from sympy import abc
from sympy.concrete.summations import Sum
from sympy.core.add import Add
from sympy.core.function import (Derivative, Function, diff)
from sympy.core.mul import Mul
from sympy.core.numbers import (Float, I, Integer, Rational, oo, pi)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, Wild, symbols)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.functions.special.hyper import meijerg
from sympy.polys.polytools import Poly
from sympy.simplify.radsimp import collect
from sympy.simplify.simplify import signsimp

from sympy.testing.pytest import XFAIL


def test_symbol():
    x = Symbol('x')
    a, b, c, p, q = map(Wild, 'abcpq')

    e = x
    assert e.match(x) == {}
    assert e.matches(x) == {}
    assert e.match(a) == {a: x}

    e = Rational(5)
    assert e.match(c) == {c: 5}
    assert e.match(e) == {}
    assert e.match(e + 1) is None


def test_add():
    x, y, a, b, c = map(Symbol, 'xyabc')
    p, q, r = map(Wild, 'pqr')

    e = a + b
    assert e.match(p + b) == {p: a}
    assert e.match(p + a) == {p: b}

    e = 1 + b
    assert e.match(p + b) == {p: 1}

    e = a + b + c
    assert e.match(a + p + c) == {p: b}
    assert e.match(b + p + c) == {p: a}

    e = a + b + c + x
    assert e.match(a + p + x + c) == {p: b}
    assert e.match(b + p + c + x) == {p: a}
    assert e.match(b) is None
    assert e.match(b + p) == {p: a + c + x}
    assert e.match(a + p + c) == {p: b + x}
    assert e.match(b + p + c) == {p: a + x}

    e = 4*x + 5
    assert e.match(4*x + p) == {p: 5}
    assert e.match(3*x + p) == {p: x + 5}
    assert e.match(p*x + 5) == {p: 4}


def test_power():
    x, y, a, b, c = map(Symbol, 'xyabc')
    p, q, r = map(Wild, 'pqr')

    e = (x + y)**a
    assert e.match(p**q) == {p: x + y, q: a}
    assert e.match(p**p) is None

    e = (x + y)**(x + y)
    assert e.match(p**p) == {p: x + y}
    assert e.match(p**q) == {p: x + y, q: x + y}

    e = (2*x)**2
    assert e.match(p*q**r) == {p: 4, q: x, r: 2}

    e = Integer(1)
    assert e.match(x**p) == {p: 0}


def test_match_exclude():
    x = Symbol('x')
    y = Symbol('y')
    p = Wild("p")
    q = Wild("q")
    r = Wild("r")

    e = Rational(6)
    assert e.match(2*p) == {p: 3}

    e = 3/(4*x + 5)
    assert e.match(3/(p*x + q)) == {p: 4, q: 5}

    e = 3/(4*x + 5)
    assert e.match(p/(q*x + r)) == {p: 3, q: 4, r: 5}

    e = 2/(x + 1)
    assert e.match(p/(q*x + r)) == {p: 2, q: 1, r: 1}

    e = 1/(x + 1)
    assert e.match(p/(q*x + r)) == {p: 1, q: 1, r: 1}

    e = 4*x + 5
    assert e.match(p*x + q) == {p: 4, q: 5}

    e = 4*x + 5*y + 6
    assert e.match(p*x + q*y + r) == {p: 4, q: 5, r: 6}

    a = Wild('a', exclude=[x])

    e = 3*x
    assert e.match(p*x) == {p: 3}
    assert e.match(a*x) == {a: 3}

    e = 3*x**2
    assert e.match(p*x) == {p: 3*x}
    assert e.match(a*x) is None

    e = 3*x + 3 + 6/x
    assert e.match(p*x**2 + p*x + 2*p) == {p: 3/x}
    assert e.match(a*x**2 + a*x + 2*a) is None


def test_mul():
    x, y, a, b, c = map(Symbol, 'xyabc')
    p, q = map(Wild, 'pq')

