Proposed resolution:

This wording is relative to N4835.

1. Change [concept.totallyordered] as follows:

   template<class T>
     concept totally_ordered =
       equality_comparable<T> && partially-ordered-with<T, T>;
       requires(const remove_reference_t<T>& a,
                const remove_reference_t<T>& b) {
         { a <  b } -> boolean;
         { a >  b } -> boolean;
         { a <= b } -> boolean;
         { a >= b } -> boolean;
       };
   

   -1- For some type T, let a, b, and c be lvalues of type const remove_reference_t<T>. T models totally_ordered only if

   (1.1) — Exactly one of bool(a < b), bool(a > b), or bool(a == b) is true.

   (1.2) — If bool(a < b) and bool(b < c), then bool(a < c).

   (1.3) — bool(a > b) == bool(b < a).

   (1.4) — bool(a <= b) == !bool(b < a).

   (1.5) — bool(a >= b) == !bool(a < b).

   template<class T, class U>
     concept totally_ordered_with =
       totally_ordered<T> && totally_ordered<U> &&
       common_reference_with<const remove_reference_t<T>&, const remove_reference_t<U>&> &&
       totally_ordered<
         common_reference_t<
           const remove_reference_t<T>&,
           const remove_reference_t<U>&>> &&
         equality_comparable_with<T, U> &&
         partially-ordered-with<T, U>;
         requires(const remove_reference_t<T>& t,
                  const remove_reference_t<U>& u) {
           { t <  u } -> boolean;
           { t >  u } -> boolean;
           { t <= u } -> boolean;
           { t >= u } -> boolean;
           { u <  t } -> boolean;
           { u >  t } -> boolean;
           { u <= t } -> boolean;
           { u >= t } -> boolean;
         };
   

   […]

[ 2019-11 Status to Ready Friday afternoon LWG in Belfast ]

Addresses GB 202

Define totally_ordered[_with] in terms of partially-ordered-with. This will simplify the definition of both concepts (particularly totally_ordered_with), and make them in-line with equality_comparable[_with]. Now that we've defined partially-ordered-with for [cmp.concept], we should consider utilising it in as many locations as possible.

Proposed change:

template<class T>
  concept totally_ordered =
    equality_comparable<T> &&
    partially-ordered-with<T, T>;

template<class T, class U>
  concept totally_ordered_with =
    totally_ordered<T> &&
    totally_ordered<U> &&
    common_reference_with<
      const remove_reference_t<T>&,
      const remove_reference_t<U>&> &&
    totally_ordered<
      common_reference_t<
        const remove_reference_t<T>&,
        const remove_reference_t<U>&>> &&
    equality_comparable_with<T, U> &&
    partially-ordered-with<T, U>;

LWG discussion in Belfast notes that 3329 also touches the definition of totally_ordered_with; the two sets of changes are consistent.