breakup value_type and scalar_type for Eigen types based on whether t…

…hey have a Scalar type trait

stan/math/prim/meta/is_eigen.hpp

@@ -20,6 +20,28 @@ template <typename T>
 struct is_eigen
     : bool_constant<is_base_pointer_convertible<Eigen::EigenBase, T>::value> {};
 
+namespace internal {
+// primary template handles types that have no nested ::type member:
+template <class, class = void>
+struct has_internal_trait : std::false_type {};
+
+// specialization recognizes types that do have a nested ::type member:
+template <class T>
+struct has_internal_trait<T, std::void_t<Eigen::internal::traits<std::decay_t<T>>>>
+    : std::true_type {};
+
+// primary template handles types that have no nested ::type member:
+template <class, class = void>
+struct has_scalar_trait : std::false_type {};
+
+// specialization recognizes types that do have a nested ::type member:
+template <class T>
+struct has_scalar_trait<T, std::void_t<typename std::decay_t<T>::Scalar>>
+    : std::true_type {};
+
+}  // namespace internal
+
+
 /**
  * Template metaprogram defining the base scalar type of
  * values stored in an Eigen matrix.
@@ -28,7 +50,7 @@ struct is_eigen
  * @ingroup type_trait
  */
 template <typename T>
-struct scalar_type<T, std::enable_if_t<is_eigen<T>::value>> {
+struct scalar_type<T, std::enable_if_t<is_eigen<T>::value && internal::has_scalar_trait<T>::value>> {
   using type = scalar_type_t<typename std::decay_t<T>::Scalar>;
 };
 
@@ -40,10 +62,36 @@ struct scalar_type<T, std::enable_if_t<is_eigen<T>::value>> {
  * @ingroup type_trait
  */
 template <typename T>
-struct value_type<T, std::enable_if_t<is_eigen<T>::value>> {
+struct value_type<T, std::enable_if_t<is_eigen<T>::value && internal::has_scalar_trait<T>::value>> {
+  using type = typename std::decay_t<T>::Scalar;
+};
+
+
+/**
+ * Template metaprogram defining the base scalar type of
+ * values stored in an Eigen matrix.
+ *
+ * @tparam T type to check.
+ * @ingroup type_trait
+ */
+template <typename T>
+struct scalar_type<T, std::enable_if_t<is_eigen<T>::value && !internal::has_scalar_trait<T>::value>> {
+  using type = scalar_type_t<typename Eigen::internal::traits<std::decay_t<T>>::Scalar>;
+};
+
+/**
+ * Template metaprogram defining the type of values stored in an
+ * Eigen matrix, vector, or row vector.
+ *
+ * @tparam T type to check
+ * @ingroup type_trait
+ */
+template <typename T>
+struct value_type<T, std::enable_if_t<is_eigen<T>::value && !internal::has_scalar_trait<T>::value>> {
   using type = typename Eigen::internal::traits<std::decay_t<T>>::Scalar;
 };
 
+
 /*! \ingroup require_eigens_types */
 /*! \defgroup eigen_types eigen  */
 /*! \addtogroup eigen_types */

stan/math/rev/core/arena_matrix.hpp

@@ -482,6 +482,7 @@ namespace internal {
 template <typename T>
 struct traits<stan::math::arena_matrix<T>> {
   using base = traits<Eigen::Map<T>>;
+  using Scalar = typename base::Scalar;
   using XprKind = typename Eigen::internal::traits<std::decay_t<T>>::XprKind;
   enum {
     PlainObjectTypeInnerSize = base::PlainObjectTypeInnerSize,