    e = 4*x
    assert e.match(p*x) == {p: 4}
    assert e.match(p*y) is None
    assert e.match(e + p*y) == {p: 0}

    e = a*x*b*c
    assert e.match(p*x) == {p: a*b*c}
    assert e.match(c*p*x) == {p: a*b}

    e = (a + b)*(a + c)
    assert e.match((p + b)*(p + c)) == {p: a}

    e = x
    assert e.match(p*x) == {p: 1}

    e = exp(x)
    assert e.match(x**p*exp(x*q)) == {p: 0, q: 1}

    e = I*Poly(x, x)
    assert e.match(I*p) == {p: x}


def test_mul_noncommutative():
    x, y = symbols('x y')
    A, B, C = symbols('A B C', commutative=False)
    u, v = symbols('u v', cls=Wild)
    w, z = symbols('w z', cls=Wild, commutative=False)

    assert (u*v).matches(x) in ({v: x, u: 1}, {u: x, v: 1})
    assert (u*v).matches(x*y) in ({v: y, u: x}, {u: y, v: x})
    assert (u*v).matches(A) is None
    assert (u*v).matches(A*B) is None
    assert (u*v).matches(x*A) is None
    assert (u*v).matches(x*y*A) is None
    assert (u*v).matches(x*A*B) is None
    assert (u*v).matches(x*y*A*B) is None

    assert (v*w).matches(x) is None
    assert (v*w).matches(x*y) is None
    assert (v*w).matches(A) == {w: A, v: 1}
    assert (v*w).matches(A*B) == {w: A*B, v: 1}
    assert (v*w).matches(x*A) == {w: A, v: x}
    assert (v*w).matches(x*y*A) == {w: A, v: x*y}
    assert (v*w).matches(x*A*B) == {w: A*B, v: x}
    assert (v*w).matches(x*y*A*B) == {w: A*B, v: x*y}

    assert (v*w).matches(-x) is None
    assert (v*w).matches(-x*y) is None
    assert (v*w).matches(-A) == {w: A, v: -1}
    assert (v*w).matches(-A*B) == {w: A*B, v: -1}
    assert (v*w).matches(-x*A) == {w: A, v: -x}
    assert (v*w).matches(-x*y*A) == {w: A, v: -x*y}
    assert (v*w).matches(-x*A*B) == {w: A*B, v: -x}
    assert (v*w).matches(-x*y*A*B) == {w: A*B, v: -x*y}

    assert (w*z).matches(x) is None
    assert (w*z).matches(x*y) is None
    assert (w*z).matches(A) is None
    assert (w*z).matches(A*B) == {w: A, z: B}
    assert (w*z).matches(B*A) == {w: B, z: A}
    assert (w*z).matches(A*B*C) in [{w: A, z: B*C}, {w: A*B, z: C}]
    assert (w*z).matches(x*A) is None
    assert (w*z).matches(x*y*A) is None
    assert (w*z).matches(x*A*B) is None
    assert (w*z).matches(x*y*A*B) is None

    assert (w*A).matches(A) is None
    assert (A*w*B).matches(A*B) is None

    assert (u*w*z).matches(x) is None
    assert (u*w*z).matches(x*y) is None
    assert (u*w*z).matches(A) is None
    assert (u*w*z).matches(A*B) == {u: 1, w: A, z: B}
    assert (u*w*z).matches(B*A) == {u: 1, w: B, z: A}
    assert (u*w*z).matches(x*A) is None
    assert (u*w*z).matches(x*y*A) is None
    assert (u*w*z).matches(x*A*B) == {u: x, w: A, z: B}
    assert (u*w*z).matches(x*B*A) == {u: x, w: B, z: A}
    assert (u*w*z).matches(x*y*A*B) == {u: x*y, w: A, z: B}
    assert (u*w*z).matches(x*y*B*A) == {u: x*y, w: B, z: A}

