Vehicle-cpp/hkpc/software/sympy/core/tests/test_subs.py

from sympy.calculus.accumulationbounds import AccumBounds
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.containers import (Dict, Tuple)
from sympy.core.function import (Derivative, Function, Lambda, Subs)
from sympy.core.mul import Mul
from sympy.core.numbers import (Float, I, Integer, Rational, oo, pi, zoo)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, Wild, symbols)
from sympy.core.sympify import SympifyError
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (atan2, cos, cot, sin, tan)
from sympy.matrices.dense import (Matrix, zeros)
from sympy.matrices.expressions.special import ZeroMatrix
from sympy.polys.polytools import factor
from sympy.polys.rootoftools import RootOf
from sympy.simplify.cse_main import cse
from sympy.simplify.simplify import nsimplify
from sympy.core.basic import _aresame
from sympy.testing.pytest import XFAIL, raises
from sympy.abc import a, x, y, z, t


def test_subs():
    n3 = Rational(3)
    e = x
    e = e.subs(x, n3)
    assert e == Rational(3)

    e = 2*x
    assert e == 2*x
    e = e.subs(x, n3)
    assert e == Rational(6)


def test_subs_Matrix():
    z = zeros(2)
    z1 = ZeroMatrix(2, 2)
    assert (x*y).subs({x:z, y:0}) in [z, z1]
    assert (x*y).subs({y:z, x:0}) == 0
    assert (x*y).subs({y:z, x:0}, simultaneous=True) in [z, z1]
    assert (x + y).subs({x: z, y: z}, simultaneous=True) in [z, z1]
    assert (x + y).subs({x: z, y: z}) in [z, z1]

    # Issue #15528
    assert Mul(Matrix([[3]]), x).subs(x, 2.0) == Matrix([[6.0]])
    # Does not raise a TypeError, see comment on the MatAdd postprocessor
    assert Add(Matrix([[3]]), x).subs(x, 2.0) == Add(Matrix([[3]]), 2.0)


def test_subs_AccumBounds():
    e = x
    e = e.subs(x, AccumBounds(1, 3))
    assert e == AccumBounds(1, 3)

    e = 2*x
    e = e.subs(x, AccumBounds(1, 3))
    assert e == AccumBounds(2, 6)

    e = x + x**2
    e = e.subs(x, AccumBounds(-1, 1))
    assert e == AccumBounds(-1, 2)


def test_trigonometric():
    n3 = Rational(3)
    e = (sin(x)**2).diff(x)
    assert e == 2*sin(x)*cos(x)
    e = e.subs(x, n3)
    assert e == 2*cos(n3)*sin(n3)

    e = (sin(x)**2).diff(x)
    assert e == 2*sin(x)*cos(x)
    e = e.subs(sin(x), cos(x))
    assert e == 2*cos(x)**2

    assert exp(pi).subs(exp, sin) == 0
    assert cos(exp(pi)).subs(exp, sin) == 1

    i = Symbol('i', integer=True)
    zoo = S.ComplexInfinity
    assert tan(x).subs(x, pi/2) is zoo
    assert cot(x).subs(x, pi) is zoo
    assert cot(i*x).subs(x, pi) is zoo
    assert tan(i*x).subs(x, pi/2) == tan(i*pi/2)
    assert tan(i*x).subs(x, pi/2).subs(i, 1) is zoo
    o = Symbol('o', odd=True)
    assert tan(o*x).subs(x, pi/2) == tan(o*pi/2)


def test_powers():
    assert sqrt(1 - sqrt(x)).subs(x, 4) == I
    assert (sqrt(1 - x**2)**3).subs(x, 2) == - 3*I*sqrt(3)
    assert (x**Rational(1, 3)).subs(x, 27) == 3
    assert (x**Rational(1, 3)).subs(x, -27) == 3*(-1)**Rational(1, 3)
    assert ((-x)**Rational(1, 3)).subs(x, 27) == 3*(-1)**Rational(1, 3)
    n = Symbol('n', negative=True)
    assert (x**n).subs(x, 0) is S.ComplexInfinity
    assert exp(-1).subs(S.Exp1, 0) is S.ComplexInfinity
    assert (x**(4.0*y)).subs(x**(2.0*y), n) == n**2.0
    assert (2**(x + 2)).subs(2, 3) == 3**(x + 3)


def test_logexppow():   # no eval()
    x = Symbol('x', real=True)
    w = Symbol('w')
    e = (3**(1 + x) + 2**(1 + x))/(3**x + 2**x)
    assert e.subs(2**x, w) != e
    assert e.subs(exp(x*log(Rational(2))), w) != e


