Trac #34374: Use cantor_product for Cartesian products of infinite en…

…umerated sets

(split out from #19195)

URL: https://trac.sagemath.org/34374
Reported by: mkoeppe
Ticket author(s): Matthias Koeppe
Reviewer(s): Travis Scrimshaw

src/sage/categories/sets_cat.py

@@ -2217,7 +2217,14 @@ def example(self):
  class ParentMethods:
  def __iter__(self):
  r"""
- Return a lexicographic iterator for the elements of this Cartesian product.
+ Return an iterator for the elements of this Cartesian product.
+
+ If all factors (except possibly the first factor) are known to be finite,
+ it uses the lexicographic order.
+
+ Otherwise, the iterator enumerates the elements in increasing
+ order of sum-of-ranks, refined by the reverse lexicographic order
+ (see :func:`~sage.misc.mrange.cantor_product`).
 
  EXAMPLES:
 
@@ -2260,29 +2267,38 @@ def __iter__(self):
  [1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
  [1, 2, 3, 4, 5, 6, 7, 8, 10, 9])
 
- .. WARNING::
-
- The elements are returned in lexicographic order,
- which gives a valid enumeration only if all
- factors, but possibly the first one, are
- finite. So the following one is fine::
-
- sage: it = iter(cartesian_product([ZZ, GF(2)]))
- sage: [next(it) for _ in range(10)]
- [(0, 0), (0, 1), (1, 0), (1, 1),
- (-1, 0), (-1, 1), (2, 0), (2, 1),
- (-2, 0), (-2, 1)]
-
- But this one is not::
-
- sage: it = iter(cartesian_product([GF(2), ZZ]))
- sage: [next(it) for _ in range(10)]
- doctest:...: UserWarning: Sage is not able to determine
- whether the factors of this Cartesian product are
- finite. The lexicographic ordering might not go through
- all elements.
- [(0, 0), (0, 1), (0, -1), (0, 2), (0, -2),
- (0, 3), (0, -3), (0, 4), (0, -4), (0, 5)]
+ When all factors (except possibly the first factor) are known to be finite, it
+ uses the lexicographic order::
+
+ sage: it = iter(cartesian_product([ZZ, GF(2)]))
+ sage: [next(it) for _ in range(10)]
+ [(0, 0), (0, 1),
+ (1, 0), (1, 1),
+ (-1, 0), (-1, 1),
+ (2, 0), (2, 1),
+ (-2, 0), (-2, 1)]
+
+ When other factors are infinite (or not known to be finite), it enumerates
+ the elements in increasing order of sum-of-ranks::
+
+ sage: NN = NonNegativeIntegers()
+ sage: it = iter(cartesian_product([NN, NN]))
+ sage: [next(it) for _ in range(10)]
+ [(0, 0),
+ (1, 0), (0, 1),
+ (2, 0), (1, 1), (0, 2),
+ (3, 0), (2, 1), (1, 2), (0, 3)]
+
+ An example with the first factor finite, the second infinite::
+
+ sage: it = iter(cartesian_product([GF(2), ZZ]))
+ sage: [next(it) for _ in range(11)]
+ [(0, 0),
+ (1, 0), (0, 1),
+ (1, 1), (0, -1),
+ (1, -1), (0, 2),
+ (1, 2), (0, -2),
+ (1, -2), (0, 3)]
 
  .. NOTE::
 
@@ -2292,17 +2308,18 @@ def __iter__(self):
 
  ALGORITHM:
 
- Recipe 19.9 in the Python Cookbook by Alex Martelli
- and David Ascher.
+ The lexicographic enumeration follows Recipe 19.9 in the Python Cookbook
+ by Alex Martelli and David Ascher.
  """
- if any(f not in Sets().Finite() for f in self.cartesian_factors()[1:]):
- from warnings import warn
- warn("Sage is not able to determine whether the factors of "
- "this Cartesian product are finite. The lexicographic "
- "ordering might not go through all elements.")
+ factors = list(self.cartesian_factors())
+ if any(f not in Sets().Finite() for f in factors[1:]):
+ from sage.misc.mrange import cantor_product
+ for t in cantor_product(*factors):
+ yield self._cartesian_product_of_elements(t)
+ return
 
+ # Lexicographic enumeration:
  # visualize an odometer, with "wheels" displaying "digits"...:
- factors = list(self.cartesian_factors())
  wheels = [iter(f) for f in factors]
  try:
  digits = [next(it) for it in wheels]