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std::totally_ordered, std::totally_ordered_with

From cppreference.com
< 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 =      // exposition only
  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) (since C++20)
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.

[edit] Semantic requirements

These concepts are modeled only if they are satisified 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

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

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).

[edit] Equality preservation

An expression is equality preserving if it results in equal outputs given equal inputs.

  • The inputs to an expression consist of its operands.
  • The outputs of an expression consist of its result and all operands modified by the expression (if any).

In specification of standard concepts, operands are defined as the largest subexpressions that include only:

The cv-qualification and value category of each operand is determined by assuming that each template type parameter denotes a cv-unqualified complete non-array object type.

Every expression required to be equality preserving is further required to be stable: two evaluations of such an expression with the same input objects must have equal outputs absent any explicit intervening modification of those input objects.

Unless noted otherwise, every expression used in a requires-expression is required to be equality preserving and stable, and the evaluation of the expression may only modify its non-constant operands. Operands that are constant must not be modified.

[edit] 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.