def test_bug():
    x1 = Symbol('x1')
    x2 = Symbol('x2')
    y = x1*x2
    assert y.subs(x1, Float(3.0)) == Float(3.0)*x2


def test_subbug1():
    # see that they don't fail
    (x**x).subs(x, 1)
    (x**x).subs(x, 1.0)


def test_subbug2():
    # Ensure this does not cause infinite recursion
    assert Float(7.7).epsilon_eq(abs(x).subs(x, -7.7))


def test_dict_set():
    a, b, c = map(Wild, 'abc')

    f = 3*cos(4*x)
    r = f.match(a*cos(b*x))
    assert r == {a: 3, b: 4}
    e = a/b*sin(b*x)
    assert e.subs(r) == r[a]/r[b]*sin(r[b]*x)
    assert e.subs(r) == 3*sin(4*x) / 4
    s = set(r.items())
    assert e.subs(s) == r[a]/r[b]*sin(r[b]*x)
    assert e.subs(s) == 3*sin(4*x) / 4

    assert e.subs(r) == r[a]/r[b]*sin(r[b]*x)
    assert e.subs(r) == 3*sin(4*x) / 4
    assert x.subs(Dict((x, 1))) == 1


def test_dict_ambigous():   # see issue 3566
    f = x*exp(x)
    g = z*exp(z)

    df = {x: y, exp(x): y}
    dg = {z: y, exp(z): y}

    assert f.subs(df) == y**2
    assert g.subs(dg) == y**2

    # and this is how order can affect the result
    assert f.subs(x, y).subs(exp(x), y) == y*exp(y)
    assert f.subs(exp(x), y).subs(x, y) == y**2

    # length of args and count_ops are the same so
    # default_sort_key resolves ordering...if one
    # doesn't want this result then an unordered
    # sequence should not be used.
    e = 1 + x*y
    assert e.subs({x: y, y: 2}) == 5
    # here, there are no obviously clashing keys or values
    # but the results depend on the order
    assert exp(x/2 + y).subs({exp(y + 1): 2, x: 2}) == exp(y + 1)


def test_deriv_sub_bug3():
    f = Function('f')
    pat = Derivative(f(x), x, x)
    assert pat.subs(y, y**2) == Derivative(f(x), x, x)
    assert pat.subs(y, y**2) != Derivative(f(x), x)


def test_equality_subs1():
    f = Function('f')
    eq = Eq(f(x)**2, x)
    res = Eq(Integer(16), x)
    assert eq.subs(f(x), 4) == res


def test_equality_subs2():
    f = Function('f')
    eq = Eq(f(x)**2, 16)
    assert bool(eq.subs(f(x), 3)) is False
    assert bool(eq.subs(f(x), 4)) is True


def test_issue_3742():
    e = sqrt(x)*exp(y)
    assert e.subs(sqrt(x), 1) == exp(y)


def test_subs_dict1():
    assert (1 + x*y).subs(x, pi) == 1 + pi*y
    assert (1 + x*y).subs({x: pi, y: 2}) == 1 + 2*pi

    c2, c3, q1p, q2p, c1, s1, s2, s3 = symbols('c2 c3 q1p q2p c1 s1 s2 s3')
    test = (c2**2*q2p*c3 + c1**2*s2**2*q2p*c3 + s1**2*s2**2*q2p*c3
            - c1**2*q1p*c2*s3 - s1**2*q1p*c2*s3)
    assert (test.subs({c1**2: 1 - s1**2, c2**2: 1 - s2**2, c3**3: 1 - s3**2})
        == c3*q2p*(1 - s2**2) + c3*q2p*s2**2*(1 - s1**2)
            - c2*q1p*s3*(1 - s1**2) + c3*q2p*s1**2*s2**2 - c2*q1p*s3*s1**2)


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) == 1
    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) != z*Rational(2, 5)
    assert (x*y*A).subs(x*y, a) == a*A
    assert (x**2*y**(x*Rational(3, 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 (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(x*Rational(3, 2))).subs(y*exp(x), 2) == 2*exp(x/2)
    assert (y*exp(x*Rational(3, 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 6361
    assert (-8*I*a).subs(-2*a, 1) == 4*I
    assert (-I*a).subs(-a, 1) == I

