def prove(Eq):
from axiom import calculus, algebra
from axiom.calculus.lt.is_continuous.is_differentiable.eq.imply.any_eq.Rolle import is_differentiable
from axiom.calculus.integral.intermediate_value_theorem import is_continuous
a = Symbol.a(real=True)
b = Symbol.b(real=True)
f = Function.f(shape=(), real=True)
Eq << apply(a <= b, is_continuous(f, a, b),
is_differentiable(f, a, b, open=False))
Eq << algebra.cond.given.suffice.split.apply(Eq[-1], cond=b > a)
Eq << Suffice(b <= a, Equal(a, b), plausible=True)
Eq << algebra.suffice.given.ou.apply(Eq[-1]).reversed
Eq <<= Eq[-2] & Eq[-1]
Eq << Eq[-1].this.rhs.apply(algebra.et.given.et.subs.eq)
Eq << algebra.all.imply.all.limits.restrict.apply(
Eq[2], Interval(a, b, left_open=True, right_open=True))
Eq << algebra.cond.imply.suffice.apply(Eq[1] & Eq[-1], cond=b > a)
Eq << algebra.suffice.imply.suffice.et.apply(Eq[-1])
Eq << Eq[-1].this.rhs.args[0].reversed
Eq << Eq[-1].this.rhs.apply(calculus.lt.is_continuous.is_differentiable.
imply.any_eq.mean_value_theorem.Lagrange)
Eq << Eq[-1].this.rhs.apply(algebra.any.imply.any.limits.relax,
domain=Interval(a, b))
def prove(Eq):
from axiom import calculus, algebra
a = Symbol.a(real=True)
b = Symbol.b(real=True)
x = Symbol.x(real=True)
f = Function.f(real=True)
from axiom.calculus.lt.is_continuous.is_differentiable.eq.imply.any_eq.Rolle import is_differentiable
Eq << apply(is_differentiable(f, a, b, open=False))
xi = Symbol.xi(domain=Interval(a, b))
Eq << Contains(Subs(Eq[0].function.lhs, x, xi), Eq[0].function.rhs, plausible=True)
Eq << Eq[-1].this.lhs.simplify()
Eq << algebra.cond.given.all.apply(Eq[-1], xi)
Eq << Eq[-2].this.lhs.apply(calculus.subs.to.limit)
Eq << Contains(Limit[x:xi](x - xi), Reals, plausible=True)
Eq << Eq[-1].this.lhs.simplify()
Eq << calculus.is_limited.is_limited.imply.eq.algebraic_limit_theorem.mul.apply(Eq[-1], Eq[-2])
Eq << Eq[-1].this.rhs.args[1].simplify()
Eq << Eq[-1].this.lhs.simplify().reversed
Eq << algebra.cond.imply.all.apply(Eq[-1], xi)
def apply(is_positive, x0, x1, x=None):
fx, (x_, d) = is_positive.of(Derivative > 0)
assert d == 2
domain = x_.domain
assert x0.domain == domain == x1.domain
from axiom.calculus.lt.is_continuous.is_differentiable.eq.imply.any_eq.Rolle import is_differentiable
f = lambda x: fx._subs(x_, x)
return is_differentiable(f, x0, x1, open=False, x=x)
def apply(all_is_positive):
(fx, (x, d)), (x_, domain) = all_is_positive.of(All[Derivative > 0])
assert x == x_
assert d == 2
assert domain.left_open and domain.right_open
a, b = domain.of(Interval)
from axiom.calculus.lt.is_continuous.is_differentiable.eq.imply.any_eq.Rolle import is_differentiable
f = lambda x: fx._subs(x_, x)
return is_differentiable(f, a, b)
def apply(contains0, contains1, all_is_positive):
x0, _domain = contains0.of(Contains)
x1, __domain = contains1.of(Contains)
(fx, (x, d)), (x_, domain) = all_is_positive.of(All[Derivative > 0])
