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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.

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

• t and t2, lvalues denoting distinct equal objects of types const std::remove_reference_t <T> and std::remove_reference_t <T> respectively, and
• u and u2, lvalues denoting distinct equal objects of types const std::remove_reference_t <U> and std::remove_reference_t <U> respectively,

let C be std::common_reference_t <const std::remove_reference_t <T>&, const std::remove_reference_t <U>&>, and, given an expression E and a type C, let CONVERT_TO<C>(E) be:

the following are true:

• bool(t < u) == bool(CONVERT_TO<C>(t2) < CONVERT_TO<C>(u2))
• bool(t > u) == bool(CONVERT_TO<C>(t2) > CONVERT_TO<C>(u2))
• bool(t <= u) == bool(CONVERT_TO<C>(t2) <= CONVERT_TO<C>(u2))
• bool(t >= u) == bool(CONVERT_TO<C>(t2) >= CONVERT_TO<C>(u2))
• bool(u < t) == bool(CONVERT_TO<C>(u2) < CONVERT_TO<C>(t2))
• bool(u > t) == bool(CONVERT_TO<C>(u2) > CONVERT_TO<C>(t2))
• bool(u <= t) == bool(CONVERT_TO<C>(u2) <= CONVERT_TO<C>(t2))
• bool(u >= t) == bool(CONVERT_TO<C>(u2) >= CONVERT_TO<C>(t2))

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).

• C++23 standard (ISO/IEC 14882:2024):

• 18.5.5 Concept totally_ordered [concept.totallyordered]

• C++20 standard (ISO/IEC 14882:2020):

• 18.5.4 Concept totally_ordered [concept.totallyordered]