# std::totally_ordered, std::totally_ordered_with

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< cpp‎ | concepts

Defined in header `<concepts>` |  |
template< class T > concept totally_ordered = std::equality_comparable<T> && __PartiallyOrderedWith<T, T>; | (1) | (since C++20)
template< class T, class U > concept totally_ordered_with = std::totally_ordered<T> && std::totally_ordered<U> && std::equality_comparable_with<T, U> && std::totally_ordered< std::common_reference_t< const std::remove_reference_t<T>&, const std::remove_reference_t<U>&>> && __PartiallyOrderedWith<T, U>; | (2) | (since C++20)
template< class T, class U > concept __PartiallyOrderedWith = requires(const std::remove_reference_t<T>& t, const std::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; }; | (3) | (exposition only*)

1) The concept `std::totally_ordered` specifies that the comparison operators `==,!=,<,>,<=,>=` on a type yield results consistent with a strict total order on the type.
2) The concept `std::totally_ordered_with` specifies that the comparison operators `==,!=,<,>,<=,>=` on (possibly mixed) `T` and `U` operands yield results consistent with a strict total order. Comparing mixed operands yields results equivalent to comparing the operands converted to their common type.
3) The exposition-only concept `__PartiallyOrderedWith` specifies that a value of type `T` and a value of type `U` can be compared in a partial order with each other (in either order) using `<`, `>`, `<=`, and `>=`, and the results of the comparisons are consistent.

### Semantic requirements

These concepts are modeled only if they are satisfied and all concepts they subsume are modeled.

1) std::totally_ordered<T> is modeled only if, given lvalues `a`, `b` and `c` of type const std::remove_reference_t<T>:

- Exactly one of bool(a < b), bool(a > b) and bool(a == b) is true;
- If bool(a < b) and bool(b < c) are both true, then bool(a < c) is true;
- bool(a > b) == bool(b < a)
- bool(a >= b) == !bool(a < b)
- bool(a <= b) == !bool(b < a)

2) std::totally_ordered_with<T, U> is modeled only if, given

- any lvalue `t` of type const std::remove_reference_t<T>, and
- any lvalue `u` of type const std::remove_reference_t<U>,

and let `C` be std::common_reference_t<const std::remove_reference_t<T>&, const std::remove_reference_t<U>&>:

- bool(t < u) == bool(C(t) < C(u))
- bool(t > u) == bool(C(t) > C(u))
- bool(t <= u) == bool(C(t) <= C(u))
- bool(t >= u) == bool(C(t) >= C(u))
- bool(u < t) == bool(C(u) < C(t))
- bool(u > t) == bool(C(u) > C(t))
- bool(u <= t) == bool(C(u) <= C(t))
- bool(u >= t) == bool(C(u) >= C(t))

3) __PartiallyOrderedWith<T, U> is modeled only if given

- any lvalue `t` of type const std::remove_reference_t<T>, and
- any lvalue `u` of type const std::remove_reference_t<U>,

the following are true:

- t < u, t <= u, t > u, t >= u, u < t, u <= t, u > t, and u >= t have the same domain;
- bool(t < u) == bool(u > t);
- bool(u < t) == bool(t > u);
- bool(t <= u) == bool(u >= t); and
- bool(u <= t) == bool(t >= u).

### Equality preservation

Expressions declared in requires-expressions of the standard library concepts are required to be equality-preserving (except where stated otherwise).

### Implicit expression variations

A requires-expression that uses an expression that is non-modifying for some constant lvalue operand also implicitly requires additional variations of that expression that accept a non-constant lvalue or (possibly constant) rvalue for the given operand unless such an expression variation is explicitly required with differing semantics. These implicit expression variations must meet the same semantic requirements of the declared expression. The extent to which an implementation validates the syntax of the variations is unspecified.

### See also

three_way_comparable three_way_comparable_with (C++20) | specifies that operator <=> produces consistent result on given types (concept)

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