    # issue 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_subs_simple():
    a = symbols('a', commutative=True)
    x = symbols('x', commutative=False)

    assert (2*a).subs(1, 3) == 2*a
    assert (2*a).subs(2, 3) == 3*a
    assert (2*a).subs(a, 3) == 6
    assert sin(2).subs(1, 3) == sin(2)
    assert sin(2).subs(2, 3) == sin(3)
    assert sin(a).subs(a, 3) == sin(3)

    assert (2*x).subs(1, 3) == 2*x
    assert (2*x).subs(2, 3) == 3*x
    assert (2*x).subs(x, 3) == 6
    assert sin(x).subs(x, 3) == sin(3)


def test_subs_constants():
    a, b = symbols('a b', commutative=True)
    x, y = symbols('x y', commutative=False)

    assert (a*b).subs(2*a, 1) == a*b
    assert (1.5*a*b).subs(a, 1) == 1.5*b
    assert (2*a*b).subs(2*a, 1) == b
    assert (2*a*b).subs(4*a, 1) == 2*a*b

    assert (x*y).subs(2*x, 1) == x*y
    assert (1.5*x*y).subs(x, 1) == 1.5*y
    assert (2*x*y).subs(2*x, 1) == y
    assert (2*x*y).subs(4*x, 1) == 2*x*y


def test_subs_commutative():
    a, b, c, d, K = symbols('a b c d K', commutative=True)

    assert (a*b).subs(a*b, K) == K
    assert (a*b*a*b).subs(a*b, K) == K**2
    assert (a*a*b*b).subs(a*b, K) == K**2
    assert (a*b*c*d).subs(a*b*c, K) == d*K
    assert (a*b**c).subs(a, K) == K*b**c
    assert (a*b**c).subs(b, K) == a*K**c
    assert (a*b**c).subs(c, K) == a*b**K
    assert (a*b*c*b*a).subs(a*b, K) == c*K**2
    assert (a**3*b**2*a).subs(a*b, K) == a**2*K**2


def test_subs_noncommutative():
    w, x, y, z, L = symbols('w x y z L', commutative=False)
    alpha = symbols('alpha', commutative=True)
    someint = symbols('someint', commutative=True, integer=True)

    assert (x*y).subs(x*y, L) == L
    assert (w*y*x).subs(x*y, L) == w*y*x
    assert (w*x*y*z).subs(x*y, L) == w*L*z
    assert (x*y*x*y).subs(x*y, L) == L**2
    assert (x*x*y).subs(x*y, L) == x*L
    assert (x*x*y*y).subs(x*y, L) == x*L*y
    assert (w*x*y).subs(x*y*z, L) == w*x*y
    assert (x*y**z).subs(x, L) == L*y**z
    assert (x*y**z).subs(y, L) == x*L**z
    assert (x*y**z).subs(z, L) == x*y**L
    assert (w*x*y*z*x*y).subs(x*y*z, L) == w*L*x*y
    assert (w*x*y*y*w*x*x*y*x*y*y*x*y).subs(x*y, L) == w*L*y*w*x*L**2*y*L

    # Check fractional power substitutions. It should not do
    # substitutions that choose a value for noncommutative log,
    # or inverses that don't already appear in the expressions.
    assert (x*x*x).subs(x*x, L) == L*x
    assert (x*x*x*y*x*x*x*x).subs(x*x, L) == L*x*y*L**2
    for p in range(1, 5):
        for k in range(10):
            assert (y * x**k).subs(x**p, L) == y * L**(k//p) * x**(k % p)
    assert (x**Rational(3, 2)).subs(x**S.Half, L) == x**Rational(3, 2)
    assert (x**S.Half).subs(x**S.Half, L) == L
    assert (x**Rational(-1, 2)).subs(x**S.Half, L) == x**Rational(-1, 2)
    assert (x**Rational(-1, 2)).subs(x**Rational(-1, 2), L) == L

    assert (x**(2*someint)).subs(x**someint, L) == L**2
    assert (x**(2*someint + 3)).subs(x**someint, L) == L**2*x**3
    assert (x**(3*someint + 3)).subs(x**someint, L) == L**3*x**3
    assert (x**(3*someint)).subs(x**(2*someint), L) == L * x**someint
    assert (x**(4*someint)).subs(x**(2*someint), L) == L**2
    assert (x**(4*someint + 1)).subs(x**(2*someint), L) == L**2 * x
    assert (x**(4*someint)).subs(x**(3*someint), L) == L * x**someint
    assert (x**(4*someint + 1)).subs(x**(3*someint), L) == L * x**(someint + 1)

    assert (x**(2*alpha)).subs(x**alpha, L) == x**(2*alpha)
    assert (x**(2*alpha + 2)).subs(x**2, L) == x**(2*alpha + 2)
    assert ((2*z)**alpha).subs(z**alpha, y) == (2*z)**alpha
    assert (x**(2*someint*alpha)).subs(x**someint, L) == x**(2*someint*alpha)
    assert (x**(2*someint + alpha)).subs(x**someint, L) == x**(2*someint + alpha)