    assert (u*A).matches(x*A) == {u: x}
    assert (u*A).matches(x*A*B) is None
    assert (u*B).matches(x*A) is None
    assert (u*A*B).matches(x*A*B) == {u: x}
    assert (u*A*B).matches(x*B*A) is None
    assert (u*A*B).matches(x*A) is None

    assert (u*w*A).matches(x*A*B) is None
    assert (u*w*B).matches(x*A*B) == {u: x, w: A}

    assert (u*v*A*B).matches(x*A*B) in [{u: x, v: 1}, {v: x, u: 1}]
    assert (u*v*A*B).matches(x*B*A) is None
    assert (u*v*A*B).matches(u*v*A*C) is None


def test_mul_noncommutative_mismatch():
    A, B, C = symbols('A B C', commutative=False)
    w = symbols('w', cls=Wild, commutative=False)

    assert (w*B*w).matches(A*B*A) == {w: A}
    assert (w*B*w).matches(A*C*B*A*C) == {w: A*C}
    assert (w*B*w).matches(A*C*B*A*B) is None
    assert (w*B*w).matches(A*B*C) is None
    assert (w*w*C).matches(A*B*C) is None


def test_mul_noncommutative_pow():
    A, B, C = symbols('A B C', commutative=False)
    w = symbols('w', cls=Wild, commutative=False)

    assert (A*B*w).matches(A*B**2) == {w: B}
    assert (A*(B**2)*w*(B**3)).matches(A*B**8) == {w: B**3}
    assert (A*B*w*C).matches(A*(B**4)*C) == {w: B**3}

    assert (A*B*(w**(-1))).matches(A*B*(C**(-1))) == {w: C}
    assert (A*(B*w)**(-1)*C).matches(A*(B*C)**(-1)*C) == {w: C}

    assert ((w**2)*B*C).matches((A**2)*B*C) == {w: A}
    assert ((w**2)*B*(w**3)).matches((A**2)*B*(A**3)) == {w: A}
    assert ((w**2)*B*(w**4)).matches((A**2)*B*(A**2)) is None

def test_complex():
    a, b, c = map(Symbol, 'abc')
    x, y = map(Wild, 'xy')

    assert (1 + I).match(x + I) == {x: 1}
    assert (a + I).match(x + I) == {x: a}
    assert (2*I).match(x*I) == {x: 2}
    assert (a*I).match(x*I) == {x: a}
    assert (a*I).match(x*y) == {x: I, y: a}
    assert (2*I).match(x*y) == {x: 2, y: I}
    assert (a + b*I).match(x + y*I) == {x: a, y: b}


def test_functions():
    from sympy.core.function import WildFunction
    x = Symbol('x')
    g = WildFunction('g')
    p = Wild('p')
    q = Wild('q')

    f = cos(5*x)
    notf = x
    assert f.match(p*cos(q*x)) == {p: 1, q: 5}
    assert f.match(p*g) == {p: 1, g: cos(5*x)}
    assert notf.match(g) is None


@XFAIL
def test_functions_X1():
    from sympy.core.function import WildFunction
    x = Symbol('x')
    g = WildFunction('g')
    p = Wild('p')
    q = Wild('q')

    f = cos(5*x)
    assert f.match(p*g(q*x)) == {p: 1, g: cos, q: 5}


def test_interface():
    x, y = map(Symbol, 'xy')
    p, q = map(Wild, 'pq')

    assert (x + 1).match(p + 1) == {p: x}
    assert (x*3).match(p*3) == {p: x}
    assert (x**3).match(p**3) == {p: x}
    assert (x*cos(y)).match(p*cos(q)) == {p: x, q: y}

    assert (x*y).match(p*q) in [{p:x, q:y}, {p:y, q:x}]
    assert (x + y).match(p + q) in [{p:x, q:y}, {p:y, q:x}]
    assert (x*y + 1).match(p*q) in [{p:1, q:1 + x*y}, {p:1 + x*y, q:1}]


