Make assumptions about indexed symbols (sympy#13992)

``Indexed`` and ``IndexedBase`` now accepts a new keyword argument
allowing the specification of assumptions on indexed symbols.

sympy/tensor/indexed.py

@@ -106,9 +106,11 @@
 
 from __future__ import print_function, division
 
+from sympy.core.assumptions import StdFactKB
 from sympy.core import Expr, Tuple, Symbol, sympify, S
 from sympy.core.compatibility import (is_sequence, string_types, NotIterable,
                                       Iterable)
+from sympy.core.logic import fuzzy_bool
 from sympy.core.sympify import _sympify
 from sympy.functions.special.tensor_functions import KroneckerDelta
 
@@ -137,7 +139,7 @@ class Indexed(Expr):
     is_symbol = True
     is_Atom = True
 
-    def __new__(cls, base, *args, **kw_args):
+    def __new__(cls, base, *args, assumptions=None, **kw_args):
         from sympy.utilities.misc import filldedent
         from sympy.tensor.array.ndim_array import NDimArray
         from sympy.matrices.matrices import MatrixBase
@@ -156,7 +158,15 @@ def __new__(cls, base, *args, **kw_args):
             else:
                 return base[args]
 
-        return Expr.__new__(cls, base, *args, **kw_args)
+
+        # Apply assumptions, with same logic as for Symbol.__new__.
+        obj = Expr.__new__(cls, base, *args, **kw_args)
+        assumptions = dict() if assumptions is None else assumptions
+        is_commutative = fuzzy_bool(assumptions.get('commutative', True))
+        assumptions['commutative'] = is_commutative
+        obj._assumptions = StdFactKB(assumptions)
+
+        return obj
 
     @property
     def name(self):
@@ -377,7 +387,7 @@ class IndexedBase(Expr, NotIterable):
     is_symbol = True
     is_Atom = True
 
-    def __new__(cls, label, shape=None, **kw_args):
+    def __new__(cls, label, shape=None, assumptions=None, **kw_args):
         from sympy import MatrixBase, NDimArray
 
         if isinstance(label, string_types):
@@ -407,6 +417,7 @@ def __new__(cls, label, shape=None, **kw_args):
         obj._offset = offset
         obj._strides = strides
         obj._name = str(label)
+        obj._assumptions = dict() if assumptions is None else assumptions
         return obj
 
     @property
@@ -418,11 +429,11 @@ def __getitem__(self, indices, **kw_args):
             # Special case needed because M[*my_tuple] is a syntax error.
             if self.shape and len(self.shape) != len(indices):
                 raise IndexException("Rank mismatch.")
-            return Indexed(self, *indices, **kw_args)
+            return Indexed(self, *indices, assumptions=self._assumptions, **kw_args)
         else:
             if self.shape and len(self.shape) != 1:
                 raise IndexException("Rank mismatch.")
-            return Indexed(self, indices, **kw_args)
+            return Indexed(self, indices, assumptions=self._assumptions, **kw_args)
 
     @property
     def shape(self):