, oo, 5) assert -5 in Range(-oo, 0, 5) assert oo not in Range(0, oo) ni = symbols('ni', integer=False) assert ni not in Range(oo) u = symbols('u', integer=None) assert Range(oo).contains(u) is not False inf = symbols('inf', infinite=True) assert inf not in Range(-oo, oo) raises(ValueError, lambda: Range(0, oo, 2)[-1]) raises(ValueError, lambda: Range(0, -oo, -2)[-1]) assert Range(-oo, 1, 1)[-1] is S.Zero assert Range(oo, 1, -1)[-1] == 2 assert inf not in Range(oo) inf = symbols('inf', infinite=True) assert inf not in Range(oo) assert Range(-oo, 1, 1)[-1] is S.Zero assert Range(oo, 1, -1)[-1] == 2 assert Range(1, 10, 1)[-1] == 9 assert all(i.is_Integer for i in Range(0, -1, 1)) it = iter(Range(-oo, 0, 2)) raises(TypeError, lambda: next(it)) assert empty.intersect(S.Integers) == empty assert Range(-1, 10, 1).intersect(S.Integers) == Range(-1, 10, 1) assert Range(-1, 10, 1).intersect(S.Naturals) == Range(1, 10, 1) assert Range(-1, 10, 1).intersect(S.Naturals0) == Range(0, 10, 1) # test slicing assert Range(1, 10, 1)[5] == 6 assert Range(1, 12, 2)[5] == 11 assert Range(1, 10, 1)[-1] == 9 assert Range(1, 10, 3)[-1] == 7 raises(ValueError, lambda: Range(oo, 0, -1)[1:3:0]) raises(ValueError, lambda: Range(oo, 0, -1)[:1]) raises(ValueError, lambda: Range(1, oo)[-2]) raises(ValueError, lambda: Range(-oo, 1)[2]) raises(IndexError, lambda: Range(10)[-20]) raises(IndexError, lambda: Range(10)[20]) raises(ValueError, lambda: Range(2, -oo, -2)[2:2:0]) assert Range(2, -oo, -2)[2:2:2] == empty assert Range(2, -oo, -2)[:2:2] == Range(2, -2, -4) raises(ValueError, lambda: Range(-oo, 4, 2)[:2:2]) assert Range(-oo, 4, 2)[::-2] == Range(2, -oo, -4) raises(ValueError, lambda: Range(-oo, 4, 2)[::2]) assert Range(oo, 2, -2)[::] == Range(oo, 2, -2) assert Range(-oo, 4, 2)[:-2:-2] == Range(2, 0, -4) assert Range(-oo, 4, 2)[:-2:2] == Range(-oo, 0, 4) raises(ValueError, lambda: Range(-oo, 4, 2)[:0:-2]) raises(ValueError, lambda: Range(-oo, 4, 2)[:2:-2]) assert Range(-oo, 4, 2)[-2::-2] == Range(0, -oo, -4) raises(ValueError, lambda: Range(-oo, 4, 2)[-2:0:-2]) raises(ValueError, lambda: Range(-oo, 4, 2)[0::2]) assert Range(oo, 2, -2)[0::] == Range(oo, 2, -2) raises(ValueError, lambda: Range(-oo, 4, 2)[0:-2:2]) assert Range(oo, 2, -2)[0:-2:] == Range(oo, 6, -2) raises(ValueError, lambda: Range(oo, 2, -2)[0:2:]) raises(ValueError, lambda: Range(-oo, 4, 2)[2::-1]) assert Range(-oo, 4, 2)[-2::2] == Range(0, 4, 4) assert Range(oo, 0, -2)[-10:0:2] == empty raises(ValueError, lambda: Range(oo, 0, -2)[-10:10:2]) raises(ValueError, lambda: Range(oo, 0, -2)[0::-2]) assert Range(oo, 0, -2)[0:-4:-2] == empty assert Range(oo, 0, -2)[:0:2] == empty raises(ValueError, lambda: Range(oo, 0, -2)[:1:-1]) # test empty Range assert Range(x, x, y) == empty assert empty.reversed == empty assert 0 not in empty assert list(empty) == [] assert len(empty) == 0 assert empty.size is S.Zero assert empty.intersect(FiniteSet(0)) is S.EmptySet assert bool(empty) is False raises(IndexError, lambda: empty[0]) assert empty[:0] == empty raises(NotImplementedError, lambda: empty.inf) raises(NotImplementedError, lambda: empty.sup) AB = [None] + list(range(12)) for R in [ Range(1, 10), Range(1, 10, 2), ]: r = list(R) for a, b, c in cartes(AB, AB, [-3, -1, None, 1, 3]): for reverse in range(2): r = list(reversed(r)) R = R.reversed result = list(R[a:b:c]) ans = r[a:b:c] txt = ('\n%s[%s:%s:%s] = %s -> %s' % (R, a, b, c, result, ans)) check = ans == result assert check, txt assert