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===Composition of pure functions===
====Original version – with language-independent (MH) comments on type semantics====
A variant of this code with '''actual Python type hints for the compiler''', in lieu of the language-independent Hindley-Milner type comments for the reader shown here, has been helpfully added below, by another user (who objected that my comments looked too much like Haskell), for comparison.
For example, where an informal MH type comment (for the central function below) reads:
<code># sumOfAnySubset :: [Int] -> Int -> Bool</code>
the additional variant, show below the original code here, gives an actual Python type hint:
<code>def sumOfAnySubset(xs: List[int]) -> Callable[[int], bool]:</code>
The affordances of these two approaches differ – one aims for clarity for the human reader, and speed of drafting,
the other for formal type-checking by a suitably equipped Python compiler, possibly at the cost of some extra work.
(Nothing, of course, excludes the option of using both)
<lang Python>'''Practical numbers'''
from itertools import accumulate, chain, groupby, product
from math import floor, sqrt
from operator import mul
from functools import reduce
# isPractical :: Int -> Bool
def isPractical(n):
'''True if n is a Practical number
(a member of OEIS A005153)
'''
ds = properDivisors(n)
return all(map(
sumOfAnySubset(ds),
range(1, n)
))
# sumOfAnySubset :: [Int] -> Int -> Bool
def sumOfAnySubset(xs):
'''True if any subset of xs sums to n.
'''
def go(n):
if n in xs:
return True
else:
total = sum(xs)
if n == total:
return True
elif n < total:
h, *t = reversed(xs)
d = n - h
return d in t or (
d > 0 and sumOfAnySubset(t)(d)
) or sumOfAnySubset(t)(n)
else:
return False
return go
# ------------------------- TEST -------------------------
# main :: IO ()
def main():
'''Practical numbers in the range [1..333],
and the OEIS A005153 membership of 666.
'''
xs = [x for x in range(1, 334) if isPractical(x)]
print(
f'{len(xs)} OEIS A005153 numbers in [1..333]\n\n' + (
spacedTable(
chunksOf(10)([
str(x) for x in xs
])
)
)
)
print("\n")
for n in [666]:
print(
f'{n} is practical :: {isPractical(n)}'
)
# ----------------------- GENERIC ------------------------
# chunksOf :: Int -> [a] -> [[a]]
def chunksOf(n):
'''A series of lists of length n, subdividing the
contents of xs. Where the length of xs is not evenly
divible, the final list will be shorter than n.
'''
def go(xs):
return [
xs[i:n + i] for i in range(0, len(xs), n)
] if 0 < n else None
return go
# primeFactors :: Int -> [Int]
def primeFactors(n):
'''A list of the prime factors of n.
'''
def f(qr):
r = qr[1]
return step(r), 1 + r
def step(x):
return 1 + (x << 2) - ((x >> 1) << 1)
def go(x):
root = floor(sqrt(x))
def p(qr):
q = qr[0]
return root < q or 0 == (x % q)
q = until(p)(f)(
(2 if 0 == x % 2 else 3, 1)
)[0]
return [x] if q > root else [q] + go(x // q)
return go(n)
# properDivisors :: Int -> [Int]
def properDivisors(n):
'''The ordered divisors of n, excluding n itself.
'''
def go(a, x):
return [a * b for a, b in product(
a,
accumulate(chain([1], x), mul)
)]
return sorted(
reduce(go, [
list(g) for _, g in groupby(primeFactors(n))
], [1])
)[:-1] if 1 < n else []
# listTranspose :: [[a]] -> [[a]]
def listTranspose(xss):
'''Transposed matrix'''
def go(xss):
if xss:
h, *t = xss
return (
[[h[0]] + [xs[0] for xs in t if xs]] + (
go([h[1:]] + [xs[1:] for xs in t])
)
) if h and isinstance(h, list) else go(t)
else:
return []
return go(xss)
# until :: (a -> Bool) -> (a -> a) -> a -> a
def until(p):
'''The result of repeatedly applying f until p holds.
The initial seed value is x.
'''
def go(f):
def g(x):
v = x
while not p(v):
v = f(v)
return v
return g
return go
# ---------------------- FORMATTING ----------------------
# spacedTable :: [[String]] -> String
def spacedTable(rows):
'''Tabulation with right-aligned cells'''
columnWidths = [
len(str(row[-1])) for row in listTranspose(rows)
]
def aligned(s, w):
return s.rjust(w, ' ')
return '\n'.join(
' '.join(
map(aligned, row, columnWidths)
) for row in rows
)
# MAIN ---
if __name__ == '__main__':
main()</lang>
{{Out}}
<pre>77 OEIS A005153 numbers in [1..333]
1 2 4 6 8 12 16 18 20 24
28 30 32 36 40 42 48 54 56 60
64 66 72 78 80 84 88 90 96 100
104 108 112 120 126 128 132 140 144 150
156 160 162 168 176 180 192 196 198 200
204 208 210 216 220 224 228 234 240 252
256 260 264 270 272 276 280 288 294 300
304 306 308 312 320 324 330
666 is practical :: True</pre>
====Variant with actual Python type hints====
<lang python>'''Practical numbers'''
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