def test_diff_wrt():
fx = f(x)
dfx = diff(f(x), x)
ddfx = diff(f(x), x, x)
assert diff(sin(fx) + fx**2, fx) == cos(fx) + 2 * fx
assert diff(sin(dfx) + dfx**2, dfx) == cos(dfx) + 2 * dfx
assert diff(sin(ddfx) + ddfx**2, ddfx) == cos(ddfx) + 2 * ddfx
assert diff(fx**2, dfx) == 0
assert diff(fx**2, ddfx) == 0
assert diff(dfx**2, fx) == 0
assert diff(dfx**2, ddfx) == 0
assert diff(ddfx**2, dfx) == 0
assert diff(fx * dfx * ddfx, fx) == dfx * ddfx
assert diff(fx * dfx * ddfx, dfx) == fx * ddfx
assert diff(fx * dfx * ddfx, ddfx) == fx * dfx
assert diff(f(x), x).diff(f(x)) == 0
assert (sin(f(x)) - cos(diff(f(x), x))).diff(f(x)) == cos(f(x))
assert diff(sin(fx), fx, x) == diff(sin(fx), x, fx)
# Chain rule cases
assert f(g(x)).diff(x) == \
Subs(Derivative(f(x), x), (x, g(x)))*Derivative(g(x), x)
assert diff(f(g(x), h(x)), x) == \
Subs(Derivative(f(y, h(x)), y), (y, g(x)))*Derivative(g(x), x) + \
Subs(Derivative(f(g(x), y), y), (y, h(x)))*Derivative(h(x), x)
assert f(sin(x)).diff(x) == Subs(Derivative(f(x), x), (x, sin(x))) * cos(x)
assert diff(f(g(x)), g(x)) == Subs(Derivative(f(x), x), (x, g(x)))
def test_trigsimp2():
assert trigsimp(cos(x)**2*sin(y)**2 + cos(x)**2*cos(y)**2 + sin(x)**2,
recursive=True) == 1
assert trigsimp(sin(x)**2*sin(y)**2 + sin(x)**2*cos(y)**2 + cos(x)**2,
recursive=True) == 1
assert trigsimp(
Subs(x, x, sin(y)**2 + cos(y)**2)) == Subs(x, x, 1)
def test_mul():
x, y, z, a, b, c = symbols('x y z a b c')
A, B, C = symbols('A B C', commutative=0)
assert (x * y * z).subs(z * x, y) == y**2
assert (z * x).subs(1 / x, z) == z * x
assert (x * y / z).subs(1 / z, a) == a * x * y
assert (x * y / z).subs(x / z, a) == a * y
assert (x * y / z).subs(y / z, a) == a * x
assert (x * y / z).subs(x / z, 1 / a) == y / a
assert (x * y / z).subs(x, 1 / a) == y / (z * a)
assert (2 * x * y).subs(5 * x * y, z) != 2 * z / 5
assert (x * y * A).subs(x * y, a) == a * A
assert Subs(x * y * A, x * y, a).is_commutative is False
assert (x**2 * y**(3 * x / 2)).subs(x * y**(x / 2), 2) == 4 * y**(x / 2)
assert (x * exp(x * 2)).subs(x * exp(x), 2) == 2 * exp(x)
assert ((x**(2 * y))**3).subs(x**y, 2) == 64
assert (x * A * B).subs(x * A, y) == y * B
assert (x * y * (1 + x) * (1 + x * y)).subs(x * y, 2) == 6 * (1 + x)
assert ((1 + A * B) * A * B).subs(A * B, x * A * B)
assert (x * a / z).subs(x / z, A) == a * A
assert Subs(x * a / z, x / z, A).is_commutative is False
assert (x**3 * A).subs(x**2 * A, a) == a * x
assert (x**2 * A * B).subs(x**2 * B, a) == a * A
assert (x**2 * A * B).subs(x**2 * A, a) == a * B
assert (b*A**3/(a**3*c**3)).subs(a**4*c**3*A**3/b**4, z) == \
b*A**3/(a**3*c**3)
assert (6 * x).subs(2 * x, y) == 3 * y
assert (y * exp(3 * x / 2)).subs(y * exp(x), 2) == 2 * exp(x / 2)
assert (y * exp(3 * x / 2)).subs(y * exp(x), 2) == 2 * exp(x / 2)
assert (A**2 * B * A**2 * B * A**2).subs(A * B * A, C) == A * C**2 * A
assert (x * A**3).subs(x * A, y) == y * A**2
assert (x**2 * A**3).subs(x * A, y) == y**2 * A
assert (x * A**3).subs(x * A, B) == B * A**2
assert (x * A * B * A * exp(x * A * B)).subs(x * A,
B) == B**2 * A * exp(B * B)
assert (x**2 * A * B * A * exp(x * A * B)).subs(x * A,
B) == B**3 * exp(B**2)
assert (x**3*A*exp(x*A*B)*A*exp(x*A*B)).subs(x*A, B) == \
x*B*exp(B**2)*B*exp(B**2)
assert (x * A * B * C * A * B).subs(x * A * B, C) == C**2 * A * B
assert (-I * a * b).subs(a * b, 2) == -2 * I
# issue sympy/sympy#6361
assert (-8 * I * a).subs(-2 * a, 1) == 4 * I
assert (-I * a).subs(-a, 1) == I
