def test_Tuple():
t = (1, 2, 3, 4)
st = Tuple(*t)
assert set(sympify(t)) == set(st)
assert len(t) == len(st)
assert set(sympify(t[:2])) == set(st[:2])
assert isinstance(st[:], Tuple)
assert st == Tuple(1, 2, 3, 4)
assert st.func(*st.args) == st
p, q, r, s = symbols('p q r s')
t2 = (p, q, r, s)
st2 = Tuple(*t2)
assert st2.atoms() == set(t2)
assert st == st2.subs({p: 1, q: 2, r: 3, s: 4})
# issue 5505
assert all(isinstance(arg, Basic) for arg in st.args)
assert Tuple(p, 1).subs(p, 0) == Tuple(0, 1)
assert Tuple(p, Tuple(p, 1)).subs(p, 0) == Tuple(0, Tuple(0, 1))
assert Tuple(t2) == Tuple(Tuple(*t2))
assert Tuple.fromiter(t2) == Tuple(*t2)
assert Tuple.fromiter(x for x in range(4)) == Tuple(0, 1, 2, 3)
assert st2.fromiter(st2.args) == st2
def __radd__(self, other):
return Tuple.__radd__(self, other + other)
def test_AlgebraicNumber():
minpoly, root = x**2 - 2, sqrt(2)
a = AlgebraicNumber(root, gen=x)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert a.coeffs() == [S(1), S(0)]
assert a.native_coeffs() == [QQ(1), QQ(0)]
a = AlgebraicNumber(root, gen=x, alias='y')
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is True
a = AlgebraicNumber(root, gen=x, alias=Symbol('y'))
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is True
assert AlgebraicNumber(sqrt(2), []).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), ()).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), (0, 0)).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), [8]).rep == DMP([QQ(8)], QQ)
assert AlgebraicNumber(sqrt(2), [S(8) / 3]).rep == DMP([QQ(8, 3)], QQ)
assert AlgebraicNumber(sqrt(2), [7, 3]).rep == DMP([QQ(7), QQ(3)], QQ)
assert AlgebraicNumber(sqrt(2), [S(7) / 9, S(3) / 2]).rep == DMP(
[QQ(7, 9), QQ(3, 2)], QQ)
assert AlgebraicNumber(sqrt(2), [1, 2, 3]).rep == DMP([QQ(2), QQ(5)], QQ)
a = AlgebraicNumber(AlgebraicNumber(root, gen=x), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert a.coeffs() == [S(1), S(2)]
assert a.native_coeffs() == [QQ(1), QQ(2)]
a = AlgebraicNumber((minpoly, root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
a = AlgebraicNumber((Poly(minpoly), root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert AlgebraicNumber(sqrt(3)).rep == DMP([QQ(1), QQ(0)], QQ)
assert AlgebraicNumber(-sqrt(3)).rep == DMP([QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(2))
assert a == b
c = AlgebraicNumber(sqrt(2), gen=x)
d = AlgebraicNumber(sqrt(2), gen=x)
assert a == b
assert a == c
a = AlgebraicNumber(sqrt(2), [1, 2])
b = AlgebraicNumber(sqrt(2), [1, 3])
assert a != b and a != sqrt(2) + 3
assert (a == x) is False and (a != x) is True
a = AlgebraicNumber(sqrt(2), [1, 0])
b = AlgebraicNumber(sqrt(2), [1, 0], alias=y)
assert a.as_poly(x) == Poly(x)
assert b.as_poly() == Poly(y)
assert a.as_expr() == sqrt(2)
assert a.as_expr(x) == x
assert b.as_expr() == sqrt(2)
assert b.as_expr(x) == x
a = AlgebraicNumber(sqrt(2), [2, 3])
b = AlgebraicNumber(sqrt(2), [2, 3], alias=y)
p = a.as_poly()
assert p == Poly(2 * p.gen + 3)
assert a.as_poly(x) == Poly(2 * x + 3)
assert b.as_poly() == Poly(2 * y + 3)
assert a.as_expr() == 2 * sqrt(2) + 3
assert a.as_expr(x) == 2 * x + 3
assert b.as_expr() == 2 * sqrt(2) + 3
assert b.as_expr(x) == 2 * x + 3
a = AlgebraicNumber(sqrt(2))
b = to_number_field(sqrt(2))
assert a.args == b.args == (sqrt(2), Tuple(1, 0))
b = AlgebraicNumber(sqrt(2), alias='alpha')
assert b.args == (sqrt(2), Tuple(1, 0), Symbol('alpha'))
a = AlgebraicNumber(sqrt(2), [1, 2, 3])
assert a.args == (sqrt(2), Tuple(1, 2, 3))
def given(expr, condition=None, **kwargs):
""" Conditional Random Expression
From a random expression and a condition on that expression creates a new
probability space from the condition and returns the same expression on that
conditional probability space.
Examples
========
>>> from sympy.stats import given, density, Die
>>> X = Die('X', 6)
>>> Y = given(X, X > 3)
>>> density(Y).dict
{4: 1/3, 5: 1/3, 6: 1/3}
Following convention, if the condition is a random symbol then that symbol
is considered fixed.
>>> from sympy.stats import Normal
>>> from sympy import pprint
>>> from sympy.abc import z
>>> X = Normal('X', 0, 1)
>>> Y = Normal('Y', 0, 1)
>>> pprint(density(X + Y, Y)(z), use_unicode=False)
2
-(-Y + z)
-----------
___ 2
\/ 2 *e
------------------
____
2*\/ pi
"""
if not random_symbols(condition) or pspace_independent(expr, condition):
return expr
if isinstance(condition, RandomSymbol):
condition = Eq(condition, condition.symbol)
condsymbols = random_symbols(condition)
if (isinstance(condition, Equality) and len(condsymbols) == 1
and not isinstance(pspace(expr).domain, ConditionalDomain)):
rv = tuple(condsymbols)[0]
results = solveset(condition, rv)
if isinstance(results, Intersection) and S.Reals in results.args:
results = list(results.args[1])
return sum(expr.subs(rv, res) for res in results)
# Get full probability space of both the expression and the condition
fullspace = pspace(Tuple(expr, condition))
# Build new space given the condition
space = fullspace.conditional_space(condition, **kwargs)
# Dictionary to swap out RandomSymbols in expr with new RandomSymbols
# That point to the new conditional space
swapdict = rs_swap(fullspace.values, space.values)
# Swap random variables in the expression
expr = expr.xreplace(swapdict)
return expr
def plot_implicit(expr, x_var=None, y_var=None, adaptive=True, depth=0,
points=300, line_color="blue", show=True, **kwargs):
"""A plot function to plot implicit equations / inequalities.
Arguments
=========
- ``expr`` : The equation / inequality that is to be plotted.
- ``x_var`` (optional) : symbol to plot on x-axis or tuple giving symbol
and range as ``(symbol, xmin, xmax)``
- ``y_var`` (optional) : symbol to plot on y-axis or tuple giving symbol
and range as ``(symbol, ymin, ymax)``
If neither ``x_var`` nor ``y_var`` are given then the free symbols in the
expression will be assigned in the order they are sorted.
The following keyword arguments can also be used:
- ``adaptive`` Boolean. The default value is set to True. It has to be
set to False if you want to use a mesh grid.
- ``depth`` integer. The depth of recursion for adaptive mesh grid.
Default value is 0. Takes value in the range (0, 4).
- ``points`` integer. The number of points if adaptive mesh grid is not
used. Default value is 300.
- ``show`` Boolean. Default value is True. If set to False, the plot will
not be shown. See ``Plot`` for further information.
- ``title`` string. The title for the plot.
- ``xlabel`` string. The label for the x-axis
- ``ylabel`` string. The label for the y-axis
Aesthetics options:
- ``line_color``: float or string. Specifies the color for the plot.
See ``Plot`` to see how to set color for the plots.
Default value is "Blue"
plot_implicit, by default, uses interval arithmetic to plot functions. If
the expression cannot be plotted using interval arithmetic, it defaults to
a generating a contour using a mesh grid of fixed number of points. By
setting adaptive to False, you can force plot_implicit to use the mesh
grid. The mesh grid method can be effective when adaptive plotting using
interval arithmetic, fails to plot with small line width.
