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// all freestanding
namespace std {
 // language-related concepts
 // concept same_as
 template<class T, class U>
 concept same_as = /* see description */;
 
 // concept derived_from
 template<class Derived, class Base>
 concept derived_from = /* see description */;
 
 // concept convertible_to
 template<class From, class To>
 concept convertible_to = /* see description */;
 
 // concept common_reference_with
 template<class T, class U>
 concept common_reference_with = /* see description */;
 
 // concept common_with
 template<class T, class U>
 concept common_with = /* see description */;
 
 // arithmetic concepts
 template<class T>
 concept integral = /* see description */;
 template<class T>
 concept signed_integral = /* see description */;
 template<class T>
 concept unsigned_integral = /* see description */;
 template<class T>
 concept floating_point = /* see description */;
 
 // concept assignable_from
 template<class LHS, class RHS>
 concept assignable_from = /* see description */;
 
 // concept swappable
 namespace ranges {
 inline namespace /* unspecified */ {
 inline constexpr /* unspecified */ swap = /* unspecified */;
 }
 }
 template<class T>
 concept swappable = /* see description */;
 template<class T, class U>
 concept swappable_with = /* see description */;
 
 // concept destructible
 template<class T>
 concept destructible = /* see description */;
 
 // concept constructible_from
 template<class T, class... Args>
 concept constructible_from = /* see description */;
 
 // concept default_initializable
 template<class T>
 concept default_initializable = /* see description */;
 
 // concept move_constructible
 template<class T>
 concept move_constructible = /* see description */;
 
 // concept copy_constructible
 template<class T>
 concept copy_constructible = /* see description */;
 
 // comparison concepts
 // concept equality_comparable
 template<class T>
 concept equality_comparable = /* see description */;
 template<class T, class U>
 concept equality_comparable_with = /* see description */;
 
 // concept totally_ordered
 template<class T>
 concept totally_ordered = /* see description */;
 template<class T, class U>
 concept totally_ordered_with = /* see description */;
 
 // object concepts
 template<class T>
 concept movable = /* see description */;
 template<class T>
 concept copyable = /* see description */;
 template<class T>
 concept semiregular = /* see description */;
 template<class T>
 concept regular = /* see description */;
 
 // callable concepts
 // concept invocable
 template<class F, class... Args>
 concept invocable = /* see description */;
 
 // concept regular_invocable
 template<class F, class... Args>
 concept regular_invocable = /* see description */;
 
 // concept predicate
 template<class F, class... Args>
 concept predicate = /* see description */;
 
 // concept relation
 template<class R, class T, class U>
 concept relation = /* see description */;
 
 // concept equivalence_relation
 template<class R, class T, class U>
 concept equivalence_relation = /* see description */;
 
 // concept strict_weak_order
 template<class R, class T, class U>
 concept strict_weak_order = /* see description */;
}

template<class T>
concept /*boolean-testable-impl*/ = convertible_to<T, bool>; // exposition only;
 
template<class T>
concept boolean-testable = // exposition only
 /*boolean-testable-impl*/<T> && requires(T&& t) {
 {
 !std::forward <T>(t)
 } -> /*boolean-testable-impl*/;
 };

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>
concept swappable = requires(T& a, T& b) { ranges::swap (a, b); };

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 description */; // 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>;

template<class T, class U>
concept /*weakly-equality-comparable-with*/ = // exposition only
 requires(const remove_reference_t<T>& t, const remove_reference_t<U>& u) {
 { t == u } -> boolean-testable ;
 { t != u } -> boolean-testable ;
 { u == t } -> boolean-testable ;
 { u != t } -> boolean-testable ;
 };
 
template<class T>
concept equality_comparable = /*weakly-equality-comparable-with*/<T, T>;

template<class T, class U, class C = common_reference_t<const T&, const U&>>
concept /*comparison-common-type-with-impl*/ = // exposition only
 same_as<common_reference_t<const T&, const U&>,
 common_reference_t<const U&, const T&>> &&
 requires {
 requires convertible_to<const T&, const C&> || convertible_to<T, const C&>;
 requires convertible_to<const U&, const C&> || convertible_to<U, const C&>;
 };
 
template<class T, class U>
concept /*comparison-common-type-with*/ = // exposition only
 /*comparison-common-type-with-impl*/<remove_cvref_t<T>, remove_cvref_t<U>>;
 
template<class T, class U>
concept equality_comparable_with =
 equality_comparable<T> && equality_comparable<U> &&
 /*comparison-common-type-with*/<T, U> &&
 equality_comparable<
 common_reference_t<const remove_reference_t<T>&, const remove_reference_t<U>&>> &&
 /*weakly-equality-comparable-with*/<T, U>;

template<class T, class U>
concept /*partially-ordered-with*/ = // exposition only
 requires(const remove_reference_t<T>& t, const remove_reference_t<U>& 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 T>
concept totally_ordered = equality_comparable<T> && /*partially-ordered-with*/<T, T>;

template<class T, class U>
concept totally_ordered_with =
 totally_ordered<T> && totally_ordered<U> && equality_comparable_with<T, U> &&
 totally_ordered<
 common_reference_t<const remove_reference_t<T>&, const remove_reference_t<U>&>> &&
 /*partially-ordered-with*/<T, U>;

template<class T>
concept movable =
 is_object_v<T> && move_constructible<T> && assignable_from<T&, T> && swappable<T>;

template<class T>
concept copyable = copy_constructible<T> && movable<T> && assignable_from<T&, T&> &&
 assignable_from<T&, const T&> && assignable_from<T&, const T>;

template<class T>
concept semiregular = copyable<T> && default_initializable<T>;

template<class T>
concept regular = semiregular<T> && equality_comparable<T>;

template<class F, class... Args>
concept invocable = requires(F&& f, Args&&... args) {
 invoke(std::forward <F>(f),
 std::forward <Args>(args)...); // not required to be equality-preserving
};

template<class F, class... Args>
 concept regular_invocable = invocable<F, Args...>;

template<class F, class... Args>
concept predicate =
 regular_invocable<F, Args...> && boolean-testable <invoke_result_t<F, Args...>>;

template<class R, class T, class U>
concept relation =
 predicate<R, T, T> && predicate<R, U, U> && predicate<R, T, U> && predicate<R, U, T>;

template<class R, class T, class U>
concept equivalence_relation = relation<R, T, U>;

template<class R, class T, class U>
concept strict_weak_order = relation<R, T, U>;