def test_derivative1():
    x, y = map(Symbol, 'xy')
    p, q = map(Wild, 'pq')

    f = Function('f', nargs=1)
    fd = Derivative(f(x), x)

    assert fd.match(p) == {p: fd}
    assert (fd + 1).match(p + 1) == {p: fd}
    assert (fd).match(fd) == {}
    assert (3*fd).match(p*fd) is not None
    assert (3*fd - 1).match(p*fd + q) == {p: 3, q: -1}


def test_derivative_bug1():
    f = Function("f")
    x = Symbol("x")
    a = Wild("a", exclude=[f, x])
    b = Wild("b", exclude=[f])
    pattern = a * Derivative(f(x), x, x) + b
    expr = Derivative(f(x), x) + x**2
    d1 = {b: x**2}
    d2 = pattern.xreplace(d1).matches(expr, d1)
    assert d2 is None


def test_derivative2():
    f = Function("f")
    x = Symbol("x")
    a = Wild("a", exclude=[f, x])
    b = Wild("b", exclude=[f])
    e = Derivative(f(x), x)
    assert e.match(Derivative(f(x), x)) == {}
    assert e.match(Derivative(f(x), x, x)) is None
    e = Derivative(f(x), x, x)
    assert e.match(Derivative(f(x), x)) is None
    assert e.match(Derivative(f(x), x, x)) == {}
    e = Derivative(f(x), x) + x**2
    assert e.match(a*Derivative(f(x), x) + b) == {a: 1, b: x**2}
    assert e.match(a*Derivative(f(x), x, x) + b) is None
    e = Derivative(f(x), x, x) + x**2
    assert e.match(a*Derivative(f(x), x) + b) is None
    assert e.match(a*Derivative(f(x), x, x) + b) == {a: 1, b: x**2}


def test_match_deriv_bug1():
    n = Function('n')
    l = Function('l')

    x = Symbol('x')
    p = Wild('p')

    e = diff(l(x), x)/x - diff(diff(n(x), x), x)/2 - \
        diff(n(x), x)**2/4 + diff(n(x), x)*diff(l(x), x)/4
    e = e.subs(n(x), -l(x)).doit()
    t = x*exp(-l(x))
    t2 = t.diff(x, x)/t
    assert e.match( (p*t2).expand() ) == {p: Rational(-1, 2)}


def test_match_bug2():
    x, y = map(Symbol, 'xy')
    p, q, r = map(Wild, 'pqr')
    res = (x + y).match(p + q + r)
    assert (p + q + r).subs(res) == x + y


def test_match_bug3():
    x, a, b = map(Symbol, 'xab')
    p = Wild('p')
    assert (b*x*exp(a*x)).match(x*exp(p*x)) is None


def test_match_bug4():
    x = Symbol('x')
    p = Wild('p')
    e = x
    assert e.match(-p*x) == {p: -1}


def test_match_bug5():
    x = Symbol('x')
    p = Wild('p')
    e = -x
    assert e.match(-p*x) == {p: 1}


def test_match_bug6():
    x = Symbol('x')
    p = Wild('p')
    e = x
    assert e.match(3*p*x) == {p: Rational(1)/3}


def test_match_polynomial():
    x = Symbol('x')
    a = Wild('a', exclude=[x])
    b = Wild('b', exclude=[x])
    c = Wild('c', exclude=[x])
    d = Wild('d', exclude=[x])

    eq = 4*x**3 + 3*x**2 + 2*x + 1
    pattern = a*x**3 + b*x**2 + c*x + d
    assert eq.match(pattern) == {a: 4, b: 3, c: 2, d: 1}
    assert (eq - 3*x**2).match(pattern) == {a: 4, b: 0, c: 2, d: 1}
    assert (x + sqrt(2) + 3).match(a + b*x + c*x**2) == \
        {b: 1, a: sqrt(2) + 3, c: 0}