Examples
========
Plot expressions:
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import plot_implicit, symbols, Eq, And
>>> x, y = symbols('x y')
Without any ranges for the symbols in the expression:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> p1 = plot_implicit(Eq(x**2 + y**2, 5))
With the range for the symbols:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> p2 = plot_implicit(
... Eq(x**2 + y**2, 3), (x, -3, 3), (y, -3, 3))
With depth of recursion as argument:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> p3 = plot_implicit(
... Eq(x**2 + y**2, 5), (x, -4, 4), (y, -4, 4), depth = 2)
Using mesh grid and not using adaptive meshing:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> p4 = plot_implicit(
... Eq(x**2 + y**2, 5), (x, -5, 5), (y, -2, 2),
... adaptive=False)
Using mesh grid without using adaptive meshing with number of points
specified:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> p5 = plot_implicit(
... Eq(x**2 + y**2, 5), (x, -5, 5), (y, -2, 2),
... adaptive=False, points=400)
Plotting regions:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> p6 = plot_implicit(y > x**2)
Plotting Using boolean conjunctions:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> p7 = plot_implicit(And(y > x, y > -x))
When plotting an expression with a single variable (y - 1, for example),
specify the x or the y variable explicitly:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> p8 = plot_implicit(y - 1, y_var=y)
>>> p9 = plot_implicit(x - 1, x_var=x)
"""
has_equality = False # Represents whether the expression contains an Equality,
#GreaterThan or LessThan
def arg_expand(bool_expr):
"""
Recursively expands the arguments of an Boolean Function
"""
for arg in bool_expr.args:
if isinstance(arg, BooleanFunction):
arg_expand(arg)
elif isinstance(arg, Relational):
arg_list.append(arg)
arg_list = []
if isinstance(expr, BooleanFunction):
arg_expand(expr)
#Check whether there is an equality in the expression provided.
if any(isinstance(e, (Equality, GreaterThan, LessThan))
for e in arg_list):
has_equality = True
elif not isinstance(expr, Relational):
expr = Eq(expr, 0)
has_equality = True
elif isinstance(expr, (Equality, GreaterThan, LessThan)):
has_equality = True
xyvar = [i for i in (x_var, y_var) if i is not None]
free_symbols = expr.free_symbols
range_symbols = Tuple(*flatten(xyvar)).free_symbols
undeclared = free_symbols - range_symbols
if len(free_symbols & range_symbols) > 2:
raise NotImplementedError("Implicit plotting is not implemented for "
"more than 2 variables")
#Create default ranges if the range is not provided.
default_range = Tuple(-5, 5)
def _range_tuple(s):
if isinstance(s, Symbol):
return Tuple(s) + default_range
if len(s) == 3:
return Tuple(*s)
raise ValueError('symbol or `(symbol, min, max)` expected but got %s' % s)
if len(xyvar) == 0:
xyvar = list(_sort_gens(free_symbols))
var_start_end_x = _range_tuple(xyvar[0])
x = var_start_end_x[0]
if len(xyvar) != 2:
if x in undeclared or not undeclared:
xyvar.append(Dummy('f(%s)' % x.name))
else:
xyvar.append(undeclared.pop())
var_start_end_y = _range_tuple(xyvar[1])
#Check whether the depth is greater than 4 or less than 0.
if depth > 4:
depth = 4
elif depth < 0:
depth = 0
series_argument = ImplicitSeries(expr, var_start_end_x, var_start_end_y,
has_equality, adaptive, depth,
points, line_color)
#set the x and y limits
kwargs['xlim'] = tuple(float(x) for x in var_start_end_x[1:])
kwargs['ylim'] = tuple(float(y) for y in var_start_end_y[1:])
# set the x and y labels
kwargs.setdefault('xlabel', var_start_end_x[0].name)
kwargs.setdefault('ylabel', var_start_end_y[0].name)
p = Plot(series_argument, **kwargs)
if show:
p.show()
return p
def eval_sum_symbolic(f, limits):
from sympy.functions import harmonic, bernoulli
f_orig = f
(i, a, b) = limits
if not f.has(i):
return f * (b - a + 1)
# Linearity
if f.is_Mul:
# Try factor out everything not including i
without_i, with_i = f.as_independent(i)
if without_i != 1:
s = eval_sum_symbolic(with_i, (i, a, b))
if s:
r = without_i * s
if r is not S.NaN:
return r
else:
# Try term by term
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR:
return L * sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL:
return sL * R
try:
f = apart(f, i) # see if it becomes an Add
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
# Try factor out everything not including i
without_i, with_i = f.as_independent(i)
if without_i != 0:
s = eval_sum_symbolic(with_i, (i, a, b))
if s:
r = without_i * (b - a + 1) + s
if r is not S.NaN:
return r
else:
# Try term by term
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
r = lsum + rsum
if r is not S.NaN:
return r
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
if (b is S.Infinity and not a is S.NegativeInfinity) or \
(a is S.NegativeInfinity and not b is S.Infinity):
return S.Infinity
return ((bernoulli(n + 1, b + 1) - bernoulli(n + 1, a)) /
(n + 1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return harmonic(b) - harmonic(a - 1)
else:
return harmonic(b, abs(n)) - harmonic(a - 1, abs(n))
if not (a.has(S.Infinity, S.NegativeInfinity)
or b.has(S.Infinity, S.NegativeInfinity)):
# Geometric terms
c1 = Wild('c1', exclude=[i])
c2 = Wild('c2', exclude=[i])
c3 = Wild('c3', exclude=[i])
wexp = Wild('wexp')
# Here we first attempt powsimp on f for easier matching with the
# exponential pattern, and attempt expansion on the exponent for easier
# matching with the linear pattern.
e = f.powsimp().match(c1**wexp)
if e is not None:
e_exp = e.pop(wexp).expand().match(c2 * i + c3)
if e_exp is not None:
e.update(e_exp)
if e is not None:
p = (c1**c3).subs(e)
q = (c1**c2).subs(e)
r = p * (q**a - q**(b + 1)) / (1 - q)
l = p * (b - a + 1)
return Piecewise((l, Eq(q, S.One)), (r, True))
r = gosper_sum(f, (i, a, b))
if isinstance(r, (Mul, Add)):
from sympy import ordered, Tuple
non_limit = r.free_symbols - Tuple(*limits[1:]).free_symbols
den = denom(together(r))
den_sym = non_limit & den.free_symbols
args = []
for v in ordered(den_sym):
try:
s = solve(den, v)
m = Eq(v, s[0]) if s else S.false
if m != False:
args.append((Sum(f_orig.subs(*m.args),
limits).doit(), m))
break
except NotImplementedError:
continue
args.append((r, True))
return Piecewise(*args)
if not r in (None, S.NaN):
return r
h = eval_sum_hyper(f_orig, (i, a, b))
if h is not None:
return h
factored = f_orig.factor()
if factored != f_orig:
return eval_sum_symbolic(factored, (i, a, b))
def plot_implicit(expr, *args, **kwargs):
"""A plot function to plot implicit equations / inequalities.
Arguments
=========
- ``expr`` : The equation / inequality that is to be plotted.
- ``(x, xmin, xmax)`` optional, 3-tuple denoting the range of symbol
``x``
- ``(y, ymin, ymax)`` optional, 3-tuple denoting the range of symbol
``y``
The following arguments can be passed as named parameters.
- ``adaptive``. Boolean. The default value is set to True. It has to be
set to False if you want to use a mesh grid.
- ``depth`` integer. The depth of recursion for adaptive mesh grid.
Default value is 0. Takes value in the range (0, 4).
- ``points`` integer. The number of points if adaptive mesh grid is not
used. Default value is 200.
- ``title`` string .The title for the plot.
- ``xlabel`` string. The label for the x - axis
- ``ylabel`` string. The label for the y - axis
plot_implicit, by default, uses interval arithmetic to plot functions. If
the expression cannot be plotted using interval arithmetic, it defaults to
a generating a contour using a mesh grid of fixed number of points. By
setting adaptive to False, you can force plot_implicit to use the mesh
grid. The mesh grid method can be effective when adaptive plotting using
interval arithmetic, fails to plot with small line width.
Examples:
=========
Plot expressions:
>>> from sympy import plot_implicit, cos, sin, symbols, Eq
>>> x, y = symbols('x y')
Without any ranges for the symbols in the expression
>>> p1 = plot_implicit(Eq(x**2 + y**2, 5)) #doctest: +SKIP
With the range for the symbols
>>> p2 = plot_implicit(Eq(x**2 + y**2, 3), (x, -3, 3), (y, -3, 3)) #doctest: +SKIP
With depth of recursion as argument.
>>> p3 = plot_implicit(Eq(x**2 + y**2, 5), (x, -4, 4), (y, -4, 4), depth = 2) #doctest: +SKIP
Using mesh grid and not using adaptive meshing.
>>> p4 = plot_implicit(Eq(x**2 + y**2, 5), (x, -5, 5), (y, -2, 2), adaptive=False) #doctest: +SKIP
Using mesh grid with number of points as input.
>>> p5 = plot_implicit(Eq(x**2 + y**2, 5), (x, -5, 5), (y, -2, 2), adaptive=False, points=400) #doctest: +SKIP
Plotting regions.
