18 Concepts library [concepts]

18.1 General [concepts.general]

This Clause describes library components that C++ programs may use to perform compile-time validation of template arguments and perform function dispatch based on properties of types.
The purpose of these concepts is to establish a foundation for equational reasoning in programs.
The following subclauses describe language-related concepts, comparison concepts, object concepts, and callable concepts as summarized in Table 41.
Table 41: Fundamental concepts library summary [tab:concepts.summary]
Subclause
Header
Equality preservation
Language-related concepts
<concepts>
Comparison concepts
Object concepts
Callable concepts

18.2 Equality preservation [concepts.equality]

An expression is equality-preserving if, given equal inputs, the expression results in equal outputs.
The inputs to an expression are the set of the expression's operands.
The output of an expression is the expression's result and all operands modified by the expression.
For the purposes of this subclause, the operands of an expression are the largest subexpressions that include only:
[Example 1: 
The operands of the expression a = std​::​move(b) are a and std​::​move(b).
— end example]
Not all input values need be valid for a given expression.
[Example 2: 
For integers a and b, the expression a / b is not well-defined when b is 0.
This does not preclude the expression a / b being equality-preserving.
— end example]
The domain of an expression is the set of input values for which the expression is required to be well-defined.
Expressions required to be equality-preserving are further required to be stable: two evaluations of such an expression with the same input objects are required to have equal outputs absent any explicit intervening modification of those input objects.
[Note 1: 
This requirement allows generic code to reason about the current values of objects based on knowledge of the prior values as observed via equality-preserving expressions.
It effectively forbids spontaneous changes to an object, changes to an object from another thread of execution, changes to an object as side effects of non-modifying expressions, and changes to an object as side effects of modifying a distinct object if those changes could be observable to a library function via an equality-preserving expression that is required to be valid for that object.
— end note]
Expressions declared in a requires-expression in the library clauses are required to be equality-preserving, except for those annotated with the comment “not required to be equality-preserving.”
An expression so annotated may be equality-preserving, but is not required to be so.
An expression that may alter the value of one or more of its inputs in a manner observable to equality-preserving expressions is said to modify those inputs.
The library clauses use a notational convention to specify which expressions declared in a requires-expression modify which inputs: except where otherwise specified, an expression operand that is a non-constant lvalue or rvalue may be modified.
Operands that are constant lvalues or rvalues are required to not be modified.
For the purposes of this subclause, the cv-qualification and value category of each operand are determined by assuming that each template type parameter denotes a cv-unqualified complete non-array object type.
Where a requires-expression declares an expression that is non-modifying for some constant lvalue operand, additional variations of that expression that accept a non-constant lvalue or (possibly constant) rvalue for the given operand are also required except where such an expression variation is explicitly required with differing semantics.
These implicit expression variations are required to meet the semantic requirements of the declared expression.
The extent to which an implementation validates the syntax of the variations is unspecified.
[Example 3: template<class T> concept C = requires(T a, T b, const T c, const T d) { c == d; // #1 a = std::move(b); // #2 a = c; // #3 };
For the above example:
  • Expression #1 does not modify either of its operands, #2 modifies both of its operands, and #3 modifies only its first operand a.
  • Expression #1 implicitly requires additional expression variations that meet the requirements for c == d (including non-modification), as if the expressions c == b; c == std::move(d); c == std::move(b); std::move(c) == d; std::move(c) == b; std::move(c) == std::move(d); std::move(c) == std::move(b); a == d; a == b; a == std::move(d); a == std::move(b); std::move(a) == d; std::move(a) == b; std::move(a) == std::move(d); std::move(a) == std::move(b); had been declared as well.
  • Expression #3 implicitly requires additional expression variations that meet the requirements for a = c (including non-modification of the second operand), as if the expressions a = b and a = std​::​move(c) had been declared.
    Expression #3 does not implicitly require an expression variation with a non-constant rvalue second operand, since expression #2 already specifies exactly such an expression explicitly.
— end example]
[Example 4: 
The following type T meets the explicitly stated syntactic requirements of concept C above but does not meet the additional implicit requirements: struct T { bool operator==(const T&) const { return true; } bool operator==(T&) = delete; };
T fails to meet the implicit requirements of C, so T satisfies but does not model C.
Since implementations are not required to validate the syntax of implicit requirements, it is unspecified whether an implementation diagnoses as ill-formed a program that requires C<T>.
— end example]