Range(1, 10, 1).boundary == Range(1, 10, 1) for r in (Range(1, 10, 2), Range(1, oo, 2)): rev = r.reversed assert r.inf == rev.inf and r.sup == rev.sup assert r.step == -rev.step # Make sure to use range in Python 3 and xrange in Python 2 (regardless of # compatibility imports above) if PY3: builtin_range = range else: builtin_range = xrange # noqa raises(TypeError, lambda: Range(builtin_range(1))) assert S(builtin_range(10)) == Range(10) if PY3: assert S(builtin_range(1000000000000)) == \ Range(1000000000000) # test Range.as_relational assert Range(1, 4).as_relational(x) == (x >= 1) & (x <= 3) & Eq(x, floor(x)) assert Range(oo, 1, -2).as_relational(x) == (x >= 3) & (x < oo) & Eq(x, floor(x)) # symbolic Range sr = Range(x, y, t) i = Symbol('i', integer=True) ip = Symbol('i', integer=True, positive=True) ir = Range(i, i + 20, 2) # args assert sr.args == (x, y, t) assert ir.args == (i, i + 20, 2) # reversed raises(ValueError, lambda: sr.reversed) assert ir.reversed == Range(i + 18, i - 2, -2) # contains assert inf not in sr assert inf not in ir assert .1 not in sr assert .1 not in ir assert i + 1 not in ir assert i + 2 in ir raises(TypeError, lambda: 1 in sr) # XXX is this what contains is supposed to do? # iter raises(ValueError, lambda: next(iter(sr))) assert next(iter(ir)) == i assert sr.intersect(S.Integers) == sr assert sr.intersect(FiniteSet(x)) == Intersection({x}, sr) raises(ValueError, lambda: sr[:2]) raises(ValueError, lambda: sr[0]) raises(ValueError, lambda: sr.as_relational(x)) # len assert len(ir) == ir.size == 10 raises(ValueError, lambda: len(sr)) raises(ValueError, lambda: sr.size) # bool assert bool(ir) == bool(sr) == True # getitem raises(ValueError, lambda: sr[0]) raises(ValueError, lambda: sr[-1]) raises(ValueError, lambda: sr[:2]) assert ir[:2] == Range(i, i + 4, 2) assert ir[0] == i assert ir[-2] == i + 16 assert ir[-1] == i + 18 raises(ValueError, lambda: Range(i)[-1]) assert Range(ip)[-1] == ip - 1 assert ir.inf == i assert ir.sup == i + 18 assert Range(ip).inf == 0 assert Range(ip).sup == ip - 1 raises(ValueError, lambda: Range(i).inf) raises(ValueError, lambda: sr.as_relational(x)) assert ir.as_relational(x) == (x >= i) & Eq(x, floor(x)) & (x <= i + 18)
def set(self): N, m, n = self.N, self.m, self.n if self.is_symbolic: return Intersection(S.Naturals0, Interval(self.low, self.high)) return {i for i in range(max(0, n + m - N), min(n, m) + 1)}
def set(self): if self.is_symbolic: return Intersection(S.Naturals0, Interval(0, self.n)) return set(map(Integer, list(range(0, self.n + 1))))
def solve_univariate_inequality(expr, gen, relational=True, domain=S.Reals, continuous=False): """Solves a real univariate inequality. Parameters ========== expr : Relational The target inequality gen : Symbol The variable for which the inequality is solved relational : bool A Relational type output is expected or not domain : Set The domain over which the equation is solved continuous: bool True if expr is known to be continuous over the given domain (and so continuous_domain() doesn't need to be called on it) Raises ====== NotImplementedError The solution of the inequality cannot be determined due to limitation in :func:`sympy.solvers.solveset.solvify`. Notes ===== Currently, we cannot solve all the inequalities due to limitations in :func:`sympy.solvers.solveset.solvify`. Also, the solution returned for trigonometric inequalities are restricted in its periodic interval. See Also ======== sympy.solvers.solveset.solvify: solver returning solveset solutions with solve's output API Examples ======== >>> from sympy import solve_univariate_inequality, Symbol, sin, Interval, S >>> x = Symbol('x') >>> solve_univariate_inequality(x**2 >= 4, x) ((2 <= x) & (x < oo)) | ((-oo < x) & (x <= -2)) >>> solve_univariate_inequality(x**2 >= 4, x, relational=False) Union(Interval(-oo, -2), Interval(2, oo)) >>> domain = Interval(0, S.Infinity) >>> solve_univariate_inequality(x**2 >= 4, x, False, domain) Interval(2, oo) >>> solve_univariate_inequality(sin(x) > 0, x, relational=False) Interval.open(0, pi) """ from sympy.solvers.solvers import denoms if domain.is_subset(S.Reals) is False: raise NotImplementedError(filldedent(''' Inequalities in the complex domain are not supported. Try the real domain by setting domain=S.Reals''')) elif domain is not S.Reals: rv = solve_univariate_inequality( expr, gen, relational=False, continuous=continuous).intersection(domain) if relational: rv = rv.as_relational(gen) return rv else: pass # continue with attempt to solve in Real domain # This keeps the function independent of the assumptions about `gen`. # `solveset` makes sure this function is called only when the domain is # real. _gen = gen _domain = domain if gen.is_extended_real is False: rv = S.EmptySet return rv if not relational else rv.as_relational(_gen) elif gen.is_extended_real is None: gen = Dummy('gen', extended_real=True) try: expr = expr.xreplace({_gen: gen}) except TypeError: raise TypeError(filldedent(''' When gen is real, the relational has a complex part which leads to an invalid comparison like I < 0. ''')) rv = None if expr is S.true: rv = domain elif expr is S.false: rv = S.EmptySet else: e = expr.lhs - expr.rhs period = periodicity(e, gen) if period == S.Zero: e = expand_mul(e) const = expr.func(e, 0) if const is S.true: rv = domain elif const is S.false: rv = S.EmptySet elif period is not None: frange = function_range(e, gen, domain) rel = expr.rel_op if rel in ('<', '<='): if expr.func(frange.sup, 0): rv = domain elif not expr.func(frange.inf, 0): rv = S.EmptySet elif rel in ('>', '>='): if expr.func(frange.inf, 0): rv = domain elif not expr.func(frange.sup, 0): rv = S.EmptySet inf, sup = domain.inf, domain.sup if sup - inf is S.Infinity: domain = Interval(0, period, False, True).intersect(_domain) _domain = domain if rv is None: n, d = e.as_numer_denom() try: if gen not in n.free_symbols and len(e.free_symbols) > 1: raise ValueError # this might raise ValueError on its own # or it might give None... solns = solvify(e, gen, domain) if solns is None: # in which case we raise ValueError raise ValueError except (ValueError, NotImplementedError): # replace gen with generic x since it's # univariate anyway raise NotImplementedError(filldedent(''' The inequality, %s, cannot be solved using solve_univariate_inequality. ''' % expr.subs(gen, Symbol('x')))) expanded_e = expand_mul(e) def valid(x): # this is used to see if gen=x satisfies the # relational by substituting it into the # expanded form and testing against 0, e.g. # if expr = x*(x + 1) < 2 then e = x*(x + 1) - 2 # and expanded_e = x**2 + x - 2; the test is # whether a given value of x satisfies # x**2 + x - 2 < 0 # # expanded_e, expr and gen used from enclosing scope v = expanded_e.subs(gen, expand_mul(x)) try: r = expr.func(v, 0) except TypeError: r = S.false if r in (S.true, S.false): return r if v.is_extended_real is False: return S.false else: v = v.n(2) if v.is_comparable: return expr.func(v, 0) # not comparable or couldn't be evaluated raise NotImplementedError( 'relationship did not evaluate: %s' % r) singularities = [] for d in denoms(expr, gen): singularities.extend(solvify(d, gen, domain)) if not continuous: domain = continuous_domain(expanded_e, gen, domain) include_x = '=' in expr.rel_op and expr.rel_op != '!