>>> p6 = plot_implicit(y > x**2) #doctest: +SKIP
Plotting Using boolean conjunctions.
>>> p7 = plot_implicit(And(y > x, y > -x)) #doctest: +SKIP
"""
assert isinstance(expr, Expr)
has_equality = False #Represents whether the expression contains an Equality,
#GreaterThan or LessThan
def arg_expand(bool_expr):
"""
Recursively expands the arguments of an Boolean Function
"""
for arg in bool_expr.args:
if isinstance(arg, BooleanFunction):
arg_expand(arg)
elif isinstance(arg, Relational):
arg_list.append(arg)
arg_list = []
if isinstance(expr, BooleanFunction):
arg_expand(expr)
#Check whether there is an equality in the expression provided.
if any(isinstance(e, (Equality, GreaterThan, LessThan))
for e in arg_list):
has_equality = True
elif not isinstance(expr, Relational):
expr = Eq(expr, 0)
has_equality = True
elif isinstance(expr, (Equality, GreaterThan, LessThan)):
has_equality = True
free_symbols = set(expr.free_symbols)
range_symbols = set([t[0] for t in args])
symbols = set_union(free_symbols, range_symbols)
if len(symbols) > 2:
raise NotImplementedError("Implicit plotting is not implemented for "
"more than 2 variables")
#Create default ranges if the range is not provided.
default_range = Tuple(-5, 5)
if len(args) == 2:
var_start_end_x = args[0]
var_start_end_y = args[1]
elif len(args) == 1:
if len(free_symbols) == 2:
var_start_end_x = args[0]
var_start_end_y, = (Tuple(e) + default_range
for e in (free_symbols - range_symbols))
else:
var_start_end_x, = (Tuple(e) + default_range for e in free_symbols)
#Create a random symbol
var_start_end_y = Tuple(Dummy()) + default_range
elif len(args) == 0:
if len(free_symbols) == 1:
var_start_end_x, = (Tuple(e) + default_range for e in free_symbols)
#create a random symbol
var_start_end_y = Tuple(Dummy()) + default_range
else:
var_start_end_x, var_start_end_y = (Tuple(e) + default_range
for e in free_symbols)
use_interval = kwargs.pop('adaptive', True)
nb_of_points = kwargs.pop('points', 300)
depth = kwargs.pop('depth', 0)
#Check whether the depth is greater than 4 or less than 0.
if depth > 4:
depth = 4
elif depth < 0:
depth = 0
series_argument = ImplicitSeries(expr, var_start_end_x, var_start_end_y,
has_equality, use_interval, depth,
nb_of_points)
show = kwargs.pop('show', True)
#set the x and y limits
kwargs['xlim'] = tuple(float(x) for x in var_start_end_x[1:])
kwargs['ylim'] = tuple(float(y) for y in var_start_end_y[1:])
p = Plot(series_argument, **kwargs)
if show:
p.show()
return p
def test_divmod():
assert divmod(S(12), S(8)) == Tuple(1, 4)
assert divmod(-S(12), S(8)) == Tuple(-2, 4)
assert divmod(S(0), S(1)) == Tuple(0, 0)
raises(ZeroDivisionError, lambda: divmod(S(0), S(0)))
raises(ZeroDivisionError, lambda: divmod(S(1), S(0)))
assert divmod(S(12), 8) == Tuple(1, 4)
assert divmod(12, S(8)) == Tuple(1, 4)
assert divmod(S("2"), S("3/2")) == Tuple(S("1"), S("1/2"))
assert divmod(S("3/2"), S("2")) == Tuple(S("0"), S("3/2"))
assert divmod(S("2"), S("3.5")) == Tuple(S("0"), S("2"))
assert divmod(S("3.5"), S("2")) == Tuple(S("1"), S("1.5"))
assert divmod(S("2"), S("1/3")) == Tuple(S("6"), S("0"))
assert divmod(S("1/3"), S("2")) == Tuple(S("0"), S("1/3"))
assert divmod(S("2"), S("0.1")) == Tuple(S("20"), S("0"))
assert divmod(S("0.1"), S("2")) == Tuple(S("0"), S("0.1"))
assert divmod(S("2"), 2) == Tuple(S("1"), S("0"))
assert divmod(2, S("2")) == Tuple(S("1"), S("0"))
assert divmod(S("2"), 1.5) == Tuple(S("1"), S("0.5"))
assert divmod(1.5, S("2")) == Tuple(S("0"), S("1.5"))
assert divmod(0.3, S("2")) == Tuple(S("0"), S("0.3"))
assert divmod(S("3/2"), S("3.5")) == Tuple(S("0"), S("3/2"))
assert divmod(S("3.5"), S("3/2")) == Tuple(S("2"), S("0.5"))
assert divmod(S("3/2"), S("1/3")) == Tuple(S("4"), Float("1/6"))
assert divmod(S("1/3"), S("3/2")) == Tuple(S("0"), S("1/3"))
assert divmod(S("3/2"), S("0.1")) == Tuple(S("15"), S("0"))
assert divmod(S("0.1"), S("3/2")) == Tuple(S("0"), S("0.1"))
assert divmod(S("3/2"), 2) == Tuple(S("0"), S("3/2"))
assert divmod(2, S("3/2")) == Tuple(S("1"), S("0.5"))
assert divmod(S("3/2"), 1.5) == Tuple(S("1"), S("0"))
assert divmod(1.5, S("3/2")) == Tuple(S("1"), S("0"))
assert divmod(S("3/2"), 0.3) == Tuple(S("5"), S("0"))
assert divmod(0.3, S("3/2")) == Tuple(S("0"), S("0.3"))
assert divmod(S("1/3"), S("3.5")) == Tuple(S("0"), S("1/3"))
assert divmod(S("3.5"), S("0.1")) == Tuple(S("35"), S("0"))
assert divmod(S("0.1"), S("3.5")) == Tuple(S("0"), S("0.1"))
assert divmod(S("3.5"), 2) == Tuple(S("1"), S("1.5"))
assert divmod(2, S("3.5")) == Tuple(S("0"), S("2"))
assert divmod(S("3.5"), 1.5) == Tuple(S("2"), S("0.5"))
assert divmod(1.5, S("3.5")) == Tuple(S("0"), S("1.5"))
assert divmod(0.3, S("3.5")) == Tuple(S("0"), S("0.3"))
assert divmod(S("0.1"), S("1/3")) == Tuple(S("0"), S("0.1"))
assert divmod(S("1/3"), 2) == Tuple(S("0"), S("1/3"))
assert divmod(2, S("1/3")) == Tuple(S("6"), S("0"))
assert divmod(S("1/3"), 1.5) == Tuple(S("0"), S("1/3"))
assert divmod(0.3, S("1/3")) == Tuple(S("0"), S("0.3"))
assert divmod(S("0.1"), 2) == Tuple(S("0"), S("0.1"))
assert divmod(2, S("0.1")) == Tuple(S("20"), S("0"))
assert divmod(S("0.1"), 1.5) == Tuple(S("0"), S("0.1"))
assert divmod(1.5, S("0.1")) == Tuple(S("15"), S("0"))
assert divmod(S("0.1"), 0.3) == Tuple(S("0"), S("0.1"))
assert str(divmod(S("2"), 0.3)) == '(6, 0.2)'
assert str(divmod(S("3.5"), S("1/3"))) == '(10, 0.166666666666667)'
assert str(divmod(S("3.5"), 0.3)) == '(11, 0.2)'
assert str(divmod(S("1/3"), S("0.1"))) == '(3, 0.0333333333333333)'
assert str(divmod(1.5, S("1/3"))) == '(4, 0.166666666666667)'
assert str(divmod(S("1/3"), 0.3)) == '(1, 0.0333333333333333)'
assert str(divmod(0.3, S("0.1"))) == '(2, 0.1)'
assert divmod(-3, S(2)) == (-2, 1)
assert divmod(S(-3), S(2)) == (-2, 1)
assert divmod(S(-3), 2) == (-2, 1)
def linear_factors(expr, *syms):
"""Reduce a Matrix Expression to a sum of linear factors
Given symbols and a matrix expression linear in those symbols return a
dict mapping symbol to the linear factor
>>> from sympy import MatrixSymbol, linear_factors, symbols
>>> n, m, l = symbols('n m l')
>>> A = MatrixSymbol('A', n, m)
>>> B = MatrixSymbol('B', m, l)
>>> C = MatrixSymbol('C', n, l)
>>> linear_factors(2*A*B + C, B, C)
{B: 2*A, C: I}
"""
expr = matrixify(expand(expr))
d = {}
if expr.is_Matrix and expr.is_Symbol:
if expr in syms:
d[expr] = Identity(expr.rows)