18.3 Header <concepts> synopsis [concepts.syn]

// all freestanding namespace std { // [concepts.lang], language-related concepts // [concept.same], concept same_as template<class T, class U> concept same_as = see below; // [concept.derived], concept derived_from template<class Derived, class Base> concept derived_from = see below; // [concept.convertible], concept convertible_to template<class From, class To> concept convertible_to = see below; // [concept.commonref], concept common_reference_with template<class T, class U> concept common_reference_with = see below; // [concept.common], concept common_with template<class T, class U> concept common_with = see below; // [concepts.arithmetic], arithmetic concepts template<class T> concept integral = see below; template<class T> concept signed_integral = see below; template<class T> concept unsigned_integral = see below; template<class T> concept floating_point = see below; // [concept.assignable], concept assignable_from template<class LHS, class RHS> concept assignable_from = see below; // [concept.swappable], concept swappable namespace ranges { inline namespace unspecified { inline constexpr unspecified swap = unspecified; } } template<class T> concept swappable = see below; template<class T, class U> concept swappable_with = see below; // [concept.destructible], concept destructible template<class T> concept destructible = see below; // [concept.constructible], concept constructible_from template<class T, class... Args> concept constructible_from = see below; // [concept.default.init], concept default_initializable template<class T> concept default_initializable = see below; // [concept.moveconstructible], concept move_constructible template<class T> concept move_constructible = see below; // [concept.copyconstructible], concept copy_constructible template<class T> concept copy_constructible = see below; // [concepts.compare], comparison concepts // [concept.equalitycomparable], concept equality_comparable template<class T> concept equality_comparable = see below; template<class T, class U> concept equality_comparable_with = see below; // [concept.totallyordered], concept totally_ordered template<class T> concept totally_ordered = see below; template<class T, class U> concept totally_ordered_with = see below; // [concepts.object], object concepts template<class T> concept movable = see below; template<class T> concept copyable = see below; template<class T> concept semiregular = see below; template<class T> concept regular = see below; // [concepts.callable], callable concepts // [concept.invocable], concept invocable template<class F, class... Args> concept invocable = see below; // [concept.regularinvocable], concept regular_invocable template<class F, class... Args> concept regular_invocable = see below; // [concept.predicate], concept predicate template<class F, class... Args> concept predicate = see below; // [concept.relation], concept relation template<class R, class T, class U> concept relation = see below; // [concept.equiv], concept equivalence_relation template<class R, class T, class U> concept equivalence_relation = see below; // [concept.strictweakorder], concept strict_weak_order template<class R, class T, class U> concept strict_weak_order = see below; }

18.4 Language-related concepts [concepts.lang]

18.4.1 General [concepts.lang.general]

Subclause [concepts.lang] contains the definition of concepts corresponding to language features.
These concepts express relationships between types, type classifications, and fundamental type properties.

