diff --git a/libcxx/include/concepts b/libcxx/include/concepts
--- a/libcxx/include/concepts
+++ b/libcxx/include/concepts
@@ -366,6 +366,34 @@
       const remove_reference_t<_Up>&>> &&
   __weakly_equality_comparable_with<_Tp, _Up>;
 
+// [concept.totallyordered]
+
+template<class _Tp, class _Up>
+concept __partially_ordered_with =
+  requires(const remove_reference_t<_Tp>& __t, const remove_reference_t<_Up>& __u) {
+    { __t <  __u } -> __boolean_testable;
+    { __t >  __u } -> __boolean_testable;
+    { __t <= __u } -> __boolean_testable;
+    { __t >= __u } -> __boolean_testable;
+    { __u <  __t } -> __boolean_testable;
+    { __u >  __t } -> __boolean_testable;
+    { __u <= __t } -> __boolean_testable;
+    { __u >= __t } -> __boolean_testable;
+  };
+
+template<class _Tp>
+concept totally_ordered = equality_comparable<_Tp> && __partially_ordered_with<_Tp, _Tp>;
+
+template<class _Tp, class _Up>
+concept totally_ordered_with =
+  totally_ordered<_Tp> && totally_ordered<_Up> &&
+  equality_comparable_with<_Tp, _Up> &&
+  totally_ordered<
+    common_reference_t<
+      const remove_reference_t<_Tp>&,
+      const remove_reference_t<_Up>&>> &&
+  __partially_ordered_with<_Tp, _Up>;
+
 // [concepts.object]
 template<class _Tp>
 concept movable =
diff --git a/libcxx/test/std/concepts/comparison/concepts.equalitycomparable/equality_comparable.compile.pass.cpp b/libcxx/test/std/concepts/comparison/concepts.equalitycomparable/equality_comparable.compile.pass.cpp
--- a/libcxx/test/std/concepts/comparison/concepts.equalitycomparable/equality_comparable.compile.pass.cpp
+++ b/libcxx/test/std/concepts/comparison/concepts.equalitycomparable/equality_comparable.compile.pass.cpp
@@ -112,14 +112,23 @@
 static_assert(std::equality_comparable<member_three_way_comparable>);
 static_assert(std::equality_comparable<friend_three_way_comparable>);
 static_assert(std::equality_comparable<explicit_operators>);
-static_assert(std::equality_comparable<eq_neq_different_return_types>);
+static_assert(std::equality_comparable<different_return_types>);
 static_assert(std::equality_comparable<one_member_one_friend>);
 static_assert(std::equality_comparable<equality_comparable_with_ec1>);
 
 static_assert(!std::equality_comparable<no_eq>);
 static_assert(!std::equality_comparable<no_neq>);
+static_assert(std::equality_comparable<no_lt>);
+static_assert(std::equality_comparable<no_gt>);
+static_assert(std::equality_comparable<no_le>);
+static_assert(std::equality_comparable<no_ge>);
+
 static_assert(!std::equality_comparable<wrong_return_type_eq>);
 static_assert(!std::equality_comparable<wrong_return_type_ne>);
+static_assert(std::equality_comparable<wrong_return_type_lt>);
+static_assert(std::equality_comparable<wrong_return_type_gt>);
+static_assert(std::equality_comparable<wrong_return_type_le>);
+static_assert(std::equality_comparable<wrong_return_type_ge>);
 static_assert(!std::equality_comparable<wrong_return_type>);
 static_assert(
     !std::equality_comparable<cxx20_member_eq_operator_with_deleted_ne>);
@@ -134,9 +143,13 @@
 static_assert(
     !std::equality_comparable<friend_three_way_comparable_with_deleted_ne>);
 
-static_assert(!std::equality_comparable<returns_explicit_bool>);
+static_assert(!std::equality_comparable<eq_returns_explicit_bool>);
+static_assert(!std::equality_comparable<ne_returns_explicit_bool>);
+static_assert(std::equality_comparable<lt_returns_explicit_bool>);
+static_assert(std::equality_comparable<gt_returns_explicit_bool>);
+static_assert(std::equality_comparable<le_returns_explicit_bool>);
+static_assert(std::equality_comparable<ge_returns_explicit_bool>);
 static_assert(std::equality_comparable<returns_true_type>);
-static_assert(std::equality_comparable<returns_false_type>);
 static_assert(std::equality_comparable<returns_int_ptr>);
 } // namespace types_fit_for_purpose
 
diff --git a/libcxx/test/std/concepts/comparison/concepts.equalitycomparable/equality_comparable_with.compile.pass.cpp b/libcxx/test/std/concepts/comparison/concepts.equalitycomparable/equality_comparable_with.compile.pass.cpp
--- a/libcxx/test/std/concepts/comparison/concepts.equalitycomparable/equality_comparable_with.compile.pass.cpp
+++ b/libcxx/test/std/concepts/comparison/concepts.equalitycomparable/equality_comparable_with.compile.pass.cpp
@@ -1097,12 +1097,12 @@
     check_equality_comparable_with<explicit_operators, explicit_operators>());
 static_assert(check_equality_comparable_with<equality_comparable_with_ec1,
                                              equality_comparable_with_ec1>());
-static_assert(check_equality_comparable_with<eq_neq_different_return_types,
-                                             eq_neq_different_return_types>());
+static_assert(check_equality_comparable_with<different_return_types,
+                                             different_return_types>());
 static_assert(check_equality_comparable_with<explicit_operators,
                                              equality_comparable_with_ec1>());
 static_assert(check_equality_comparable_with<explicit_operators,
-                                             eq_neq_different_return_types>());
+                                             different_return_types>());
 