# issue sympy/sympy#6441
assert (4 * x**2).subs(2 * x, y) == y**2
assert (2 * 4 * x**2).subs(2 * x, y) == 2 * y**2
assert (-x**3 / 9).subs(-x / 3, z) == -z**2 * x
assert (-x**3 / 9).subs(x / 3, z) == -z**2 * x
assert (-2 * x**3 / 9).subs(x / 3, z) == -2 * x * z**2
assert (-2 * x**3 / 9).subs(-x / 3, z) == -2 * x * z**2
assert (-2 * x**3 / 9).subs(-2 * x, z) == z * x**2 / 9
assert (-2 * x**3 / 9).subs(2 * x, z) == -z * x**2 / 9
assert (2 * (3 * x / 5 / 7)**2).subs(3 * x / 5,
z) == 2 * (Rational(1, 7))**2 * z**2
assert (4 * x).subs(-2 * x, z) == 4 * x # try keep subs literal
def test_diofantissue_376():
f = symbols('f', cls=Function)
e1 = Subs(Derivative(f(x), x), (x, y*z))
e2 = Subs(Derivative(f(t), t), (t, y*z))
assert (x*e1).subs({e2: y}) == x*y
e3 = Subs(Derivative(f(t), t), (t, y*z**2))
assert e3.subs({e2: y}) == e3
e4 = Subs(Derivative(f(t), t, t), (t, y*z))
assert e4.subs({e2: y}) == e4
def test_sympyissue_12005():
e1 = Subs(Derivative(f(x), x), (x, x))
assert e1.diff(x) == Derivative(f(x), x, x)
e2 = Subs(Derivative(f(x), x), (x, x**2 + 1))
assert e2.diff(x) == 2*x*Subs(Derivative(f(x), x, x), (x, x**2 + 1))
e3 = Subs(Derivative(f(x) + y**2 - y, y), (y, y**2))
assert e3.diff(y) == 4*y
e4 = Subs(Derivative(f(x + y), y), (y, x**2))
assert e4.diff(y) == 0
e5 = Subs(Derivative(f(x), x), (y, y), (z, z))
assert e5.diff(x) == Derivative(f(x), x, x)
assert f(g(x)).diff(g(x), g(x)) == Subs(Derivative(f(y), y, y), (y, g(x)))
def test_sympyissue_7068():
f = Function('f')
y1 = Dummy('y')
y2 = Dummy('y')
func1 = f(a + y1 * b)
func2 = f(a + y2 * b)
func1_y = func1.diff(y1)
func2_y = func2.diff(y2)
assert func1_y != func2_y
z1 = Subs(f(a), (a, y1))
z2 = Subs(f(a), (a, y2))
assert z1 != z2
def test_sympyissue_7231():
ans1 = f(x).series(x, a)
_xi_1 = ans1.atoms(Dummy).pop()
res = (f(a) + (-a + x)*Subs(Derivative(f(_xi_1), _xi_1), (_xi_1, a)) +
(-a + x)**2*Subs(Derivative(f(_xi_1), _xi_1, _xi_1), (_xi_1, a))/2 +
(-a + x)**3*Subs(Derivative(f(_xi_1), _xi_1, _xi_1, _xi_1), (_xi_1, a))/6 +
(-a + x)**4*Subs(Derivative(f(_xi_1), _xi_1, _xi_1, _xi_1, _xi_1), (_xi_1, a))/24 +
(-a + x)**5*Subs(Derivative(f(_xi_1), _xi_1, _xi_1, _xi_1, _xi_1, _xi_1),
(_xi_1, a))/120 + O((-a + x)**6, (x, a)))
assert res == ans1
ans2 = f(x).series(x, a)
assert res == ans2
def test_sympyissue_7068():
from diofant.abc import a, b
f = Function('f')
y1 = Dummy('y')
y2 = Dummy('y')
func1 = f(a + y1 * b)
func2 = f(a + y2 * b)
func1_y = func1.diff(y1)
func2_y = func2.diff(y2)
assert func1_y != func2_y
z1 = Subs(f(a), a, y1)
z2 = Subs(f(a), a, y2)
assert z1 != z2
def check_solutions(eq):
"""
Determines whether solutions returned by diophantine() satisfy the original
equation. Hope to generalize this so we can remove functions like check_ternay_quadratic,
check_solutions_normal, check_solutions()
"""
s = diophantine(eq)
terms = factor_list(eq)[1]
var = list(eq.free_symbols)
var.sort(key=default_sort_key)
okay = True
while len(s) and okay:
solution = s.pop()
okay = False
for term in terms:
subeq = term[0]
if simplify(_mexpand(Subs(subeq, var, solution).doit())) == 0:
okay = True
break
return okay
def is_pell_transformation_ok(eq):
"""
Test whether X*Y, X, or Y terms are present in the equation
after transforming the equation using the transformation returned
by transformation_to_pell(). If they are not present we are good.
Moreover, coefficient of X**2 should be a divisor of coefficient of
Y**2 and the constant term.