    # This could in principle be substituted, but is not currently
    # because it requires recognizing that someint**2 is divisible by
    # someint.
    assert (x**(someint**2 + 3)).subs(x**someint, L) == x**(someint**2 + 3)

    # alpha**z := exp(log(alpha) z) is usually well-defined
    assert (4**z).subs(2**z, y) == y**2

    # Negative powers
    assert (x**(-1)).subs(x**3, L) == x**(-1)
    assert (x**(-2)).subs(x**3, L) == x**(-2)
    assert (x**(-3)).subs(x**3, L) == L**(-1)
    assert (x**(-4)).subs(x**3, L) == L**(-1) * x**(-1)
    assert (x**(-5)).subs(x**3, L) == L**(-1) * x**(-2)

    assert (x**(-1)).subs(x**(-3), L) == x**(-1)
    assert (x**(-2)).subs(x**(-3), L) == x**(-2)
    assert (x**(-3)).subs(x**(-3), L) == L
    assert (x**(-4)).subs(x**(-3), L) == L * x**(-1)
    assert (x**(-5)).subs(x**(-3), L) == L * x**(-2)

    assert (x**1).subs(x**(-3), L) == x
    assert (x**2).subs(x**(-3), L) == x**2
    assert (x**3).subs(x**(-3), L) == L**(-1)
    assert (x**4).subs(x**(-3), L) == L**(-1) * x
    assert (x**5).subs(x**(-3), L) == L**(-1) * x**2


def test_subs_basic_funcs():
    a, b, c, d, K = symbols('a b c d K', commutative=True)
    w, x, y, z, L = symbols('w x y z L', commutative=False)

    assert (x + y).subs(x + y, L) == L
    assert (x - y).subs(x - y, L) == L
    assert (x/y).subs(x, L) == L/y
    assert (x**y).subs(x, L) == L**y
    assert (x**y).subs(y, L) == x**L
    assert ((a - c)/b).subs(b, K) == (a - c)/K
    assert (exp(x*y - z)).subs(x*y, L) == exp(L - z)
    assert (a*exp(x*y - w*z) + b*exp(x*y + w*z)).subs(z, 0) == \
        a*exp(x*y) + b*exp(x*y)
    assert ((a - b)/(c*d - a*b)).subs(c*d - a*b, K) == (a - b)/K
    assert (w*exp(a*b - c)*x*y/4).subs(x*y, L) == w*exp(a*b - c)*L/4


def test_subs_wild():
    R, S, T, U = symbols('R S T U', cls=Wild)

    assert (R*S).subs(R*S, T) == T
    assert (S*R).subs(R*S, T) == T
    assert (R + S).subs(R + S, T) == T
    assert (R**S).subs(R, T) == T**S
    assert (R**S).subs(S, T) == R**T
    assert (R*S**T).subs(R, U) == U*S**T
    assert (R*S**T).subs(S, U) == R*U**T
    assert (R*S**T).subs(T, U) == R*S**U


def test_subs_mixed():
    a, b, c, d, K = symbols('a b c d K', commutative=True)
    w, x, y, z, L = symbols('w x y z L', commutative=False)
    R, S, T, U = symbols('R S T U', cls=Wild)

    assert (a*x*y).subs(x*y, L) == a*L
    assert (a*b*x*y*x).subs(x*y, L) == a*b*L*x
    assert (R*x*y*exp(x*y)).subs(x*y, L) == R*L*exp(L)
    assert (a*x*y*y*x - x*y*z*exp(a*b)).subs(x*y, L) == a*L*y*x - L*z*exp(a*b)
    e = c*y*x*y*x**(R*S - a*b) - T*(a*R*b*S)
    assert e.subs(x*y, L).subs(a*b, K).subs(R*S, U) == \
        c*y*L*x**(U - K) - T*(U*K)


def test_division():
    a, b, c = symbols('a b c', commutative=True)
    x, y, z = symbols('x y z', commutative=True)

    assert (1/a).subs(a, c) == 1/c
    assert (1/a**2).subs(a, c) == 1/c**2
    assert (1/a**2).subs(a, -2) == Rational(1, 4)
    assert (-(1/a**2)).subs(a, -2) == Rational(-1, 4)

    assert (1/x).subs(x, z) == 1/z
    assert (1/x**2).subs(x, z) == 1/z**2
    assert (1/x**2).subs(x, -2) == Rational(1, 4)
    assert (-(1/x**2)).subs(x, -2) == Rational(-1, 4)

