Trac #15919: unrank via R[i] conflicts with notation for constructing polynomial rings in CartesianProduct

{{{
FF = IntegerModRing(29)
tester = FF._tester()
list(tester.some_elements(CartesianProduct(FF, FF, FF)))
}}}
This should give a list of 3-arrays of elements of Z/29. Instead I get:
{{{
sage: FF = IntegerModRing(29)
sage: tester = FF._tester()
sage: list(tester.some_elements(CartesianProduct(FF, FF, FF)))
------------------------------------------------------------------------
---
ValueError                                Traceback (most recent call
last)
<ipython-input-3-5d7e71a13340> in <module>()
----> 1 list(tester.some_elements(CartesianProduct(FF, FF, FF)))

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/misc/sage_unittest.pyc in some_elements(self, S)
    498             if n > self._max_runs:
    499                 from random import sample
--> 500                 S = sample(S, self._max_runs)
    501         except (TypeError, AttributeError):
    502             # We already can't tell what the length of n is, so

/home/darij/gitsage/sage-5.13.beta1/local/lib/python/random.pyc in
sample(self, population, k)
    342                         j = _int(random() * n)
    343                     selected_add(j)
--> 344                     result[i] = population[j]
    345             except (TypeError, KeyError):   # handle (at least)
sets
    346                 if isinstance(population, list):

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/combinat/combinat.pyc in __getitem__(self, i)
   1220             ValueError: the value must be between 0 and 2
inclusive
   1221         """
-> 1222         return self.unrank(i)
   1223
   1224     def __str__(self):

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/combinat/cartesian_product.pyc in unrank(self, x)
    258             raise IndexError("x larger than the size of the
Cartesian Product")
    259         positions.reverse()
--> 260         return [L[i] for L,i in zip(self.iters, positions)]
    261
    262     def random_element(self):

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/structure/parent.so in
sage.structure.parent.Parent.__getitem__
(sage/structure/parent.c:11060)()

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/categories/rings.pyc in __getitem__(self, arg)
    859
    860             from
sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
--> 861             return PolynomialRing(self, elts)
    862
    863     class ElementMethods:

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/rings/polynomial/polynomial_ring_constructor.pyc in
PolynomialRing(base_ring, arg1, arg2, sparse, order, names, name,
var_array, implementation)
    467                 raise TypeError("if second arguments is a string
with no commas, then there must be no other non-optional arguments")
    468             name = arg1
--> 469             R = _single_variate(base_ring, name, sparse,
implementation)
    470         else:
    471             # 2-4. PolynomialRing(base_ring, names,
order='degrevlex'):

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/rings/polynomial/polynomial_ring_constructor.pyc in
_single_variate(base_ring, name, sparse, implementation)
    518 def _single_variate(base_ring, name, sparse, implementation):
    519     import sage.rings.polynomial.polynomial_ring as m
--> 520     name = normalize_names(1, name)
    521     key = (base_ring, name, sparse, implementation if not sparse
else None)
    522     R = _get_from_cache(key)

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/structure/parent_gens.so in
sage.structure.parent_gens.normalize_names
(sage/structure/parent_gens.c:2759)()

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/structure/parent_gens.so in
sage.structure.parent_gens._certify_names
(sage/structure/parent_gens.c:2322)()

ValueError: first letter of variable name must be a letter
}}}
And 29 is the smallest number where this seems to throw this error;
also, it doesn't fail if I only take the cartesian product of two copies
of the field.

I get a similar error if I do `CartesianProduct(FF, FF).unrank(3)`, and
this has been around for a while (not just since beta4):
{{{
sage: FF = IntegerModRing(3)
sage: CartesianProduct(FF, FF).unrank(3)
------------------------------------------------------------------------
---
ValueError                                Traceback (most recent call
last)
<ipython-input-15-98d0a571cc61> in <module>()
----> 1 CartesianProduct(FF, FF).unrank(Integer(3))

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/combinat/cartesian_product.pyc in unrank(self, x)
    258             raise IndexError("x larger than the size of the
Cartesian Product")
    259         positions.reverse()
--> 260         return [L[i] for L,i in zip(self.iters, positions)]
    261
    262     def random_element(self):

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/structure/parent.so in
sage.structure.parent.Parent.__getitem__
(sage/structure/parent.c:11060)()

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/categories/rings.pyc in __getitem__(self, arg)
    859
    860             from
sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
--> 861             return PolynomialRing(self, elts)
    862
    863     class ElementMethods:

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/rings/polynomial/polynomial_ring_constructor.pyc in
PolynomialRing(base_ring, arg1, arg2, sparse, order, names, name,
var_array, implementation)
    467                 raise TypeError("if second arguments is a string
with no commas, then there must be no other non-optional arguments")
    468             name = arg1
--> 469             R = _single_variate(base_ring, name, sparse,
implementation)
    470         else:
    471             # 2-4. PolynomialRing(base_ring, names,
order='degrevlex'):

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/rings/polynomial/polynomial_ring_constructor.pyc in
_single_variate(base_ring, name, sparse, implementation)
    518 def _single_variate(base_ring, name, sparse, implementation):
    519     import sage.rings.polynomial.polynomial_ring as m
--> 520     name = normalize_names(1, name)
    521     key = (base_ring, name, sparse, implementation if not sparse
else None)
    522     R = _get_from_cache(key)

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/structure/parent_gens.so in
sage.structure.parent_gens.normalize_names
(sage/structure/parent_gens.c:2759)()

/home/darij/gitsage/sage-5.13.beta1/local/lib/python2.7/site-
packages/sage/structure/parent_gens.so in
sage.structure.parent_gens._certify_names
(sage/structure/parent_gens.c:2322)()

ValueError: first letter of variable name must be a letter
}}}
So it seems that both bugs are due to the fancy `R[x]` syntax for
polynomial rings conflicting with the `R[i]` syntax for the i-th element
of an enumerated set `R`, and #8389 didn't cause the error, but at most
exposed it better.