if expr.is_Add:
for sym in syms:
total_factor = 0
for arg in expr.args:
factor = arg.coeff(sym)
if not factor:
# .coeff fails when powers are in the expression
if sym in arg.free_symbols:
raise ValueError("Expression not linear in symbols")
else:
factor = 0
factor = sympify(factor)
if not factor.is_Matrix:
if factor.is_zero:
factor = ZeroMatrix(expr.rows, sym.rows)
if not sym.cols == expr.cols:
raise ShapeError(
"%s not compatible as factor of %s"%(sym, expr))
else:
factor = Identity(sym.rows)*factor
total_factor += factor
d[sym] = total_factor
elif expr.is_Mul:
for sym in syms:
factor = expr.coeff(sym)
if not factor:
# .coeff fails when powers are in the expression
if sym in expr.free_symbols:
raise ValueError("Expression not linear in symbols")
else:
factor = 0
factor = sympify(factor)
if not factor.is_Matrix:
if factor.is_zero:
factor = ZeroMatrix(expr.rows, sym.rows)
if not sym.cols == expr.cols:
raise ShapeError("%s not compatible as factor of %s"%
(sym, expr))
else:
factor = Identity(sym.rows)*factor
d[sym] = factor
if any(sym in matrix_symbols(Tuple(*d.values())) for sym in syms):
raise ValueError("Expression not linear in symbols")
return d
def _eval_subs(self, old, new):
# only do substitutions in shape
shape = Tuple(*self.shape)._subs(old, new)
return MatrixSymbol(self.name, *shape)
def add_variables_and_ranges(plot):
"""make sure all limits are in the form (symbol, a, b)"""
# find where the limits begin and expressions end
for i in range(len(plot)):
if isinstance(plot[i], Tuple):
break
else:
i = len(plot) + 1
exprs = list(plot[:i])
assert all(isinstance(e, Expr) for e in exprs)
assert all(isinstance(t, Tuple) for t in plot[i:])
ranges = set([i for i in plot[i:] if isinstance(i, Tuple) and len(i) > 1])
range_variables = set([t[0] for t in ranges if len(t) == 3])
expr_free = set_union(*[e.free_symbols for e in exprs if isinstance(e, Expr)])
default_range = Tuple(-10, 10)
# unambiguous cases for limits
# no ranges
if not ranges:
plot = exprs + [Tuple(e) + default_range for e in expr_free or [Dummy()]]
# all limits of length 3
elif all(len(i) == 3 for i in ranges):
pass
# all ranges the same
elif len(ranges) == 1:
range1 = ranges.pop()
if len(range1) == 2:
plot = exprs + [Tuple(x) + range1 for x in expr_free]
# ranges cover free variables of expression
elif expr_free < range_variables:
plot = exprs + [i if len(i) == 3 else Tuple(Dummy()) + i for i in ranges]
# ranges cover all but 1 free variable
elif len(expr_free - range_variables) == 1:
x = (expr_free - range_variables).pop()
ranges = list(ranges)
for i, ri in enumerate(ranges):
if len(ri) == 2:
ranges[i] = Tuple(x) + ri
break
else:
ranges.append(Tuple(x) + default_range)
for i, ri in enumerate(ranges):
if len(ri) == 2:
ranges[i] = Tuple(Dummy()) + ri
plot = exprs + ranges
# all implicit ranges
elif all(len(i) == 2 for i in ranges):
more = len(ranges) - len(expr_free)
all_free = list(expr_free) + [Dummy() for i in range(more)]
ranges = list(ranges)
ranges.extend(-more*[default_range])
plot = exprs + [Tuple(all_free[i]) + ri for i, ri in enumerate(ranges)]
else:
raise ValueError('erroneous or unanticipated range input')
return plot
def plot(*args, **kwargs):
"""A plot function for interactive use.
It implements many heuristics in order to guess what the user wants on
incomplete input.
There is also the 'show' argument that defaults to True (immediately
showing the plot).
The input arguments can be:
- lists with coordinates for ploting a line in 2D or 3D
- the expressions and variable lists with ranges in order to plot any of
the following: 2d line, 2d parametric line, 3d parametric line,
surface, parametric surface
- if the variable lists do not provide ranges a default range is used
- if the variables are not provided, the free variables are
automatically supplied in the order they are sorted (e.g. x, y, z)
- if neither variables nor ranges are provided, both are guessed
- if multiple expressions are provided in a list all of them are
plotted
- an instance of BaseSeries() subclass
- another Plot() instance
- tuples containing any of the above mentioned options, for plotting them
together
In the case of 2D line and parametric plots, an adaptive sampling algorithm
is used. The adaptive sampling algorithm recursively selects a random point
in between two previously sampled points, which might lead to slightly
different plots being rendered each time.
Examples:
---------
Plot expressions:
>>> from sympy import plot, cos, sin, symbols
>>> x,y,u,v = symbols('x y u v')
>>> p1 = plot(x**2, show=False) # with default [-10,+10] range
>>> p2 = plot(x**2, (0, 5), show=False) # it finds the free variable itself
>>> p3 = plot(x**2, (x, 0, 5), show=False) # fully explicit
Fully implicit examples (finding the free variable and using default
range). For the explicit versions just add the tuples with ranges:
>>> p4 = plot(x**2, show=False) # cartesian line
>>> p5 = plot(cos(u), sin(u), show=False) # parametric line
>>> p6 = plot(cos(u), sin(u), u, show=False) # parametric line in 3d
>>> p7 = plot(x**2 + y**2, show=False) # cartesian surface
>>> p8 = plot(u, v, u+v, show=False) # parametric surface
Multiple plots per figure:
>>> p9 = plot((x**2, ), (cos(u), sin(u)), show=False) # cartesian and parametric lines
Set title or other options:
>>> p10 = plot(x**2, title='second order polynomial', show=False)
Plot a list of expressions:
>>> p11 = plot([x, x**2, x**3], show=False)
>>> p12 = plot([x, x**2, x**3], (0,2), show=False) # explicit range
>>> p13 = plot([x*y, -x*y], show=False) # list of surfaces
And you can even plot a Plot or a Series object:
>>> a = plot(x, show=False)
>>> p14 = plot(a, show=False) # plotting a plot object
>>> p15 = plot(a[0], show=False) # plotting a series object
"""
plot_arguments = [p for p in args if isinstance(p, Plot)]
series_arguments = [p for p in args if isinstance(p, BaseSeries)]
args = sympify([np for np in args if not isinstance(np, (Plot, BaseSeries))])
# Are the arguments for only one plot or are they tuples with arguments
# for many plots. If it is the latter make the Tuples into list so they are
# mutable.
# args = (x, (x, 10, 20)) vs args = ((x, (x, 10, 20)), (y, (y, 20, 30)))
if all([isinstance(a, Tuple) for a in args]):
list_of_plots = map(list, args)
else:
list_of_plots = [list(args), ]
