def apply(le, is_continuous, is_differentiable):
a, b = le.of(LessEqual)
from axiom.calculus.lt.is_continuous.is_differentiable.eq.imply.any_eq.Rolle import of_differentiable, of_continuous
fz, (z, _a, _b) = of_continuous(is_continuous)
_fz, (_z, __a, __b) = of_differentiable(is_differentiable, open=False)
assert _fz == fz
assert _z == z
assert _a == a == __a
assert _b == b == __b
fa = fz._subs(z, a)
fb = fz._subs(z, b)
return Any[z:Interval(a, b)](Equal(fb - fa, (b - a) * Derivative(fz, z)))
def apply(*given):
is_continuous, is_differentiable, equal = given
from axiom.calculus.lt.is_continuous.is_differentiable.eq.imply.any_eq.Rolle import of_continuous, of_differentiable
fz, (z, a, b) = of_continuous(is_continuous)
_fz, (_z, _a, _b) = of_differentiable(is_differentiable)
assert _fz == fz
assert _z == z
assert _a == a
assert _b == b
assert a < b
fa, fb = equal.of(Equal)
assert fz._subs(z, a) == fa
assert fz._subs(z, b) == fb
return Any[z:Interval(a, b, left_open=True, right_open=True)](Equal(
Derivative(fz, z), 0))
def apply(*given):
from axiom.calculus.lt.is_continuous.is_differentiable.eq.imply.any_eq.Rolle import of_differentiable, of_continuous
is_continuous, is_differentiable = given
fz, (z, a, b) = of_continuous(is_continuous)
_fz, (_z, _a, _b) = of_differentiable(is_differentiable)
assert _fz == fz
assert _z == z
assert _a == a
assert _b == b
assert a < b
fa = fz._subs(z, a)
fb = fz._subs(z, b)
return Any[z:Interval(a, b, left_open=True, right_open=True)](Equal(
fb - fa, (b - a) * Derivative(fz, z)))
def apply(lt, is_continuous, is_differentiable):
from axiom.calculus.lt.is_continuous.is_differentiable.eq.imply.any_eq.Rolle import of_differentiable, of_continuous
a, b = lt.of(Less)
if is_continuous.function.is_Contains:
is_continuous, is_differentiable = is_differentiable, is_continuous
fz, (z, _a, _b) = of_continuous(is_continuous)
_fz, (_z, __a, __b) = of_differentiable(is_differentiable)
assert _fz == fz
assert _z == z
assert _a == a == __a
assert _b == b == __b
fa = fz._subs(z, a)
fb = fz._subs(z, b)
return Any[z:Interval(a, b, left_open=True, right_open=True)](Equal(fb - fa, (b - a) * Derivative(fz, z)))