 static_assert(check_equality_comparable_with<one_way_eq, one_way_eq>());
 static_assert(
diff --git a/libcxx/test/std/concepts/comparison/concepts.totallyordered/totally_ordered.compile.pass.cpp b/libcxx/test/std/concepts/comparison/concepts.totallyordered/totally_ordered.compile.pass.cpp
new file mode 100644
--- /dev/null
+++ b/libcxx/test/std/concepts/comparison/concepts.totallyordered/totally_ordered.compile.pass.cpp
@@ -0,0 +1,171 @@
+//===----------------------------------------------------------------------===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+// UNSUPPORTED: c++03, c++11, c++14, c++17
+// UNSUPPORTED: libcpp-no-concepts
+
+// template<class T>
+// concept totally_ordered = // see below
+
+#include <concepts>
+
+#include <array>
+#include <deque>
+#include <forward_list>
+#include <list>
+#include <map>
+#include <memory>
+#include <optional>
+#include <set>
+#include <unordered_map>
+#include <unordered_set>
+#include <vector>
+
+#include "../types.h"
+
+// `models_totally_ordered` checks that `std::totally_ordered` subsumes
+// `std::equality_comparable`. This overload should *never* be called.
+template <std::equality_comparable T>
+constexpr bool models_totally_ordered() noexcept {
+  return false;
+}
+
+template <std::totally_ordered T>
+constexpr bool models_totally_ordered() noexcept {
+  return true;
+}
+
+namespace fundamentals {
+static_assert(models_totally_ordered<int>());
+static_assert(models_totally_ordered<double>());
+static_assert(models_totally_ordered<void*>());
+static_assert(models_totally_ordered<char*>());
+static_assert(models_totally_ordered<char const*>());
+static_assert(models_totally_ordered<char volatile*>());
+static_assert(models_totally_ordered<char const volatile*>());
+static_assert(models_totally_ordered<wchar_t&>());
+static_assert(models_totally_ordered<char8_t const&>());
+static_assert(models_totally_ordered<char16_t volatile&>());
+static_assert(models_totally_ordered<char32_t const volatile&>());
+static_assert(models_totally_ordered<unsigned char&&>());
+static_assert(models_totally_ordered<unsigned short const&&>());
+static_assert(models_totally_ordered<unsigned int volatile&&>());
+static_assert(models_totally_ordered<unsigned long const volatile&&>());
+static_assert(models_totally_ordered<int[5]>());
+static_assert(models_totally_ordered<int (*)(int)>());
+static_assert(models_totally_ordered<int (&)(int)>());
+static_assert(models_totally_ordered<int (*)(int) noexcept>());
+static_assert(models_totally_ordered<int (&)(int) noexcept>());
+static_assert(!std::totally_ordered<std::nullptr_t>);
+
+struct S {};
+static_assert(!std::totally_ordered<S>);
+static_assert(!std::totally_ordered<int S::*>);
+static_assert(!std::totally_ordered<int (S::*)()>);
+static_assert(!std::totally_ordered<int (S::*)() noexcept>);
+static_assert(!std::totally_ordered<int (S::*)() &>);
+static_assert(!std::totally_ordered<int (S::*)() & noexcept>);
+static_assert(!std::totally_ordered<int (S::*)() &&>);
+static_assert(!std::totally_ordered < int (S::*)() && noexcept >);
+static_assert(!std::totally_ordered<int (S::*)() const>);
+static_assert(!std::totally_ordered<int (S::*)() const noexcept>);
+static_assert(!std::totally_ordered<int (S::*)() const&>);
+static_assert(!std::totally_ordered<int (S::*)() const & noexcept>);
+static_assert(!std::totally_ordered<int (S::*)() const&&>);
+static_assert(!std::totally_ordered < int (S::*)() const&& noexcept >);
+static_assert(!std::totally_ordered<int (S::*)() volatile>);
+static_assert(!std::totally_ordered<int (S::*)() volatile noexcept>);
+static_assert(!std::totally_ordered<int (S::*)() volatile&>);
+static_assert(!std::totally_ordered<int (S::*)() volatile & noexcept>);
+static_assert(!std::totally_ordered<int (S::*)() volatile&&>);
+static_assert(!std::totally_ordered < int (S::*)() volatile&& noexcept >);
+static_assert(!std::totally_ordered<int (S::*)() const volatile>);
+static_assert(!std::totally_ordered<int (S::*)() const volatile noexcept>);
+static_assert(!std::totally_ordered<int (S::*)() const volatile&>);
+static_assert(!std::totally_ordered<int (S::*)() const volatile & noexcept>);
+static_assert(!std::totally_ordered<int (S::*)() const volatile&&>);
+static_assert(!std::totally_ordered < int (S::*)() const volatile&& noexcept >);
+
+static_assert(!std::totally_ordered<void>);
+} // namespace fundamentals
+
+namespace standard_types {
+static_assert(models_totally_ordered<std::array<int, 10> >());
+static_assert(models_totally_ordered<std::deque<int> >());
+static_assert(models_totally_ordered<std::forward_list<int> >());
+static_assert(models_totally_ordered<std::list<int> >());
+static_assert(models_totally_ordered<std::optional<int> >());
+static_assert(models_totally_ordered<std::set<int> >());
+static_assert(models_totally_ordered<std::vector<bool> >());
+static_assert(models_totally_ordered<std::vector<int> >());
+
+static_assert(!std::totally_ordered<std::unordered_map<int, void*> >);
+static_assert(!std::totally_ordered<std::unordered_set<int> >);
+
+struct A {};
+// FIXME(cjdb): uncomment when operator<=> is implemented for each of these types.
+// static_assert(!std::totally_ordered<std::array<A, 10> >);
+// static_assert(!std::totally_ordered<std::deque<A> >);
+// static_assert(!std::totally_ordered<std::forward_list<A> >);
+// static_assert(!std::totally_ordered<std::list<A> >);
+static_assert(!std::totally_ordered<std::optional<A> >);
+// static_assert(!std::totally_ordered<std::set<A> >);
+// static_assert(!std::totally_ordered<std::vector<A> >);
+} // namespace standard_types
+
+namespace types_fit_for_purpose {
+static_assert(models_totally_ordered<member_three_way_comparable>());
+static_assert(models_totally_ordered<friend_three_way_comparable>());
+static_assert(models_totally_ordered<explicit_operators>());
+static_assert(models_totally_ordered<different_return_types>());
+static_assert(!std::totally_ordered<cxx20_member_eq>);
+static_assert(!std::totally_ordered<cxx20_friend_eq>);
+static_assert(!std::totally_ordered<one_member_one_friend>);
+static_assert(!std::totally_ordered<equality_comparable_with_ec1>);
+
+static_assert(!std::totally_ordered<no_eq>);
+static_assert(!std::totally_ordered<no_neq>);
+static_assert(!std::totally_ordered<no_lt>);
+static_assert(!std::totally_ordered<no_gt>);
+static_assert(!std::totally_ordered<no_le>);
+static_assert(!std::totally_ordered<no_ge>);
+
+static_assert(!std::totally_ordered<wrong_return_type_eq>);
+static_assert(!std::totally_ordered<wrong_return_type_ne>);
+static_assert(!std::totally_ordered<wrong_return_type_lt>);
+static_assert(!std::totally_ordered<wrong_return_type_gt>);
+static_assert(!std::totally_ordered<wrong_return_type_le>);
+static_assert(!std::totally_ordered<wrong_return_type_ge>);
+static_assert(!std::totally_ordered<wrong_return_type>);
+
+static_assert(!std::totally_ordered<cxx20_member_eq_operator_with_deleted_ne>);
+static_assert(!std::totally_ordered<cxx20_friend_eq_operator_with_deleted_ne>);
+static_assert(
+    !std::totally_ordered<member_three_way_comparable_with_deleted_eq>);
+static_assert(
+    !std::totally_ordered<member_three_way_comparable_with_deleted_ne>);
+static_assert(
+    !std::totally_ordered<friend_three_way_comparable_with_deleted_eq>);
+static_assert(
+    !std::totally_ordered<friend_three_way_comparable_with_deleted_ne>);
+
+static_assert(!std::totally_ordered<eq_returns_explicit_bool>);
+static_assert(!std::totally_ordered<ne_returns_explicit_bool>);
+static_assert(!std::totally_ordered<lt_returns_explicit_bool>);
+static_assert(!std::totally_ordered<gt_returns_explicit_bool>);
+static_assert(!std::totally_ordered<le_returns_explicit_bool>);
+static_assert(!std::totally_ordered<ge_returns_explicit_bool>);
+static_assert(std::totally_ordered<returns_true_type>);
+static_assert(std::totally_ordered<returns_int_ptr>);
+
+static_assert(std::totally_ordered<partial_ordering_totally_ordered_with>);
+static_assert(std::totally_ordered<weak_ordering_totally_ordered_with>);
+static_assert(std::totally_ordered<strong_ordering_totally_ordered_with>);
+} // namespace types_fit_for_purpose
+
+int main(int, char**) { return 0; }
diff --git a/libcxx/test/std/concepts/comparison/concepts.totallyordered/totally_ordered_with.compile.pass.cpp b/libcxx/test/std/concepts/comparison/concepts.totallyordered/totally_ordered_with.compile.pass.cpp
new file mode 100644
--- /dev/null
+++ b/libcxx/test/std/concepts/comparison/concepts.totallyordered/totally_ordered_with.compile.pass.cpp
@@ -0,0 +1,1136 @@
+//===----------------------------------------------------------------------===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+// UNSUPPORTED: c++03, c++11, c++14, c++17
+// UNSUPPORTED: libcpp-no-concepts
+
+// template<class T>
+// concept totally_ordered_with = // see below
+
+#include <concepts>
+
+#include <array>
+#include <deque>
+#include <forward_list>
+#include <list>
+#include <map>
+#include <memory>
+#include <optional>
+#include <set>
+#include <unordered_map>
+#include <unordered_set>
+#include <vector>
+
+#include "../types.h"
+
+template <class T, class U>
+constexpr bool check_totally_ordered_with() noexcept {
+  constexpr bool result = std::totally_ordered_with<T, U>;
+  static_assert(std::totally_ordered_with<U, T> == result);
+  static_assert(std::totally_ordered_with<T, U const> == result);
+  static_assert(std::totally_ordered_with<T const, U const> == result);
+  static_assert(std::totally_ordered_with<T, U const&> == result);
+  static_assert(std::totally_ordered_with<T const, U const&> == result);
+  static_assert(std::totally_ordered_with<T&, U const> == result);
+  static_assert(std::totally_ordered_with<T const&, U const> == result);
+  static_assert(std::totally_ordered_with<T&, U const&> == result);
+  static_assert(std::totally_ordered_with<T const&, U const&> == result);
+  static_assert(std::totally_ordered_with<T, U const&&> == result);
+  static_assert(std::totally_ordered_with<T const, U const&&> == result);
+  static_assert(std::totally_ordered_with<T&, U const&&> == result);
+  static_assert(std::totally_ordered_with<T const&, U const&&> == result);
+  static_assert(std::totally_ordered_with<T&&, U const> == result);
+  static_assert(std::totally_ordered_with<T const&&, U const> == result);
+  static_assert(std::totally_ordered_with<T&&, U const&> == result);
+  static_assert(std::totally_ordered_with<T const&&, U const&> == result);
+  static_assert(std::totally_ordered_with<T&&, U const&&> == result);
+  static_assert(std::totally_ordered_with<T const&&, U const&&> == result);
+  return result;
+}
+
+namespace fundamentals {
+static_assert(check_totally_ordered_with<int, int>());
+static_assert(check_totally_ordered_with<int, bool>());
+static_assert(check_totally_ordered_with<int, char>());
+static_assert(check_totally_ordered_with<int, wchar_t>());
+static_assert(check_totally_ordered_with<int, double>());
+static_assert(!check_totally_ordered_with<int, int*>());
+static_assert(!check_totally_ordered_with<int, int[5]>());
+static_assert(!check_totally_ordered_with<int, int (*)()>());
+static_assert(!check_totally_ordered_with<int, int (&)()>());
+
+struct S {};
+static_assert(!check_totally_ordered_with<int, int S::*>());
+static_assert(!check_totally_ordered_with<int, int (S::*)()>());
+static_assert(!check_totally_ordered_with<int, int (S::*)() noexcept>());
+static_assert(!check_totally_ordered_with<int, int (S::*)() const>());
+static_assert(!check_totally_ordered_with<int, int (S::*)() const noexcept>());
+static_assert(!check_totally_ordered_with<int, int (S::*)() volatile>());
+static_assert(
+    !check_totally_ordered_with<int, int (S::*)() volatile noexcept>());
+static_assert(!check_totally_ordered_with<int, int (S::*)() const volatile>());
+static_assert(
+    !check_totally_ordered_with<int, int (S::*)() const volatile noexcept>());
+static_assert(!check_totally_ordered_with<int, int (S::*)() &>());
+static_assert(!check_totally_ordered_with<int, int (S::*)() & noexcept>());
+static_assert(!check_totally_ordered_with<int, int (S::*)() const&>());
+static_assert(
+    !check_totally_ordered_with<int, int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int, int (S::*)() volatile&>());
+static_assert(
+    !check_totally_ordered_with<int, int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int, int (S::*)() const volatile&>());
+static_assert(
+    !check_totally_ordered_with<int, int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int, int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int, int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int, int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int, int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int, int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(check_totally_ordered_with<int*, int*>());
+static_assert(check_totally_ordered_with<int*, int[5]>());
+static_assert(!check_totally_ordered_with<int*, int (*)()>());
+static_assert(!check_totally_ordered_with<int*, int (&)()>());
+static_assert(!check_totally_ordered_with<int*, int (S::*)()>());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() noexcept>());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() const>());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() const noexcept>());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() volatile>());
+static_assert(
+    !check_totally_ordered_with<int*, int (S::*)() volatile noexcept>());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() const volatile>());
+static_assert(
+    !check_totally_ordered_with<int*, int (S::*)() const volatile noexcept>());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() &>());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() & noexcept>());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() const&>());
+static_assert(
+    !check_totally_ordered_with<int*, int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() volatile&>());
+static_assert(
+    !check_totally_ordered_with<int*, int (S::*)() volatile & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int*, int (S::*)() const volatile&>());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() const volatile &
+                                                    noexcept>());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int*,
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int*,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int*, int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int*,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int*, int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int*,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(check_totally_ordered_with<int[5], int[5]>());
