template<class T, class U>
  concept same-as-impl = is_same_v<T, U>;       // exposition only
template<class T, class U>
  concept same_as = same-as-impl<T, U> && same-as-impl<U, T>;
template<class Derived, class Base>
  concept derived_from =
    is_base_of_v<Base, Derived> &&
    is_convertible_v<const volatile Derived*, const volatile Base*>;
template<class From, class To>
  concept convertible_to =
    is_convertible_v<From, To> &&
    requires {
      static_cast<To>(declval<From>());
    };
template<class T, class U>
  concept common_reference_with =
    same_as<common_reference_t<T, U>, common_reference_t<U, T>> &&
    convertible_to<T, common_reference_t<T, U>> &&
    convertible_to<U, common_reference_t<T, U>>;
template<class T, class U>
  concept common_with =
    same_as<common_type_t<T, U>, common_type_t<U, T>> &&
    requires {
      static_cast<common_type_t<T, U>>(declval<T>());
      static_cast<common_type_t<T, U>>(declval<U>());
    } &&
    common_reference_with<
      add_lvalue_reference_t<const T>,
      add_lvalue_reference_t<const U>> &&
    common_reference_with<
      add_lvalue_reference_t<common_type_t<T, U>>,
      common_reference_t<
        add_lvalue_reference_t<const T>,
        add_lvalue_reference_t<const U>>>;
template<class T>
  concept integral = is_integral_v<T>;
template<class T>
  concept signed_integral = integral<T> && is_signed_v<T>;
template<class T>
  concept unsigned_integral = integral<T> && !signed_integral<T>;
template<class T>
  concept floating_point = is_floating_point_v<T>;
template<class LHS, class RHS>
  concept assignable_from =
    is_lvalue_reference_v<LHS> &&
    common_reference_with<const remove_reference_t<LHS>&, const remove_reference_t<RHS>&> &&
    requires(LHS lhs, RHS&& rhs) {
      { lhs = std::forward<RHS>(rhs) } -> same_as<LHS>;
    };
template<class T, class U>
  concept swappable_with =
    common_reference_with<T, U> &&
    requires(T&& t, U&& u) {
      ranges::swap(std::forward<T>(t), std::forward<T>(t));
      ranges::swap(std::forward<U>(u), std::forward<U>(u));
      ranges::swap(std::forward<T>(t), std::forward<U>(u));
      ranges::swap(std::forward<U>(u), std::forward<T>(t));
    };
template<class T>
  concept destructible = is_nothrow_destructible_v<T>;
template<class T, class... Args>
  concept constructible_from = destructible<T> && is_constructible_v<T, Args...>;
template<class T>
  constexpr bool is-default-initializable = see below;         // exposition only
template<class T>
  concept default_initializable = constructible_from<T> &&
                                  requires { T{}; } &&
                                  is-default-initializable<T>;
template<class T>
  concept move_constructible = constructible_from<T, T> && convertible_to<T, T>;
template<class T>
  concept copy_constructible =
    move_constructible<T> &&
    constructible_from<T, T&> && convertible_to<T&, T> &&
    constructible_from<T, const T&> && convertible_to<const T&, T> &&
    constructible_from<T, const T> && convertible_to<const T, T>;