# Check for arguments containing lists of expressions for the same ranges.
# args = (x**2, (x, 10, 20)) vs args = ([x**3, x**2], (x, 10, 20))
list_arguments = [p for p in list_of_plots
if any(isinstance(a, list) for a in p)]
list_of_plots = [p for p in list_of_plots
if not any(isinstance(a, list) for a in p)]
def expand(plot):
"""
>> expand(([1,2,3],(1,2),(4,5)))
[[1, (1, 2), (4, 5)], [2, (1, 2), (4, 5)], [3, (1, 2), (4, 5)]]
"""
lists = [i for i in plot if isinstance(i, list)]
not_lists = [i for i in plot if not isinstance(i, list)]
return [list(l) + not_lists for l in zip(*lists)]
for a in list_arguments:
list_of_plots.extend(expand(a))
def add_variables_and_ranges(plot):
"""make sure all limits are in the form (symbol, a, b)"""
# find where the limits begin and expressions end
for i in range(len(plot)):
if isinstance(plot[i], Tuple):
break
else:
i = len(plot) + 1
exprs = list(plot[:i])
assert all(isinstance(e, Expr) for e in exprs)
assert all(isinstance(t, Tuple) for t in plot[i:])
ranges = set([i for i in plot[i:] if isinstance(i, Tuple) and len(i) > 1])
range_variables = set([t[0] for t in ranges if len(t) == 3])
expr_free = set_union(*[e.free_symbols for e in exprs if isinstance(e, Expr)])
default_range = Tuple(-10, 10)
# unambiguous cases for limits
# no ranges
if not ranges:
plot = exprs + [Tuple(e) + default_range for e in expr_free or [Dummy()]]
# all limits of length 3
elif all(len(i) == 3 for i in ranges):
pass
# all ranges the same
elif len(ranges) == 1:
range1 = ranges.pop()
if len(range1) == 2:
plot = exprs + [Tuple(x) + range1 for x in expr_free]
# ranges cover free variables of expression
elif expr_free < range_variables:
plot = exprs + [i if len(i) == 3 else Tuple(Dummy()) + i for i in ranges]
# ranges cover all but 1 free variable
elif len(expr_free - range_variables) == 1:
x = (expr_free - range_variables).pop()
ranges = list(ranges)
for i, ri in enumerate(ranges):
if len(ri) == 2:
ranges[i] = Tuple(x) + ri
break
else:
ranges.append(Tuple(x) + default_range)
for i, ri in enumerate(ranges):
if len(ri) == 2:
ranges[i] = Tuple(Dummy()) + ri
plot = exprs + ranges
# all implicit ranges
elif all(len(i) == 2 for i in ranges):
more = len(ranges) - len(expr_free)
all_free = list(expr_free) + [Dummy() for i in range(more)]
ranges = list(ranges)
ranges.extend(-more*[default_range])
plot = exprs + [Tuple(all_free[i]) + ri for i, ri in enumerate(ranges)]
else:
raise ValueError('erroneous or unanticipated range input')
return plot
list_of_plots = [Tuple(*add_variables_and_ranges(pl)) for pl in list_of_plots]
series_arguments.extend([OverloadedSeriesFactory(*pl) for pl in list_of_plots])
show = kwargs.pop('show', True)
p = Plot(*series_arguments, **kwargs)
for plot_argument in plot_arguments:
p.extend(plot_argument)
if show:
p.show()
return p
def eval(cls, expr):
types = (VectorFunction, ScalarFunction, DifferentialOperator)
if isinstance(expr, _logical_partial_derivatives):
atom = get_atom_logical_derivatives(expr)
indices = get_index_logical_derivatives(expr)
if cls in _logical_partial_derivatives:
indices[cls.coordinate] += 1
for i,n in enumerate(list(indices.values())[::-1]):
d = _logical_partial_derivatives[-i-1]
for _ in range(n):
atom = d(atom, evaluate=False)
if cls in _partial_derivatives:
atom = cls(atom, evaluate=False)
elif cls not in _logical_partial_derivatives:
raise NotImplementedError('TODO')
return atom
if isinstance(expr, VectorFunction):
n = expr.shape[0]
args = [cls(expr[i], evaluate=False) for i in range(0, n)]
args = Tuple(*args)
return Matrix([args])
elif isinstance(expr, (list, tuple, Tuple)):
args = [cls(i, evaluate=True) for i in expr]
args = Tuple(*args)
return Matrix([args])
elif isinstance(expr, (Matrix, ImmutableDenseMatrix)):
newexpr = Matrix.zeros(*expr.shape)
for i in range(expr.shape[0]):
for j in range(expr.shape[1]):
newexpr[i,j] = cls(expr[i,j], evaluate=True)
return type(expr)(newexpr)
elif isinstance(expr, (IndexedVectorFunction, DifferentialOperator)):
return cls(expr, evaluate=False)
elif isinstance(expr, ScalarFunction):
return cls(expr, evaluate=False)
elif isinstance(expr, (minus, plus)):
return cls(expr, evaluate=False)
elif isinstance(expr, Indexed) and isinstance(expr.base, BasicMapping):
return cls(expr, evaluate=False)
elif not has(expr, types):
if expr.is_number:
return S.Zero
elif isinstance(expr, Expr):
x = Symbol(cls.coordinate)
if cls.logical:
M = expr.atoms(Mapping)
if len(M)>0:
M = list(M)[0]
expr_primes = [diff(expr, M[i]) for i in range(M.pdim)]
Jj = Jacobian(M)[:,cls.grad_index]
expr_prime = sum([ei*Jji for ei,Jji in zip(expr_primes, Jj)])
return expr_prime + diff(expr, x)
return diff(expr, x)
if isinstance(expr, Add):
args = [cls(a, evaluate=True) for a in expr.args]
v = Add(*args)
return v
elif isinstance(expr, Mul):
coeffs = [a for a in expr.args if isinstance(a, _coeffs_registery)]
vectors = [a for a in expr.args if not(a in coeffs)]
c = S.One
if coeffs:
c = Mul(*coeffs)
V = S.Zero
if vectors:
if len(vectors) == 1:
# do we need to use Mul?
V = cls(vectors[0], evaluate=True)
elif len(vectors) == 2:
a = vectors[0]
b = vectors[1]
fa = cls(a, evaluate=True)
fb = cls(b, evaluate=True)
V = a * fb + fa * b
else:
a = vectors[0]
b = Mul(*vectors[1:])
fa = cls(a, evaluate=True)
fb = cls(b, evaluate=True)
V = a * fb + fa * b
v = Mul(c, V)
return v
elif isinstance(expr, Pow):
b = expr.base
e = expr.exp
v = (log(b)*cls(e, evaluate=True) + e*cls(b, evaluate=True)/b) * b**e
return v
else:
msg = '{expr} of type {type}'.format(expr=expr, type=type(expr))
raise NotImplementedError(msg)
def test_Tuple_tuple_count():
assert Tuple(0, 1, 2, 3).tuple_count(4) == 0
assert Tuple(0, 4, 1, 2, 3).tuple_count(4) == 1
assert Tuple(0, 4, 1, 4, 2, 3).tuple_count(4) == 2
assert Tuple(0, 4, 1, 4, 2, 4, 3).tuple_count(4) == 3
def test_Tuple_comparision():
assert (Tuple(1, 3) >= Tuple(-10, 30)) is True
assert (Tuple(1, 3) <= Tuple(-10, 30)) is False
assert (Tuple(1, 3) >= Tuple(1, 3)) is True
assert (Tuple(1, 3) <= Tuple(1, 3)) is True
def test_Tuple():
t = (1, 2, 3, 4)
st = Tuple(*t)
assert set(sympify(t)) == set(st)
assert len(t) == len(st)
assert set(sympify(t[:2])) == set(st[:2])
assert isinstance(st[:], Tuple)
assert st == Tuple(1, 2, 3, 4)
assert st.func(*st.args) == st
p, q, r, s = symbols('p q r s')
t2 = (p, q, r, s)
st2 = Tuple(*t2)
assert st2.atoms() == set(t2)
assert st == st2.subs({p: 1, q: 2, r: 3, s: 4})
# issue 2406
assert all(isinstance(arg, Basic) for arg in st.args)
assert Tuple(p, 1).subs(p, 0) == Tuple(0, 1)
assert Tuple(p, Tuple(p, 1)).subs(p, 0) == Tuple(0, Tuple(0, 1))
assert Tuple(t2) == Tuple(Tuple(*t2))
assert Tuple.fromiter(t2) == Tuple(*t2)
assert Tuple.fromiter(x for x in range(4)) == Tuple(0, 1, 2, 3)
assert st2.fromiter(st2.args) == st2
def _sort_contraction_indices(pairing_indices):
pairing_indices = [Tuple(*sorted(i)) for i in pairing_indices]
pairing_indices.sort(key=lambda x: min(x))
return pairing_indices
def _construct_template_types(args):
return Tuple(*args)
def test_as_Basic():
assert as_Basic(1) is S.One
assert as_Basic(()) == Tuple()
raises(TypeError, lambda: as_Basic([]))
def as_static_function(func, name=None):
assert(isinstance(func, FunctionDef))
args = list(func.arguments)
results = list(func.results)
body = func.body
arguments_inout = func.arguments_inout
functions = func.functions
_results = []
interfaces = func.interfaces
# Convert array results to inout arguments
for r in results:
if r.rank > 0 and r not in args:
args += [r]
arguments_inout += [False]
elif r.rank == 0:
_results += [r]
if name is None:
name = 'bind_c_{}'.format(func.name).lower()
# ...
results_names = [i.name for i in _results]
_args = []
_arguments_inout = []
for i_a, a in enumerate(args):
if not isinstance(a, (Variable, FunctionAddress)):
raise TypeError('Expecting a Variable or FunctionAddress type for {}'.format(a))
if not isinstance(a, FunctionAddress) and a.rank > 0:
# ...
additional_args = []
for i in range(a.rank):
n_name = 'n{i}_{name}'.format(name=str(a.name), i=i)
n_arg = Variable('int', n_name, precision=4)
additional_args += [n_arg]
shape_new = Tuple(*additional_args, sympify=False)
# ...
_args += additional_args
_arguments_inout += [False] * len(additional_args)
a_new = Variable( a.dtype, a.name,
allocatable = a.allocatable,
is_pointer = a.is_pointer,
is_target = a.is_target,
is_optional = a.is_optional,
shape = shape_new,
rank = a.rank,
order = a.order,
precision = a.precision)
if not( a.name in results_names ):
_args += [a_new]
else:
_results += [a_new]
else:
_args += [a]
intent = arguments_inout[i_a]
_arguments_inout += [intent]
args = _args
results = _results
arguments_inout = _arguments_inout
# ...
return BindCFunctionDef( name, list(args), results, body,
local_vars = func.local_vars,
is_static = True,
arguments_inout = arguments_inout,
functions = functions,
interfaces = interfaces,
imports = func.imports,
original_function = func,
doc_string = func.doc_string,
)
def test_issue_3982():
a = [3, 2.0]
assert sympify(a) == [Integer(3), Float(2.0)]
assert sympify(tuple(a)) == Tuple(Integer(3), Float(2.0))
assert sympify(set(a)) == set([Integer(3), Float(2.0)])
def __new__(cls, name, transformation=None, parent=None, location=None,
rotation_matrix=None, vector_names=None, variable_names=None):
"""
The orientation/location parameters are necessary if this system
is being defined at a certain orientation or location wrt another.
