sagemathgh-38013: DualMatroid: Add is_valid and relabel methods

    
Implement class-specific `is_valid` and `relabel` methods.
Previously, such functionality was achieved through the generic and very
slow rank-based algorithms.

The PR also does a touch-up of the docstrings of `dual_matroid.py`.
    
URL: sagemath#38013
Reported by: gmou3
Reviewer(s): gmou3, Travis Scrimshaw

src/sage/matroids/basis_exchange_matroid.pyx

@@ -193,13 +193,13 @@ cdef class BasisExchangeMatroid(Matroid):
 
     cdef _relabel(self, mapping):
         """
-        Relabel each element ``e`` as ``mapping[e]``, where ``mapping`` is a
+        Relabel each element `e` as ``mapping[e]``, where ``mapping`` is a
         given injective map.
 
         INPUT:
 
-        - ``mapping`` -- a python object such that ``mapping[e]`` is the new
-          label of ``e``
+        - ``mapping`` -- a Python object such that ``mapping[e]`` is the new
+          label of `e`
 
         OUTPUT: ``None``
 

src/sage/matroids/basis_matroid.pyx

@@ -525,16 +525,16 @@ cdef class BasisMatroid(BasisExchangeMatroid):
         r"""
         Return an isomorphic matroid with relabeled groundset.
 
-        The output is obtained by relabeling each element ``e`` by
+        The output is obtained by relabeling each element `e` by
         ``mapping[e]``, where ``mapping`` is a given injective map. If
         ``mapping[e]`` is not defined, then the identity map is assumed.
 
         INPUT:
 
-        - ``mapping`` -- a python object such that ``mapping[e]`` is the new
-          label of ``e``
+        - ``mapping`` -- a Python object such that ``mapping[e]`` is the new
+          label of `e`
 
-        OUTPUT: a matroid
+        OUTPUT: matroid
 
         EXAMPLES::
 

src/sage/matroids/circuit_closures_matroid.pyx

@@ -527,16 +527,16 @@ cdef class CircuitClosuresMatroid(Matroid):
         r"""
         Return an isomorphic matroid with relabeled groundset.
 
-        The output is obtained by relabeling each element ``e`` by
+        The output is obtained by relabeling each element `e` by
         ``mapping[e]``, where ``mapping`` is a given injective map. If
         ``mapping[e]`` is not defined, then the identity map is assumed.
 
         INPUT:
 
-        - ``mapping`` -- a python object such that ``mapping[e]`` is the new
-          label of ``e``
+        - ``mapping`` -- a Python object such that ``mapping[e]`` is the new
+          label of `e`
 
-        OUTPUT: a matroid
+        OUTPUT: matroid
 
         EXAMPLES::
 

src/sage/matroids/circuits_matroid.pyx

@@ -406,14 +406,14 @@ cdef class CircuitsMatroid(Matroid):
         r"""
         Return an isomorphic matroid with relabeled groundset.
 
-        The output is obtained by relabeling each element ``e`` by
+        The output is obtained by relabeling each element `e` by
         ``mapping[e]``, where ``mapping`` is a given injective map. If
         ``mapping[e]`` is not defined, then the identity map is assumed.
 
         INPUT:
 
-        - ``mapping`` -- a python object such that ``mapping[e]`` is the new
-          label of ``e``
+        - ``mapping`` -- a Python object such that ``mapping[e]`` is the new
+          label of `e`
 
         OUTPUT: matroid
 

src/sage/matroids/database_matroids.py

@@ -1489,7 +1489,7 @@ def K5dual(groundset='abcdefghij'):
     EXAMPLES::
 
         sage: M = matroids.catalog.K5dual(); M
-        M*(K5): Matroid of rank 6 on 10 elements with 15 circuits
+        M*(K5): Dual of 'Graphic matroid of rank 4 on 10 elements'
         sage: M.is_3connected()
         True
         sage: G1 = M.automorphism_group()
@@ -1502,7 +1502,6 @@ def K5dual(groundset='abcdefghij'):
     [Oxl2011]_, p. 656.
     """
     M = CompleteGraphic(5).dual()
-    M = Matroid(circuits=list(M.circuits()))
     M = _rename_and_relabel(M, "M*(K5)", groundset)
     return M
 
@@ -2067,7 +2066,7 @@ def Z(r, t=True, groundset=None):
     - ``t`` -- boolean (default: ``True``); whether the spike is tipped
     - ``groundset`` -- a string (optional); the groundset of the matroid
 
-    OUTPUT: a matroid; the unique rank-`r` binary spike (tipped or tipless)
+    OUTPUT: matroid; the unique rank-`r` binary spike (tipped or tipless)
 
     EXAMPLES::
 
@@ -2173,7 +2172,7 @@ def Spike(r, t=True, C3=[], groundset=None):
       The default (i.e. the empty list) results in a free `r`-spike
     - ``groundset`` -- a string (optional); the groundset of the matroid
 
-    OUTPUT: a matroid; a rank-`r` spike (tipped or tipless)
+    OUTPUT: matroid; a rank-`r` spike (tipped or tipless)
 
     EXAMPLES::
 
@@ -2291,7 +2290,7 @@ def Theta(n, groundset=None):
     - ``n`` -- an integer (`n \ge 2`); the rank of the matroid
     - ``groundset`` -- a string (optional); the groundset of the matroid
 
-    OUTPUT: a matroid (`\Theta_n`)
+    OUTPUT: matroid (`\Theta_n`)
 
     EXAMPLES::
 
@@ -2364,7 +2363,7 @@ def Psi(r, groundset=None):
     - ``r`` -- an integer (`r \ge 3`); the rank of the matroid
     - ``groundset`` -- a string (optional); the groundset of the matroid
 
-    OUTPUT: a matroid (`\Psi_r`)
+    OUTPUT: matroid (`\Psi_r`)
 
     EXAMPLES::
 
@@ -5234,7 +5233,7 @@ def _rename_and_relabel(M, name=None, groundset=None):
     - ``name`` -- a string (optional)
     - ``groundset`` -- a string (optional)
 
-    OUTPUT: a matroid
+    OUTPUT: matroid
     """
     if groundset is not None:
         if len(groundset) != len(M.groundset()):