+static_assert(!check_totally_ordered_with<int[5], int (*)()>());
+static_assert(!check_totally_ordered_with<int[5], int (&)()>());
+static_assert(!check_totally_ordered_with<int[5], int (S::*)()>());
+static_assert(!check_totally_ordered_with<int[5], int (S::*)() noexcept>());
+static_assert(!check_totally_ordered_with<int[5], int (S::*)() const>());
+static_assert(
+    !check_totally_ordered_with<int[5], int (S::*)() const noexcept>());
+static_assert(!check_totally_ordered_with<int[5], int (S::*)() volatile>());
+static_assert(
+    !check_totally_ordered_with<int[5], int (S::*)() volatile noexcept>());
+static_assert(
+    !check_totally_ordered_with<int[5], int (S::*)() const volatile>());
+static_assert(!check_totally_ordered_with<
+              int[5], int (S::*)() const volatile noexcept>());
+static_assert(!check_totally_ordered_with<int[5], int (S::*)() &>());
+static_assert(!check_totally_ordered_with<int[5], int (S::*)() & noexcept>());
+static_assert(!check_totally_ordered_with<int[5], int (S::*)() const&>());
+static_assert(
+    !check_totally_ordered_with<int[5], int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int[5], int (S::*)() volatile&>());
+static_assert(
+    !check_totally_ordered_with<int[5], int (S::*)() volatile & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int[5], int (S::*)() const volatile&>());
+static_assert(!check_totally_ordered_with<int[5], int (S::*)() const volatile &
+                                                      noexcept>());
+static_assert(!check_totally_ordered_with<int[5], int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int[5],
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int[5], int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int[5],
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int[5], int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int[5],
+              int (S::*)() volatile&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int[5], int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int[5],
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(check_totally_ordered_with<int (*)(), int (*)()>());
+static_assert(check_totally_ordered_with<int (*)(), int (&)()>());
+static_assert(!check_totally_ordered_with<int (*)(), int (S::*)()>());
+static_assert(!check_totally_ordered_with<int (*)(), int (S::*)() noexcept>());
+static_assert(!check_totally_ordered_with<int (*)(), int (S::*)() const>());
+static_assert(
+    !check_totally_ordered_with<int (*)(), int (S::*)() const noexcept>());
+static_assert(!check_totally_ordered_with<int (*)(), int (S::*)() volatile>());
+static_assert(
+    !check_totally_ordered_with<int (*)(), int (S::*)() volatile noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (*)(), int (S::*)() const volatile>());
+static_assert(!check_totally_ordered_with<
+              int (*)(), int (S::*)() const volatile noexcept>());
+static_assert(!check_totally_ordered_with<int (*)(), int (S::*)() &>());
+static_assert(
+    !check_totally_ordered_with<int (*)(), int (S::*)() & noexcept>());
+static_assert(!check_totally_ordered_with<int (*)(), int (S::*)() const&>());
+static_assert(
+    !check_totally_ordered_with<int (*)(), int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int (*)(), int (S::*)() volatile&>());
+static_assert(
+    !check_totally_ordered_with<int (*)(), int (S::*)() volatile & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (*)(), int (S::*)() const volatile&>());
+static_assert(!check_totally_ordered_with<
+              int (*)(), int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (*)(), int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (*)(),
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int (*)(), int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (*)(),
+              int (S::*)() const&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (*)(), int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (*)(),
+              int (S::*)() volatile&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (*)(), int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (*)(),
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(check_totally_ordered_with<int (&)(), int (&)()>());
+static_assert(!check_totally_ordered_with<int (&)(), int (S::*)()>());
+static_assert(!check_totally_ordered_with<int (&)(), int (S::*)() noexcept>());
+static_assert(!check_totally_ordered_with<int (&)(), int (S::*)() const>());
+static_assert(
+    !check_totally_ordered_with<int (&)(), int (S::*)() const noexcept>());
+static_assert(!check_totally_ordered_with<int (&)(), int (S::*)() volatile>());
+static_assert(
+    !check_totally_ordered_with<int (&)(), int (S::*)() volatile noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (&)(), int (S::*)() const volatile>());
+static_assert(!check_totally_ordered_with<
+              int (&)(), int (S::*)() const volatile noexcept>());
+static_assert(!check_totally_ordered_with<int (&)(), int (S::*)() &>());
+static_assert(
+    !check_totally_ordered_with<int (&)(), int (S::*)() & noexcept>());
+static_assert(!check_totally_ordered_with<int (&)(), int (S::*)() const&>());
+static_assert(
+    !check_totally_ordered_with<int (&)(), int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int (&)(), int (S::*)() volatile&>());
+static_assert(
+    !check_totally_ordered_with<int (&)(), int (S::*)() volatile & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (&)(), int (S::*)() const volatile&>());
+static_assert(!check_totally_ordered_with<
+              int (&)(), int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (&)(), int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (&)(),
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int (&)(), int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (&)(),
+              int (S::*)() const&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (&)(), int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (&)(),
+              int (S::*)() volatile&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (&)(), int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (&)(),
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)(), int (S::*)()>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)(), int (S::*)() noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)(), int (S::*)() const>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)(), int (S::*)() const noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)(), int (S::*)() volatile>());
+static_assert(!check_totally_ordered_with<int (S::*)(),
+                                          int (S::*)() volatile noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)(), int (S::*)() const volatile>());
+static_assert(!check_totally_ordered_with<
+              int (S::*)(), int (S::*)() const volatile noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)(), int (S::*)() &>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)(), int (S::*)() & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)(), int (S::*)() const&>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)(), int (S::*)() const & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)(), int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)(),
+                                          int (S::*)() volatile & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)(), int (S::*)() const volatile&>());
+static_assert(!check_totally_ordered_with<
+              int (S::*)(), int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)(), int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)(),
+              int (S::*)() && noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)(), int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)(),
+              int (S::*)() const&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)(), int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)(),
+              int (S::*)() volatile&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)(), int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)(),
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() noexcept,
+                                          int (S::*)() noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() noexcept, int (S::*)() const>());
+static_assert(!check_totally_ordered_with<int (S::*)() noexcept,
+                                          int (S::*)() const noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() noexcept,
+                                          int (S::*)() volatile>());
+static_assert(!check_totally_ordered_with<int (S::*)() noexcept,
+                                          int (S::*)() volatile noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() noexcept,
+                                          int (S::*)() const volatile>());
+static_assert(!check_totally_ordered_with<
+              int (S::*)() noexcept, int (S::*)() const volatile noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() noexcept, int (S::*)() &>());
+static_assert(!check_totally_ordered_with<int (S::*)() noexcept,
+                                          int (S::*)() & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() noexcept, int (S::*)() const&>());
+static_assert(!check_totally_ordered_with<int (S::*)() noexcept,
+                                          int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() noexcept,
+                                          int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)() noexcept,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() noexcept,
+                                          int (S::*)() const volatile&>());
+static_assert(!check_totally_ordered_with<
+              int (S::*)() noexcept, int (S::*)() const volatile & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() noexcept, int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() noexcept,
+              int (S::*)() && noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() noexcept, int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() noexcept,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() noexcept,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() noexcept,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() noexcept,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() noexcept,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const, int (S::*)() const>());
+static_assert(!check_totally_ordered_with<int (S::*)() const,
+                                          int (S::*)() const noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const, int (S::*)() volatile>());
+static_assert(!check_totally_ordered_with<int (S::*)() const,
+                                          int (S::*)() volatile noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const,
+                                          int (S::*)() const volatile>());
+static_assert(!check_totally_ordered_with<
+              int (S::*)() const, int (S::*)() const volatile noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const, int (S::*)() &>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const, int (S::*)() & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const, int (S::*)() const&>());
+static_assert(!check_totally_ordered_with<int (S::*)() const,
+                                          int (S::*)() const & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const, int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)() const,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const,
+                                          int (S::*)() const volatile&>());
+static_assert(!check_totally_ordered_with<
+              int (S::*)() const, int (S::*)() const volatile & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const, int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const,
+              int (S::*)() && noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const, int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const,
+              int (S::*)() const&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const, int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() const noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() volatile>());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() volatile noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() const volatile>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const noexcept,
+                                int (S::*)() const volatile noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const noexcept, int (S::*)() &>());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() const&>());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() const volatile&>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const noexcept,
+                                int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const noexcept,
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const noexcept,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const noexcept,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const noexcept,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const noexcept,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() volatile,
+                                          int (S::*)() volatile>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile,
+                                          int (S::*)() volatile noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile,
+                                          int (S::*)() const volatile>());
+static_assert(!check_totally_ordered_with<
+              int (S::*)() volatile, int (S::*)() const volatile noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() volatile, int (S::*)() &>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile,
+                                          int (S::*)() & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() volatile, int (S::*)() const&>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile,
+                                          int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile,
+                                          int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile,
+                                          int (S::*)() const volatile&>());
+static_assert(!check_totally_ordered_with<
+              int (S::*)() volatile, int (S::*)() const volatile & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() volatile, int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile,