    #issue 5360
    assert (1/x).subs(x, 0) == 1/S.Zero


def test_add():
    a, b, c, d, x, y, t = symbols('a b c d x y t')

    assert (a**2 - b - c).subs(a**2 - b, d) in [d - c, a**2 - b - c]
    assert (a**2 - c).subs(a**2 - c, d) == d
    assert (a**2 - b - c).subs(a**2 - c, d) in [d - b, a**2 - b - c]
    assert (a**2 - x - c).subs(a**2 - c, d) in [d - x, a**2 - x - c]
    assert (a**2 - b - sqrt(a)).subs(a**2 - sqrt(a), c) == c - b
    assert (a + b + exp(a + b)).subs(a + b, c) == c + exp(c)
    assert (c + b + exp(c + b)).subs(c + b, a) == a + exp(a)
    assert (a + b + c + d).subs(b + c, x) == a + d + x
    assert (a + b + c + d).subs(-b - c, x) == a + d - x
    assert ((x + 1)*y).subs(x + 1, t) == t*y
    assert ((-x - 1)*y).subs(x + 1, t) == -t*y
    assert ((x - 1)*y).subs(x + 1, t) == y*(t - 2)
    assert ((-x + 1)*y).subs(x + 1, t) == y*(-t + 2)

    # this should work every time:
    e = a**2 - b - c
    assert e.subs(Add(*e.args[:2]), d) == d + e.args[2]
    assert e.subs(a**2 - c, d) == d - b

    # the fallback should recognize when a change has
    # been made; while .1 == Rational(1, 10) they are not the same
    # and the change should be made
    assert (0.1 + a).subs(0.1, Rational(1, 10)) == Rational(1, 10) + a

    e = (-x*(-y + 1) - y*(y - 1))
    ans = (-x*(x) - y*(-x)).expand()
    assert e.subs(-y + 1, x) == ans

    #Test issue 18747
    assert (exp(x) + cos(x)).subs(x, oo) == oo
    assert Add(*[AccumBounds(-1, 1), oo]) == oo
    assert Add(*[oo, AccumBounds(-1, 1)]) == oo


def test_subs_issue_4009():
    assert (I*Symbol('a')).subs(1, 2) == I*Symbol('a')


def test_functions_subs():
    f, g = symbols('f g', cls=Function)
    l = Lambda((x, y), sin(x) + y)
    assert (g(y, x) + cos(x)).subs(g, l) == sin(y) + x + cos(x)
    assert (f(x)**2).subs(f, sin) == sin(x)**2
    assert (f(x, y)).subs(f, log) == log(x, y)
    assert (f(x, y)).subs(f, sin) == f(x, y)
    assert (sin(x) + atan2(x, y)).subs([[atan2, f], [sin, g]]) == \
        f(x, y) + g(x)
    assert (g(f(x + y, x))).subs([[f, l], [g, exp]]) == exp(x + sin(x + y))


def test_derivative_subs():
    f = Function('f')
    g = Function('g')
    assert Derivative(f(x), x).subs(f(x), y) != 0
    # need xreplace to put the function back, see #13803
    assert Derivative(f(x), x).subs(f(x), y).xreplace({y: f(x)}) == \
        Derivative(f(x), x)
    # issues 5085, 5037
    assert cse(Derivative(f(x), x) + f(x))[1][0].has(Derivative)
    assert cse(Derivative(f(x, y), x) +
               Derivative(f(x, y), y))[1][0].has(Derivative)
    eq = Derivative(g(x), g(x))
    assert eq.subs(g, f) == Derivative(f(x), f(x))
    assert eq.subs(g(x), f(x)) == Derivative(f(x), f(x))
    assert eq.subs(g, cos) == Subs(Derivative(y, y), y, cos(x))


def test_derivative_subs2():
    f_func, g_func = symbols('f g', cls=Function)
    f, g = f_func(x, y, z), g_func(x, y, z)
    assert Derivative(f, x, y).subs(Derivative(f, x, y), g) == g
    assert Derivative(f, y, x).subs(Derivative(f, x, y), g) == g
    assert Derivative(f, x, y).subs(Derivative(f, x), g) == Derivative(g, y)
    assert Derivative(f, x, y).subs(Derivative(f, y), g) == Derivative(g, x)
    assert (Derivative(f, x, y, z).subs(
                Derivative(f, x, z), g) == Derivative(g, y))
    assert (Derivative(f, x, y, z).subs(
                Derivative(f, z, y), g) == Derivative(g, x))
    assert (Derivative(f, x, y, z).subs(
                Derivative(f, z, y, x), g) == g)