=' try: discontinuities = set(domain.boundary - FiniteSet(domain.inf, domain.sup)) # remove points that are not between inf and sup of domain critical_points = FiniteSet(*(solns + singularities + list( discontinuities))).intersection( Interval(domain.inf, domain.sup, domain.inf not in domain, domain.sup not in domain)) if all(r.is_number for r in critical_points): reals = _nsort(critical_points, separated=True)[0] else: sifted = sift(critical_points, lambda x: x.is_extended_real) if sifted[None]: # there were some roots that weren't known # to be real raise NotImplementedError try: reals = sifted[True] if len(reals) > 1: reals = list(sorted(reals)) except TypeError: raise NotImplementedError except NotImplementedError: raise NotImplementedError('sorting of these roots is not supported') # If expr contains imaginary coefficients, only take real # values of x for which the imaginary part is 0 make_real = S.Reals if im(expanded_e) != S.Zero: check = True im_sol = FiniteSet() try: a = solveset(im(expanded_e), gen, domain) if not isinstance(a, Interval): for z in a: if z not in singularities and valid(z) and z.is_extended_real: im_sol += FiniteSet(z) else: start, end = a.inf, a.sup for z in _nsort(critical_points + FiniteSet(end)): valid_start = valid(start) if start != end: valid_z = valid(z) pt = _pt(start, z) if pt not in singularities and pt.is_extended_real and valid(pt): if valid_start and valid_z: im_sol += Interval(start, z) elif valid_start: im_sol += Interval.Ropen(start, z) elif valid_z: im_sol += Interval.Lopen(start, z) else: im_sol += Interval.open(start, z) start = z for s in singularities: im_sol -= FiniteSet(s) except (TypeError): im_sol = S.Reals check = False if im_sol is S.EmptySet: raise ValueError(filldedent(''' %s contains imaginary parts which cannot be made 0 for any value of %s satisfying the inequality, leading to relations like I < 0. ''' % (expr.subs(gen, _gen), _gen))) make_real = make_real.intersect(im_sol) sol_sets = [S.EmptySet] start = domain.inf if start in domain and valid(start) and start.is_finite: sol_sets.append(FiniteSet(start)) for x in reals: end = x if valid(_pt(start, end)): sol_sets.append(Interval(start, end, True, True)) if x in singularities: singularities.remove(x) else: if x in discontinuities: discontinuities.remove(x) _valid = valid(x) else: # it's a solution _valid = include_x if _valid: sol_sets.append(FiniteSet(x)) start = end end = domain.sup if end in domain and valid(end) and end.is_finite: sol_sets.append(FiniteSet(end)) if valid(_pt(start, end)): sol_sets.append(Interval.open(start, end)) if im(expanded_e) != S.Zero and check: rv = (make_real).intersect(_domain) else: rv = Intersection( (Union(*sol_sets)), make_real, _domain).subs(gen, _gen) return rv if not relational else rv.as_relational(_gen)
def _union(self, other): intersect = Intersection(self, other) if intersect == self: return other elif intersect == other: return self