assert x == x_
assert d == 2
assert domain == _domain == __domain
assert domain.left_open and domain.right_open
a, b = domain.of(Interval)
from axiom.calculus.lt.is_continuous.is_differentiable.eq.imply.any_eq.Rolle import is_differentiable
f = lambda x: fx._subs(x_, x)
return is_differentiable(f, x0, x1, open=False)
def prove(Eq):
from axiom import calculus
from axiom.calculus.lt.is_continuous.is_differentiable.eq.imply.any_eq.Rolle import is_differentiable
from axiom.calculus.integral.intermediate_value_theorem import is_continuous
a = Symbol.a(real=True)
b = Symbol.b(domain=Interval(a, oo))
f = Function.f(shape=(), real=True)
Eq << apply(is_continuous(f, a, b), is_differentiable(f, a, b, open=False))
Eq << LessEqual(a, b, plausible=True)
Eq << calculus.le.is_continuous.is_differentiable.imply.any_eq.mean_value_theorem.Lagrange.close.apply(Eq[-1], Eq[0], Eq[1])
def prove(Eq):
from axiom import calculus, algebra, sets
from axiom.calculus.lt.is_continuous.is_differentiable.eq.imply.any_eq.Rolle import is_differentiable
from axiom.calculus.integral.intermediate_value_theorem import is_continuous
a = Symbol.a(real=True, given=True)
b = Symbol.b(real=True, given=True)
f = Function.f(shape=(), real=True)
Eq << apply(a < b, is_continuous(f, a, b), is_differentiable(f, a, b))
def g(x):
return (b - a) * f(x) - (f(b) - f(a)) * x
g = Function.g(eval=g)
x = Symbol.x(real=True)
Eq.g_definition = g(x).this.defun()
Eq << Eq.g_definition.subs(x, a)
Eq << Eq.g_definition.subs(x, b)
Eq << Eq[-1] - Eq[-2]
Eq.equal = Eq[-1].this.rhs.expand()
Eq.is_continuous = Eq[1]._subs(f, g).copy(plausible=True)
Eq << Eq.is_continuous.this.function.lhs.expr.defun()
Eq << Eq[-1].this.function.rhs.defun()
Eq << Eq[-1].this.function.lhs.apply(calculus.limit.to.add)
Eq << Eq[-1].this.function.lhs.args[0].apply(calculus.limit.to.mul)
Eq << Eq[-1].this.find(Limit).apply(calculus.limit.to.mul)
Eq <<= Eq[1] & Eq[-1]
Eq <<= Eq[-1].this.function.apply(algebra.et.given.et.subs.eq)
Eq << algebra.all_et.given.all.apply(Eq[-1])
Eq << Eq[-1].this.function.simplify()
Eq << Eq[-1].this.function.rhs.apply(algebra.mul.distribute)
Eq.is_differentiable = Eq[2]._subs(f, g).copy(plausible=True)
Eq << Eq.is_differentiable.this.function.lhs.expr.defun()
Eq << Eq[-1].this.function.lhs.apply(calculus.derivative.to.add)
Eq << Eq[-1].this.function.apply(sets.contains.given.contains.add, f(b) - f(a))
Eq << sets.all.imply.all_et.apply(Eq[2], simplify=None)
Eq << Eq[-1].this.find(Unequal).apply(sets.interval_is_nonemptyset.imply.is_positive, simplify=None)
Eq << Eq[-1].this.function.apply(sets.is_positive.contains.imply.contains.mul)
Eq << calculus.lt.is_continuous.is_differentiable.eq.imply.any_eq.Rolle.apply(Eq[0], Eq.is_continuous, Eq.is_differentiable, Eq.equal)
Eq << Eq[-1].this.function.lhs.expr.defun()
Eq << Eq[-1].this.function.lhs.apply(calculus.derivative.to.add)
Eq << Eq[-1].this.function - f(a)