    # Issue 9135
    assert (Derivative(f, x, x, y).subs(
                Derivative(f, y, y), g) == Derivative(f, x, x, y))
    assert (Derivative(f, x, y, y, z).subs(
                Derivative(f, x, y, y, y), g) == Derivative(f, x, y, y, z))

    assert Derivative(f, x, y).subs(Derivative(f_func(x), x, y), g) == Derivative(f, x, y)


def test_derivative_subs3():
    dex = Derivative(exp(x), x)
    assert Derivative(dex, x).subs(dex, exp(x)) == dex
    assert dex.subs(exp(x), dex) == Derivative(exp(x), x, x)


def test_issue_5284():
    A, B = symbols('A B', commutative=False)
    assert (x*A).subs(x**2*A, B) == x*A
    assert (A**2).subs(A**3, B) == A**2
    assert (A**6).subs(A**3, B) == B**2


def test_subs_iter():
    assert x.subs(reversed([[x, y]])) == y
    it = iter([[x, y]])
    assert x.subs(it) == y
    assert x.subs(Tuple((x, y))) == y


def test_subs_dict():
    a, b, c, d, e = symbols('a b c d e')

    assert (2*x + y + z).subs({"x": 1, "y": 2}) == 4 + z

    l = [(sin(x), 2), (x, 1)]
    assert (sin(x)).subs(l) == \
           (sin(x)).subs(dict(l)) == 2
    assert sin(x).subs(reversed(l)) == sin(1)

    expr = sin(2*x) + sqrt(sin(2*x))*cos(2*x)*sin(exp(x)*x)
    reps = {sin(2*x): c,
               sqrt(sin(2*x)): a,
               cos(2*x): b,
               exp(x): e,
               x: d,}
    assert expr.subs(reps) == c + a*b*sin(d*e)

    l = [(x, 3), (y, x**2)]
    assert (x + y).subs(l) == 3 + x**2
    assert (x + y).subs(reversed(l)) == 12

    # If changes are made to convert lists into dictionaries and do
    # a dictionary-lookup replacement, these tests will help to catch
    # some logical errors that might occur
    l = [(y, z + 2), (1 + z, 5), (z, 2)]
    assert (y - 1 + 3*x).subs(l) == 5 + 3*x
    l = [(y, z + 2), (z, 3)]
    assert (y - 2).subs(l) == 3


def test_no_arith_subs_on_floats():
    assert (x + 3).subs(x + 3, a) == a
    assert (x + 3).subs(x + 2, a) == a + 1

    assert (x + y + 3).subs(x + 3, a) == a + y
    assert (x + y + 3).subs(x + 2, a) == a + y + 1

    assert (x + 3.0).subs(x + 3.0, a) == a
    assert (x + 3.0).subs(x + 2.0, a) == x + 3.0

    assert (x + y + 3.0).subs(x + 3.0, a) == a + y
    assert (x + y + 3.0).subs(x + 2.0, a) == x + y + 3.0


def test_issue_5651():
    a, b, c, K = symbols('a b c K', commutative=True)
    assert (a/(b*c)).subs(b*c, K) == a/K
    assert (a/(b**2*c**3)).subs(b*c, K) == a/(c*K**2)
    assert (1/(x*y)).subs(x*y, 2) == S.Half
    assert ((1 + x*y)/(x*y)).subs(x*y, 1) == 2
    assert (x*y*z).subs(x*y, 2) == 2*z
    assert ((1 + x*y)/(x*y)/z).subs(x*y, 1) == 2/z


def test_issue_6075():
    assert Tuple(1, True).subs(1, 2) == Tuple(2, True)


def test_issue_6079():
    # since x + 2.0 == x + 2 we can't do a simple equality test
    assert _aresame((x + 2.0).subs(2, 3), x + 2.0)
    assert _aresame((x + 2.0).subs(2.0, 3), x + 3)
    assert not _aresame(x + 2, x + 2.0)
    assert not _aresame(Basic(cos(x), S(1)), Basic(cos(x), S(1.)))
    assert _aresame(cos, cos)
    assert not _aresame(1, S.One)
    assert not _aresame(x, symbols('x', positive=True))


def test_issue_4680():
    N = Symbol('N')
    assert N.subs({"N": 3}) == 3


def test_issue_6158():
    assert (x - 1).subs(1, y) == x - y
    assert (x - 1).subs(-1, y) == x + y
    assert (x - oo).subs(oo, y) == x - y
    assert (x - oo).subs(-oo, y) == x + y


def test_Function_subs():
    f, g, h, i = symbols('f g h i', cls=Function)
    p = Piecewise((g(f(x, y)), x < -1), (g(x), x <= 1))
    assert p.subs(g, h) == Piecewise((h(f(x, y)), x < -1), (h(x), x <= 1))
    assert (f(y) + g(x)).subs({f: h, g: i}) == i(x) + h(y)