"""
A, B = transformation_to_DN(eq)
u = (A*Matrix([X, Y]) + B)[0]
v = (A*Matrix([X, Y]) + B)[1]
simplified = _mexpand(Subs(eq, (x, y), (u, v)).doit())
coeff = {val: key for key, val in (t.as_independent(X, Y)
for t in simplified.args)}
for term in [X*Y, X, Y]:
if term in coeff.keys():
return False
for term in [X**2, Y**2, Integer(1)]:
if term not in coeff.keys():
coeff[term] = Integer(0)
if coeff[X**2] != 0:
return isinstance(coeff[Y**2]/coeff[X**2], Integer) and isinstance(coeff[Integer(1)]/coeff[X**2], Integer)
return True
def test_diofantissue_376():
f = symbols('f', cls=Function)
e1 = Subs(Derivative(f(x), x), (x, y * z))
e2 = Subs(Derivative(f(t), t), (t, y * z))
assert (x * e1).subs({e2: y}) == x * y
e3 = Subs(Derivative(f(t), t), (t, y * z**2))
assert e3.subs({e2: y}) == e3
e4 = Subs(Derivative(f(t), t, t), (t, y * z))
assert e4.subs({e2: y}) == e4
def test_diofantissue_376():
x, y, z, t = symbols('x y z t')
f = symbols('f', cls=Function)
e1 = Subs(Derivative(f(x), x), x, y * z)
e2 = Subs(Derivative(f(t), t), t, y * z)
assert (x * e1).subs(e2, y) == x * y
e3 = Subs(Derivative(f(t), t), t, y * z**2)
assert e3.subs(e2, y) == e3
e4 = Subs(Derivative(f(t), t, t), t, y * z)
assert e4.subs(e2, y) == e4
def test_deriv1():
# These all requre derivatives evaluated at a point (issue sympy/sympy#4719) to work.
# See issue sympy/sympy#4624
assert f(2*x).diff(x) == 2*Subs(Derivative(f(x), x), (x, 2*x))
assert (f(x)**3).diff(x) == 3*f(x)**2*f(x).diff(x)
assert (f(2*x)**3).diff(x) == 6*f(2*x)**2*Subs(Derivative(f(x), x), (x, 2*x))
assert f(2 + x).diff(x) == Subs(Derivative(f(x), x), (x, x + 2))
assert f(2 + 3*x).diff(x) == 3*Subs(Derivative(f(x), x), (x, 3*x + 2))
assert f(3*sin(x)).diff(x) == 3*cos(x)*Subs(Derivative(f(x), x), (x, 3*sin(x)))
# See issue sympy/sympy#8510
assert f(x, x + z).diff(x) == Subs(Derivative(f(y, x + z), y), (y, x)) \
+ Subs(Derivative(f(x, y), y), (y, x + z))
assert f(x, x**2).diff(x) == Subs(Derivative(f(y, x**2), y), (y, x)) \
+ 2*x*Subs(Derivative(f(x, y), y), (y, x**2))
def test_Subs():
assert Subs(x, (x, 0)) == Subs(y, (y, 0))
assert Subs(x, (x, 0)).subs({x: 1}) == Subs(x, (x, 0))
assert Subs(y, (x, 0)).subs({y: 1}) == Subs(1, (x, 0))
assert Subs(f(x), (x, 0)).doit() == f(0)
assert Subs(f(x**2), (x**2, 0)).doit() == f(0)
assert Subs(f(x, y, z), (x, 0), (y, 1), (z, 1)) != \
Subs(f(x, y, z), (x, 0), (y, 0), (z, 1))
assert Subs(f(x, y), (x, 0), (y, 1), (z, 1)) == \
Subs(f(x, y), (x, 0), (y, 1), (z, 2))
assert Subs(f(x, y), (x, 0), (y, 1), (z, 1)) != \
Subs(f(x, y) + z, (x, 0), (y, 1), (z, 0))
assert Subs(f(x, y), (x, 0), (y, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), (x, 0)), (y, 1)).doit() == f(0, 1)
pytest.raises(ValueError, lambda: Subs(f(x, y), x))
pytest.raises(ValueError, lambda: Subs(f(x, y), (x, 0), (x, 0), (y, 1)))
assert len(Subs(f(x, y), (x, 0), (y, 1)).variables) == 2
assert Subs(f(x, y), (x, 0), (y, 1)).point == Tuple(0, 1)
assert Subs(f(x), (x, 0)) == Subs(f(y), (y, 0))
assert Subs(f(x, y), (x, 0), (y, 1)) == Subs(f(x, y), (y, 1), (x, 0))
assert Subs(f(x)*y, (x, 0), (y, 1)) == Subs(f(y)*x, (y, 0), (x, 1))
assert Subs(f(x)*y, (x, 1), (y, 1)) == Subs(f(y)*x, (x, 1), (y, 1))
assert Subs(f(x), (x, 0)).subs({x: 1}).doit() == f(0)
assert Subs(f(x), (x, y)).subs({y: 0}) == Subs(f(x), (x, 0))
assert Subs(y*f(x), (x, y)).subs({y: 2}) == Subs(2*f(x), (x, 2))
assert (2 * Subs(f(x), (x, 0))).subs({Subs(f(x), (x, 0)): y}) == 2*y
assert Subs(f(x), (x, 0)).free_symbols == set()
assert Subs(f(x, y), (x, z)).free_symbols == {y, z}
assert Subs(f(x).diff(x), (x, 0)).doit(), Subs(f(x).diff(x), (x, 0))
assert Subs(1 + f(x).diff(x), (x, 0)).doit(), 1 + Subs(f(x).diff(x), (x, 0))
assert Subs(y*f(x, y).diff(x), (x, 0), (y, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), (x, 0))
assert Subs(y**2*f(x), (x, 0)).diff(y) == 2*y*f(0)