def test_exclude():
    x, y, a = map(Symbol, 'xya')
    p = Wild('p', exclude=[1, x])
    q = Wild('q')
    r = Wild('r', exclude=[sin, y])

    assert sin(x).match(r) is None
    assert cos(y).match(r) is None

    e = 3*x**2 + y*x + a
    assert e.match(p*x**2 + q*x + r) == {p: 3, q: y, r: a}

    e = x + 1
    assert e.match(x + p) is None
    assert e.match(p + 1) is None
    assert e.match(x + 1 + p) == {p: 0}

    e = cos(x) + 5*sin(y)
    assert e.match(r) is None
    assert e.match(cos(y) + r) is None
    assert e.match(r + p*sin(q)) == {r: cos(x), p: 5, q: y}


def test_floats():
    a, b = map(Wild, 'ab')

    e = cos(0.12345, evaluate=False)**2
    r = e.match(a*cos(b)**2)
    assert r == {a: 1, b: Float(0.12345)}


def test_Derivative_bug1():
    f = Function("f")
    x = abc.x
    a = Wild("a", exclude=[f(x)])
    b = Wild("b", exclude=[f(x)])
    eq = f(x).diff(x)
    assert eq.match(a*Derivative(f(x), x) + b) == {a: 1, b: 0}


def test_match_wild_wild():
    p = Wild('p')
    q = Wild('q')
    r = Wild('r')

    assert p.match(q + r) in [ {q: p, r: 0}, {q: 0, r: p} ]
    assert p.match(q*r) in [ {q: p, r: 1}, {q: 1, r: p} ]

    p = Wild('p')
    q = Wild('q', exclude=[p])
    r = Wild('r')

    assert p.match(q + r) == {q: 0, r: p}
    assert p.match(q*r) == {q: 1, r: p}

    p = Wild('p')
    q = Wild('q', exclude=[p])
    r = Wild('r', exclude=[p])

    assert p.match(q + r) is None
    assert p.match(q*r) is None


def test__combine_inverse():
    x, y = symbols("x y")
    assert Mul._combine_inverse(x*I*y, x*I) == y
    assert Mul._combine_inverse(x*x**(1 + y), x**(1 + y)) == x
    assert Mul._combine_inverse(x*I*y, y*I) == x
    assert Mul._combine_inverse(oo*I*y, y*I) is oo
    assert Mul._combine_inverse(oo*I*y, oo*I) == y
    assert Mul._combine_inverse(oo*I*y, oo*I) == y
    assert Mul._combine_inverse(oo*y, -oo) == -y
    assert Mul._combine_inverse(-oo*y, oo) == -y
    assert Mul._combine_inverse((1-exp(x/y)),(exp(x/y)-1)) == -1
    assert Add._combine_inverse(oo, oo) is S.Zero
    assert Add._combine_inverse(oo*I, oo*I) is S.Zero
    assert Add._combine_inverse(x*oo, x*oo) is S.Zero
    assert Add._combine_inverse(-x*oo, -x*oo) is S.Zero
    assert Add._combine_inverse((x - oo)*(x + oo), -oo)


def test_issue_3773():
    x = symbols('x')
    z, phi, r = symbols('z phi r')
    c, A, B, N = symbols('c A B N', cls=Wild)
    l = Wild('l', exclude=(0,))

    eq = z * sin(2*phi) * r**7
    matcher = c * sin(phi*N)**l * r**A * log(r)**B

    assert eq.match(matcher) == {c: z, l: 1, N: 2, A: 7, B: 0}
    assert (-eq).match(matcher) == {c: -z, l: 1, N: 2, A: 7, B: 0}
    assert (x*eq).match(matcher) == {c: x*z, l: 1, N: 2, A: 7, B: 0}
    assert (-7*x*eq).match(matcher) == {c: -7*x*z, l: 1, N: 2, A: 7, B: 0}