Parameters
==========
name : str
The name of the new CoordSys3D instance.
transformation : Lambda, Tuple, str
Transformation defined by transformation equations or choosen
from predefined ones.
location : Vector
The position vector of the new system's origin wrt the parent
instance.
rotation_matrix : SymPy ImmutableMatrix
The rotation matrix of the new coordinate system with respect
to the parent. In other words, the output of
new_system.rotation_matrix(parent).
parent : CoordSys3D
The coordinate system wrt which the orientation/location
(or both) is being defined.
vector_names, variable_names : iterable(optional)
Iterables of 3 strings each, with custom names for base
vectors and base scalars of the new system respectively.
Used for simple str printing.
"""
name = str(name)
Vector = sympy.vector.Vector
BaseVector = sympy.vector.BaseVector
Point = sympy.vector.Point
if not isinstance(name, string_types):
raise TypeError("name should be a string")
if transformation is not None:
if (location is not None) or (rotation_matrix is not None):
raise ValueError("specify either `transformation` or "
"`location`/`rotation_matrix`")
if isinstance(transformation, (Tuple, tuple, list)):
if isinstance(transformation[0], MatrixBase):
rotation_matrix = transformation[0]
location = transformation[1]
else:
transformation = Lambda(transformation[0],
transformation[1])
elif isinstance(transformation, collections.Callable):
x1, x2, x3 = symbols('x1 x2 x3', cls=Dummy)
transformation = Lambda((x1, x2, x3),
transformation(x1, x2, x3))
elif isinstance(transformation, string_types):
transformation = Symbol(transformation)
elif isinstance(transformation, (Symbol, Lambda)):
pass
else:
raise TypeError("transformation: "
"wrong type {0}".format(type(transformation)))
# If orientation information has been provided, store
# the rotation matrix accordingly
if rotation_matrix is None:
rotation_matrix = ImmutableDenseMatrix(eye(3))
else:
if not isinstance(rotation_matrix, MatrixBase):
raise TypeError("rotation_matrix should be an Immutable" +
"Matrix instance")
rotation_matrix = rotation_matrix.as_immutable()
# If location information is not given, adjust the default
# location as Vector.zero
if parent is not None:
if not isinstance(parent, CoordSys3D):
raise TypeError("parent should be a " +
"CoordSys3D/None")
if location is None:
location = Vector.zero
else:
if not isinstance(location, Vector):
raise TypeError("location should be a Vector")
# Check that location does not contain base
# scalars
for x in location.free_symbols:
if isinstance(x, BaseScalar):
raise ValueError("location should not contain" +
" BaseScalars")
origin = parent.origin.locate_new(name + '.origin',
location)
else:
location = Vector.zero
origin = Point(name + '.origin')
if transformation is None:
transformation = Tuple(rotation_matrix, location)
if isinstance(transformation, Tuple):
lambda_transformation = CoordSys3D._compose_rotation_and_translation(
transformation[0],
transformation[1],
parent
)
r, l = transformation
l = l._projections
lambda_lame = CoordSys3D._get_lame_coeff('cartesian')
lambda_inverse = lambda x, y, z: r.inv()*Matrix(
[x-l[0], y-l[1], z-l[2]])
elif isinstance(transformation, Symbol):
trname = transformation.name
lambda_transformation = CoordSys3D._get_transformation_lambdas(trname)
if parent is not None:
if parent.lame_coefficients() != (S(1), S(1), S(1)):
raise ValueError('Parent for pre-defined coordinate '
'system should be Cartesian.')
lambda_lame = CoordSys3D._get_lame_coeff(trname)
lambda_inverse = CoordSys3D._set_inv_trans_equations(trname)
elif isinstance(transformation, Lambda):
if not CoordSys3D._check_orthogonality(transformation):
raise ValueError("The transformation equation does not "
"create orthogonal coordinate system")
lambda_transformation = transformation
lambda_lame = CoordSys3D._calculate_lame_coeff(lambda_transformation)
lambda_inverse = None
else:
lambda_transformation = lambda x, y, z: transformation(x, y, z)
lambda_lame = CoordSys3D._get_lame_coeff(transformation)
lambda_inverse = None
if variable_names is None:
if isinstance(transformation, Lambda):
variable_names = ["x1", "x2", "x3"]
elif isinstance(transformation, Symbol):
if transformation.name is 'spherical':
variable_names = ["r", "theta", "phi"]
elif transformation.name is 'cylindrical':
variable_names = ["r", "theta", "z"]
else:
variable_names = ["x", "y", "z"]
else:
variable_names = ["x", "y", "z"]
if vector_names is None:
vector_names = ["i", "j", "k"]
# All systems that are defined as 'roots' are unequal, unless
# they have the same name.
# Systems defined at same orientation/position wrt the same
# 'parent' are equal, irrespective of the name.
# This is true even if the same orientation is provided via
# different methods like Axis/Body/Space/Quaternion.
# However, coincident systems may be seen as unequal if
# positioned/oriented wrt different parents, even though
# they may actually be 'coincident' wrt the root system.
if parent is not None:
obj = super(CoordSys3D, cls).__new__(
cls, Symbol(name), transformation, parent)
else:
obj = super(CoordSys3D, cls).__new__(
cls, Symbol(name), transformation)
obj._name = name
# Initialize the base vectors
_check_strings('vector_names', vector_names)
vector_names = list(vector_names)
latex_vects = [(r'\mathbf{\hat{%s}_{%s}}' % (x, name)) for
x in vector_names]
pretty_vects = [(name + '_' + x) for x in vector_names]
obj._vector_names = vector_names
v1 = BaseVector(0, obj, pretty_vects[0], latex_vects[0])
v2 = BaseVector(1, obj, pretty_vects[1], latex_vects[1])
v3 = BaseVector(2, obj, pretty_vects[2], latex_vects[2])
obj._base_vectors = (v1, v2, v3)
# Initialize the base scalars
_check_strings('variable_names', vector_names)
variable_names = list(variable_names)
latex_scalars = [(r"\mathbf{{%s}_{%s}}" % (x, name)) for
x in variable_names]
pretty_scalars = [(name + '_' + x) for x in variable_names]
obj._variable_names = variable_names
obj._vector_names = vector_names
x1 = BaseScalar(0, obj, pretty_scalars[0], latex_scalars[0])
x2 = BaseScalar(1, obj, pretty_scalars[1], latex_scalars[1])
x3 = BaseScalar(2, obj, pretty_scalars[2], latex_scalars[2])
obj._base_scalars = (x1, x2, x3)
obj._transformation = transformation
obj._transformation_lambda = lambda_transformation
obj._lame_coefficients = lambda_lame(x1, x2, x3)
obj._transformation_from_parent_lambda = lambda_inverse
setattr(obj, variable_names[0], x1)
setattr(obj, variable_names[1], x2)
setattr(obj, variable_names[2], x3)
setattr(obj, vector_names[0], v1)
setattr(obj, vector_names[1], v2)
setattr(obj, vector_names[2], v3)
# Assign params
obj._parent = parent
if obj._parent is not None:
obj._root = obj._parent._root
else:
obj._root = obj
obj._parent_rotation_matrix = rotation_matrix
obj._origin = origin
# Return the instance
return obj
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 test_CodeBlock():
c = CodeBlock(Assignment(x, 1), Assignment(y, x + 1))
assert c.func(*c.args) == c
assert c.left_hand_sides == Tuple(x, y)
assert c.right_hand_sides == Tuple(1, x + 1)
def _range_tuple(s):
if isinstance(s, Symbol):
return Tuple(s) + default_range
if len(s) == 3:
return Tuple(*s)
raise ValueError('symbol or `(symbol, min, max)` expected but got %s' % s)
def test_deriv1():