18.4.2 Concept same_as [concept.same]

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>;
[Note 1: 
same_as<T, U> subsumes same_as<U, T> and vice versa.
— end note]

18.4.3 Concept derived_from [concept.derived]

template<class Derived, class Base> concept derived_from = is_base_of_v<Base, Derived> && is_convertible_v<const volatile Derived*, const volatile Base*>;
[Note 1: 
derived_from<Derived, Base> is satisfied if and only if Derived is publicly and unambiguously derived from Base, or Derived and Base are the same class type ignoring cv-qualifiers.
— end note]

18.4.4 Concept convertible_to [concept.convertible]

Given types From and To and an expression E whose type and value category are the same as those of declval<From>(), convertible_to<From, To> requires E to be both implicitly and explicitly convertible to type To.
The implicit and explicit conversions are required to produce equal results.
template<class From, class To> concept convertible_to = is_convertible_v<From, To> && requires { static_cast<To>(declval<From>()); };
Let FromR be add_rvalue_reference_t<From> and test be the invented function: To test(FromR (&f)()) { return f(); } and let f be a function with no arguments and return type FromR such that f() is equality-preserving.
Types From and To model convertible_to<From, To> only if:
  • To is not an object or reference-to-object type, or static_cast<To>(f()) is equal to test(f).
  • FromR is not a reference-to-object type, or
    • If FromR is an rvalue reference to a non const-qualified type, the resulting state of the object referenced by f() after either above expression is valid but unspecified ([lib.types.movedfrom]).
    • Otherwise, the object referred to by f() is not modified by either above expression.

18.4.5 Concept common_reference_with [concept.commonref]

For two types T and U, if common_reference_t<T, U> is well-formed and denotes a type C such that both convertible_to<T, C> and convertible_to<U, C> are modeled, then T and U share a common reference type, C.
[Note 1: 
C can be the same as T or U, or can be a different type.
C can be a reference type.
— end note]
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>>;
Let C be common_reference_t<T, U>.
Let t1 and t2 be equality-preserving expressions ([concepts.equality]) such that decltype((t1)) and decltype((t2)) are each T, and let u1 and u2 be equality-preserving expressions such that decltype((u1)) and decltype((u2)) are each U.
T and U model common_reference_with<T, U> only if:
  • C(t1) equals C(t2) if and only if t1 equals t2, and
  • C(u1) equals C(u2) if and only if u1 equals u2.
[Note 2: 
Users can customize the behavior of common_reference_with by specializing the basic_common_reference class template ([meta.trans.other]).
— end note]

18.4.6 Concept common_with [concept.common]

If T and U can both be explicitly converted to some third type, C, then T and U share a common type, C.
[Note 1: 
C can be the same as T or U, or can be a different type.
C is not necessarily unique.
— end note]
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>>>;
Let C be common_type_t<T, U>.
Let t1 and t2 be equality-preserving expressions ([concepts.equality]) such that decltype((t1)) and decltype((t2)) are each T, and let u1 and u2 be equality-preserving expressions such that decltype((u1)) and decltype((u2)) are each U.
T and U model common_with<T, U> only if:
  • C(t1) equals C(t2) if and only if t1 equals t2, and
  • C(u1) equals C(u2) if and only if u1 equals u2.
[Note 2: 
Users can customize the behavior of common_with by specializing the common_type class template ([meta.trans.other]).
— end note]

18.4.7 Arithmetic concepts [concepts.arithmetic]

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>;
[Note 1: 
signed_integral can be modeled even by types that are not signed integer types ([basic.fundamental]); for example, char.
— end note]
[Note 2: 
unsigned_integral can be modeled even by types that are not unsigned integer types ([basic.fundamental]); for example, bool.
— end note]

18.4.8 Concept assignable_from [concept.assignable]

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>; };
Let:
  • lhs be an lvalue that refers to an object lcopy such that decltype((lhs)) is LHS,
  • rhs be an expression such that decltype((rhs)) is RHS, and
  • rcopy be a distinct object that is equal to rhs.
LHS and RHS model assignable_from<LHS, RHS> only if
  • addressof(lhs = rhs) == addressof(lcopy).
  • After evaluating lhs = rhs:
    • lhs is equal to rcopy, unless rhs is a non-const xvalue that refers to lcopy.
    • If rhs is a non-const xvalue, the resulting state of the object to which it refers is valid but unspecified ([lib.types.movedfrom]).
    • Otherwise, if rhs is a glvalue, the object to which it refers is not modified.
[Note 1: 
Assignment need not be a total function ([structure.requirements]); in particular, if assignment to an object x can result in a modification of some other object y, then x = y is likely not in the domain of =.
— end note]