    matcher = c*sin(phi*N)**l * r**A

    assert eq.match(matcher) == {c: z, l: 1, N: 2, A: 7}
    assert (-eq).match(matcher) == {c: -z, l: 1, N: 2, A: 7}
    assert (x*eq).match(matcher) == {c: x*z, l: 1, N: 2, A: 7}
    assert (-7*x*eq).match(matcher) == {c: -7*x*z, l: 1, N: 2, A: 7}


def test_issue_3883():
    from sympy.abc import gamma, mu, x
    f = (-gamma * (x - mu)**2 - log(gamma) + log(2*pi))/2
    a, b, c = symbols('a b c', cls=Wild, exclude=(gamma,))

    assert f.match(a * log(gamma) + b * gamma + c) == \
        {a: Rational(-1, 2), b: -(-mu + x)**2/2, c: log(2*pi)/2}
    assert f.expand().collect(gamma).match(a * log(gamma) + b * gamma + c) == \
        {a: Rational(-1, 2), b: (-(x - mu)**2/2).expand(), c: (log(2*pi)/2).expand()}
    g1 = Wild('g1', exclude=[gamma])
    g2 = Wild('g2', exclude=[gamma])
    g3 = Wild('g3', exclude=[gamma])
    assert f.expand().match(g1 * log(gamma) + g2 * gamma + g3) == \
    {g3: log(2)/2 + log(pi)/2, g1: Rational(-1, 2), g2: -mu**2/2 + mu*x - x**2/2}


def test_issue_4418():
    x = Symbol('x')
    a, b, c = symbols('a b c', cls=Wild, exclude=(x,))
    f, g = symbols('f g', cls=Function)

    eq = diff(g(x)*f(x).diff(x), x)

    assert eq.match(
        g(x).diff(x)*f(x).diff(x) + g(x)*f(x).diff(x, x) + c) == {c: 0}
    assert eq.match(a*g(x).diff(
        x)*f(x).diff(x) + b*g(x)*f(x).diff(x, x) + c) == {a: 1, b: 1, c: 0}


def test_issue_4700():
    f = Function('f')
    x = Symbol('x')
    a, b = symbols('a b', cls=Wild, exclude=(f(x),))

    p = a*f(x) + b
    eq1 = sin(x)
    eq2 = f(x) + sin(x)
    eq3 = f(x) + x + sin(x)
    eq4 = x + sin(x)

    assert eq1.match(p) == {a: 0, b: sin(x)}
    assert eq2.match(p) == {a: 1, b: sin(x)}
    assert eq3.match(p) == {a: 1, b: x + sin(x)}
    assert eq4.match(p) == {a: 0, b: x + sin(x)}


def test_issue_5168():
    a, b, c = symbols('a b c', cls=Wild)
    x = Symbol('x')
    f = Function('f')

    assert x.match(a) == {a: x}
    assert x.match(a*f(x)**c) == {a: x, c: 0}
    assert x.match(a*b) == {a: 1, b: x}
    assert x.match(a*b*f(x)**c) == {a: 1, b: x, c: 0}

    assert (-x).match(a) == {a: -x}
    assert (-x).match(a*f(x)**c) == {a: -x, c: 0}
    assert (-x).match(a*b) == {a: -1, b: x}
    assert (-x).match(a*b*f(x)**c) == {a: -1, b: x, c: 0}

    assert (2*x).match(a) == {a: 2*x}
    assert (2*x).match(a*f(x)**c) == {a: 2*x, c: 0}
    assert (2*x).match(a*b) == {a: 2, b: x}
    assert (2*x).match(a*b*f(x)**c) == {a: 2, b: x, c: 0}

    assert (-2*x).match(a) == {a: -2*x}
    assert (-2*x).match(a*f(x)**c) == {a: -2*x, c: 0}
    assert (-2*x).match(a*b) == {a: -2, b: x}
    assert (-2*x).match(a*b*f(x)**c) == {a: -2, b: x, c: 0}


def test_issue_4559():
    x = Symbol('x')
    e = Symbol('e')
    w = Wild('w', exclude=[x])
    y = Wild('y')