# These all requre derivatives evaluated at a point (issue 4719) to work.
# See issue 4624
assert f(
2 * x).diff(x) == 2 * Subs(Derivative(f(x), x), Tuple(x), Tuple(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), Tuple(x), Tuple(2 * x))
assert f(2 + x).diff(x) == Subs(Derivative(f(x), x), Tuple(x),
Tuple(x + 2))
assert f(2 + 3 * x).diff(x) == 3 * Subs(Derivative(f(x), x), Tuple(x),
Tuple(3 * x + 2))
assert f(3 * sin(x)).diff(x) == 3 * cos(x) * Subs(Derivative(
f(x), x), Tuple(x), Tuple(3 * sin(x)))
# See issue 8510
assert f(x, x + z).diff(x) == Subs(Derivative(f(y, x + z), y), Tuple(y), Tuple(x)) \
+ Subs(Derivative(f(x, y), y), Tuple(y), Tuple(x + z))
assert f(x, x**2).diff(x) == Subs(Derivative(f(y, x**2), y), Tuple(y), Tuple(x)) \
+ 2*x*Subs(Derivative(f(x, y), y), Tuple(y), Tuple(x**2))
def __new__(cls,
base_dims,
derived_dims=[],
dimensional_dependencies={},
name=None,
descr=None):
dimensional_dependencies = dict(dimensional_dependencies)
if (name is not None) or (descr is not None):
SymPyDeprecationWarning(
deprecated_since_version="1.2",
issue=13336,
useinstead="do not define a `name` or `descr`",
).warn()
def parse_dim(dim):
if isinstance(dim, string_types):
dim = Dimension(Symbol(dim))
elif isinstance(dim, Dimension):
pass
elif isinstance(dim, Symbol):
dim = Dimension(dim)
else:
raise TypeError("%s wrong type" % dim)
return dim
base_dims = [parse_dim(i) for i in base_dims]
derived_dims = [parse_dim(i) for i in derived_dims]
for dim in base_dims:
dim = dim.name
if (dim in dimensional_dependencies and
(len(dimensional_dependencies[dim]) != 1
or dimensional_dependencies[dim].get(dim, None) != 1)):
raise IndexError("Repeated value in base dimensions")
dimensional_dependencies[dim] = Dict({dim: 1})
def parse_dim_name(dim):
if isinstance(dim, Dimension):
return dim.name
elif isinstance(dim, string_types):
return Symbol(dim)
elif isinstance(dim, Symbol):
return dim
else:
raise TypeError("unrecognized type %s for %s" %
(type(dim), dim))
for dim in dimensional_dependencies.keys():
dim = parse_dim(dim)
if (dim not in derived_dims) and (dim not in base_dims):
derived_dims.append(dim)
def parse_dict(d):
return Dict({parse_dim_name(i): j for i, j in d.items()})
# Make sure everything is a SymPy type:
dimensional_dependencies = {
parse_dim_name(i): parse_dict(j)
for i, j in dimensional_dependencies.items()
}
for dim in derived_dims:
if dim in base_dims:
raise ValueError("Dimension %s both in base and derived" % dim)
if dim.name not in dimensional_dependencies:
# TODO: should this raise a warning?
dimensional_dependencies[dim] = Dict({dim.name: 1})
base_dims.sort(key=default_sort_key)
derived_dims.sort(key=default_sort_key)
base_dims = Tuple(*base_dims)
derived_dims = Tuple(*derived_dims)
dimensional_dependencies = Dict(
{i: Dict(j)
for i, j in dimensional_dependencies.items()})
obj = Basic.__new__(cls, base_dims, derived_dims,
dimensional_dependencies)
return obj
def sample_iter(expr,
condition=None,
size=(1, ),
library='scipy',
numsamples=S.Infinity,
**kwargs):
"""
Returns an iterator of realizations from the expression given a condition
Parameters
==========
expr: Expr
Random expression to be realized
condition: Expr, optional
A conditional expression
size : int, tuple
Represents size of each sample in numsamples
numsamples: integer, optional
Length of the iterator (defaults to infinity)
Examples
========
>>> from sympy.stats import Normal, sample_iter
>>> X = Normal('X', 0, 1)
>>> expr = X*X + 3
>>> iterator = sample_iter(expr, numsamples=3) # doctest: +SKIP
>>> list(iterator) # doctest: +SKIP
[12, 4, 7]
Returns
=======
sample_iter: iterator object
iterator object containing the sample/samples of given expr
See Also
========
sample
sampling_P
sampling_E
"""
if not import_module(library):
raise ValueError("Failed to import %s" % library)
if condition is not None:
ps = pspace(Tuple(expr, condition))
else:
ps = pspace(expr)
rvs = list(ps.values)
if library == 'pymc3':
# Currently unable to lambdify in pymc3
# TODO : Remove 'pymc3' when lambdify accepts 'pymc3' as module
fn = lambdify(rvs, expr, **kwargs)
else:
fn = lambdify(rvs, expr, modules=library, **kwargs)
if condition is not None:
given_fn = lambdify(rvs, condition, **kwargs)
def return_generator():
count = 0
while count < numsamples:
d = ps.sample(
size=size,
library=library) # a dictionary that maps RVs to values
args = [d[rv] for rv in rvs]
if condition is not None: # Check that these values satisfy the condition
gd = given_fn(*args)
if gd != True and gd != False:
raise ValueError(
"Conditions must not contain free symbols")
if not gd: # If the values don't satisfy then try again
continue
yield fn(*args)
count += 1
return return_generator()
def balance_stoichiometry(reactants, products, substances=None,
substance_factory=Substance.from_formula,
parametric_symbols=None, underdetermined=True, allow_duplicates=False):
""" Balances stoichiometric coefficients of a reaction
Parameters
----------
reactants : iterable of reactant keys
products : iterable of product keys
substances : OrderedDict or string or None
Mapping reactant/product keys to instances of :class:`Substance`.
substance_factory : callback
parametric_symbols : generator of symbols
Used to generate symbols for parametric solution for
under-determined system of equations. Default is numbered "x-symbols" starting
from 1.
underdetermined : bool
Allows to find a non-unique solution (in addition to a constant factor
across all terms). Set to ``False`` to disallow (raise ValueError) on
e.g. "C + O2 -> CO + CO2". Set to ``None`` if you want the symbols replaced
so that the coefficients are the smallest possible positive (non-zero) integers.
allow_duplicates : bool
If False: raises an excpetion if keys appear in both ``reactants`` and ``products``.
Examples
--------
>>> ref = {'C2H2': 2, 'O2': 3}, {'CO': 4, 'H2O': 2}
>>> balance_stoichiometry({'C2H2', 'O2'}, {'CO', 'H2O'}) == ref
True
>>> ref2 = {'H2': 1, 'O2': 1}, {'H2O2': 1}
>>> balance_stoichiometry('H2 O2'.split(), ['H2O2'], 'H2 O2 H2O2') == ref2
True
>>> reac, prod = 'CuSCN KIO3 HCl'.split(), 'CuSO4 KCl HCN ICl H2O'.split()
>>> Reaction(*balance_stoichiometry(reac, prod)).string()
'4 CuSCN + 7 KIO3 + 14 HCl -> 4 CuSO4 + 7 KCl + 4 HCN + 7 ICl + 5 H2O'
>>> balance_stoichiometry({'Fe', 'O2'}, {'FeO', 'Fe2O3'}, underdetermined=False)
Traceback (most recent call last):
...