18.4.9 Concept swappable [concept.swappable]

Let t1 and t2 be equality-preserving expressions that denote distinct equal objects of type T, and let u1 and u2 similarly denote distinct equal objects of type U.
[Note 1: 
t1 and u1 can denote distinct objects, or the same object.
— end note]
An operation exchanges the values denoted by t1 and u1 if and only if the operation modifies neither t2 nor u2 and:
  • If T and U are the same type, the result of the operation is that t1 equals u2 and u1 equals t2.
  • If T and U are different types and common_reference_with<decltype((t1)), decltype((u1))> is modeled, the result of the operation is that C(t1) equals C(u2) and C(u1) equals C(t2) where C is common_reference_t<decltype((t1)), decltype((u1))>.
The name ranges​::​swap denotes a customization point object ([customization.point.object]).
The expression ranges​::​swap(E1, E2) for subexpressions E1 and E2 is expression-equivalent to an expression S determined as follows:
  • S is (void)swap(E1, E2)202 if E1 or E2 has class or enumeration type ([basic.compound]) and that expression is valid, with overload resolution performed in a context that includes the declaration template<class T> void swap(T&, T&) = delete; and does not include a declaration of ranges​::​swap.
    If the function selected by overload resolution does not exchange the values denoted by E1 and E2, the program is ill-formed, no diagnostic required.
    [Note 2: 
    This precludes calling unconstrained program-defined overloads of swap.
    When the deleted overload is viable, program-defined overloads need to be more specialized ([temp.func.order]) to be selected.
    — end note]
  • Otherwise, if E1 and E2 are lvalues of array types ([basic.compound]) with equal extent and ranges​::​swap(*E1, *E2) is a valid expression, S is (void)ranges​::​swap_ranges(E1, E2), except that noexcept(S) is equal to noexcept(​ranges​::​swap(*E1, *E2)).
  • Otherwise, if E1 and E2 are lvalues of the same type T that models move_constructible<T> and assignable_from<T&, T>, S is an expression that exchanges the denoted values.
    S is a constant expression if noexcept(S) is equal to is_nothrow_move_constructible_v<T> && is_nothrow_move_assignable_v<T>.
  • Otherwise, ranges​::​swap(E1, E2) is ill-formed.
    [Note 3: 
    This case can result in substitution failure when ranges​::​swap(E1, E2) appears in the immediate context of a template instantiation.
    — end note]
[Note 4: 
Whenever ranges​::​swap(E1, E2) is a valid expression, it exchanges the values denoted by E1 and E2 and has type void.
— end note]
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)); };
[Note 5: 
The semantics of the swappable and swappable_with concepts are fully defined by the ranges​::​swap customization point object.
— end note]
[Example 1: 
User code can ensure that the evaluation of swap calls is performed in an appropriate context under the various conditions as follows: #include <cassert> #include <concepts> #include <utility> namespace ranges = std::ranges; template<class T, std::swappable_with<T> U> void value_swap(T&& t, U&& u) { ranges::swap(std::forward<T>(t), std::forward<U>(u)); } template<std::swappable T> void lv_swap(T& t1, T& t2) { ranges::swap(t1, t2); } namespace N { struct A { int m; }; struct Proxy { A* a; Proxy(A& a) : a{&a} {} friend void swap(Proxy x, Proxy y) { ranges::swap(*x.a, *y.a); } }; Proxy proxy(A& a) { return Proxy{ a }; } } int main() { int i = 1, j = 2; lv_swap(i, j); assert(i == 2 && j == 1); N::A a1 = { 5 }, a2 = { -5 }; value_swap(a1, proxy(a2)); assert(a1.m == -5 && a2.m == 5); }
— end example]
202)202)
The name swap is used here unqualified.