    # this is as it should be

    assert (3/x).match(w/y) == {w: 3, y: x}
    assert (3*x).match(w*y) == {w: 3, y: x}
    assert (x/3).match(y/w) == {w: 3, y: x}
    assert (3*x).match(y/w) == {w: S.One/3, y: x}
    assert (3*x).match(y/w) == {w: Rational(1, 3), y: x}

    # these could be allowed to fail

    assert (x/3).match(w/y) == {w: S.One/3, y: 1/x}
    assert (3*x).match(w/y) == {w: 3, y: 1/x}
    assert (3/x).match(w*y) == {w: 3, y: 1/x}

    # Note that solve will give
    # multiple roots but match only gives one:
    #
    # >>> solve(x**r-y**2,y)
    # [-x**(r/2), x**(r/2)]

    r = Symbol('r', rational=True)
    assert (x**r).match(y**2) == {y: x**(r/2)}
    assert (x**e).match(y**2) == {y: sqrt(x**e)}

    # since (x**i = y) -> x = y**(1/i) where i is an integer
    # the following should also be valid as long as y is not
    # zero when i is negative.

    a = Wild('a')

    e = S.Zero
    assert e.match(a) == {a: e}
    assert e.match(1/a) is None
    assert e.match(a**.3) is None

    e = S(3)
    assert e.match(1/a) == {a: 1/e}
    assert e.match(1/a**2) == {a: 1/sqrt(e)}
    e = pi
    assert e.match(1/a) == {a: 1/e}
    assert e.match(1/a**2) == {a: 1/sqrt(e)}
    assert (-e).match(sqrt(a)) is None
    assert (-e).match(a**2) == {a: I*sqrt(pi)}

# The pattern matcher doesn't know how to handle (x - a)**2 == (a - x)**2. To
# avoid ambiguity in actual applications, don't put a coefficient (including a
# minus sign) in front of a wild.
@XFAIL
def test_issue_4883():
    a = Wild('a')
    x = Symbol('x')

    e = [i**2 for i in (x - 2, 2 - x)]
    p = [i**2 for i in (x - a, a- x)]
    for eq in e:
        for pat in p:
            assert eq.match(pat) == {a: 2}


def test_issue_4319():
    x, y = symbols('x y')

    p = -x*(S.One/8 - y)
    ans = {S.Zero, y - S.One/8}

    def ok(pat):
        assert set(p.match(pat).values()) == ans

    ok(Wild("coeff", exclude=[x])*x + Wild("rest"))
    ok(Wild("w", exclude=[x])*x + Wild("rest"))
    ok(Wild("coeff", exclude=[x])*x + Wild("rest"))
    ok(Wild("w", exclude=[x])*x + Wild("rest"))
    ok(Wild("e", exclude=[x])*x + Wild("rest"))
    ok(Wild("ress", exclude=[x])*x + Wild("rest"))
    ok(Wild("resu", exclude=[x])*x + Wild("rest"))


def test_issue_3778():
    p, c, q = symbols('p c q', cls=Wild)
    x = Symbol('x')

    assert (sin(x)**2).match(sin(p)*sin(q)*c) == {q: x, c: 1, p: x}
    assert (2*sin(x)).match(sin(p) + sin(q) + c) == {q: x, c: 0, p: x}


def test_issue_6103():
    x = Symbol('x')
    a = Wild('a')
    assert (-I*x*oo).match(I*a*oo) == {a: -x}


def test_issue_3539():
    a = Wild('a')
    x = Symbol('x')
    assert (x - 2).match(a - x) is None
    assert (6/x).match(a*x) is None
    assert (6/x**2).match(a/x) == {a: 6/x}

