def apply(given):
(x, _j), (j, n1) = given.of(All[Indexed >= 1, Tuple[1, Expr]])
n = n1 - 1
assert _j == j
return GreaterEqual(K(x[:n + 1]), K(x[:n]))
def apply(x):
n = x.shape[0]
n -= 1
assert x.is_positive
assert n > 0
return Equal(
K(x[1:n + 1]) / H(x[1:n + 1]),
H(x[:n + 1]) / K(x[:n + 1]) - x[0])
def apply(given):
(x, _j), (j, n) = given.of(All[Indexed > 0, Tuple[0, Expr]])
offset = _j - j
if offset != 0:
assert not offset._has(j)
x = x[offset:]
n = n - 1
assert n > 0
return Equal(
K(x[1:n + 1]) / H(x[1:n + 1]),
H(x[:n + 1]) / K(x[:n + 1]) - x[0])
def apply(given):
(x, _j), (j, n) = given.of(All[Indexed > 0, Tuple[0, Expr]])
offset = _j - j
if offset != 0:
assert not offset._has(j)
x = x[offset:]
return Equal(alpha(x[:n]), H(x[:n]) / K(x[:n]))
def apply(self):
assert self.is_alpha
x = self.arg
assert x.is_positive
assert x.shape[0].is_finite
return Equal(self, H(x) / K(x))
def prove(Eq):
from axiom import discrete, algebra
x = Symbol.x(real=True, positive=True, shape=(oo, ))
n = Symbol.n(integer=True)
Eq.hypothesis = apply(alpha(x[:n + 1]))
Eq.n2 = Suffice(n >= 2, Eq.hypothesis, plausible=True)
Eq << Eq.n2.this.apply(algebra.suffice.to.all)
_n = Symbol.n(domain=Range(2, oo))
Eq << discrete.alpha.to.mul.HK.induct.apply(alpha(x[:_n + 1]))
Eq << algebra.cond.imply.all.apply(Eq[-1], _n)
Eq.n1 = Suffice(Equal(n, 1), Eq.hypothesis, plausible=True)
Eq << Eq.n1.this.apply(algebra.suffice.subs)
Eq << Eq[-1].this.rhs.subs(
alpha(x[:2]).this.defun(),
alpha(x[1]).this.defun(),
H(x[:2]).this.defun(),
K(x[:2]).this.defun())
Eq << Eq[-1].this.rhs.rhs.apart(x=x[1])
Eq.n1 = algebra.suffice.suffice.imply.suffice.ou.apply(Eq.n1, Eq.n2)
Eq.n0 = Suffice(Equal(n, 0), Eq.hypothesis, plausible=True)
Eq << Eq.n0.this.apply(algebra.suffice.subs)
Eq << Eq[-1].this.rhs.subs(
alpha(x[0]).this.defun(),
H(x[0]).this.defun(),
K(x[0]).this.defun())
Eq << algebra.suffice.suffice.imply.suffice.ou.apply(Eq.n1, Eq.n0)
Eq << Eq[-1].this.apply(algebra.suffice.to.all, wrt=n)
Eq << Eq[-1].simplify()
def apply(x, wrt=None):
n = x.shape[0]
assert n.is_finite
assert n >= 3
if wrt is None:
i = x.generate_var(integer=True, var='i')
else:
i = wrt
return Suffice(All[i:1:n](x[i] > 0), Equal(alpha(x), H(x) / K(x)))
def apply(x):
n = x.shape[0]
n -= 1
assert n >= 2
return Equal(alpha(x[:n + 1]), alpha(x[:n - 1]) + (-1) ** n * x[n] / (K(x[:n + 1]) * K(x[:n - 1])))
def apply(x, n):
return All[x[:n]:CartesianSpace(Range(1, oo), n)](Greater(K(x[:n]), 0))
def apply(given):
from axiom.discrete.K.to.add.definition import K
(x, _j), (j, n) = given.of(All[Indexed > 0, Tuple[1, Expr]])
assert _j == j
return Greater(K(x[:n]), 0)
def apply(x):
n = x.shape[0]
assert x.is_positive
n -= 1
assert n >= 1
return Equal(alpha(reverse(x[1:n + 1])), K(x[:n + 1]) / K(x[:n]))
def apply(x):
n = x.shape[0]
n -= 1
assert n > 0
return Equal(H(x[:n + 1]) * K(x[:n]) - H(x[:n]) * K(x[:n + 1]), (-1) ** (n + 1))
def apply(given):
(x, _j), (j, n) = given.of(All[Indexed > 0, Tuple[1, Expr]])
assert _j == j
return Equal(alpha(x[:n]), H(x[:n]) / K(x[:n]))
def apply(x):
n = x.shape[0]
n -= 1
assert n >= 1
return Equal(alpha(x[:n + 1]),
alpha(x[:n]) + (-1)**(n + 1) / (K(x[:n + 1]) * K(x[:n])))
def apply(x, i=None):
if i is None:
i = x.generate_var(integer=True, var='i')
n = x.shape[0]
return Suffice(All[i:1:n](x[i] > 0), Greater(K(x), 0))
def apply(x):
assert x.is_positive
return Greater(K(x), 0)
def apply(x):
n = x.shape[0]
n -= 1
return Equal(H(x[:n + 1]) * K(x[:n - 1]) - H(x[:n - 1]) * K(x[:n + 1]), (-1) ** n * x[n])