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_issue_6419_6421():
    assert (1/(1 + x/y)).subs(x/y, x) == 1/(1 + x)
    assert (-2*I).subs(2*I, x) == -x
    assert (-I*x).subs(I*x, x) == -x
    assert (-3*I*y**4).subs(3*I*y**2, x) == -x*y**2


def test_issue_6559():
    assert (-12*x + y).subs(-x, 1) == 12 + y
    # though this involves cse it generated a failure in Mul._eval_subs
    x0, x1 = symbols('x0 x1')
    e = -log(-12*sqrt(2) + 17)/24 - log(-2*sqrt(2) + 3)/12 + sqrt(2)/3
    # XXX modify cse so x1 is eliminated and x0 = -sqrt(2)?
    assert cse(e) == (
        [(x0, sqrt(2))], [x0/3 - log(-12*x0 + 17)/24 - log(-2*x0 + 3)/12])


def test_issue_5261():
    x = symbols('x', real=True)
    e = I*x
    assert exp(e).subs(exp(x), y) == y**I
    assert (2**e).subs(2**x, y) == y**I
    eq = (-2)**e
    assert eq.subs((-2)**x, y) == eq


def test_issue_6923():
    assert (-2*x*sqrt(2)).subs(2*x, y) == -sqrt(2)*y


def test_2arg_hack():
    N = Symbol('N', commutative=False)
    ans = Mul(2, y + 1, evaluate=False)
    assert (2*x*(y + 1)).subs(x, 1, hack2=True) == ans
    assert (2*(y + 1 + N)).subs(N, 0, hack2=True) == ans


@XFAIL
def test_mul2():
    """When this fails, remove things labelled "2-arg hack"
    1) remove special handling in the fallback of subs that
    was added in the same commit as this test
    2) remove the special handling in Mul.flatten
    """
    assert (2*(x + 1)).is_Mul


def test_noncommutative_subs():
    x,y = symbols('x,y', commutative=False)
    assert (x*y*x).subs([(x, x*y), (y, x)], simultaneous=True) == (x*y*x**2*y)


def test_issue_2877():
    f = Float(2.0)
    assert (x + f).subs({f: 2}) == x + 2

    def r(a, b, c):
        return factor(a*x**2 + b*x + c)
    e = r(5.0/6, 10, 5)
    assert nsimplify(e) == 5*x**2/6 + 10*x + 5


def test_issue_5910():
    t = Symbol('t')
    assert (1/(1 - t)).subs(t, 1) is zoo
    n = t
    d = t - 1
    assert (n/d).subs(t, 1) is zoo
    assert (-n/-d).subs(t, 1) is zoo


def test_issue_5217():
    s = Symbol('s')
    z = (1 - 2*x*x)
    w = (1 + 2*x*x)
    q = 2*x*x*2*y*y
    sub = {2*x*x: s}
    assert w.subs(sub) == 1 + s
    assert z.subs(sub) == 1 - s
    assert q == 4*x**2*y**2
    assert q.subs(sub) == 2*y**2*s


def test_issue_10829():
    assert (4**x).subs(2**x, y) == y**2
    assert (9**x).subs(3**x, y) == y**2


def test_pow_eval_subs_no_cache():
    # Tests pull request 9376 is working
    from sympy.core.cache import clear_cache

    s = 1/sqrt(x**2)
    # This bug only appeared when the cache was turned off.
    # We need to approximate running this test without the cache.
    # This creates approximately the same situation.
    clear_cache()

    # This used to fail with a wrong result.
    # It incorrectly returned 1/sqrt(x**2) before this pull request.
    result = s.subs(sqrt(x**2), y)
    assert result == 1/y


def test_RootOf_issue_10092():
    x = Symbol('x', real=True)
    eq = x**3 - 17*x**2 + 81*x - 118
    r = RootOf(eq, 0)
    assert (x < r).subs(x, r) is S.false


def test_issue_8886():
    from sympy.physics.mechanics import ReferenceFrame as R
    # if something can't be sympified we assume that it
    # doesn't play well with SymPy and disallow the
    # substitution
    v = R('A').x
    raises(SympifyError, lambda: x.subs(x, v))
    raises(SympifyError, lambda: v.subs(v, x))
    assert v.__eq__(x) is False


def test_issue_12657():
    # treat -oo like the atom that it is
    reps = [(-oo, 1), (oo, 2)]
    assert (x < -oo).subs(reps) == (x < 1)
    assert (x < -oo).subs(list(reversed(reps))) == (x < 1)
    reps = [(-oo, 2), (oo, 1)]
    assert (x < oo).subs(reps) == (x < 1)
    assert (x < oo).subs(list(reversed(reps))) == (x < 1)


