def factorise_term(term):
"""
Write a single transfinite ordinal term (addend=0) as a product of primes.
For example:
w**(w**w*7 + w*3 + 2)
becomes
[(w**w**w, 7), (w**w, 3), (w, 2)]
"""
factors_ = []
for t in ordinal_terms(term.exponent):
if is_finite_ordinal(t):
factors_.append((Ordinal(), t))
else:
factors_.append((Ordinal(exponent=Ordinal(t.exponent)), t.copies))
if term.copies > 1:
factors_.append((term.copies, 1))
return factors_
def divide_terms_by_ordinal(terms, ordinal):
"""
Divide each term in the list by the specified ordinal.
"""
return [
Ordinal(subtract(t.exponent, ordinal.exponent), t.copies)
for t in terms
]
def factorise_term_successor(ordinal_term):
"""
Given a term (w**e * c) return the factors of its successor.
Note that since (w**e + 1) is prime, the method returns this
ordinal as a factor, as well as the copies if not 1:
term -> successor -> factors
w -> w + 1 -> [(w + 1, 1)]
w*7 -> w*7 + 1 -> [(w + 1, 1), (7, 1)]
"""
factors_ = [(Ordinal(exponent=ordinal_term.exponent, addend=1), 1)]
if ordinal_term.copies > 1:
factors_.append((ordinal_term.copies, 1))
return factors_
def subtract(a, b):
"""
Given ordinals a > b, return the ordinal c such that b + c == a
"""
if a <= b:
raise ValueError("First argument must be greater than second argument")
if is_finite_ordinal(a):
return a - b
if is_finite_ordinal(b) or a.exponent > b.exponent:
return a
if a.exponent == b.exponent and a.copies == b.copies:
return subtract(a.addend, b.addend)
# Here we know that a.copies > b.copies
return Ordinal(a.exponent, a.copies - b.copies, a.addend)
def ordinal_terms(ordinal):
"""
Return a list containing all terms of the ordinal.
For example:
w**w**w + w**3 + w*7 + 9
becomes the list
[w**w**w, w**3, w*7, 9]
"""
terms = []
while not is_finite_ordinal(ordinal):
term = Ordinal(exponent=ordinal.exponent, copies=ordinal.copies)
terms.append(term)
ordinal = ordinal.addend
if ordinal:
terms.append(ordinal)
return terms
def test_invalid_args_to_class(kwargs):
with pytest.raises(OrdinalConstructionError):
Ordinal(**kwargs)
def test_invalid_types_operation(op, a, expected_error):
with pytest.raises(expected_error):
op(Ordinal(), a)
op(a, Ordinal())
"exponent": "0"
},
{
"copies": 0
},
{
"copies": -1
},
{
"copies": -3.14
},
{
"copies": "0"
},
{
"copies": Ordinal()
},
{
"addend": -1
},
{
"addend": -3.14
},
{
"addend": "0"
},
{
"exponent": 1,
"addend": Ordinal(exponent=1)
},
{