18.4.10 Concept destructible [concept.destructible]

The destructible concept specifies properties of all types, instances of which can be destroyed at the end of their lifetime, or reference types.
template<class T> concept destructible = is_nothrow_destructible_v<T>;
[Note 1: 
Unlike the Cpp17Destructible requirements (Table 35), this concept forbids destructors that are potentially throwing, even if a particular invocation of the destructor does not actually throw.
— end note]

18.4.11 Concept constructible_from [concept.constructible]

The constructible_from concept constrains the initialization of a variable of a given type with a particular set of argument types.
template<class T, class... Args> concept constructible_from = destructible<T> && is_constructible_v<T, Args...>;

18.4.12 Concept default_initializable [concept.default.init]

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>;
For a type T, is-default-initializable<T> is true if and only if the variable definition T t; is well-formed for some invented variable t; otherwise it is false.
Access checking is performed as if in a context unrelated to T.
Only the validity of the immediate context of the variable initialization is considered.

18.4.13 Concept move_constructible [concept.moveconstructible]

template<class T> concept move_constructible = constructible_from<T, T> && convertible_to<T, T>;
If T is an object type, then let rv be an rvalue of type T and u2 a distinct object of type T equal to rv.
T models move_constructible only if
  • After the definition T u = rv;, u is equal to u2.
  • T(rv) is equal to u2.
  • If T is not const, rv's resulting state is valid but unspecified ([lib.types.movedfrom]); otherwise, it is unchanged.

18.4.14 Concept copy_constructible [concept.copyconstructible]

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>;
If T is an object type, then let v be an lvalue of type T or const T or an rvalue of type const T.
T models copy_constructible only if

18.5 Comparison concepts [concepts.compare]

18.5.1 General [concepts.compare.general]

Subclause [concepts.compare] describes concepts that establish relationships and orderings on values of possibly differing object types.
Given an expression E and a type C, let CONVERT_TO_LVALUE<C>(E) be:
  • static_cast<const C&>(as_const(E)) if that is a valid expression, and
  • static_cast<const C&>(std​::​move(E)) otherwise.

18.5.2 Boolean testability [concept.booleantestable]

The exposition-only boolean-testable concept specifies the requirements on expressions that are convertible to bool and for which the logical operators ([expr.log.and], [expr.log.or], [expr.unary.op]) have the conventional semantics.
template<class T> concept boolean-testable-impl = convertible_to<T, bool>; // exposition only
Let e be an expression such that decltype((e)) is T.
T models boolean-testable-impl only if:
  • either remove_cvref_t<T> is not a class type, or a search for the names operator&& and operator|| in the scope of remove_cvref_t<T> finds nothing; and
  • argument-dependent lookup ([basic.lookup.argdep]) for the names operator&& and operator|| with T as the only argument type finds no disqualifying declaration (defined below).
A disqualifying parameter is a function parameter whose declared type P
  • is not dependent on a template parameter, and there exists an implicit conversion sequence ([over.best.ics]) from e to P; or
  • is dependent on one or more template parameters, and either
    • P contains no template parameter that participates in template argument deduction ([temp.deduct.type]), or
    • template argument deduction using the rules for deducing template arguments in a function call ([temp.deduct.call]) and e as the argument succeeds.
A key parameter of a function template D is a function parameter of type cv X or reference thereto, where X names a specialization of a class template that has the same innermost enclosing non-inline namespace as D, and X contains at least one template parameter that participates in template argument deduction.
[Example 1: 
In namespace Z { template<class> struct C {}; template<class T> void operator&&(C<T> x, T y); template<class T> void operator||(C<type_identity_t<T>> x, T y); } the declaration of Z​::​operator&& contains one key parameter, C<T> x, and the declaration of Z​::​operator|| contains no key parameters.
— end example]
A disqualifying declaration is
  • a (non-template) function declaration that contains at least one disqualifying parameter; or
  • a function template declaration that contains at least one disqualifying parameter, where
    • at least one disqualifying parameter is a key parameter; or
    • the declaration contains no key parameters; or
    • the declaration declares a function template to which no name is bound ([dcl.meaning]).
[Note 1: 
The intention is to ensure that given two types T1 and T2 that each model boolean-testable-impl, the && and || operators within the expressions declval<T1>() && declval<T2>() and declval<T1>() || declval<T2>() resolve to the corresponding built-in operators.
— end note]
template<class T> concept boolean-testable = // exposition only boolean-testable-impl<T> && requires(T&& t) { { !std::forward<T>(t) } -> boolean-testable-impl; };
Let e be an expression such that decltype((e)) is T.
T models boolean-testable only if bool(e) == !bool(!e).
[Example 2: 
The types bool, true_type ([meta.type.synop]), int*, and bitset<N>​::​reference ([template.bitset]) model boolean-testable.
— end example]