This reworks the `unrank` function in `sage.combinat.ranker` and uses it
for the Cartesian product (this doesn't work for `ZZ`, see #16239). In
followup tickets, we also want to be more systematic about using unrank
in `TestSuite` and do better construction of `some_elements()` #16244.

URL: http://trac.sagemath.org/15919
Reported by: darij
Ticket author(s): Nicolas M. Thiéry
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/combinat/cartesian_product.py

@@ -18,11 +18,10 @@
 
 from inspect import isgenerator
 import sage.misc.prandom as rnd
-import __builtin__
+from sage.misc.mrange import xmrange_iter, _is_finite, _len
 from combinat import CombinatorialClass
-from sage.rings.integer import Integer
+from ranker import unrank
 from sage.rings.infinity import infinity
-from sage.misc.mrange import xmrange_iter, _is_finite, _len
 
 def CartesianProduct(*iters):
     """
@@ -241,6 +240,13 @@ def unrank(self, x):
             sage: C = CartesianProduct(xrange(1000), xrange(1000), xrange(1000))
             sage: C[238792368]
             [238, 792, 368]
+
+        Check for :trac:`15919`::
+
+            sage: FF = IntegerModRing(29)
+            sage: C = CartesianProduct(FF, FF, FF)
+            sage: C.unrank(0)
+            [0, 0, 0]
         """
         try:
             lens = [_len(it) for it in self.iters]
@@ -257,7 +263,7 @@ def unrank(self, x):
         if x != 0:
             raise IndexError("x larger than the size of the Cartesian Product")
         positions.reverse()
-        return [L[i] for L,i in zip(self.iters, positions)]
+        return [unrank(L, i) for L,i in zip(self.iters, positions)]
 
     def random_element(self):
         """

src/sage/combinat/ranker.py

@@ -18,7 +18,10 @@
 #                  http://www.gnu.org/licenses/
 #*****************************************************************************
 
-from sage.misc.all import cached_function
+from collections import Iterable, Sequence
+from sage.misc.cachefunc import cached_function
+from sage.structure.parent import Parent
+from sage.categories.enumerated_sets import EnumeratedSets
 
 def from_list(l):
     """
@@ -134,3 +137,113 @@ def unrank(i):
         return None
 
     return [rank, unrank]
+
+def unrank(L, i):
+    r"""
+    Return the `i`-th element of `L`.
+
+    INPUT:
+
+    - ``L`` -- a list, tuple, finite enumerated set, ...
+    - ``i`` -- an int or :class:`Integer`
+
+    The purpose of this utility is to give a uniform idiom to recover
+    the `i`-th element of an object ``L``, whether ``L`` is a list,
+    tuple (or more generally a :class:`collections.Sequence`), an
+    enumerated set, some old parent of Sage still implementing
+    unranking in the method ``__getitem__``, or an iterable (see
+    :class:`collections.Iterable`). See :trac:`15919`.
+
+    EXAMPLES:
+
+    Lists, tuples, and other :class:`sequences <collections.Sequence>`::
+
+        sage: from sage.combinat.ranker import unrank
+        sage: unrank(['a','b','c'], 2)
+        'c'
+        sage: unrank(('a','b','c'), 1)
+        'b'
+        sage: unrank(xrange(3,13,2), 1)
+        5
+
+    Enumerated sets::
+
+        sage: unrank(GF(7), 2)
+        2
+        sage: unrank(IntegerModRing(29), 10)
+        10
+
+    An old parent with unranking implemented in ``__getitem__``::
+
+        sage: M = MatrixSpace(GF(3), 2, 2)
+        sage: hasattr(M, "unrank")
+        False
+        sage: M[42]
+        [1 0]
+        [2 1]
+        sage: unrank(M, 42)
+        [1 0]
+        [2 1]
+
+    An iterable::
+
+        sage: unrank(NN,4)
+        4
+
+    An iterator::
+
+        sage: unrank(('a{}'.format(i) for i in range(20)), 0)
+        'a0'
+        sage: unrank(('a{}'.format(i) for i in range(20)), 2)
+        'a2'
+
+    .. WARNING::
+
+        When unranking an iterator, it returns the``i``-th element
+        beyond where it is currently at::
+
+            sage: from sage.combinat.ranker import unrank
+            sage: it = iter(range(20))
+            sage: unrank(it, 2)
+            2
+            sage: unrank(it, 2)
+            5
+
+    TESTS::
+
+        sage: from sage.combinat.ranker import unrank
+        sage: unrank(range(3), 10)
+        Traceback (most recent call last):
+        ...
+        IndexError: list index out of range
+
+        sage: unrank(('a{}'.format(i) for i in range(20)), 22)
+        Traceback (most recent call last):
+        ...
+        IndexError: index out of range
+
+        sage: M[100]
+        Traceback (most recent call last):
+        ...
+        IndexError: list index out of range
+    """
+    if L in EnumeratedSets:
+        return L.unrank(i)
+    if isinstance(L, Sequence):
+        return L[i]
+    if isinstance(L, Parent):
+        # handle parents still implementing unranking in __getitem__
+        try:
+            return L[i]
+        except (AttributeError, TypeError, ValueError):
+            pass
+    if isinstance(L, Iterable):
+        try:
+            it = iter(L)
+            for _ in range(i):
+                it.next()
+            return it.next()
+        except StopIteration as e:
+            raise IndexError("index out of range")
+    raise ValueError("Don't know how to unrank on {}".format(L))
+