std::three_way_comparable,std::three_way_comparable_with(3)

NAME
std::three_way_comparable,std::three_way_comparable_with - std::three_way_comparable,std::three_way_comparable_with

Synopsis
   Defined in header <compare>
   template<class T, class Cat = std::partial_ordering>
   concept three_way_comparable =
   __WeaklyEqualityComparableWith<T, T> &&
   __PartiallyOrderedWith<T, T> && (1) (since C++20)
   requires(const std::remove_reference_t<T>& a,
   const std::remove_reference_t<T>& b) {
   { a <=> b } -> __ComparesAs<Cat>;
   };
   template<class T, class U, class Cat = std::partial_ordering>
   concept three_way_comparable_with =
   std::three_way_comparable<T, Cat> &&
   std::three_way_comparable<U, Cat> &&
   std::common_reference_with<
   const std::remove_reference_t<T>&,
   const std::remove_reference_t<U>&> &&
   std::three_way_comparable<
   std::common_reference_t< (2) (since C++20)
   const std::remove_reference_t<T>&,
   const std::remove_reference_t<U>&>, Cat> &&
   __WeaklyEqualityComparableWith<T, U> &&
   __PartiallyOrderedWith<T, U> &&
   requires(const std::remove_reference_t<T>& t,
   const std::remove_reference_t<U>& u) {
   { t <=> u } -> __ComparesAs<Cat>;
   { u <=> t } -> __ComparesAs<Cat>;
   };
   template<class T, class Cat>
   concept __ComparesAs = // exposition only (3) (since C++20)
   std::same_as<std::common_comparison_category_t<T, Cat>, Cat>;
   1) The concept std::three_way_comparable specifies that the three way comparison
   operator <=> on T yield results consistent with the comparison category implied by
   Cat.
   2) The concept std::three_way_comparable_with specifies that the three way
   comparison operator <=> on (possibly mixed) T and U operands yield results
   consistent with the comparison category implied by Cat. Comparing mixed operands
   yields results equivalent to comparing the operands converted to their common type.
   In both definitions, __WeaklyEqualityComparableWith and __PartiallyOrderedWith are
   exposition-only concepts also used by equality_comparable and totally_ordered,
   respectively.
  Semantic requirements
   These concepts are modeled only if they are satisified and all concepts they subsume
   are modeled.
   1) T and Cat model std::three_way_comparable<T, Cat> only if, given lvalues a and b
   of type const std::remove_reference_t<T>, following are true:
     * (a <=> b == 0) == bool(a == b),
     * (a <=> b != 0) == bool(a != b),
     * ((a <=> b) <=> 0) and (0 <=> (b <=> a)) are equal,
     * bool(a > b) == bool(b < a),
     * bool(a >= b) == !bool(a < b),
     * bool(a <= b) == !bool(b < a),
     * (a <=> b < 0) == bool(a < b),
     * (a <=> b > 0) == bool(a > b),
     * (a <=> b <= 0) == bool(a <= b), and
     * (a <=> b >= 0) == bool(a >= b); and
     * if Cat is convertible to std::strong_ordering, T models totally_ordered.
   2) T, U, and Cat model std::three_way_comparable_with<T, U, Cat> only if given
     * t, an lvalue of type const std::remove_reference_t<T> and
     * u, an lvalue of type const std::remove_reference_t<U>,
   Let C be std::common_reference_t<const std::remove_reference_t<T>&, const
   std::remove_reference_t<U>&>, the following are true:
     * t <=> u and u <=> t have the same domain;
     * ((t <=> u) <=> 0) and (0 <=> (u <=> t)) are equal;
     * (t <=> u == 0) == bool(t == u),
     * (t <=> u != 0) == bool(t != u),
     * Cat(t <=> u) == Cat(C(t) <=> C(u)),
     * (t <=> u < 0) == bool(t < u),
     * (t <=> u > 0) == bool(t > u),
     * (t <=> u <= 0) == bool(t <= u),
     * (t <=> u >= 0) == bool(t >= u); and
     * if Cat is convertible to std::strong_ordering, T and U model
       std::totally_ordered_with<T, U>.
  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:
     * an id-expression, and
     * invocations of std::move, std::forward, and std::declval.
   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
   modify only its non-constant operands. Operands that are constant must not be
   modified.
  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
   equality_comparable specifies that operator == is an equivalence relation
   equality_comparable_with (concept)
   (C++20)
   totally_ordered specifies that the comparison operators on the type yield a
   totally_ordered_with total order
   (C++20) (concept)