Alternative proposal sketch.

src/sage/rings/lazy_series.py

@@ -923,21 +923,21 @@ def _richcmp_(self, other, op):
             False
 
             sage: fz = L(lambda n: 0, valuation=0)
-            sage: L.zero() == fz
-            False
-            sage: fz == L.zero()
-            False
+            sage: (L.zero() == fz) is None
+            True
+            sage: (fz == L.zero()) is None
+            True
 
         With using :class:`Unknown`::
 
             sage: L.options.use_unknown = True
             sage: fz = L(lambda n: 0, valuation=0)
-            sage: L.zero() == fz
-            Unknown
-            sage: fz == L.zero()
-            Unknown
-            sage: fz != L.zero()
-            Unknown
+            sage: (L.zero() == fz) is None
+            True
+            sage: (fz == L.zero()) is None
+            True
+            sage: (fz != L.zero()) is None
+            True
 
         With using finite halting precision::
 
@@ -993,7 +993,8 @@ def _richcmp_(self, other, op):
             # undecidable otherwise
             prec = self.parent().options['halting_precision']
             if prec is None:
-                return Unknown
+                return None
+                #return Unknown
                 # raise UnknownError("undecidable")
             # at least one of the approximate orders is not infinity
             m = min(self._coeff_stream._approximate_order,
@@ -1002,7 +1003,7 @@ def _richcmp_(self, other, op):
 
         if op is op_NE:
             ret = (self == other)
-            if ret is Unknown:
+            if ret is None:
                 return ret
             return not ret
 
@@ -1140,7 +1141,8 @@ def __bool__(self):
                 return True
 
         if prec is None:
-            raise UnknownError("undecidable")
+            return True
+            #raise UnknownError("undecidable")
         v = self._coeff_stream._approximate_order
         return any(self[i] for i in range(v, v + prec))
 
@@ -1884,12 +1886,12 @@ def _acted_upon_(self, scalar, self_on_left):
 
         Different scalars potentially give different series::
 
-            sage: 2 * M == 3 * M
-            False
+            sage: (2 * M == 3 * M) is None
+            True
 
             sage: L.options.use_unknown = True
-            sage: 2 * M == 3 * M
-            Unknown
+            sage: (2 * M == 3 * M) is None
+            True
 
             sage: L.options.halting_precision = 30
             sage: 2 * M == 3 * M

src/sage/rings/lazy_series_ring.py

@@ -677,7 +677,7 @@ class options(GlobalOptions):
               - constant_length:   3
               - display_length:    7
               - halting_precision: None
-              - use_unknown:     False
+              - use_unknown:     True
 
             sage: LLS.options.display_length
             7
@@ -713,7 +713,7 @@ class options(GlobalOptions):
         halting_precision = dict(default=None,
                                  description='the number of coefficients, beginning with the approximate valuation, to check in equality tests',
                                  checker=lambda x: x is None or x in ZZ and x > 0)
-        use_unknown = dict(default=False,
+        use_unknown = dict(default=True,
                              description='whether to raise an error when a comparison is unknown',
                              checker=lambda x: x is True or x is False)
 
@@ -1160,8 +1160,8 @@ class LazyLaurentSeriesRing(LazySeriesRing):
     be equal are considered to be different::
 
         sage: f = L(lambda n: 0, valuation=0)
-        sage: f == 0
-        False
+        sage: (f == 0) is None
+        True
 
     .. WARNING::
 
@@ -1170,7 +1170,7 @@ class LazyLaurentSeriesRing(LazySeriesRing):
         series are actually different::
 
             sage: g = L.zero()
-            sage: f != g
+            sage: (f != g) is None
             True
 
         This can be verified by :meth:`~sage.rings.lazy_series.is_nonzero()`,
@@ -1205,27 +1205,25 @@ class LazyLaurentSeriesRing(LazySeriesRing):
         z^-1 - 1 + z - z^2 + z^3 - z^4 + z^5 + O(z^6)
         sage: f2 = f * 2  # currently no coefficients computed
         sage: f3 = f * 3  # currently no coefficients computed
-        sage: f2 == f3
-        Unknown
+        sage: (f2 == f3) is None
+        True
         sage: f2  # computes some of the coefficients of f2
         2*z^-1 - 2 + 2*z - 2*z^2 + 2*z^3 - 2*z^4 + 2*z^5 + O(z^6)
         sage: f3  # computes some of the coefficients of f3
         3*z^-1 - 3 + 3*z - 3*z^2 + 3*z^3 - 3*z^4 + 3*z^5 + O(z^6)
         sage: f2 == f3
         False
         sage: f2a = f + f
-        sage: f2 == f2a
-        Unknown
+        sage: (f2 == f2a) is None
+        True
         sage: zf = L(lambda n: 0, valuation=0)
-        sage: zf == 0
-        Unknown
+        sage: (zf == 0) is None
+        True
 
     For boolean checks, an error is raised when it is not known to be nonzero::
 
         sage: bool(zf)
-        Traceback (most recent call last):
-        ...
-        UnknownError: undecidable
+        True
 
     If the halting precision is set to a finite number `p` (for unlimited
     precision, it is set to ``None``), then it will check up to `p` values