e = Subs(y**2*f(x), (x, y))
assert e.diff(y) == e.doit().diff(y) == y**2*Derivative(f(y), y) + 2*y*f(y)
assert Subs(f(x), (x, 0)) + Subs(f(x), (x, 0)) == 2*Subs(f(x), (x, 0))
e1 = Subs(z*f(x), (x, 1))
e2 = Subs(z*f(y), (y, 1))
assert e1 + e2 == 2*e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z*f(x + 1), (x, 1)) not in (e1, e2)
assert Derivative(f(x), x).subs({x: g(x)}) == Derivative(f(g(x)), g(x))
assert Derivative(f(x), x).subs({x: x + y}) == Subs(Derivative(f(x), x),
(x, x + y))
assert Subs(f(x)*cos(y) + z, (x, 0), (y, pi/3)).evalf(2, strict=False) == \
Subs(f(x)*cos(y) + z, (x, 0), (y, pi/3)).evalf(2, strict=False) == \
z + Rational('1/2').evalf(2)*f(0)
assert f(x).diff(x).subs({x: 0}).subs({x: y}) == f(x).diff(x).subs({x: 0})
assert (x*f(x).diff(x).subs({x: 0})).subs({x: y}) == y*f(x).diff(x).subs({x: 0})
def is_normal_transformation_ok(eq):
A = transformation_to_normal(eq)
X, Y, Z = A*Matrix([x, y, z])
simplified = _mexpand(Subs(eq, (x, y, z), (X, Y, Z)).doit())
coeff = dict(reversed(t.as_independent(*[X, Y, Z])) for t in simplified.args)
for term in [X*Y, Y*Z, X*Z]:
if term in coeff.keys():
return False
return True
def test_simultaneous_subs():
reps = {x: 0, y: 0}
assert (x / y).subs(reps) != (y / x).subs(reps)
assert (x/y).subs(reps, simultaneous=True) == \
(y/x).subs(reps, simultaneous=True)
reps = reps.items()
assert (x / y).subs(reps) != (y / x).subs(reps)
assert (x/y).subs(reps, simultaneous=True) == \
(y/x).subs(reps, simultaneous=True)
assert Derivative(x, y, z).subs(reps, simultaneous=True) == \
Subs(Derivative(0, y, z), (y, 0))
def test_sympyissue_11799():
n = 2
M = Manifold('M', n)
P = Patch('P', M)
coord = CoordSystem('coord', P, ['x%s' % i for i in range(n)])
x = coord.coord_functions()
dx = coord.base_oneforms()
f = Function('f')
g = [[f(x[0], x[1])**2, 0], [0, f(x[0], x[1])**2]]
metric = sum(g[i][j] * TP(dx[i], dx[j]) for i in range(n)
for j in range(n))
R = metric_to_Riemann_components(metric)
d = Symbol('d')
assert (R[0, 1, 0, 1] == -Subs(Derivative(f(d, x[1]), d, d),
(d, x[0])) / f(x[0], x[1]) -
Subs(Derivative(f(x[0], d), d, d), (d, x[1])) / f(x[0], x[1]) +
Subs(Derivative(f(d, x[1]), d), (d, x[0]))**2 / f(x[0], x[1])**2 +
Subs(Derivative(f(x[0], d), d), (d, x[1]))**2 / f(x[0], x[1])**2)
def test_dont_cse_tuples():
f = Function("f")
g = Function("g")
name_val, (expr,) = cse(Subs(f(x, y), (x, 0), (y, 1)) +
Subs(g(x, y), (x, 0), (y, 1)))
assert name_val == []
assert expr == (Subs(f(x, y), (x, 0), (y, 1))
+ Subs(g(x, y), (x, 0), (y, 1)))
name_val, (expr,) = cse(Subs(f(x, y), (x, 0), (y, x + y)) +
Subs(g(x, y), (x, 0), (y, x + y)))
assert name_val == [(x0, x + y)]
assert expr == Subs(f(x, y), (x, 0), (y, x0)) + Subs(g(x, y), (x, 0), (y, x0))
def test_dont_cse_tuples():
from diofant import Subs
f = Function("f")
g = Function("g")
name_val, (expr, ) = cse(
Subs(f(x, y), (x, y), (0, 1)) + Subs(g(x, y), (x, y), (0, 1)))
assert name_val == []
assert expr == (Subs(f(x, y), (x, y),
(0, 1)) + Subs(g(x, y), (x, y), (0, 1)))
name_val, (expr, ) = cse(
Subs(f(x, y), (x, y), (0, x + y)) + Subs(g(x, y), (x, y), (0, x + y)))
assert name_val == [(x0, x + y)]
assert expr == Subs(f(x, y), (x, y), (0, x0)) + \
Subs(g(x, y), (x, y), (0, x0))
def test_sympyissue_3978():
assert f(x).series(x, 0, 3, dir=+1) == \
f(0) + x*Subs(Derivative(f(x), x), (x, 0)) + \
x**2*Subs(Derivative(f(x), x, x), (x, 0))/2 + O(x**3)
assert f(x).series(x, 0, 3) == \
f(0) + x*Subs(Derivative(f(x), x), (x, 0)) + \
x**2*Subs(Derivative(f(x), x, x), (x, 0))/2 + O(x**3)
assert f(x**2).series(x, 0, 3) == \
f(0) + x**2*Subs(Derivative(f(x), x), (x, 0)) + O(x**3)
assert f(x**2+1).series(x, 0, 3) == \
f(1) + x**2*Subs(Derivative(f(x), x), (x, 1)) + O(x**3)
class TestF(Function):
pass
assert TestF(x).series(x, 0, 3) == TestF(0) + \