+              int (S::*)() && noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() volatile, int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() volatile noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() const volatile>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                int (S::*)() const volatile noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() &>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() const&>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() const volatile&>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile noexcept,
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile noexcept,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile noexcept,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile noexcept,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile noexcept,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile,
+                                          int (S::*)() const volatile>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const volatile,
+                                int (S::*)() const volatile noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const volatile, int (S::*)() &>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile,
+                                          int (S::*)() & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile,
+                                          int (S::*)() const&>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile,
+                                          int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile,
+                                          int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile,
+                                          int (S::*)() const volatile&>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const volatile,
+                                int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile,
+                                          int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const volatile,
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile,
+                                          int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const volatile,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const volatile,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const volatile,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                int (S::*)() const volatile noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                          int (S::*)() &>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                          int (S::*)() & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                          int (S::*)() const&>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                          int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                          int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                          int (S::*)() const volatile&>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                          int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)()
+                                                const volatile noexcept,
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                          int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)()
+                                                const volatile noexcept,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)()
+                                                const volatile noexcept,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile noexcept,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)()
+                                                const volatile noexcept,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() &, int (S::*)() &>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() &, int (S::*)() & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() &, int (S::*)() const&>());
+static_assert(!check_totally_ordered_with<int (S::*)() &,
+                                          int (S::*)() const & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() &, int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)() &,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() &,
+                                          int (S::*)() const volatile&>());
+static_assert(!check_totally_ordered_with<
+              int (S::*)() &, int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() &, int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() &,
+              int (S::*)() && noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() &, int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() &,
+              int (S::*)() const&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() &, int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() &,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() &,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() &,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() & noexcept,
+                                          int (S::*)() & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() & noexcept,
+                                          int (S::*)() const&>());
+static_assert(!check_totally_ordered_with<int (S::*)() & noexcept,
+                                          int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() & noexcept,
+                                          int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)() & noexcept,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() & noexcept,
+                                          int (S::*)() const volatile&>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() & noexcept,
+                                int (S::*)() const volatile & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() & noexcept, int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() & noexcept,
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() & noexcept,
+                                          int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() & noexcept,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() & noexcept,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() & noexcept,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() & noexcept,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() & noexcept,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const&, int (S::*)() const&>());
+static_assert(!check_totally_ordered_with<int (S::*)() const&,
+                                          int (S::*)() const & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const&, int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)() const&,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const&,
+                                          int (S::*)() const volatile&>());
+static_assert(!check_totally_ordered_with<
+              int (S::*)() const&, int (S::*)() const volatile & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const&, int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const&,
+              int (S::*)() && noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const&, int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const&,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const&,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const&,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const&,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const&,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() const & noexcept,
+                                          int (S::*)() const & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const & noexcept,
+                                          int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)() const & noexcept,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const & noexcept,
+                                          int (S::*)() const volatile&>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const & noexcept,
+                                int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const & noexcept,
+                                          int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const& noexcept,
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const & noexcept,
+                                          int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const& noexcept,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const & noexcept,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const& noexcept,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const & noexcept,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const& noexcept,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() volatile&,
+                                          int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile&,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile&,
+                                          int (S::*)() const volatile&>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() volatile&,
+                                int (S::*)() const volatile & noexcept>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() volatile&, int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile&,
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile&,
+                                          int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile&,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile&,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile&,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile&,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile&,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() volatile & noexcept,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile & noexcept,
+                                          int (S::*)() const volatile&>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() volatile & noexcept,
+                                int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile & noexcept,
+                                          int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile& noexcept,
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile & noexcept,
+                                          int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile& noexcept,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile & noexcept,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile& noexcept,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile & noexcept,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile& noexcept,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile&,
+                                          int (S::*)() const volatile&>());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const volatile&,
+                                int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile&,
+                                          int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const volatile&,
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile&,
+                                          int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const volatile&,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile&,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const volatile&,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile&,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const volatile&,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const volatile & noexcept,
+                                int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<
+              int (S::*)() const volatile & noexcept, int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)()
+                                                const volatile& noexcept,
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with<
+              int (S::*)() const volatile & noexcept, int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)()
+                                                const volatile& noexcept,
+              int (S::*)() const&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const volatile & noexcept,
+                                int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)()
+                                                const volatile& noexcept,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const volatile & noexcept,
+                                int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)()
+                                                const volatile& noexcept,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() &&, int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < int (S::*)() &&,
+              int (S::*)() && noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() &&, int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() &&,
+              int (S::*)() const&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<int (S::*)() &&, int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() &&,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() &&,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() &&,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with < int (S::*)() && noexcept,
+              int (S::*)() && noexcept > ());
+static_assert(!check_totally_ordered_with < int (S::*)() && noexcept,
+              int (S::*)() const&& > ());
+static_assert(!check_totally_ordered_with < int (S::*)() && noexcept,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with < int (S::*)() && noexcept,
+              int (S::*)() volatile&& > ());
+static_assert(!check_totally_ordered_with < int (S::*)() && noexcept,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with < int (S::*)() && noexcept,
+              int (S::*)() const volatile&& > ());
+static_assert(!check_totally_ordered_with < int (S::*)() && noexcept,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(
+    !check_totally_ordered_with<int (S::*)() const&&, int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const&&,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const&&,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const&&,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() const&&,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const&&,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with < int (S::*)() const&& noexcept,
+              int (S::*)() const&& noexcept > ());
+static_assert(!check_totally_ordered_with < int (S::*)() const&& noexcept,
