sagemathgh-38203: Change inheritance order in UniqueRepresentation

    
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Currently in `UniqueRepresentation`, the MRO is constructed by
```class UniqueRepresentation(CachedRepresentation,
WithEqualityById):```
However, in order to maximize the utility of the `WithEqualityById`
class, in particular with subclasses of `UniqueRepresentation`, we
should place it *first* in the MRO. This is useful in sagemath#37128, where this
resolves MRO resolution issues and not having to reimplement (or set
aliases) the behavior of `WithEqualityById`.

However, this change, while seemingly innocuous, does cause a problem
with one doctest dealing with very old pickles (Sage version <= 4.0) in
`combinat/root_system/cartan_type.py` concerning `class
CartanType_simple_finite:`. Currently, it is unclear why this change is
resulting in the error:
```
sage: from sage.misc.fpickle import unpickleModule
sage: pg_make_integer = unpickle_global('sage.rings.integer',
'make_integer')
sage: si2 = pg_make_integer('4')
sage: unpickle_build(si1,
{'tools':unpickleModule('sage.combinat.root_system.type_A'), 't':['A',
si2], 'letter':'A', 'n':si2})
Traceback (most recent call last):
  File "/sage/src/sage/doctest/forker.py", line 715, in _run
    self.compile_and_execute(example, compiler, test.globs)
  File "/sage/src/sage/doctest/forker.py", line 1147, in
compile_and_execute
    exec(compiled, globs)
  File "<doctest sage.combinat.root_system.cartan_type.CartanType_simple
_finite.__setstate__[5]>", line 1, in <module>
    unpickle_build(si1,
{'tools':unpickleModule('sage.combinat.root_system.type_A'), 't':['A',
si2], 'letter':'A', 'n':si2})
  File "/sage/src/sage/misc/explain_pickle.py", line 2442, in
unpickle_build
    setstate(state)
  File "/sage/src/sage/combinat/root_system/cartan_type.py", line 3106,
in __setstate__
    self.__class__ = T.__class__
TypeError: __class__ assignment: 'CartanType' object layout differs from
'CartanType_simple_finite'
```
However, since these are very old pickles and we cannot reproduce the
failure otherwise, we feel that it is prudent to drop support for this
and remove the corresponding class.

### 📝 Checklist

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- [ ] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

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URL: sagemath#38203
Reported by: Enrique Manuel Artal Bartolo
Reviewer(s): Travis Scrimshaw

src/sage/categories/category_singleton.pyx

@@ -218,9 +218,9 @@ class Category_singleton(Category):
  <class 'sage.categories.category_singleton.Category_singleton'>,
  <class 'sage.categories.category.Category'>,
  <class 'sage.structure.unique_representation.UniqueRepresentation'>,
+ <class 'sage.misc.fast_methods.WithEqualityById'>,
  <class 'sage.structure.unique_representation.CachedRepresentation'>,
  <class 'sage.structure.unique_representation.WithPicklingByInitArgs'>,
- <class 'sage.misc.fast_methods.WithEqualityById'>,
  <class 'sage.structure.sage_object.SageObject'>,
  <class '__main__.R.subcategory_class'>,
  <class 'sage.categories.sets_cat.Sets.subcategory_class'>,

src/sage/categories/sets_cat.py

@@ -136,9 +136,9 @@ class Sets(Category_singleton):
  <class 'sage.categories.examples.sets_cat.PrimeNumbers_Inherits'>
  <class 'sage.categories.examples.sets_cat.PrimeNumbers_Abstract'>
  <class 'sage.structure.unique_representation.UniqueRepresentation'>
+ <class 'sage.misc.fast_methods.WithEqualityById'>
  <class 'sage.structure.unique_representation.CachedRepresentation'>
  <class 'sage.structure.unique_representation.WithPicklingByInitArgs'>
- <class 'sage.misc.fast_methods.WithEqualityById'>
  <class 'sage.structure.parent.Parent'>
  <class 'sage.structure.category_object.CategoryObject'>
  <class 'sage.structure.sage_object.SageObject'>

src/sage/combinat/root_system/cartan_type.py

@@ -3069,39 +3069,3 @@ def __getitem__(self, i):
  raise IndexError("index out of range")
 
  options = CartanType.options
-
-##############################################################################
-# For backward compatibility
-
-
-class CartanType_simple_finite:
- def __setstate__(self, dict):
- """
- Implements the unpickling of Cartan types pickled by Sage <= 4.0.
-
- EXAMPLES:
-
- This is the pickle for CartanType(["A", 4])::
-
- sage: pg_CartanType_simple_finite = unpickle_global('sage.combinat.root_system.cartan_type', 'CartanType_simple_finite')
- sage: si1 = unpickle_newobj(pg_CartanType_simple_finite, ())
- sage: from sage.misc.fpickle import unpickleModule
- sage: pg_make_integer = unpickle_global('sage.rings.integer', 'make_integer')
- sage: si2 = pg_make_integer('4')
- sage: unpickle_build(si1, {'tools':unpickleModule('sage.combinat.root_system.type_A'), 't':['A', si2], 'letter':'A', 'n':si2})
-
- sage: si1
- ['A', 4]
- sage: si1.dynkin_diagram() # needs sage.graphs
- O---O---O---O
- 1 2 3 4
- A4
-
- This is quite hacky; in particular unique representation is not preserved::
-
- sage: si1 == CartanType(["A", 4]) # todo: not implemented
- True
- """
- T = CartanType([dict['letter'], dict['n']])
- self.__class__ = T.__class__
- self.__dict__ = T.__dict__

src/sage/structure/unique_representation.py

@@ -1201,7 +1201,7 @@ def _clear_cache_(cls):
  del cache[k]
 
 
-class UniqueRepresentation(CachedRepresentation, WithEqualityById):
+class UniqueRepresentation(WithEqualityById, CachedRepresentation):
  r"""
  Classes derived from ``UniqueRepresentation`` inherit a unique
  representation behavior for their instances.