def test_recurse_Application_args():
    F = Lambda((x, y), exp(2*x + 3*y))
    f = Function('f')
    A = f(x, f(x, x))
    C = F(x, F(x, x))
    assert A.subs(f, F) == A.replace(f, F) == C


def test_Subs_subs():
    assert Subs(x*y, x, x).subs(x, y) == Subs(x*y, x, y)
    assert Subs(x*y, x, x + 1).subs(x, y) == \
        Subs(x*y, x, y + 1)
    assert Subs(x*y, y, x + 1).subs(x, y) == \
        Subs(y**2, y, y + 1)
    a = Subs(x*y*z, (y, x, z), (x + 1, x + z, x))
    b = Subs(x*y*z, (y, x, z), (x + 1, y + z, y))
    assert a.subs(x, y) == b and \
        a.doit().subs(x, y) == a.subs(x, y).doit()
    f = Function('f')
    g = Function('g')
    assert Subs(2*f(x, y) + g(x), f(x, y), 1).subs(y, 2) == Subs(
        2*f(x, y) + g(x), (f(x, y), y), (1, 2))


def test_issue_13333():
    eq = 1/x
    assert eq.subs({"x": '1/2'}) == 2
    assert eq.subs({"x": '(1/2)'}) == 2


def test_issue_15234():
    x, y = symbols('x y', real=True)
    p = 6*x**5 + x**4 - 4*x**3 + 4*x**2 - 2*x + 3
    p_subbed = 6*x**5 - 4*x**3 - 2*x + y**4 + 4*y**2 + 3
    assert p.subs([(x**i, y**i) for i in [2, 4]]) == p_subbed
    x, y = symbols('x y', complex=True)
    p = 6*x**5 + x**4 - 4*x**3 + 4*x**2 - 2*x + 3
    p_subbed = 6*x**5 - 4*x**3 - 2*x + y**4 + 4*y**2 + 3
    assert p.subs([(x**i, y**i) for i in [2, 4]]) == p_subbed


def test_issue_6976():
    x, y = symbols('x y')
    assert (sqrt(x)**3 + sqrt(x) + x + x**2).subs(sqrt(x), y) == \
        y**4 + y**3 + y**2 + y
    assert (x**4 + x**3 + x**2 + x + sqrt(x)).subs(x**2, y) == \
        sqrt(x) + x**3 + x + y**2 + y
    assert x.subs(x**3, y) == x
    assert x.subs(x**Rational(1, 3), y) == y**3

    # More substitutions are possible with nonnegative symbols
    x, y = symbols('x y', nonnegative=True)
    assert (x**4 + x**3 + x**2 + x + sqrt(x)).subs(x**2, y) == \
        y**Rational(1, 4) + y**Rational(3, 2) + sqrt(y) + y**2 + y
    assert x.subs(x**3, y) == y**Rational(1, 3)


def test_issue_11746():
    assert (1/x).subs(x**2, 1) == 1/x
    assert (1/(x**3)).subs(x**2, 1) == x**(-3)
    assert (1/(x**4)).subs(x**2, 1) == 1
    assert (1/(x**3)).subs(x**4, 1) == x**(-3)
    assert (1/(y**5)).subs(x**5, 1) == y**(-5)


def test_issue_17823():
    from sympy.physics.mechanics import dynamicsymbols
    q1, q2 = dynamicsymbols('q1, q2')
    expr = q1.diff().diff()**2*q1 + q1.diff()*q2.diff()
    reps={q1: a, q1.diff(): a*x*y, q1.diff().diff(): z}
    assert expr.subs(reps) == a*x*y*Derivative(q2, t) + a*z**2


def test_issue_19326():
    x, y = [i(t) for i in map(Function, 'xy')]
    assert (x*y).subs({x: 1 + x, y: x}) == (1 + x)*x


def test_issue_19558():
    e = (7*x*cos(x) - 12*log(x)**3)*(-log(x)**4 + 2*sin(x) + 1)**2/ \
    (2*(x*cos(x) - 2*log(x)**3)*(3*log(x)**4 - 7*sin(x) + 3)**2)

    assert e.subs(x, oo) == AccumBounds(-oo, oo)
    assert (sin(x) + cos(x)).subs(x, oo) == AccumBounds(-2, 2)


def test_issue_22033():
    xr = Symbol('xr', real=True)
    e = (1/xr)
    assert e.subs(xr**2, y) == e


def test_guard_against_indeterminate_evaluation():
    eq = x**y
    assert eq.subs([(x, 1), (y, oo)]) == 1  # because 1**y == 1
    assert eq.subs([(y, oo), (x, 1)]) is S.NaN
    assert eq.subs({x: 1, y: oo}) is S.NaN
    assert eq.subs([(x, 1), (y, oo)], simultaneous=True) is S.NaN