gh-35485: New comparison mode to lazy series and better undefined check

    
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This fixes #35071 by:

1. providing a method to streams to see if they are unitinalized
2. includes a new mode for comparisons in the lazy series ring based on
the proposal in #35429.

The new comparison mode `secure` simply returns `False` when it cannot
verify that `f == g` and makes sure that `(f == g) == not (f != g)`, and
this will become the new default. In particular, it will return `True`
for `f != g` both when it cannot show that `f == g` and when they are
genuinely different. In order to verify when the comparison is unknown,
we expose the `is_nonzero()` that only returns `True` when the series is
_known_ to be nonzero. Thus, we verify by `(f - g).is_nonzero()`.

When a finite halting precision is given, then that takes priority.

For the infinite halting precision in the "old" version (`secure =
True`), it will raise a `ValueError` when it cannot verify the result.

**NOTE:** `f.is_zero()` is still the default `not f` for speed and the
assumption these are in agreement elsewhere in Sage.

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### ⌛ Dependencies

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URL: #35485
Reported by: Travis Scrimshaw
Reviewer(s): Martin Rubey, Travis Scrimshaw

src/sage/combinat/species/recursive_species.py

@@ -230,7 +230,7 @@ def _gs(self, series_ring, base_ring):
 
             sage: F = CombinatorialSpecies()
             sage: F.generating_series()
-            Uninitialized Lazy Laurent Series
+            Uninitialized Lazy Series
         """
         if base_ring not in self._generating_series:
             self._generating_series[base_ring] = series_ring.undefined(valuation=(0 if self._min is None else self._min))
@@ -247,7 +247,7 @@ def _itgs(self, series_ring, base_ring):
 
             sage: F = CombinatorialSpecies()
             sage: F.isotype_generating_series()
-            Uninitialized Lazy Laurent Series
+            Uninitialized Lazy Series
         """
         if base_ring not in self._isotype_generating_series:
             self._isotype_generating_series[base_ring] = series_ring.undefined(valuation=(0 if self._min is None else self._min))
@@ -264,7 +264,7 @@ def _cis(self, series_ring, base_ring):
 
             sage: F = CombinatorialSpecies()
             sage: F.cycle_index_series()
-            Uninitialized Lazy Laurent Series
+            Uninitialized Lazy Series
         """
         if base_ring not in self._cycle_index_series:
             self._cycle_index_series[base_ring] = series_ring.undefined(valuation=(0 if self._min is None else self._min))
@@ -401,6 +401,15 @@ def define(self, x):
             [1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
             sage: F.isotype_generating_series()[0:10]
             [1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
+
+        Check that :issue:`35071` is fixed::
+
+            sage: X = species.SingletonSpecies()
+            sage: E = species.SetSpecies(max=3)
+            sage: B = species.CombinatorialSpecies(min=1)
+            sage: B.define(X*E(B))
+            sage: B.generating_series()
+            z + z^2 + 3/2*z^3 + 5/2*z^4 + 9/2*z^5 + 17/2*z^6 + 133/8*z^7 + O(z^8)
         """
         if not isinstance(x, GenericCombinatorialSpecies):
             raise TypeError("x must be a combinatorial species")