18.5.3 Comparison common types [concept.comparisoncommontype]

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>>;
Let C be common_reference_t<const T&, const U&>.
Let t1 and t2 be equality-preserving expressions that are lvalues of type remove_cvref_t<T>, and let u1 and u2 be equality-preserving expressions that are lvalues of type remove_cvref_t<U>.
T and U model comparison-common-type-with<T, U> only if:
  • CONVERT_TO_LVALUE<C>(t1) equals CONVERT_TO_LVALUE<C>(t2) if and only if t1 equals t2, and
  • CONVERT_TO_LVALUE<C>(u1) equals CONVERT_TO_LVALUE<C>(u2) if and only if u1 equals u2

18.5.4 Concept equality_comparable [concept.equalitycomparable]

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; };
Given types T and U, let t and u be lvalues of types const remove_reference_t<T> and const remove_reference_t<U> respectively.
T and U model weakly-equality-comparable-with<T, U> only if
  • t == u, u == t, t != u, and u != t have the same domain.
  • bool(u == t) == bool(t == u).
  • bool(t != u) == !bool(t == u).
  • bool(u != t) == bool(t != u).
Let a and b be objects of type T.
T models equality_comparable only if bool(a == b) is true when a is equal to b ([concepts.equality]), and false otherwise.
[Note 1: 
The requirement that the expression a == b is equality-preserving implies that == is transitive and symmetric.
— end note]
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>;
Given types T and U, let t and t2 be lvalues denoting distinct equal objects of types const remove_reference_t<T> and remove_cvref_t<T>, respectively, let u and u2 be lvalues denoting distinct equal objects of types const remove_reference_t<U> and remove_cvref_t<U>, respectively, and let C be: common_reference_t<const remove_reference_t<T>&, const remove_reference_t<U>&> T and U model equality_comparable_with<T, U> only if bool(t == u) == bool(CONVERT_TO_LVALUE<C>(t2) == CONVERT_TO_LVALUE<C>(u2))