+              int (S::*)() volatile&& > ());
+static_assert(!check_totally_ordered_with < int (S::*)() const&& noexcept,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with < int (S::*)() const&& noexcept,
+              int (S::*)() const volatile&& > ());
+static_assert(!check_totally_ordered_with < int (S::*)() const&& noexcept,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() volatile&&,
+                                          int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile&&,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<int (S::*)() volatile&&,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile&&,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with < int (S::*)() volatile&& noexcept,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile&& noexcept,
+              int (S::*)() const volatile&& > ());
+static_assert(!check_totally_ordered_with < int (S::*)() volatile&& noexcept,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<int (S::*)() const volatile&&,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < int (S::*)() const volatile&&,
+              int (S::*)() const volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with < int (S::*)()
+                                                const volatile&& noexcept,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!check_totally_ordered_with<std::nullptr_t, int>());
+static_assert(!check_totally_ordered_with<std::nullptr_t, int*>());
+static_assert(!check_totally_ordered_with<std::nullptr_t, int[5]>());
+static_assert(!check_totally_ordered_with<std::nullptr_t, int (*)()>());
+static_assert(!check_totally_ordered_with<std::nullptr_t, int (&)()>());
+static_assert(!check_totally_ordered_with<std::nullptr_t, int (S::*)()>());
+static_assert(
+    !check_totally_ordered_with<std::nullptr_t, int (S::*)() noexcept>());
+static_assert(
+    !check_totally_ordered_with<std::nullptr_t, int (S::*)() const>());
+static_assert(
+    !check_totally_ordered_with<std::nullptr_t, int (S::*)() const noexcept>());
+static_assert(
+    !check_totally_ordered_with<std::nullptr_t, int (S::*)() volatile>());
+static_assert(!check_totally_ordered_with<std::nullptr_t,
+                                          int (S::*)() volatile noexcept>());
+static_assert(
+    !check_totally_ordered_with<std::nullptr_t, int (S::*)() const volatile>());
+static_assert(!check_totally_ordered_with<
+              std::nullptr_t, int (S::*)() const volatile noexcept>());
+static_assert(!check_totally_ordered_with<std::nullptr_t, int (S::*)() &>());
+static_assert(
+    !check_totally_ordered_with<std::nullptr_t, int (S::*)() & noexcept>());
+static_assert(
+    !check_totally_ordered_with<std::nullptr_t, int (S::*)() const&>());
+static_assert(!check_totally_ordered_with<std::nullptr_t,
+                                          int (S::*)() const & noexcept>());
+static_assert(
+    !check_totally_ordered_with<std::nullptr_t, int (S::*)() volatile&>());
+static_assert(!check_totally_ordered_with<std::nullptr_t,
+                                          int (S::*)() volatile & noexcept>());
+static_assert(!check_totally_ordered_with<std::nullptr_t,
+                                          int (S::*)() const volatile&>());
+static_assert(!check_totally_ordered_with<
+              std::nullptr_t, int (S::*)() const volatile & noexcept>());
+static_assert(!check_totally_ordered_with<std::nullptr_t, int (S::*)() &&>());
+static_assert(!check_totally_ordered_with < std::nullptr_t,
+              int (S::*)() && noexcept > ());
+static_assert(
+    !check_totally_ordered_with<std::nullptr_t, int (S::*)() const&&>());
+static_assert(!check_totally_ordered_with < std::nullptr_t,
+              int (S::*)() const&& noexcept > ());
+static_assert(
+    !check_totally_ordered_with<std::nullptr_t, int (S::*)() volatile&&>());
+static_assert(!check_totally_ordered_with < std::nullptr_t,
+              int (S::*)() volatile&& noexcept > ());
+static_assert(!check_totally_ordered_with<std::nullptr_t,
+                                          int (S::*)() const volatile&&>());
+static_assert(!check_totally_ordered_with < std::nullptr_t,
+              int (S::*)() const volatile&& noexcept > ());
+
+static_assert(!std::equality_comparable_with<void, int>);
+static_assert(!std::equality_comparable_with<void, int*>);
+static_assert(!std::equality_comparable_with<void, std::nullptr_t>);
+static_assert(!std::equality_comparable_with<void, int[5]>);
+static_assert(!std::equality_comparable_with<void, int (*)()>);
+static_assert(!std::equality_comparable_with<void, int (&)()>);
+static_assert(!std::equality_comparable_with<void, int S::*>);
+static_assert(!std::equality_comparable_with<void, int (S::*)()>);
+static_assert(!std::equality_comparable_with<void, int (S::*)() noexcept>);
+static_assert(!std::equality_comparable_with<void, int (S::*)() const>);
+static_assert(
+    !std::equality_comparable_with<void, int (S::*)() const noexcept>);
+static_assert(!std::equality_comparable_with<void, int (S::*)() volatile>);
+static_assert(
+    !std::equality_comparable_with<void, int (S::*)() volatile noexcept>);
+static_assert(
+    !std::equality_comparable_with<void, int (S::*)() const volatile>);
+static_assert(
+    !std::equality_comparable_with<void, int (S::*)() const volatile noexcept>);
+static_assert(!std::equality_comparable_with<void, int (S::*)() &>);
+static_assert(!std::equality_comparable_with<void, int (S::*)() & noexcept>);
+static_assert(!std::equality_comparable_with<void, int (S::*)() const&>);
+static_assert(
+    !std::equality_comparable_with<void, int (S::*)() const & noexcept>);
+static_assert(!std::equality_comparable_with<void, int (S::*)() volatile&>);
+static_assert(
+    !std::equality_comparable_with<void, int (S::*)() volatile & noexcept>);
+static_assert(
+    !std::equality_comparable_with<void, int (S::*)() const volatile&>);
+static_assert(!std::equality_comparable_with<void, int (S::*)() const volatile &
+                                                       noexcept>);
+static_assert(!std::equality_comparable_with<void, int (S::*)() &&>);
+static_assert(!std::equality_comparable_with < void,
+              int (S::*)() && noexcept >);
+static_assert(!std::equality_comparable_with<void, int (S::*)() const&&>);
+static_assert(!std::equality_comparable_with < void,
+              int (S::*)() const&& noexcept >);
+static_assert(!std::equality_comparable_with<void, int (S::*)() volatile&&>);
+static_assert(!std::equality_comparable_with < void,
+              int (S::*)() volatile&& noexcept >);
+static_assert(
+    !std::equality_comparable_with<void, int (S::*)() const volatile&&>);
+static_assert(!std::equality_comparable_with < void,
+              int (S::*)() const volatile&& noexcept >);
+} // namespace fundamentals
+
+namespace standard_types {
+static_assert(
+    check_totally_ordered_with<std::array<int, 10>, std::array<int, 10> >());
+static_assert(!check_totally_ordered_with<std::array<int, 10>,
+                                          std::array<double, 10> >());
+static_assert(check_totally_ordered_with<std::deque<int>, std::deque<int> >());
+static_assert(
+    !check_totally_ordered_with<std::deque<int>, std::vector<int> >());
+static_assert(check_totally_ordered_with<std::forward_list<int>,
+                                         std::forward_list<int> >());
+static_assert(
+    !check_totally_ordered_with<std::forward_list<int>, std::vector<int> >());
+static_assert(check_totally_ordered_with<std::list<int>, std::list<int> >());
+static_assert(!check_totally_ordered_with<std::list<int>, std::vector<int> >());
+
+static_assert(
+    check_totally_ordered_with<std::map<int, void*>, std::map<int, void*> >());
+static_assert(
+    !check_totally_ordered_with<std::map<int, void*>, std::vector<int> >());
+static_assert(check_totally_ordered_with<std::optional<std::vector<int> >,
+                                         std::optional<std::vector<int> > >());
+static_assert(check_totally_ordered_with<std::optional<std::vector<int> >,
+                                         std::vector<int> >());
+static_assert(
+    check_totally_ordered_with<std::vector<int>, std::vector<int> >());
+static_assert(!check_totally_ordered_with<std::vector<int>, int>());
+
+struct A {};
+// FIXME(cjdb): uncomment when operator<=> is implemented for each of these types.
+// static_assert(!check_totally_ordered_with<std::optional<std::vector<A> >,
+//                                          std::optional<std::vector<A> > >());
+// static_assert(!check_totally_ordered_with<std::optional<std::vector<A> >,
+//                                          std::vector<A> >());
+struct B {};
+static_assert(!check_totally_ordered_with<std::vector<A>, std::vector<B> >());
+static_assert(
+    !check_totally_ordered_with<std::optional<A>, std::optional<B> >());
+} // namespace standard_types
+
+namespace types_fit_for_purpose {
+static_assert(!check_totally_ordered_with<cxx20_member_eq, cxx20_member_eq>());
+static_assert(!check_totally_ordered_with<cxx20_friend_eq, cxx20_friend_eq>());
+static_assert(!check_totally_ordered_with<cxx20_member_eq, cxx20_friend_eq>());
+
+static_assert(check_totally_ordered_with<member_three_way_comparable,
+                                         member_three_way_comparable>());
+static_assert(check_totally_ordered_with<friend_three_way_comparable,
+                                         friend_three_way_comparable>());
+static_assert(!check_totally_ordered_with<member_three_way_comparable,
+                                          friend_three_way_comparable>());
+
+static_assert(
+    check_totally_ordered_with<explicit_operators, explicit_operators>());
+static_assert(!check_totally_ordered_with<equality_comparable_with_ec1,
+                                          equality_comparable_with_ec1>());
+static_assert(check_totally_ordered_with<different_return_types,
+                                         different_return_types>());
+static_assert(!check_totally_ordered_with<explicit_operators,
+                                          equality_comparable_with_ec1>());
+static_assert(
+    check_totally_ordered_with<explicit_operators, different_return_types>());
+
+static_assert(!check_totally_ordered_with<one_way_eq, one_way_eq>());
+static_assert(
+    std::common_reference_with<one_way_eq const&, explicit_operators const&> &&
+    !check_totally_ordered_with<one_way_eq, explicit_operators>());
+
+static_assert(!check_totally_ordered_with<one_way_ne, one_way_ne>());
+static_assert(
+    std::common_reference_with<one_way_ne const&, explicit_operators const&> &&
+    !check_totally_ordered_with<one_way_ne, explicit_operators>());
+
+static_assert(
+    check_totally_ordered_with<totally_ordered_with_others,
+                               partial_ordering_totally_ordered_with>());
+static_assert(check_totally_ordered_with<totally_ordered_with_others,
+                                         weak_ordering_totally_ordered_with>());
+static_assert(
+    check_totally_ordered_with<totally_ordered_with_others,
+                               strong_ordering_totally_ordered_with>());
+
+static_assert(!check_totally_ordered_with<totally_ordered_with_others,
+                                          eq_returns_explicit_bool>());
+static_assert(!check_totally_ordered_with<totally_ordered_with_others,
+                                          ne_returns_explicit_bool>());
+static_assert(std::equality_comparable_with<totally_ordered_with_others,
+                                            lt_returns_explicit_bool> &&
+              !check_totally_ordered_with<totally_ordered_with_others,
+                                          lt_returns_explicit_bool>());
+static_assert(std::equality_comparable_with<totally_ordered_with_others,
+                                            gt_returns_explicit_bool> &&
+              !check_totally_ordered_with<totally_ordered_with_others,
+                                          gt_returns_explicit_bool>());
+static_assert(std::equality_comparable_with<totally_ordered_with_others,
+                                            le_returns_explicit_bool> &&
+              !check_totally_ordered_with<totally_ordered_with_others,
+                                          le_returns_explicit_bool>());
+static_assert(std::equality_comparable_with<totally_ordered_with_others,
+                                            ge_returns_explicit_bool> &&
+              !check_totally_ordered_with<totally_ordered_with_others,
+                                          ge_returns_explicit_bool>());
+static_assert(check_totally_ordered_with<totally_ordered_with_others,
+                                         returns_true_type>());
+static_assert(
+    check_totally_ordered_with<totally_ordered_with_others, returns_int_ptr>());
+
+static_assert(
+    std::totally_ordered<no_lt_not_totally_ordered_with>&&
+        std::equality_comparable_with<totally_ordered_with_others,
+                                      no_lt_not_totally_ordered_with> &&
+    !check_totally_ordered_with<totally_ordered_with_others,
+                                no_lt_not_totally_ordered_with>());
+static_assert(
+    std::totally_ordered<no_gt_not_totally_ordered_with>&&
+        std::equality_comparable_with<totally_ordered_with_others,
+                                      no_gt_not_totally_ordered_with> &&
+    !check_totally_ordered_with<totally_ordered_with_others,
+                                no_gt_not_totally_ordered_with>());
+static_assert(
+    std::totally_ordered<no_le_not_totally_ordered_with>&&
+        std::equality_comparable_with<totally_ordered_with_others,
+                                      no_le_not_totally_ordered_with> &&
+    !check_totally_ordered_with<totally_ordered_with_others,
+                                no_le_not_totally_ordered_with>());
+static_assert(
+    std::totally_ordered<no_ge_not_totally_ordered_with>&&
+        std::equality_comparable_with<totally_ordered_with_others,
+                                      no_ge_not_totally_ordered_with> &&
+    !check_totally_ordered_with<totally_ordered_with_others,
+                                no_ge_not_totally_ordered_with>());
+} // namespace types_fit_for_purpose
+
+int main(int, char**) { return 0; }
diff --git a/libcxx/test/std/concepts/comparison/types.h b/libcxx/test/std/concepts/comparison/types.h
--- a/libcxx/test/std/concepts/comparison/types.h
+++ b/libcxx/test/std/concepts/comparison/types.h
@@ -10,6 +10,7 @@
 
