Trac #21448: Avoid underscored arithmetic methods in Python

In categories, it is better to write `x + y` instead of `x._add_(y)`
since the latter can be a lot slower if `_add_` is implemented in
Cython.

We also remove several redundant implementations of `_sub_` where they
coincide with the default implementation from `ModuleElement`.

URL: https://trac.sagemath.org/21448
Reported by: jdemeyer
Ticket author(s): Jeroen Demeyer
Reviewer(s): Nicolas M. Thiéry

src/sage/categories/additive_magmas.py

@@ -199,7 +199,7 @@ def summation(self, x, y):
 
             .. TODO:: Add an example.
             """
-            return x._add_(y)
+            return x + y
 
         summation_from_element_class_add = summation
 
@@ -819,7 +819,7 @@ def _sub_(left, right):
                     sage: C.one() - C.one()
                     (0, 0)
                 """
-                return left._add_(-right)
+                return left + (-right)
 
             def __neg__(self):
                 """

src/sage/categories/magmas.py

@@ -564,7 +564,7 @@ def _div_(left, right):
                     sage: c1._div_.__module__
                     'sage.categories.magmas'
                 """
-                return left._mul_(~right)
+                return left * ~right
 
         class SubcategoryMethods:
 
@@ -785,7 +785,7 @@ def product(self, x, y):
                 From: (S x S)
                 To:   S
             """
-            return x._mul_(y)
+            return x * y
 
         product_from_element_class_mul = product
 

src/sage/rings/fraction_field_element.pyx

@@ -507,6 +507,12 @@ cdef class FractionFieldElement(FieldElement):
             (y + 1)/(x*y)
             sage: Frac(CDF['x']).gen() + 3
             x + 3.0
+
+        Subtraction is implemented by adding the negative::
+
+            sage: K.<t> = Frac(GF(7)['t'])
+            sage: t - 1/t # indirect doctest
+            (t^2 + 6)/t
         """
         rnum = self.__numerator
         rden = self.__denominator
@@ -563,26 +569,6 @@ cdef class FractionFieldElement(FieldElement):
         return self.__class__(self._parent, rnum*sden + rden*snum, rden*sden,
             coerce=False, reduce=False)
 
-    cpdef _sub_(self, right):
-        """
-        Computes the difference of ``self`` and ``right``.
-
-        INPUT:
-
-        - ``right`` - ``ModuleElement`` to subtract from ``self``
-
-        OUTPUT:
-
-        - Difference of ``self`` and ``right``
-
-        EXAMPLES::
-
-            sage: K.<t> = Frac(GF(7)['t'])
-            sage: t - 1/t # indirect doctest
-            (t^2 + 6)/t
-        """
-        return self._add_(-right)
-
     cpdef _mul_(self, right):
         """
         Computes the product of ``self`` and ``right``.

src/sage/rings/infinity.py

@@ -1281,6 +1281,20 @@ def _add_(self, other):
             Traceback (most recent call last):
             ...
             SignError: cannot add positive finite value to negative finite value
+
+        Subtraction is implemented by adding the negative::
+
+            sage: P = InfinityRing
+            sage: 4 - oo # indirect doctest
+            -Infinity
+            sage: 5 - -oo
+            +Infinity
+            sage: P(44) - P(4)
+            Traceback (most recent call last):
+            ...
+            SignError: cannot add positive finite value to negative finite value
+            sage: P(44) - P(-1)
+            A positive finite number
         """
         if isinstance(other, InfinityElement):
             return other
@@ -1343,24 +1357,6 @@ def _div_(self, other):
         """
         return self * ~other
 
-    def _sub_(self, other):
-        """
-        EXAMPLES::
-
-            sage: P = InfinityRing
-            sage: 4 - oo # indirect doctest
-            -Infinity
-            sage: 5 - -oo
-            +Infinity
-            sage: P(44) - P(4)
-            Traceback (most recent call last):
-            ...
-            SignError: cannot add positive finite value to negative finite value
-            sage: P(44) - P(-1)
-            A positive finite number
-        """
-        return self._add_(-other)
-
     def __invert__(self):
         """
         EXAMPLES::

src/sage/rings/universal_cyclotomic_field.py

@@ -829,7 +829,7 @@ def _add_(self, other):
             1/2*E(3) - 1/2*E(3)^2
         """
         P = self.parent()
-        return P.element_class(P, self._obj._add_(other._obj))
+        return P.element_class(P, self._obj + other._obj)
 
     def _sub_(self, other):
         r"""
@@ -839,7 +839,7 @@ def _sub_(self, other):
             -E(15)^2 - E(15)^11 + E(15)^13 - E(15)^14
         """
         P = self.parent()
-        return P.element_class(P, self._obj._sub_(other._obj))
+        return P.element_class(P, self._obj - other._obj)
 
     def __neg__(self):
         r"""
@@ -865,7 +865,7 @@ def _mul_(self, other):
             3*E(4)
         """
         P = self.parent()
-        return P.element_class(P, self._obj._mul_(other._obj))
+        return P.element_class(P, self._obj * other._obj)
 
     def _div_(self, other):
         r"""
@@ -878,7 +878,7 @@ def _div_(self, other):
         """
         P = self.parent()
         try:
-            return P.element_class(P, self._obj._div_(other._obj))
+            return P.element_class(P, self._obj / other._obj)
         except ValueError:
             raise ZeroDivisionError("division by zero")
 

src/sage/tensor/differential_form_element.py

@@ -651,31 +651,7 @@ def _add_(self, other):
             ...
             TypeError: Cannot add forms of degree 1 and 2
 
-        """
-
-        if self.is_zero():
-            return other
-        if other.is_zero():
-            return self
-
-        if self._degree != other._degree:
-            raise TypeError("Cannot add forms of degree %s and %s" % \
-                    (self._degree, other._degree))
-
-        sumform = DifferentialForm(self.parent(), self._degree)
-        sumform._components = self._components.copy()
-        for comp, fun in other._components.items():
-            sumform[comp] += fun
-
-        sumform._cleanup()
-        return sumform
-
-
-    def _sub_(self, other):
-        r"""
-        Subtract other from self.
-
-        EXAMPLES::
+        Subtraction is implemented by adding the negative::
 
             sage: x, y, z = var('x, y, z')
             sage: F = DifferentialForms()
@@ -699,10 +675,24 @@ def _sub_(self, other):
             Traceback (most recent call last):
             ...
             TypeError: Cannot add forms of degree 1 and 2
-
         """
-        return self._add_(-other)
 
+        if self.is_zero():
+            return other
+        if other.is_zero():
+            return self
+
+        if self._degree != other._degree:
+            raise TypeError("Cannot add forms of degree %s and %s" % \
+                    (self._degree, other._degree))
+
+        sumform = DifferentialForm(self.parent(), self._degree)
+        sumform._components = self._components.copy()
+        for comp, fun in other._components.items():
+            sumform[comp] += fun
+
+        sumform._cleanup()
+        return sumform
 
     def _cleanup(self):
         r"""