def apply(n, k, A=None):
j = Symbol.j(domain=Interval(0, k, integer=True))
if A is None:
x = Symbol.x(shape=(oo, ), dtype=dtype.integer, finite=True)
i = Symbol.i(integer=True)
s1_quote = Symbol("s'_1",
definition=Stirling.conditionset(n, k + 1, x))
x_quote = Symbol("x'",
definition=LAMBDA[i:k + 1](Piecewise(
({n} | x[i], Equality(i, j)), (x[i], True))))
A = Symbol.A(definition=LAMBDA[j](UNION[x[:k + 1]:s1_quote]
({x_quote.set_comprehension()})))
return Equality(abs(UNION[j](A[j])), Sum[j](abs(A[j])))
def apply(n, k, s2=None, A=None):
j = Symbol.j(domain=Interval(0, k, integer=True))
if s2 is None:
x = Symbol.x(shape=(oo, ), dtype=dtype.integer, finite=True)
s2 = Symbol.s2(
definition=UNION[x[:k + 1]:Stirling.conditionset(n + 1, k + 1, x)](
x[:k + 1].set_comprehension().set))
e = Symbol.e(**s2.element_type.dict)
if A is None:
x = s2.definition.variable.base
i = Symbol.i(integer=True)
s1_quote = Symbol("s'_1",
definition=Stirling.conditionset(n, k + 1, x))
x_quote = Symbol("x'",
definition=LAMBDA[i:k + 1](Piecewise(
({n} | x[i], Equality(i, j)), (x[i], True))))
A = Symbol.A(definition=LAMBDA[j](UNION[x[:k + 1]:s1_quote]
({x_quote.set_comprehension()})))
return Equality(conditionset(e, NotContains({n}, e), s2), UNION[j](A[j]))
