Trac #16351: Move RecursivelyEnumeratedSet_forest code to sage/sets/r…

…ecursively_enumerated_set.pyx, remove deprecated classes SearchForest, TransitiveIdeal, TransitiveIdealGraded

Move the code of `SearchForest` from `sage/combinat/backtrack.py` to
`sage/sets/recursively_enumerated_set.pyx` into a class named
`RecursivelyEnumeratedSet_forest`.

This is a follow up of #6637.

URL: https://trac.sagemath.org/16351
Reported by: slabbe
Ticket author(s): Matthias Koeppe
Reviewer(s): Vincent Delecroix

src/sage/categories/affine_weyl_groups.py

@@ -104,7 +104,7 @@ def affine_grassmannian_elements_of_given_length(self, k):
 
                 :meth:`AffineWeylGroups.ElementMethods.is_affine_grassmannian`
             """
-            from sage.combinat.backtrack import SearchForest
+            from sage.sets.recursively_enumerated_set import RecursivelyEnumeratedSet_forest
 
             def select_length(pair):
                 u, length = pair
@@ -119,9 +119,9 @@ def succ(pair):
                             u1.is_affine_grassmannian()):
                         yield (u1, length + 1)
                 return
-            return SearchForest(((self.one(), 0),), succ, algorithm='breadth',
-                                category=FiniteEnumeratedSets(),
-                                post_process=select_length)
+            return RecursivelyEnumeratedSet_forest(((self.one(), 0),), succ, algorithm='breadth',
+                                                   category=FiniteEnumeratedSets(),
+                                                   post_process=select_length)
 
     class ElementMethods:
         def is_affine_grassmannian(self):

src/sage/categories/coxeter_groups.py

@@ -455,7 +455,7 @@ def weak_order_ideal(self, predicate, side="right", category=None):
 
             .. rubric:: Background
 
-            The weak order is returned as a :class:`SearchForest`.
+            The weak order is returned as a :class:`RecursivelyEnumeratedSet_forest`.
             This is achieved by assigning to each element `u1` of the
             ideal a single ancestor `u=u1 s_i`, where `i` is the
             smallest descent of `u`.
@@ -474,7 +474,7 @@ def weak_order_ideal(self, predicate, side="right", category=None):
                 sage: [x.length() for x in W]
                 [0, 1, 1, 2, 2, 3]
             """
-            from sage.combinat.backtrack import SearchForest
+            from sage.sets.recursively_enumerated_set import RecursivelyEnumeratedSet_forest
 
             def succ(u):
                 for i in u.descents(positive=True, side=side):
@@ -484,7 +484,7 @@ def succ(u):
                 return
             from sage.categories.finite_coxeter_groups import FiniteCoxeterGroups
             default_category = FiniteEnumeratedSets() if self in FiniteCoxeterGroups() else EnumeratedSets()
-            return SearchForest((self.one(),), succ, algorithm='breadth',
+            return RecursivelyEnumeratedSet_forest((self.one(),), succ, algorithm='breadth',
                                 category=default_category.or_subcategory(category))
 
         @cached_method
@@ -1844,7 +1844,7 @@ def binary_factorizations(self, predicate=ConstantFunction(True)):
                 sage: w0.binary_factorizations().category()
                 Category of finite enumerated sets
             """
-            from sage.combinat.backtrack import SearchForest
+            from sage.sets.recursively_enumerated_set import RecursivelyEnumeratedSet_forest
             W = self.parent()
             if not predicate(W.one()):
                 from sage.sets.finite_enumerated_set import FiniteEnumeratedSet
@@ -1857,8 +1857,8 @@ def succ(u_v):
                     u1 = u * s[i]
                     if i == u1.first_descent() and predicate(u1):
                         yield (u1, s[i] * v)
-            return SearchForest(((W.one(), self),), succ,
-                                category=FiniteEnumeratedSets())
+            return RecursivelyEnumeratedSet_forest(((W.one(), self),), succ,
+                                                   category=FiniteEnumeratedSets())
 
         @cached_in_parent_method
         def bruhat_lower_covers(self):