ValueError: The system was under-determined
>>> r, p = balance_stoichiometry({'Fe', 'O2'}, {'FeO', 'Fe2O3'})
>>> list(set.union(*[v.free_symbols for v in r.values()]))
[x1]
>>> b = balance_stoichiometry({'Fe', 'O2'}, {'FeO', 'Fe2O3'}, underdetermined=None)
>>> b == ({'Fe': 3, 'O2': 2}, {'FeO': 1, 'Fe2O3': 1})
True
>>> d = balance_stoichiometry({'C', 'CO'}, {'C', 'CO', 'CO2'}, underdetermined=None, allow_duplicates=True)
>>> d == ({'CO': 2}, {'C': 1, 'CO2': 1})
True
Returns
-------
balanced reactants : dict
balanced products : dict
"""
import sympy
from sympy import (
MutableDenseMatrix, gcd, zeros, linsolve, numbered_symbols, Wild, Symbol,
Integer, Tuple, preorder_traversal as pre
)
_intersect = sorted(set.intersection(*map(set, (reactants, products))))
if _intersect:
if allow_duplicates:
if underdetermined is not None:
raise NotImplementedError("allow_duplicates currently requires underdetermined=None")
if set(reactants) == set(products):
raise ValueError("cannot balance: reactants and products identical")
# For each duplicate, try to drop it completely:
for dupl in _intersect:
try:
result = balance_stoichiometry(
[sp for sp in reactants if sp != dupl],
[sp for sp in products if sp != dupl],
substances=substances, substance_factory=substance_factory,
underdetermined=underdetermined, allow_duplicates=True)
except Exception:
continue
else:
return result
for perm in product(*[(False, True)]*len(_intersect)): # brute force (naive)
r = set(reactants)
p = set(products)
for remove_reac, dupl in zip(perm, _intersect):
if remove_reac:
r.remove(dupl)
else:
p.remove(dupl)
try:
result = balance_stoichiometry(
r, p, substances=substances, substance_factory=substance_factory,
parametric_symbols=parametric_symbols, underdetermined=underdetermined,
allow_duplicates=False)
except ValueError:
continue
else:
return result
else:
raise ValueError("Failed to remove duplicate keys: %s" % _intersect)
else:
raise ValueError("Substances on both sides: %s" % str(_intersect))
if substances is None:
substances = OrderedDict([(k, substance_factory(k)) for k in chain(reactants, products)])
if isinstance(substances, str):
substances = OrderedDict([(k, substance_factory(k)) for k in substances.split()])
if type(reactants) == set: # we don't want isinstance since it might be "OrderedSet"
reactants = sorted(reactants)
if type(products) == set:
products = sorted(products)
subst_keys = list(reactants) + list(products)
cks = Substance.composition_keys(substances.values())
if parametric_symbols is None:
parametric_symbols = numbered_symbols('x', start=1, integer=True, positive=True)
# ?C2H2 + ?O2 -> ?CO + ?H2O
# Ax = 0
# A: x:
#
# C2H2 O2 CO H2O
# C -2 0 1 0 x0 = 0
# H -2 0 0 2 x1 0
# O 0 -2 1 1 x2 0
# x3
def _get(ck, sk):
return substances[sk].composition.get(ck, 0) * (-1 if sk in reactants else 1)
for ck in cks: # check that all components are present on reactant & product sides
for rk in reactants:
if substances[rk].composition.get(ck, 0) != 0:
break
else:
any_pos = any(substances[pk].composition.get(ck, 0) > 0 for pk in products)
any_neg = any(substances[pk].composition.get(ck, 0) < 0 for pk in products)
if any_pos and any_neg:
pass # negative and positive parts among products, no worries
else:
raise ValueError("Component '%s' not among reactants" % ck)
for pk in products:
if substances[pk].composition.get(ck, 0) != 0:
break
else:
any_pos = any(substances[pk].composition.get(ck, 0) > 0 for pk in reactants)
any_neg = any(substances[pk].composition.get(ck, 0) < 0 for pk in reactants)
if any_pos and any_neg:
pass # negative and positive parts among reactants, no worries
else:
raise ValueError("Component '%s' not among products" % ck)
A = MutableDenseMatrix([[_get(ck, sk) for sk in subst_keys] for ck in cks])
symbs = list(reversed([next(parametric_symbols) for _ in range(len(subst_keys))]))
sol, = linsolve((A, zeros(len(cks), 1)), symbs)
wi = Wild('wi', properties=[lambda k: not k.has(Symbol)])
cd = reduce(gcd, [1] + [1/m[wi] for m in map(
lambda n: n.match(symbs[-1]/wi), pre(sol)) if m is not None])
sol = sol.func(*[arg/cd for arg in sol.args])
def remove(cont, symb, remaining):
subsd = dict(zip(remaining/symb, remaining))
cont = cont.func(*[(arg/symb).expand().subs(subsd) for arg in cont.args])
if cont.has(symb):
raise ValueError("Bug, please report an issue at https://github.com/bjodah/chempy")
return cont
done = False
for idx, symb in enumerate(symbs):
for expr in sol:
iterable = expr.args if expr.is_Add else [expr]
for term in iterable:
if term.is_number:
done = True
break
if done:
break
if done:
break
for expr in sol:
if (expr/symb).is_number:
sol = remove(sol, symb, MutableDenseMatrix(symbs[idx+1:]))
break
for symb in symbs:
cd = 1
for expr in sol:
iterable = expr.args if expr.is_Add else [expr]
for term in iterable:
if term.is_Mul and term.args[0].is_number and term.args[1] == symb:
cd = gcd(cd, term.args[0])
if cd != 1:
sol = sol.func(*[arg.subs(symb, symb/cd) for arg in sol.args])
integer_one = 1 # need 'is' check, SyntaxWarning when checking against literal
if underdetermined is integer_one:
from ._release import __version__
if int(__version__.split('.')[1]) > 6:
warnings.warn( # deprecated because comparison with ``1`` problematic (True==1)
("Pass underdetermined == None instead of ``1`` (depreacted since 0.7.0,"
" will_be_missing_in='0.9.0')"), ChemPyDeprecationWarning)
underdetermined = None
if underdetermined is None:
sol = Tuple(*[Integer(x) for x in _solve_balancing_ilp_pulp(A)])
fact = gcd(sol)
sol = MutableDenseMatrix([e/fact for e in sol]).reshape(len(sol), 1)
sol /= reduce(gcd, sol)
if 0 in sol:
raise ValueError("Superfluous species given.")
if underdetermined:
if any(x == sympy.nan for x in sol):
raise ValueError("Failed to balance reaction")
else:
for x in sol:
if len(x.free_symbols) != 0:
raise ValueError("The system was under-determined")
if not all(residual == 0 for residual in A * sol):
raise ValueError("Failed to balance reaction")
def _x(k):
coeff = sol[subst_keys.index(k)]
return int(coeff) if underdetermined is None else coeff
return (
OrderedDict([(k, _x(k)) for k in reactants]),
OrderedDict([(k, _x(k)) for k in products])
)
def given(expr, condition=None, **kwargs):
r""" Conditional Random Expression
From a random expression and a condition on that expression creates a new
probability space from the condition and returns the same expression on that
conditional probability space.
Examples
========
>>> from sympy.stats import given, density, Die
>>> X = Die('X', 6)
>>> Y = given(X, X > 3)
>>> density(Y).dict
{4: 1/3, 5: 1/3, 6: 1/3}
Following convention, if the condition is a random symbol then that symbol
is considered fixed.
>>> from sympy.stats import Normal
>>> from sympy import pprint
>>> from sympy.abc import z
>>> X = Normal('X', 0, 1)
>>> Y = Normal('Y', 0, 1)
>>> pprint(density(X + Y, Y)(z), use_unicode=False)
2
-(-Y + z)
-----------
___ 2
\/ 2 *e
------------------
____
2*\/ pi
"""
if not is_random(condition) or pspace_independent(expr, condition):
return expr
if isinstance(condition, RandomSymbol):
condition = Eq(condition, condition.symbol)
condsymbols = random_symbols(condition)
if (isinstance(condition, Equality) and len(condsymbols) == 1
and not isinstance(pspace(expr).domain, ConditionalDomain)):
rv = tuple(condsymbols)[0]
results = solveset(condition, rv)
if isinstance(results, Intersection) and S.Reals in results.args:
results = list(results.args[1])
sums = 0
for res in results:
temp = expr.subs(rv, res)
if temp == True:
return True
if temp != False:
# XXX: This seems nonsensical but preserves existing behaviour
# after the change that Relational is no longer a subclass of
# Expr. Here expr is sometimes Relational and sometimes Expr
# but we are trying to add them with +=. This needs to be
# fixed somehow.
if sums == 0 and isinstance(expr, Relational):
sums = expr.subs(rv, res)
else:
sums += expr.subs(rv, res)
if sums == 0:
return False
return sums
# Get full probability space of both the expression and the condition
fullspace = pspace(Tuple(expr, condition))
# Build new space given the condition
space = fullspace.conditional_space(condition, **kwargs)
# Dictionary to swap out RandomSymbols in expr with new RandomSymbols
# That point to the new conditional space
swapdict = rs_swap(fullspace.values, space.values)
# Swap random variables in the expression
expr = expr.xreplace(swapdict)
return expr
def test_sympyify_iterables():
ans = [Rational(3, 10), Rational(1, 5)]
assert sympify(['.3', '.2'], rational=True) == ans
assert sympify(tuple(['.3', '.2']), rational=True) == Tuple(*ans)
assert sympify(dict(x=0, y=1)) == {x: 0, y: 1}
assert sympify(['1', '2', ['3', '4']]) == [S(1), S(2), [S(3), S(4)]]
def test_Tuple_equality():
assert Tuple(1, 2) is not (1, 2)
assert (Tuple(1, 2) == (1, 2)) is True
assert (Tuple(1, 2) != (1, 2)) is False
assert (Tuple(1, 2) == (1, 3)) is False
assert (Tuple(1, 2) != (1, 3)) is True
assert (Tuple(1, 2) == Tuple(1, 2)) is True
assert (Tuple(1, 2) != Tuple(1, 2)) is False
assert (Tuple(1, 2) == Tuple(1, 3)) is False
assert (Tuple(1, 2) != Tuple(1, 3)) is True