 #include <compare>
 #include <concepts>
+#include <string>
 #include <type_traits>
 
 // `noexcept` specifiers deliberately imperfect since not all programmers bother to put the
@@ -39,6 +40,11 @@
 struct explicit_operators {
   friend bool operator==(explicit_operators, explicit_operators) noexcept;
   friend bool operator!=(explicit_operators, explicit_operators) noexcept;
+  friend bool operator<(explicit_operators, explicit_operators) noexcept;
+  friend bool operator>(explicit_operators, explicit_operators) noexcept;
+  friend bool operator<=(explicit_operators, explicit_operators) noexcept;
+  friend bool operator>=(explicit_operators, explicit_operators) noexcept;
+  friend bool operator<=>(explicit_operators, explicit_operators) noexcept;
 
   friend bool operator==(explicit_operators const&,
                          equality_comparable_with_ec1 const&) noexcept;
@@ -50,15 +56,18 @@
                          explicit_operators const&) noexcept;
 };
 
-struct eq_neq_different_return_types {
-  int operator==(eq_neq_different_return_types) const noexcept;
-  friend long operator!=(eq_neq_different_return_types,
-                         eq_neq_different_return_types) noexcept;
+struct different_return_types {
+  bool operator==(different_return_types) const noexcept;
+  char operator!=(different_return_types) const noexcept;
+  short operator<(different_return_types) const noexcept;
+  int operator>(different_return_types) const noexcept;
+  long operator<=(different_return_types) const noexcept;
+  long long operator>=(different_return_types) const noexcept;
 
-  friend int operator==(explicit_operators, eq_neq_different_return_types);
-  friend int operator==(eq_neq_different_return_types, explicit_operators);
-  friend long operator!=(explicit_operators, eq_neq_different_return_types);
-  friend long operator!=(eq_neq_different_return_types, explicit_operators);
+  friend signed char operator==(explicit_operators, different_return_types);
+  friend unsigned char operator==(different_return_types, explicit_operators);
+  friend float operator!=(explicit_operators, different_return_types);
+  friend double operator!=(different_return_types, explicit_operators);
 
   operator explicit_operators() const;
 };
@@ -73,7 +82,7 @@
   boolean operator!=(one_member_one_friend) const noexcept;
 
   operator explicit_operators() const noexcept;
-  operator eq_neq_different_return_types() const noexcept;
+  operator different_return_types() const noexcept;
 };
 
 struct equality_comparable_with_ec1 {
@@ -83,27 +92,120 @@
 };
 
 struct no_eq {
+  friend bool operator==(no_eq, no_eq) = delete;
   friend bool operator!=(no_eq, no_eq) noexcept;
+  friend bool operator<(no_eq, no_eq) noexcept;
+  friend bool operator>(no_eq, no_eq) noexcept;
+  friend bool operator>=(no_eq, no_eq) noexcept;
+  friend bool operator<=(no_eq, no_eq) noexcept;
 };
 
 struct no_neq {
   friend bool operator==(no_neq, no_neq) noexcept;
   friend bool operator!=(no_neq, no_neq) = delete;
+  friend bool operator<(no_eq, no_eq) noexcept;
+  friend bool operator>(no_eq, no_eq) noexcept;
+  friend bool operator>=(no_eq, no_eq) noexcept;
+  friend bool operator<=(no_eq, no_eq) noexcept;
+};
+
+struct no_lt {
+  friend bool operator==(no_lt, no_lt) noexcept;
+  friend bool operator!=(no_lt, no_lt) noexcept;
+  friend bool operator<(no_lt, no_lt) = delete;
+  friend bool operator>(no_lt, no_lt) noexcept;
+  friend bool operator>=(no_lt, no_lt) noexcept;
+  friend bool operator<=(no_lt, no_lt) noexcept;
+};
+
+struct no_gt {
+  friend bool operator==(no_gt, no_gt) noexcept;
+  friend bool operator!=(no_gt, no_gt) noexcept;
+  friend bool operator<(no_gt, no_gt) noexcept;
+  friend bool operator>(no_gt, no_gt) = delete;
+  friend bool operator>=(no_gt, no_gt) noexcept;
+  friend bool operator<=(no_gt, no_gt) noexcept;
+};
+
+struct no_le {
+  friend bool operator==(no_le, no_le) noexcept;
+  friend bool operator!=(no_le, no_le) noexcept;
+  friend bool operator<(no_le, no_le) noexcept;
+  friend bool operator>(no_le, no_le) noexcept;
+  friend bool operator>=(no_le, no_le) = delete;
+  friend bool operator<=(no_le, no_le) noexcept;
+};
+
+struct no_ge {
+  friend bool operator==(no_ge, no_ge) noexcept;
+  friend bool operator!=(no_ge, no_ge) noexcept;
+  friend bool operator<(no_ge, no_ge) noexcept;
+  friend bool operator>(no_ge, no_ge) noexcept;
+  friend bool operator>=(no_ge, no_ge) noexcept;
+  friend bool operator<=(no_ge, no_ge) = delete;
 };
 
 struct wrong_return_type_eq {
   void operator==(wrong_return_type_eq) const noexcept;
   bool operator!=(wrong_return_type_eq) const noexcept;
+  bool operator<(wrong_return_type_eq) const noexcept;
+  bool operator>(wrong_return_type_eq) const noexcept;
+  bool operator>=(wrong_return_type_eq) const noexcept;
+  bool operator<=(wrong_return_type_eq) const noexcept;
 };
 