x*Subs(Derivative(TestF(x), x), (x, 0)) + \
x**2*Subs(Derivative(TestF(x), x, x), (x, 0))/2 + O(x**3)
def test_issue_3978():
f = Function('f')
assert f(x).series(x, 0, 3, dir='-') == \
f(0) + x*Subs(Derivative(f(x), x), (x,), (0,)) + \
x**2*Subs(Derivative(f(x), x, x), (x,), (0,))/2 + O(x**3)
assert f(x).series(x, 0, 3) == \
f(0) + x*Subs(Derivative(f(x), x), (x,), (0,)) + \
x**2*Subs(Derivative(f(x), x, x), (x,), (0,))/2 + O(x**3)
assert f(x**2).series(x, 0, 3) == \
f(0) + x**2*Subs(Derivative(f(x), x), (x,), (0,)) + O(x**3)
assert f(x**2+1).series(x, 0, 3) == \
f(1) + x**2*Subs(Derivative(f(x), x), (x,), (1,)) + O(x**3)
class TestF(Function):
pass
assert TestF(x).series(x, 0, 3) == TestF(0) + \
x*Subs(Derivative(TestF(x), x), (x,), (0,)) + \
x**2*Subs(Derivative(TestF(x), x, x), (x,), (0,))/2 + O(x**3)
def test_underscores():
_0, _1 = symbols('_0 _1')
e = Subs(_0 + _1, (_0, 1), (_1, 0))
assert e._expr == Symbol('__0') + Symbol('__1')
def test_mul():
A, B, C = symbols('A B C', commutative=False)
assert (x * y * z).subs({z * x: y}) == y**2
assert (z * x).subs({1 / x: z}) == z * x
assert (x * y / z).subs({1 / z: a}) == a * x * y
assert (x * y / z).subs({x / z: a}) == a * y
assert (x * y / z).subs({y / z: a}) == a * x
assert (x * y / z).subs({x / z: 1 / a}) == y / a
assert (x * y / z).subs({x: 1 / a}) == y / (z * a)
assert (2 * x * y).subs({5 * x * y: z}) != 2 * z / 5
assert (x * y * A).subs({x * y: a}) == a * A
assert Subs(x * y * A, (x * y, a)).is_commutative is False
assert (x**2 * y**(3 * x / 2)).subs({x * y**(x / 2): 2}) == 4 * y**(x / 2)
assert (x * exp(x * 2)).subs({x * exp(x): 2}) == 2 * exp(x)
assert ((x**(2 * y))**3).subs({x**y: 2}) == 64
assert (x * A * B).subs({x * A: y}) == y * B
assert (x * y * (1 + x) * (1 + x * y)).subs({x * y: 2}) == 6 * (1 + x)
assert ((1 + A * B) * A * B).subs({A * B: x * A * B})
assert (x * a / z).subs({x / z: A}) == a * A
assert Subs(x * a / z, (x / z, A)).is_commutative is False
assert (x**3 * A).subs({x**2 * A: a}) == a * x
assert (x**2 * A * B).subs({x**2 * B: a}) == a * A
assert (x**2 * A * B).subs({x**2 * A: a}) == a * B
assert (b*A**3/(a**3*c**3)).subs({a**4*c**3*A**3/b**4: z}) == \
b*A**3/(a**3*c**3)
assert (6 * x).subs({2 * x: y}) == 3 * y
assert (y * exp(3 * x / 2)).subs({y * exp(x): 2}) == 2 * exp(x / 2)
assert (y * exp(3 * x / 2)).subs({y * exp(x): 2}) == 2 * exp(x / 2)
assert (A**2 * B * A**2 * B * A**2).subs({A * B * A: C}) == A * C**2 * A
assert (x * A**3).subs({x * A: y}) == y * A**2
assert (x**2 * A**3).subs({x * A: y}) == y**2 * A
assert (x * A**3).subs({x * A: B}) == B * A**2
assert (x * A * B * A * exp(x * A * B)).subs({x * A:
B}) == B**2 * A * exp(B * B)
assert (x**2 * A * B * A * exp(x * A * B)).subs({x * A:
B}) == B**3 * exp(B**2)
assert (x**3*A*exp(x*A*B)*A*exp(x*A*B)).subs({x*A: B}) == \
x*B*exp(B**2)*B*exp(B**2)
assert (x * A * B * C * A * B).subs({x * A * B: C}) == C**2 * A * B
assert (-I * a * b).subs({a * b: 2}) == -2 * I
# issue sympy/sympy#6361
assert (-8 * I * a).subs({-2 * a: 1}) == 4 * I
assert (-I * a).subs({-a: 1}) == I
# issue sympy/sympy#6441
assert (4 * x**2).subs({2 * x: y}) == y**2
assert (2 * 4 * x**2).subs({2 * x: y}) == 2 * y**2
assert (-x**3 / 9).subs({-x / 3: z}) == -z**2 * x
assert (-x**3 / 9).subs({x / 3: z}) == -z**2 * x
assert (-2 * x**3 / 9).subs({x / 3: z}) == -2 * x * z**2
assert (-2 * x**3 / 9).subs({-x / 3: z}) == -2 * x * z**2
assert (-2 * x**3 / 9).subs({-2 * x: z}) == z * x**2 / 9
assert (-2 * x**3 / 9).subs({2 * x: z}) == -z * x**2 / 9
assert (2 * (3 * x / 5 / 7)**2).subs({3 * x / 5:
z}) == 2 * Rational(1, 7)**2 * z**2
assert (4 * x).subs({-2 * x: z}) == 4 * x # try keep subs literal
assert (A**2 * B**2).subs({A * B**3: C}) == A**2 * B**2
assert (A**Rational(5, 3) * B**3).subs({sqrt(A) * B:
C}) == A**Rational(5, 3) * B**3