def solve_univariate_inequality(expr, gen, relational=True, domain=S.Reals, continuous=False): """Solves a real univariate inequality. Parameters ========== expr : Relational The target inequality gen : Symbol The variable for which the inequality is solved relational : bool A Relational type output is expected or not domain : Set The domain over which the equation is solved continuous: bool True if expr is known to be continuous over the given domain (and so continuous_domain() doesn't need to be called on it) Raises ====== NotImplementedError The solution of the inequality cannot be determined due to limitation in `solvify`. Notes ===== Currently, we cannot solve all the inequalities due to limitations in `solvify`. Also, the solution returned for trigonometric inequalities are restricted in its periodic interval. See Also ======== solvify: solver returning solveset solutions with solve's output API Examples ======== >>> from sympy.solvers.inequalities import solve_univariate_inequality >>> from sympy import Symbol, sin, Interval, S >>> x = Symbol('x') >>> solve_univariate_inequality(x**2 >= 4, x) ((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x)) >>> solve_univariate_inequality(x**2 >= 4, x, relational=False) Union(Interval(-oo, -2), Interval(2, oo)) >>> domain = Interval(0, S.Infinity) >>> solve_univariate_inequality(x**2 >= 4, x, False, domain) Interval(2, oo) >>> solve_univariate_inequality(sin(x) > 0, x, relational=False) Interval.open(0, pi) """ from sympy import im from sympy.calculus.util import (continuous_domain, periodicity, function_range) from sympy.solvers.solvers import denoms from sympy.solvers.solveset import solveset_real, solvify, solveset from sympy.solvers.solvers import solve # This keeps the function independent of the assumptions about `gen`. # `solveset` makes sure this function is called only when the domain is # real. _gen = gen _domain = domain if gen.is_real is False: rv = S.EmptySet return rv if not relational else rv.as_relational(_gen) elif gen.is_real is None: gen = Dummy('gen', real=True) try: expr = expr.xreplace({_gen: gen}) except TypeError: raise TypeError(filldedent(''' When gen is real, the relational has a complex part which leads to an invalid comparison like I < 0. ''')) rv = None if expr is S.true: rv = domain elif expr is S.false: rv = S.EmptySet else: e = expr.lhs - expr.rhs period = periodicity(e, gen) if period is not None: frange = function_range(e, gen, domain) rel = expr.rel_op if rel == '<' or rel == '<=': if expr.func(frange.sup, 0): rv = domain elif not expr.func(frange.inf, 0): rv = S.EmptySet elif rel == '>' or rel == '>=': if expr.func(frange.inf, 0): rv = domain elif not expr.func(frange.sup, 0): rv = S.EmptySet inf, sup = domain.inf, domain.sup if sup - inf is S.Infinity: domain = Interval(0, period, False, True) if rv is None: n, d = e.as_numer_denom() try: if gen not in n.free_symbols and len(e.free_symbols) > 1: raise ValueError # this might raise ValueError on its own # or it might give None... solns = solvify(e, gen, domain) if solns is None: # in which case we raise ValueError raise ValueError except (ValueError, NotImplementedError): raise NotImplementedError(filldedent(''' The inequality cannot be solved using solve_univariate_inequality. ''')) expanded_e = expand_mul(e) def valid(x): # this is used to see if gen=x satisfies the # relational by substituting it into the # expanded form and testing against 0, e.g. # if expr = x*(x + 1) < 2 then e = x*(x + 1) - 2 # and expanded_e = x**2 + x - 2; the test is # whether a given value of x satisfies # x**2 + x - 2 < 0 # # expanded_e, expr and gen used from enclosing scope v = expanded_e.subs(gen, x) try: r = expr.func(v, 0) except TypeError: r = S.false if r in (S.true, S.false): return r if v.is_real is False: return S.false else: v = v.n(2) if v.is_comparable: return expr.func(v, 0) # not comparable or couldn't be evaluated raise NotImplementedError( 'relationship did not evaluate: %s' % r) singularities = [] for d in denoms(expr, gen): singularities.extend(solvify(d, gen, domain)) if not continuous: domain = continuous_domain(e, gen, domain) include_x = '=' in expr.rel_op and expr.rel_op != '!=' try: discontinuities = set(domain.boundary - FiniteSet(domain.inf, domain.sup)) # remove points that are not between inf and sup of domain critical_points = FiniteSet(*(solns + singularities + list( discontinuities))).intersection( Interval(domain.inf, domain.sup, domain.inf not in domain, domain.sup not in domain)) if all(r.is_number for r in critical_points): reals = _nsort(critical_points, separated=True)[0] else: from sympy.utilities.iterables import sift sifted = sift(critical_points, lambda x: x.is_real) if sifted[None]: # there were some roots that weren't known # to be real raise NotImplementedError try: reals = sifted[True] if len(reals) > 1: reals = list(sorted(reals)) except TypeError: raise NotImplementedError except NotImplementedError: raise NotImplementedError('sorting of these roots is not supported') #If expr contains imaginary coefficients #Only real values of x for which the imaginary part is 0 are taken make_real = S.Reals if im(expanded_e) != S.Zero: check = True im_sol = FiniteSet() try: a = solveset(im(expanded_e), gen, domain) if not isinstance(a, Interval): for z in a: if z not in singularities and valid(z) and z.is_real: im_sol += FiniteSet(z) else: start, end = a.inf, a.sup for z in _nsort(critical_points + FiniteSet(end)): valid_start = valid(start) if start != end: valid_z = valid(z) pt = _pt(start, z) if pt not in singularities and pt.is_real and valid(pt): if valid_start and valid_z: im_sol += Interval(start, z) elif valid_start: im_sol += Interval.Ropen(start, z) elif valid_z: im_sol += Interval.Lopen(start, z) else: im_sol += Interval.open(start, z) start = z for s in singularities: im_sol -= FiniteSet(s) except (TypeError): im_sol = S.Reals check = False if isinstance(im_sol, EmptySet): raise ValueError(filldedent(''' %s contains imaginary parts which cannot be made 0 for any value of %s satisfying the inequality, leading to relations like I < 0. ''' % (expr.subs(gen, _gen), _gen))) make_real = make_real.intersect(im_sol) empty = sol_sets = [S.EmptySet] start = domain.inf if valid(start) and start.is_finite: sol_sets.append(FiniteSet(start)) for x in reals: end = x if valid(_pt(start, end)): sol_sets.append(Interval(start, end, True, True)) if x in singularities: singularities.remove(x) else: if x in discontinuities: discontinuities.remove(x) _valid = valid(x) else: # it's a solution _valid = include_x if _valid: sol_sets.append(FiniteSet(x)) start = end end = domain.sup if valid(end) and end.is_finite: sol_sets.append(FiniteSet(end)) if valid(_pt(start, end)): sol_sets.append(Interval.open(start, end)) if im(expanded_e) != S.Zero and check: rv = (make_real).intersect(_domain) else: rv = Intersection( (Union(*sol_sets)), make_real, _domain).subs(gen, _gen) return rv if not relational else rv.as_relational(_gen)
def _intersect(self, other): if not isinstance(other, ConditionSet): return ConditionSet(self.sym, self.condition, Intersection(self.base_set, other))