 struct wrong_return_type_ne {
   bool operator==(wrong_return_type_ne) const noexcept;
   void operator!=(wrong_return_type_ne) const noexcept;
+  bool operator<(wrong_return_type_ne) const noexcept;
+  bool operator>(wrong_return_type_ne) const noexcept;
+  bool operator>=(wrong_return_type_ne) const noexcept;
+  bool operator<=(wrong_return_type_ne) const noexcept;
+};
+
+struct wrong_return_type_lt {
+  bool operator==(wrong_return_type_lt) const noexcept;
+  bool operator!=(wrong_return_type_lt) const noexcept;
+  void operator<(wrong_return_type_lt) const noexcept;
+  bool operator>(wrong_return_type_lt) const noexcept;
+  bool operator>=(wrong_return_type_lt) const noexcept;
+  bool operator<=(wrong_return_type_lt) const noexcept;
+};
+
+struct wrong_return_type_gt {
+  bool operator==(wrong_return_type_gt) const noexcept;
+  bool operator!=(wrong_return_type_gt) const noexcept;
+  bool operator<(wrong_return_type_gt) const noexcept;
+  void operator>(wrong_return_type_gt) const noexcept;
+  bool operator>=(wrong_return_type_gt) const noexcept;
+  bool operator<=(wrong_return_type_gt) const noexcept;
+};
+
+struct wrong_return_type_le {
+  bool operator==(wrong_return_type_le) const noexcept;
+  bool operator!=(wrong_return_type_le) const noexcept;
+  bool operator<(wrong_return_type_le) const noexcept;
+  bool operator>(wrong_return_type_le) const noexcept;
+  void operator>=(wrong_return_type_le) const noexcept;
+  bool operator<=(wrong_return_type_le) const noexcept;
+};
+
+struct wrong_return_type_ge {
+  bool operator==(wrong_return_type_ge) const noexcept;
+  bool operator!=(wrong_return_type_ge) const noexcept;
+  bool operator<(wrong_return_type_ge) const noexcept;
+  bool operator>(wrong_return_type_ge) const noexcept;
+  bool operator>=(wrong_return_type_ge) const noexcept;
+  void operator<=(wrong_return_type_ge) const noexcept;
 };
 
 struct wrong_return_type {
   void operator==(wrong_return_type) const noexcept;
   void operator!=(wrong_return_type) const noexcept;
+  void operator<(wrong_return_type) const noexcept;
+  void operator>(wrong_return_type) const noexcept;
+  void operator>=(wrong_return_type) const noexcept;
+  void operator<=(wrong_return_type_ge) const noexcept;
 };
 
 struct cxx20_member_eq_operator_with_deleted_ne {
@@ -177,24 +279,296 @@
   explicit operator bool() const noexcept;
 };
 
-struct returns_explicit_bool {
-  friend explicit_bool operator==(returns_explicit_bool, returns_explicit_bool);
-  friend explicit_bool operator!=(returns_explicit_bool, returns_explicit_bool);
+struct totally_ordered_with_others {
+  auto operator<=>(totally_ordered_with_others const&) const = default;
+};
+
+struct no_lt_not_totally_ordered_with {
+  [[nodiscard]] bool
+  operator==(no_lt_not_totally_ordered_with const&) const = default;
+  [[nodiscard]] auto
+  operator<=>(no_lt_not_totally_ordered_with const&) const = default;
+  operator totally_ordered_with_others() const noexcept;
+
+  [[nodiscard]] bool operator==(totally_ordered_with_others const&) const;
+  [[nodiscard]] auto operator<=>(totally_ordered_with_others const&) const;
+  [[nodiscard]] auto operator<(totally_ordered_with_others const&) const;
+};
+
+struct no_gt_not_totally_ordered_with {
+  [[nodiscard]] bool
+  operator==(no_gt_not_totally_ordered_with const&) const = default;
+  [[nodiscard]] auto
+  operator<=>(no_gt_not_totally_ordered_with const&) const = default;
+  operator totally_ordered_with_others() const noexcept;
+
+  [[nodiscard]] bool operator==(totally_ordered_with_others const&) const;
+  [[nodiscard]] auto operator<=>(totally_ordered_with_others const&) const;
+  [[nodiscard]] auto operator>(totally_ordered_with_others const&) const;
+};
+
+struct no_le_not_totally_ordered_with {
+  [[nodiscard]] bool
+  operator==(no_le_not_totally_ordered_with const&) const = default;
+  [[nodiscard]] auto
+  operator<=>(no_le_not_totally_ordered_with const&) const = default;
+  operator totally_ordered_with_others() const noexcept;
+
+  [[nodiscard]] bool operator==(totally_ordered_with_others const&) const;
+  [[nodiscard]] auto operator<=>(totally_ordered_with_others const&) const;
+  [[nodiscard]] auto operator<=(totally_ordered_with_others const&) const;
+};
+
+struct no_ge_not_totally_ordered_with {
+  [[nodiscard]] bool
+  operator==(no_ge_not_totally_ordered_with const&) const = default;
+  [[nodiscard]] auto
+  operator<=>(no_ge_not_totally_ordered_with const&) const = default;
+  operator totally_ordered_with_others() const noexcept;
+
+  [[nodiscard]] bool operator==(totally_ordered_with_others const&) const;
+  [[nodiscard]] auto operator<=>(totally_ordered_with_others const&) const;
+  [[nodiscard]] auto operator>=(totally_ordered_with_others const&) const;
+};
+
+struct partial_ordering_totally_ordered_with {
+  [[nodiscard]] auto operator<=>(
+      partial_ordering_totally_ordered_with const&) const noexcept = default;
+  [[nodiscard]] std::partial_ordering
+  operator<=>(totally_ordered_with_others const&) const noexcept;
+
+  operator totally_ordered_with_others() const;
+};
+
+struct weak_ordering_totally_ordered_with {
+  [[nodiscard]] auto operator<=>(
+      weak_ordering_totally_ordered_with const&) const noexcept = default;
+  [[nodiscard]] std::weak_ordering
+  operator<=>(totally_ordered_with_others const&) const noexcept;
+
+  operator totally_ordered_with_others() const;
+};
+
+struct strong_ordering_totally_ordered_with {
+  [[nodiscard]] auto operator<=>(
+      strong_ordering_totally_ordered_with const&) const noexcept = default;
+  [[nodiscard]] std::strong_ordering
+  operator<=>(totally_ordered_with_others const&) const noexcept;
+
+  operator totally_ordered_with_others() const;
+};
+
+struct eq_returns_explicit_bool {
+  friend explicit_bool operator==(eq_returns_explicit_bool,
+                                  eq_returns_explicit_bool);
+  friend bool operator!=(eq_returns_explicit_bool, eq_returns_explicit_bool);
+  friend bool operator<(eq_returns_explicit_bool, eq_returns_explicit_bool);
+  friend bool operator>(eq_returns_explicit_bool, eq_returns_explicit_bool);
+  friend bool operator<=(eq_returns_explicit_bool, eq_returns_explicit_bool);
+  friend bool operator>=(eq_returns_explicit_bool, eq_returns_explicit_bool);
+
+  operator totally_ordered_with_others() const;
+
+  friend explicit_bool operator==(eq_returns_explicit_bool,
+                                  totally_ordered_with_others);
+  friend explicit_bool operator==(totally_ordered_with_others,
+                                  eq_returns_explicit_bool);
+  friend bool operator!=(eq_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator!=(totally_ordered_with_others, eq_returns_explicit_bool);
+  friend bool operator<(eq_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator<(totally_ordered_with_others, eq_returns_explicit_bool);
+  friend bool operator>(eq_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator>(totally_ordered_with_others, eq_returns_explicit_bool);
+  friend bool operator<=(eq_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator<=(totally_ordered_with_others, eq_returns_explicit_bool);
+  friend bool operator>=(eq_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator>=(totally_ordered_with_others, eq_returns_explicit_bool);
+};
+
+struct ne_returns_explicit_bool {
+  friend bool operator==(ne_returns_explicit_bool, ne_returns_explicit_bool);
+  friend explicit_bool operator!=(ne_returns_explicit_bool,
+                                  ne_returns_explicit_bool);
+  friend bool operator<(ne_returns_explicit_bool, ne_returns_explicit_bool);
+  friend bool operator>(ne_returns_explicit_bool, ne_returns_explicit_bool);
+  friend bool operator<=(ne_returns_explicit_bool, ne_returns_explicit_bool);
+  friend bool operator>=(ne_returns_explicit_bool, ne_returns_explicit_bool);
+
+  operator totally_ordered_with_others() const;
+
+  friend bool operator==(ne_returns_explicit_bool, totally_ordered_with_others);
+  friend explicit_bool operator!=(ne_returns_explicit_bool,
+                                  totally_ordered_with_others);
+  friend explicit_bool operator!=(totally_ordered_with_others,
+                                  ne_returns_explicit_bool);
+  friend bool operator<(ne_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator<(totally_ordered_with_others, ne_returns_explicit_bool);
+  friend bool operator>(ne_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator>(totally_ordered_with_others, ne_returns_explicit_bool);
+  friend bool operator<=(ne_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator<=(totally_ordered_with_others, ne_returns_explicit_bool);
+  friend bool operator>=(ne_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator>=(totally_ordered_with_others, ne_returns_explicit_bool);
+};
+
+struct lt_returns_explicit_bool {
+  friend bool operator==(lt_returns_explicit_bool, lt_returns_explicit_bool);
+  friend bool operator!=(lt_returns_explicit_bool, lt_returns_explicit_bool);
+  friend explicit_bool operator<(lt_returns_explicit_bool,
+                                 lt_returns_explicit_bool);
+  friend bool operator>(lt_returns_explicit_bool, lt_returns_explicit_bool);
+  friend bool operator<=(lt_returns_explicit_bool, lt_returns_explicit_bool);
+  friend bool operator>=(lt_returns_explicit_bool, lt_returns_explicit_bool);
+
+  operator totally_ordered_with_others() const;
+
+  friend bool operator==(lt_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator!=(lt_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator!=(totally_ordered_with_others, lt_returns_explicit_bool);
+  friend explicit_bool operator<(lt_returns_explicit_bool,
+                                 totally_ordered_with_others);
+  friend bool operator<(totally_ordered_with_others, lt_returns_explicit_bool);
+  friend bool operator>(lt_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator>(totally_ordered_with_others, lt_returns_explicit_bool);
+  friend bool operator<=(lt_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator<=(totally_ordered_with_others, lt_returns_explicit_bool);
+  friend bool operator>=(lt_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator>=(totally_ordered_with_others, lt_returns_explicit_bool);
+};
+
+struct gt_returns_explicit_bool {
+  friend bool operator==(gt_returns_explicit_bool, gt_returns_explicit_bool);
+  friend bool operator!=(gt_returns_explicit_bool, gt_returns_explicit_bool);
+  friend bool operator<(gt_returns_explicit_bool, gt_returns_explicit_bool);
+  friend explicit_bool operator>(gt_returns_explicit_bool,
+                                 gt_returns_explicit_bool);
+  friend bool operator<=(gt_returns_explicit_bool, gt_returns_explicit_bool);
+  friend bool operator>=(gt_returns_explicit_bool, gt_returns_explicit_bool);
+
+  operator totally_ordered_with_others() const;
+
+  friend bool operator==(gt_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator!=(gt_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator!=(totally_ordered_with_others, gt_returns_explicit_bool);
+  friend bool operator<(gt_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator<(totally_ordered_with_others, gt_returns_explicit_bool);
+  friend explicit_bool operator>(gt_returns_explicit_bool,
+                                 totally_ordered_with_others);
+  friend bool operator>(totally_ordered_with_others, gt_returns_explicit_bool);
+  friend bool operator<=(gt_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator<=(totally_ordered_with_others, gt_returns_explicit_bool);
+  friend bool operator>=(gt_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator>=(totally_ordered_with_others, gt_returns_explicit_bool);
+};
+
+struct le_returns_explicit_bool {
+  friend bool operator==(le_returns_explicit_bool, le_returns_explicit_bool);
+  friend bool operator!=(le_returns_explicit_bool, le_returns_explicit_bool);
+  friend bool operator<(le_returns_explicit_bool, le_returns_explicit_bool);
+  friend bool operator>(le_returns_explicit_bool, le_returns_explicit_bool);
+  friend explicit_bool operator<=(le_returns_explicit_bool,
+                                  le_returns_explicit_bool);
+  friend bool operator>=(le_returns_explicit_bool, le_returns_explicit_bool);
+
+  operator totally_ordered_with_others() const;
+
+  friend bool operator==(le_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator!=(le_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator!=(totally_ordered_with_others, le_returns_explicit_bool);
+  friend bool operator<(le_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator<(totally_ordered_with_others, le_returns_explicit_bool);
+  friend bool operator>(le_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator>(totally_ordered_with_others, le_returns_explicit_bool);
+  friend bool operator<=(le_returns_explicit_bool, totally_ordered_with_others);
+  friend explicit_bool operator<=(totally_ordered_with_others,
+                                  le_returns_explicit_bool);
+  friend bool operator>=(le_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator>=(totally_ordered_with_others, le_returns_explicit_bool);
+};
+
+struct ge_returns_explicit_bool {
+  friend bool operator==(ge_returns_explicit_bool, ge_returns_explicit_bool);
+  friend bool operator!=(ge_returns_explicit_bool, ge_returns_explicit_bool);
+  friend bool operator<(ge_returns_explicit_bool, ge_returns_explicit_bool);
+  friend bool operator>(ge_returns_explicit_bool, ge_returns_explicit_bool);
+  friend bool operator<=(ge_returns_explicit_bool, ge_returns_explicit_bool);
+  friend explicit_bool operator>=(ge_returns_explicit_bool,
+                                  ge_returns_explicit_bool);
+
+  operator totally_ordered_with_others() const;
+
+  friend bool operator==(ge_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator!=(ge_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator!=(totally_ordered_with_others, ge_returns_explicit_bool);
+  friend bool operator<(ge_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator<(totally_ordered_with_others, ge_returns_explicit_bool);
+  friend bool operator>(ge_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator>(totally_ordered_with_others, ge_returns_explicit_bool);
+  friend bool operator<=(ge_returns_explicit_bool, totally_ordered_with_others);
+  friend bool operator<=(totally_ordered_with_others, ge_returns_explicit_bool);
+  friend bool operator>=(ge_returns_explicit_bool, totally_ordered_with_others);
+  friend explicit_bool operator>=(totally_ordered_with_others,
+                                  ge_returns_explicit_bool);
 };
 
