def _main(expr):
     _new = []
     for a in expr.args:
         is_V = False
         if isinstance(a, V):
             is_V = True
             a = a.expr
         if a.is_Function:
             name = a.__class__.__name__
             for i in a.args:
                 if i.is_Function:
                     func = i
                     break
             if ',' in name:
                 variables = [eval(i) for i in name.split(',')[1]]
                 a = Derivative(func, *variables)
             if 'V' in name:
                 #TODO remove this V and use class
                 a = V(a)
                 a.function = func
         #TODO add more, maybe all that have args
         elif a.is_Add or a.is_Mul or a.is_Pow:
             a = _main(a)
         if is_V:
             a = V(a)
             a.function = func
         _new.append( a )
     return expr.func(*tuple(_new))

def test_deriv_sub_bug3():
    x = Symbol("x")
    y = Symbol("y")
    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_subs_in_derivative():
    expr = sin(x*exp(y))
    u = Function('u')
    v = Function('v')
    assert Derivative(expr, y).subs(expr, y) == Derivative(y, y)
    assert Derivative(expr, y).subs(y, x).doit() == \
        Derivative(expr, y).doit().subs(y, x)
    assert Derivative(f(x, y), y).subs(y, x) == Subs(Derivative(f(x, y), y), y, x)
    assert Derivative(f(x, y), y).subs(x, y) == Subs(Derivative(f(x, y), y), x, y)
    assert Derivative(f(x, y), y).subs(y, g(x, y)) == Subs(Derivative(f(x, y), y), y, g(x, y)).doit()
    assert Derivative(f(x, y), y).subs(x, g(x, y)) == Subs(Derivative(f(x, y), y), x, g(x, y))
    assert Derivative(f(x, y), g(y)).subs(x, g(x, y)) == Derivative(f(g(x, y), y), g(y))
    assert Derivative(f(u(x), h(y)), h(y)).subs(h(y), g(x, y)) == \
        Subs(Derivative(f(u(x), h(y)), h(y)), h(y), g(x, y)).doit()
    assert Derivative(f(x, y), y).subs(y, z) == Derivative(f(x, z), z)
    assert Derivative(f(x, y), y).subs(y, g(y)) == Derivative(f(x, g(y)), g(y))
    assert Derivative(f(g(x), h(y)), h(y)).subs(h(y), u(y)) == \
        Derivative(f(g(x), u(y)), u(y))
    assert Derivative(f(x, f(x, x)), f(x, x)).subs(
        f, Lambda((x, y), x + y)) == Subs(
        Derivative(z + x, z), z, 2*x)
    assert Subs(Derivative(f(f(x)), x), f, cos).doit() == sin(x)*sin(cos(x))
    assert Subs(Derivative(f(f(x)), f(x)), f, cos).doit() == -sin(cos(x))
    # Issue 13791. No comparison (it's a long formula) but this used to raise an exception.
    assert isinstance(v(x, y, u(x, y)).diff(y).diff(x).diff(y), Expr)
    # This is also related to issues 13791 and 13795; issue 15190
    F = Lambda((x, y), exp(2*x + 3*y))
    abstract = f(x, f(x, x)).diff(x, 2)
    concrete = F(x, F(x, x)).diff(x, 2)
    assert (abstract.subs(f, F).doit() - concrete).simplify() == 0
    # don't introduce a new symbol if not necessary
    assert x in f(x).diff(x).subs(x, 0).atoms()
    # case (4)
    assert Derivative(f(x,f(x,y)), x, y).subs(x, g(y)
        ) == Subs(Derivative(f(x, f(x, y)), x, y), x, g(y))

    assert Derivative(f(x, x), x).subs(x, 0
        ) == Subs(Derivative(f(x, x), x), x, 0)
    # issue 15194
    assert Derivative(f(y, g(x)), (x, z)).subs(z, x
        ) == Derivative(f(y, g(x)), (x, x))

    df = f(x).diff(x)
    assert df.subs(df, 1) is S.One
    assert df.diff(df) is S.One
    dxy = Derivative(f(x, y), x, y)
    dyx = Derivative(f(x, y), y, x)
    assert dxy.subs(Derivative(f(x, y), y, x), 1) is S.One
    assert dxy.diff(dyx) is S.One
    assert Derivative(f(x, y), x, 2, y, 3).subs(
        dyx, g(x, y)) == Derivative(g(x, y), x, 1, y, 2)
    assert Derivative(f(x, x - y), y).subs(x, x + y) == Subs(
        Derivative(f(x, x - y), y), x, x + y)

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_ODE_1():
    l = Function('l')
    r = Symbol('r')

    e = Derivative(l(r),r)/r+Derivative(l(r),r,r)/2- \
        Derivative(l(r),r)**2/2
    sol = dsolve(e, [l(r)])
    assert (e.subs(l(r), sol)).expand() == 0

    e = e*exp(-l(r))/exp(l(r))
    sol = dsolve(e, [l(r)])
    assert (e.subs(l(r), sol)).expand() == 0

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) != None
    p = Wild("p", exclude=[x])
    q = Wild("q", exclude=[x])
    assert (3*fd-1).match(p*fd + q) == {p: 3, q: -1}

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_doit():
    n = Symbol('n', integer = True)
    f = Sum(2 * n * x, (n, 1, 3))
    d = Derivative(f, x)
    assert d.doit() == 12
    assert d.doit(deep = False) == Sum(2*n, (n, 1, 3))

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)) == None
    e = Derivative(f(x), x, x)
    assert e.match(Derivative(f(x), x)) == 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) == None
    e = Derivative(f(x), x, x)+x**2
    assert e.match(a*Derivative(f(x), x) + b) == None
    assert e.match(a*Derivative(f(x), x, x) + b) == {a: 1, b: x**2}

def test_doitdoit():
    done = Derivative(f(x, g(x)), x, g(x)).doit()
    assert done == done.doit()

def test_derivative_subs3():
    x = Symbol('x')
    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_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_issue_15893():
    f = Function('f', real=True)
    x = Symbol('x', real=True)
    eq = Derivative(Abs(f(x)), f(x))
    assert eq.doit() == sign(f(x))