assert (A**2 * B**2 * A).subs({A**2 * B * A: C}) == A**2 * B**2 * A
def test_sympyissue_12005():
e1 = Subs(Derivative(f(x), x), (x, ), (x, ))
assert e1.diff(x) == Derivative(f(x), x, x)
e2 = Subs(Derivative(f(x), x), (x, ), (x**2 + 1, ))
assert e2.diff(x) == 2 * x * Subs(Derivative(f(x), x, x), (x, ),
(x**2 + 1, ))
e3 = Subs(Derivative(f(x) + y**2 - y, y), (y, ), (y**2, ))
assert e3.diff(y) == 4 * y
e4 = Subs(Derivative(f(x + y), y), (y, ), (x**2))
assert e4.diff(y) == S.Zero
e5 = Subs(Derivative(f(x), x), (y, z), (y, z))
assert e5.diff(x) == Derivative(f(x), x, x)
assert f(g(x)).diff(g(x), g(x)) == Subs(Derivative(Function('f')(y), y, y),
(y, ), (g(x), ))
def test_sympyissue_12005():
e1 = Subs(Derivative(f(x), x), (x, x))
assert e1.diff(x) == Derivative(f(x), x, x)
e2 = Subs(Derivative(f(x), x), (x, x**2 + 1))
assert e2.diff(x) == 2 * x * Subs(Derivative(f(x), x, x), (x, x**2 + 1))
e3 = Subs(Derivative(f(x) + y**2 - y, y), (y, y**2))
assert e3.diff(y) == 4 * y
e4 = Subs(Derivative(f(x + y), y), (y, x**2))
assert e4.diff(y) == 0
e5 = Subs(Derivative(f(x), x), (y, y), (z, z))
assert e5.diff(x) == Derivative(f(x), x, x)
assert f(g(x)).diff(g(x), g(x)) == Subs(Derivative(f(y), y, y), (y, g(x)))
def test_Subs():
assert Subs(x, (x, 0)) == Subs(y, (y, 0))
assert Subs(x, (x, 0)).subs({x: 1}) == Subs(x, (x, 0))
assert Subs(y, (x, 0)).subs({y: 1}) == Subs(1, (x, 0))
assert Subs(f(x), (x, 0)).doit() == f(0)
assert Subs(f(x**2), (x**2, 0)).doit() == f(0)
assert Subs(f(x, y, z), (x, 0), (y, 1), (z, 1)) != \
Subs(f(x, y, z), (x, 0), (y, 0), (z, 1))
assert Subs(f(x, y), (x, 0), (y, 1), (z, 1)) == \
Subs(f(x, y), (x, 0), (y, 1), (z, 2))
assert Subs(f(x, y), (x, 0), (y, 1), (z, 1)) != \
Subs(f(x, y) + z, (x, 0), (y, 1), (z, 0))
assert Subs(f(x, y), (x, 0), (y, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), (x, 0)), (y, 1)).doit() == f(0, 1)
pytest.raises(ValueError, lambda: Subs(f(x, y), x))
pytest.raises(ValueError, lambda: Subs(f(x, y), (x, 0), (x, 0), (y, 1)))
assert len(Subs(f(x, y), (x, 0), (y, 1)).variables) == 2
assert Subs(f(x, y), (x, 0), (y, 1)).point == Tuple(0, 1)
assert Subs(f(x), (x, 0)) == Subs(f(y), (y, 0))
assert Subs(f(x, y), (x, 0), (y, 1)) == Subs(f(x, y), (y, 1), (x, 0))
assert Subs(f(x) * y, (x, 0), (y, 1)) == Subs(f(y) * x, (y, 0), (x, 1))
assert Subs(f(x) * y, (x, 1), (y, 1)) == Subs(f(y) * x, (x, 1), (y, 1))
assert Subs(f(x), (x, 0)).subs({x: 1}).doit() == f(0)
assert Subs(f(x), (x, y)).subs({y: 0}) == Subs(f(x), (x, 0))
assert Subs(y * f(x), (x, y)).subs({y: 2}) == Subs(2 * f(x), (x, 2))
assert (2 * Subs(f(x), (x, 0))).subs({Subs(f(x), (x, 0)): y}) == 2 * y
assert Subs(f(x), (x, 0)).free_symbols == set()
assert Subs(f(x, y), (x, z)).free_symbols == {y, z}
assert Subs(f(x).diff(x), (x, 0)).doit(), Subs(f(x).diff(x), (x, 0))
assert Subs(1 + f(x).diff(x),
(x, 0)).doit(), 1 + Subs(f(x).diff(x), (x, 0))
assert Subs(y*f(x, y).diff(x), (x, 0), (y, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), (x, 0))
assert Subs(y**2 * f(x), (x, 0)).diff(y) == 2 * y * f(0)
e = Subs(y**2 * f(x), (x, y))
assert e.diff(y) == e.doit().diff(
y) == y**2 * Derivative(f(y), y) + 2 * y * f(y)
assert Subs(f(x), (x, 0)) + Subs(f(x), (x, 0)) == 2 * Subs(f(x), (x, 0))
e1 = Subs(z * f(x), (x, 1))
e2 = Subs(z * f(y), (y, 1))
assert e1 + e2 == 2 * e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z * f(x + 1), (x, 1)) not in (e1, e2)
assert Derivative(f(x), x).subs({x: g(x)}) == Derivative(f(g(x)), g(x))
assert Derivative(f(x), x).subs({x:
x + y}) == Subs(Derivative(f(x), x),
(x, x + y))
assert Subs(f(x)*cos(y) + z, (x, 0), (y, pi/3)).evalf(2, strict=False) == \
Subs(f(x)*cos(y) + z, (x, 0), (y, pi/3)).evalf(2, strict=False) == \
z + Rational('1/2').evalf(2)*f(0)
assert f(x).diff(x).subs({x: 0}).subs({x: y}) == f(x).diff(x).subs({x: 0})
assert (x * f(x).diff(x).subs({x: 0})).subs(