 struct returns_true_type {
   friend std::true_type operator==(returns_true_type, returns_true_type);
   friend std::true_type operator!=(returns_true_type, returns_true_type);
-};
-
-struct returns_false_type {
-  friend std::false_type operator==(returns_false_type, returns_false_type);
-  friend std::false_type operator!=(returns_false_type, returns_false_type);
+  friend std::true_type operator<(returns_true_type, returns_true_type);
+  friend std::true_type operator>(returns_true_type, returns_true_type);
+  friend std::true_type operator<=(returns_true_type, returns_true_type);
+  friend std::true_type operator>=(returns_true_type, returns_true_type);
+
+  operator totally_ordered_with_others() const;
+
+  friend std::true_type operator==(returns_true_type,
+                                   totally_ordered_with_others);
+  friend std::true_type operator==(totally_ordered_with_others,
+                                   returns_true_type);
+  friend std::true_type operator!=(returns_true_type,
+                                   totally_ordered_with_others);
+  friend std::true_type operator!=(totally_ordered_with_others,
+                                   returns_true_type);
+  friend std::true_type operator<(returns_true_type,
+                                  totally_ordered_with_others);
+  friend std::true_type operator<(totally_ordered_with_others,
+                                  returns_true_type);
+  friend std::true_type operator>(returns_true_type,
+                                  totally_ordered_with_others);
+  friend std::true_type operator>(totally_ordered_with_others,
+                                  returns_true_type);
+  friend std::true_type operator<=(returns_true_type,
+                                   totally_ordered_with_others);
+  friend std::true_type operator<=(totally_ordered_with_others,
+                                   returns_true_type);
+  friend std::true_type operator>=(returns_true_type,
+                                   totally_ordered_with_others);
+  friend std::true_type operator>=(totally_ordered_with_others,
+                                   returns_true_type);
 };
 
 struct returns_int_ptr {
   friend int* operator==(returns_int_ptr, returns_int_ptr);
   friend int* operator!=(returns_int_ptr, returns_int_ptr);
+  friend int* operator<(returns_int_ptr, returns_int_ptr);
+  friend int* operator>(returns_int_ptr, returns_int_ptr);
+  friend int* operator<=(returns_int_ptr, returns_int_ptr);
+  friend int* operator>=(returns_int_ptr, returns_int_ptr);
+
+  operator totally_ordered_with_others() const;
+
+  friend int* operator==(returns_int_ptr, totally_ordered_with_others);
+  friend int* operator==(totally_ordered_with_others, returns_int_ptr);
+  friend int* operator!=(returns_int_ptr, totally_ordered_with_others);
+  friend int* operator!=(totally_ordered_with_others, returns_int_ptr);
+  friend int* operator<(returns_int_ptr, totally_ordered_with_others);
+  friend int* operator<(totally_ordered_with_others, returns_int_ptr);
+  friend int* operator>(returns_int_ptr, totally_ordered_with_others);
+  friend int* operator>(totally_ordered_with_others, returns_int_ptr);
+  friend int* operator<=(returns_int_ptr, totally_ordered_with_others);
+  friend int* operator<=(totally_ordered_with_others, returns_int_ptr);
+  friend int* operator>=(returns_int_ptr, totally_ordered_with_others);
+  friend int* operator>=(totally_ordered_with_others, returns_int_ptr);
 };
 
 #endif // TEST_STD_CONCEPTS_COMPARISON_EQUALITYCOMPARABLE_H