def test_gh_issue_2711():
    x = Symbol('x')
    f = meijerg(((), ()), ((0,), ()), x)
    a = Wild('a')
    b = Wild('b')

    assert f.find(a) == {(S.Zero,), ((), ()), ((S.Zero,), ()), x, S.Zero,
                             (), meijerg(((), ()), ((S.Zero,), ()), x)}
    assert f.find(a + b) == \
        {meijerg(((), ()), ((S.Zero,), ()), x), x, S.Zero}
    assert f.find(a**2) == {meijerg(((), ()), ((S.Zero,), ()), x), x}


def test_issue_17354():
    from sympy.core.symbol import (Wild, symbols)
    x, y = symbols("x y", real=True)
    a, b = symbols("a b", cls=Wild)
    assert ((0 <= x).reversed | (y <= x)).match((1/a <= b) | (a <= b)) is None


def test_match_issue_17397():
    f = Function("f")
    x = Symbol("x")
    a3 = Wild('a3', exclude=[f(x), f(x).diff(x), f(x).diff(x, 2)])
    b3 = Wild('b3', exclude=[f(x), f(x).diff(x), f(x).diff(x, 2)])
    c3 = Wild('c3', exclude=[f(x), f(x).diff(x), f(x).diff(x, 2)])
    deq = a3*(f(x).diff(x, 2)) + b3*f(x).diff(x) + c3*f(x)

    eq = (x-2)**2*(f(x).diff(x, 2)) + (x-2)*(f(x).diff(x)) + ((x-2)**2 - 4)*f(x)
    r = collect(eq, [f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq)
    assert r == {a3: (x - 2)**2, c3: (x - 2)**2 - 4, b3: x - 2}

    eq =x*f(x) + x*Derivative(f(x), (x, 2)) - 4*f(x) + Derivative(f(x), x) \
        - 4*Derivative(f(x), (x, 2)) - 2*Derivative(f(x), x)/x + 4*Derivative(f(x), (x, 2))/x
    r = collect(eq, [f(x).diff(x, 2), f(x).diff(x), f(x)]).match(deq)
    assert r == {a3: x - 4 + 4/x, b3: 1 - 2/x, c3: x - 4}


def test_match_issue_21942():
    a, r, w = symbols('a, r, w', nonnegative=True)
    p = symbols('p', positive=True)
    g_ = Wild('g')
    pattern = g_ ** (1 / (1 - p))
    eq = (a * r ** (1 - p) + w ** (1 - p) * (1 - a)) ** (1 / (1 - p))
    m = {g_: a * r ** (1 - p) + w ** (1 - p) * (1 - a)}
    assert pattern.matches(eq) == m
    assert (-pattern).matches(-eq) == m
    assert pattern.matches(signsimp(eq)) is None


def test_match_terms():
    X, Y = map(Wild, "XY")
    x, y, z = symbols('x y z')
    assert (5*y - x).match(5*X - Y) == {X: y, Y: x}
    # 15907
    assert (x + (y - 1)*z).match(x + X*z) == {X: y - 1}
    # 20747
    assert (x - log(x/y)*(1-exp(x/y))).match(x - log(X/y)*(1-exp(x/y))) == {X: x}


def test_match_bound():
    V, W = map(Wild, "VW")
    x, y = symbols('x y')
    assert Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, W))) is None
    assert Sum(x, (x, 1, 2)).match(Sum(V, (V, 1, W))) == {W: 2, V:x}
    assert Sum(x, (x, 1, 2)).match(Sum(V, (V, 1, 2))) == {V:x}


def test_issue_22462():
    x, f = symbols('x'), Function('f')
    n, Q = symbols('n Q', cls=Wild)
    pattern = -Q*f(x)**n
    eq = 5*f(x)**2
    assert pattern.matches(eq) == {n: 2, Q: -5}