{x: y}) == y * f(x).diff(x).subs({x: 0})
def test_sign():
assert sign(1.2) == 1
assert sign(-1.2) == -1
assert sign(3 * I) == I
assert sign(-3 * I) == -I
assert sign(0) == 0
assert sign(nan) == nan
assert sign(2 + 2 * I).doit() == sqrt(2) * (2 + 2 * I) / 4
assert sign(2 + 3 * I).simplify() == sign(2 + 3 * I)
assert sign(2 + 2 * I).simplify() == sign(1 + I)
assert sign(im(sqrt(1 - sqrt(3)))) == 1
assert sign(sqrt(1 - sqrt(3))) == I
x = Symbol('x')
assert sign(x).is_finite is True
assert sign(x).is_complex is True
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_extended_real is None
assert sign(x).is_zero is None
assert sign(x).doit() == sign(x)
assert sign(1.2 * x) == sign(x)
assert sign(2 * x) == sign(x)
assert sign(I * x) == I * sign(x)
assert sign(-2 * I * x) == -I * sign(x)
assert sign(conjugate(x)) == conjugate(sign(x))
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
m = Symbol('m', negative=True)
assert sign(2 * p * x) == sign(x)
assert sign(n * x) == -sign(x)
assert sign(n * m * x) == sign(x)
x = Symbol('x', imaginary=True)
xn = Symbol('xn', imaginary=True, nonzero=True)
assert sign(x).is_imaginary is True
assert sign(x).is_integer is None
assert sign(x).is_extended_real is None
assert sign(x).is_zero is None
assert sign(x).diff(x) == 2 * DiracDelta(-I * x)
assert sign(xn).doit() == xn / abs(xn)
assert conjugate(sign(x)) == -sign(x)
x = Symbol('x', extended_real=True)
assert sign(x).is_imaginary is None
assert sign(x).is_integer is True
assert sign(x).is_extended_real is True
assert sign(x).is_zero is None
assert sign(x).diff(x) == 2 * DiracDelta(x)
assert sign(x).doit() == sign(x)
assert conjugate(sign(x)) == sign(x)
assert sign(sin(x)).series(x) == 1
y = Symbol('y')
assert sign(x * y).series(x).removeO() == sign(y)
assert sign(I + x).series(
x, n=2) == I + x * Subs(sign(x + I).diff(x), (x, 0)) + O(x**2)
x = Symbol('x', nonzero=True)
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_extended_real is None
assert sign(x).is_nonzero is True
assert sign(x).doit() == x / abs(x)
assert sign(abs(x)) == 1
assert abs(sign(x)) == 1
x = Symbol('x', positive=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_extended_real is True
assert sign(x).is_nonzero is True
assert sign(x).doit() == x / abs(x)
assert sign(abs(x)) == 1
assert abs(sign(x)) == 1
x = 0
assert sign(x).is_imaginary is True
assert sign(x).is_integer is True
assert sign(x).is_extended_real is True
assert sign(x).is_zero is True
assert sign(x).doit() == 0
assert sign(abs(x)) == 0
assert abs(sign(x)) == 0
nz = Symbol('nz', nonzero=True, integer=True)
assert sign(nz).is_imaginary is False
assert sign(nz).is_integer is True
assert sign(nz).is_extended_real is True
assert sign(nz).is_nonzero is True
assert sign(nz)**2 == 1
assert (sign(nz)**3).args == (sign(nz), 3)
assert sign(Symbol('x', nonnegative=True)).is_nonnegative
assert sign(Symbol('x', nonnegative=True)).is_nonpositive is None
assert sign(Symbol('x', nonpositive=True)).is_nonnegative is None
assert sign(Symbol('x', nonpositive=True)).is_nonpositive
assert sign(Symbol('x', extended_real=True)).is_nonnegative is None
assert sign(Symbol('x', extended_real=True)).is_nonpositive is None
assert sign(Symbol('x', extended_real=True,
zero=False)).is_nonpositive is None
x, y = Symbol('x', extended_real=True), Symbol('y')
assert sign(x).rewrite(Piecewise) == \
Piecewise((1, x > 0), (-1, x < 0), (0, True))
assert sign(y).rewrite(Piecewise) == sign(y)
assert sign(x).rewrite(Heaviside) == 2 * Heaviside(x) - 1
assert sign(y).rewrite(Heaviside) == sign(y)
assert sign(1 + I).rewrite(exp) == exp(I * pi / 4)
# evaluate what can be evaluated
assert sign(exp_polar(I * pi) * pi) is Integer(-1)
eq = -sqrt(10 + 6 * sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3 * sqrt(3))