18.5.5 Concept totally_ordered [concept.totallyordered]

template<class T> concept totally_ordered = equality_comparable<T> && partially-ordered-with<T, T>;
Given a type T, let a, b, and c be lvalues of type const remove_reference_t<T>.
T models totally_ordered only if
  • Exactly one of bool(a < b), bool(a > b), or bool(a == b) is true.
  • If bool(a < b) and bool(b < c), then bool(a < c).
  • bool(a <= b) == !bool(b < a).
  • bool(a >= b) == !bool(a < b).
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>;
Given types T and U, let t and t2 be lvalues denoting distinct equal objects of types const remove_reference_t<T> and remove_cvref_t<T>, respectively, let u and u2 be lvalues denoting distinct equal objects of types const remove_reference_t<U> and remove_cvref_t<U>, respectively, and let C be: common_reference_t<const remove_reference_t<T>&, const remove_reference_t<U>&> T and U model totally_ordered_with<T, U> only if
  • bool(t < u) == bool(CONVERT_TO_LVALUE<C>(t2) < CONVERT_TO_LVALUE<C>(u2)).
  • bool(t > u) == bool(CONVERT_TO_LVALUE<C>(t2) > CONVERT_TO_LVALUE<C>(u2)).
  • bool(t <= u) == bool(CONVERT_TO_LVALUE<C>(t2) <= CONVERT_TO_LVALUE<C>(u2)).
  • bool(t >= u) == bool(CONVERT_TO_LVALUE<C>(t2) >= CONVERT_TO_LVALUE<C>(u2)).
  • bool(u < t) == bool(CONVERT_TO_LVALUE<C>(u2) < CONVERT_TO_LVALUE<C>(t2)).
  • bool(u > t) == bool(CONVERT_TO_LVALUE<C>(u2) > CONVERT_TO_LVALUE<C>(t2)).
  • bool(u <= t) == bool(CONVERT_TO_LVALUE<C>(u2) <= CONVERT_TO_LVALUE<C>(t2)).
  • bool(u >= t) == bool(CONVERT_TO_LVALUE<C>(u2) >= CONVERT_TO_LVALUE<C>(t2)).

18.6 Object concepts [concepts.object]

This subclause describes concepts that specify the basis of the value-oriented programming style on which the library is based.
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>;
[Note 1: 
The semiregular concept is modeled by types that behave similarly to fundamental types like int, except that they need not be comparable with ==.
— end note]
[Note 2: 
The regular concept is modeled by types that behave similarly to fundamental types like int and that are comparable with ==.
— end note]

18.7 Callable concepts [concepts.callable]

18.7.1 General [concepts.callable.general]

The concepts in subclause [concepts.callable] describe the requirements on function objects ([function.objects]) and their arguments.

18.7.2 Concept invocable [concept.invocable]

The invocable concept specifies a relationship between a callable type ([func.def]) F and a set of argument types Args... which can be evaluated by the library function invoke ([func.invoke]).
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 };
[Example 1: 
A function that generates random numbers can model invocable, since the invoke function call expression is not required to be equality-preserving ([concepts.equality]).
— end example]

18.7.3 Concept regular_invocable [concept.regularinvocable]

template<class F, class... Args> concept regular_invocable = invocable<F, Args...>;
The invoke function call expression shall be equality-preserving ([concepts.equality]) and shall not modify the function object or the arguments.
[Note 1: 
This requirement supersedes the annotation in the definition of invocable.
— end note]
[Example 1: 
A random number generator does not model regular_invocable.
— end example]
[Note 2: 
The distinction between invocable and regular_invocable is purely semantic.
— end note]

18.7.4 Concept predicate [concept.predicate]

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

18.7.5 Concept relation [concept.relation]

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>;

18.7.6 Concept equivalence_relation [concept.equiv]

template<class R, class T, class U> concept equivalence_relation = relation<R, T, U>;
A relation models equivalence_relation only if it imposes an equivalence relation on its arguments.

18.7.7 Concept strict_weak_order [concept.strictweakorder]

template<class R, class T, class U> concept strict_weak_order = relation<R, T, U>;
A relation models strict_weak_order only if it imposes a strict weak ordering on its arguments.
The term strict refers to the requirement of an irreflexive relation (!comp(x, x) for all x), and the term weak to requirements that are not as strong as those for a total ordering, but stronger than those for a partial ordering.
If we define equiv(a, b) as !comp(a, b) && !comp(b, a), then the requirements are that comp and equiv both be transitive relations:
  • comp(a, b) && comp(b, c) implies comp(a, c)
  • equiv(a, b) && equiv(b, c) implies equiv(a, c)
[Note 1: 
Under these conditions, it can be shown that
  • equiv is an equivalence relation,
  • comp induces a well-defined relation on the equivalence classes determined by equiv, and
  • the induced relation is a strict total ordering.
— end note]