# if there is a fast way to know when and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert sign(eq) == 0
q = 1 + sqrt(2) - 2 * sqrt(3) + 1331 * sqrt(6)
p = cbrt(expand(q**3))
d = p - q
assert sign(d) == 0
assert abs(sign(z)) == Abs(sign(z), evaluate=False)
def test_Subs2():
# this reflects a limitation of subs(), probably won't fix
assert Subs(f(x), (x**2, x)).doit() == f(sqrt(x))
def test_series_of_Subs():
subs1 = Subs(sin(x), (x, y))
subs2 = Subs(sin(x) * cos(z), (x, y))
subs3 = Subs(sin(x * z), (x, z), (y, x))
subs4 = Subs(x, (x, z))
res1 = Subs(x, (x, y)) + Subs(-x**3 / 6,
(x, y)) + Subs(x**5 / 120, (x, y)) + O(y**6)
res2 = Subs(z**4 * sin(x) / 24,
(x, y)) + Subs(-z**2 * sin(x) / 2,
(x, y)) + Subs(sin(x), (x, y)) + O(z**6)
res3 = Subs(x * z, (x, z), (y, x)) + O(z**6)
assert subs1.series(x) == subs1
assert subs1.series(y) == res1
assert subs1.series(z) == subs1
assert subs2.series(z) == res2
assert subs3.series(x) == subs3
assert subs3.series(z) == res3
assert subs4.series(z) == subs4
assert subs1.doit().series(x) == subs1.doit()
assert subs1.doit().series(y) == res1.doit()
assert subs1.doit().series(z) == subs1.doit()
assert subs2.doit().series(z) == res2.doit()
assert subs3.doit().series(x) == subs3.doit()
assert subs3.doit().series(z) == res3.doit()
assert subs4.doit().series(z) == subs4.doit()
def test_Subs():
assert Subs(x, x, 0) == Subs(y, y, 0)
assert Subs(x, x, 0).subs(x, 1) == Subs(x, x, 0)
assert Subs(y, x, 0).subs(y, 1) == Subs(1, x, 0)
assert Subs(f(x), x, 0).doit() == f(0)
assert Subs(f(x**2), x**2, 0).doit() == f(0)
assert Subs(f(x, y, z), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y, z), (x, y, z), (0, 0, 1))
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) == \
Subs(f(x, y), (x, y, z), (0, 1, 2))
assert Subs(f(x, y), (x, y, z), (0, 1, 1)) != \
Subs(f(x, y) + z, (x, y, z), (0, 1, 0))
assert Subs(f(x, y), (x, y), (0, 1)).doit() == f(0, 1)
assert Subs(Subs(f(x, y), x, 0), y, 1).doit() == f(0, 1)
pytest.raises(ValueError, lambda: Subs(f(x, y), (x, y), (0, 0, 1)))
pytest.raises(ValueError, lambda: Subs(f(x, y), (x, x, y), (0, 0, 1)))
assert len(Subs(f(x, y), (x, y), (0, 1)).variables) == 2
assert Subs(f(x, y), (x, y), (0, 1)).point == Tuple(0, 1)
assert Subs(f(x), x, 0) == Subs(f(y), y, 0)
assert Subs(f(x, y), (x, y), (0, 1)) == Subs(f(x, y), (y, x), (1, 0))
assert Subs(f(x) * y, (x, y), (0, 1)) == Subs(f(y) * x, (y, x), (0, 1))
assert Subs(f(x) * y, (x, y), (1, 1)) == Subs(f(y) * x, (x, y), (1, 1))
assert Subs(f(x), x, 0).subs(x, 1).doit() == f(0)
assert Subs(f(x), x, y).subs(y, 0) == Subs(f(x), x, 0)
assert Subs(y * f(x), x, y).subs(y, 2) == Subs(2 * f(x), x, 2)
assert (2 * Subs(f(x), x, 0)).subs(Subs(f(x), x, 0), y) == 2 * y
assert Subs(f(x), x, 0).free_symbols == set()
assert Subs(f(x, y), x, z).free_symbols == {y, z}
assert Subs(f(x).diff(x), x, 0).doit(), Subs(f(x).diff(x), x, 0)
assert Subs(1 + f(x).diff(x), x, 0).doit(), 1 + Subs(f(x).diff(x), x, 0)
assert Subs(y*f(x, y).diff(x), (x, y), (0, 2)).doit() == \
2*Subs(Derivative(f(x, 2), x), x, 0)
assert Subs(y**2 * f(x), x, 0).diff(y) == 2 * y * f(0)
e = Subs(y**2 * f(x), x, y)
assert e.diff(y) == e.doit().diff(
y) == y**2 * Derivative(f(y), y) + 2 * y * f(y)
assert Subs(f(x), x, 0) + Subs(f(x), x, 0) == 2 * Subs(f(x), x, 0)
e1 = Subs(z * f(x), x, 1)
e2 = Subs(z * f(y), y, 1)
assert e1 + e2 == 2 * e1
assert e1.__hash__() == e2.__hash__()
assert Subs(z * f(x + 1), x, 1) not in [e1, e2]
assert Derivative(f(x), x).subs(x, g(x)) == Subs(Derivative(f(x), x),
(x, ), (g(x), ))
assert Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).n(2) == \
Subs(f(x)*cos(y) + z, (x, y), (0, pi/3)).evalf(2) == \
z + Rational('1/2').n(2)*f(0)
assert f(x).diff(x).subs(x, 0).subs(x, y) == f(x).diff(x).subs(x, 0)
assert (x * f(x).diff(x).subs(x, 0)).subs(
x, y) == y * f(x).diff(x).subs(x, 0)
