1 Revision History

[P2322R2] used the names fold_left and fold_right to refer to the left- and right- folds and used the same names for the initial value and no-initial value algorithms. LEWG took the following polls [P2322-minutes]:

• fold with an initial value, and fold with no initial value and non-empty range should have different names (presumably fold and fold_first)

• Rename fold_left to fold

This revision uses different names for the initial value and no-initial value algorithms, although rather than using fold and fold_right (and coming up with how to name the no-initial value versions), this paper uses the names foldl and foldr and then foldl1 and foldr1. This revision also changes the no-initial value versions from having a non-empty range as a precondition to instead returning optional<T>.

There was also discussion around having these algorithms return an end iterator.

• Return the end iterator in addition to the result

• Have a version of the fold algorithms that return the end iterator in addition to the result (as either an additional set of functions, or having some of the versions have different return values)

But the primary algorithms (foldl and foldl1 for left-fold) will definitely solely return a value. This revision adds further discussion on the flavors of fold that can be provided, and ultimately adds another pair that satisfies this desire.

[P2322R1] used weakly-assignable-from as the constraint, this elevates it to assignable_from. This revision also changes the return type of fold to no longer be the type of the initial value, see the discussion.

[P2322R0] used regular_invocable as the constraint in the foldable concept, but we don’t need that for this algorithm and it prohibits reasonable uses like a mutating operation. invocable is the sufficient constraint here (in the same way that it is for for_each). Also restructured the API to overload on providing the initial value instead of having differently named algorithms.

2 Introduction

As described in [P2214R0], there is one very important rangified algorithm missing from the standard library: fold.

While we do have an iterator-based version of fold in the standard library, it is currently named accumulate, defaults to performing + on its operands, and is found in the header <numeric>. But fold is much more than addition, so as described in the linked paper, it’s important to give it the more generic name and to avoid a default operator.

Also as described in the linked paper, it is important to avoid over-constraining fold in a way that prevents using it for heterogeneous folds. As such, the fold specified in this paper only requires one particular invocation of the binary operator and there is no common_reference requirement between any of the types involved.

Lastly, the fold here is proposed to go into <algorithm> rather than <numeric> since there is nothing especially numeric about it.

3 Return Type

Consider the example:

std::vector<double> v = {0.25, 0.75};
auto r = ranges::fold(v, 1, std::plus());

What is the type and value of r? There are two choices, which I’ll demonstrate with implementations (with incomplete constraints).

3.1 Always return T

We implement like so:

template <range R, movable T, typename F>
auto fold(R&& r, T init, F op) -> T
{
    ranges::iterator_t<R> first = ranges::begin(r);
    ranges::sentinel_t<R> last = ranges::end(r);
    for (; first != last; ++first) {
        init = invoke(op, move(init), *first);
    }
    return init;
}

Here, fold(v, 1, std::plus()) is an int because the initial value is 1. Since our accumulator is an int, the result here is 1. This is a consistent with std::accumulate and is simple to reason about and specify. But it is also a common gotcha with std::accumulate.

Note that if we use assignable_from<T&, invoke_result_t<F&, T, range_reference_t<R>>> as the constraint on this algorithm, in this example this becomes assignable_from<int&, double>. We would be violating the semantic requirements of assignable_from, which state 18.4.8 [concept.assignable]/1.5:

(1.5) After evaluating lhs = rhs:

• (1.5.1) lhs is equal to rcopy, unless rhs is a non-const xvalue that refers to lcopy.

This only holds if all the doubles happen to be whole numbers, which is not the case for our example. This invocation would be violating the semantic constraints of the algorithm.

3.2 Return the result of the initial invocation

When we talk about the mathematical definition of fold, that’s f(f(f(f(init, x1), x2), ...), xn). If we actually evaluate this expression in this context, that’s ((1 + 0.25) + 0.75) which would be 2.0.

We cannot in general get this type correctly. A hypothetical f could actually change its type every time which we cannot possibly implement, so we can’t exactly mirror the mathematical definition regardless. But let’s just put that case aside as being fairly silly.

We could at least address the gotcha from std::accumulate by returning the decayed result of invoking the binary operation with T (the initial value) and the reference type of the range. That is, U = decay_t<invoke_result_t<F&, T, ranges::range_reference_t<R>>>. There are two possible approaches to implementing a fold that returns U instead of T:

template <range R, movable T, typename F,
            typename U = /* ... */>
auto fold(R&& r, T init, F f) -> U
{
    ranges::iterator_t<R> first = ranges::begin(r);
    ranges::sentinel_t<R> last = ranges::end(r);
    
    if (first == last) {
        return move(init);
    }
    
    U accum = invoke(f, move(init), *first);
    for (++first; first != last; ++first) {
        accum = invoke(f, move(accum), *first);
    }
    return accum;
}

template <range R, movable T, typename F,
            typename U = /* ... */>
auto fold(R&& r, T init, F f) -> U
{
    ranges::iterator_t<R> first = ranges::begin(r);
    ranges::sentinel_t<R> last = ranges::end(r);
    
    U accum = std::move(init);
    for (; first != last; ++first) {
        accum = invoke(f, move(accum), *first);
    }
    return accum;
}

Either way, our set of requirements is:

• invocable<F&, T, range_reference_t<R>> (even though the implementation on the right does not actually invoke the function using these arguments, we still need this to determine the type U)
• invocable<F&, U, range_reference_t<R>>
• convertible_to<T, U>
• assignable_from<U&, invoke_result_t<F&, U, range_reference_t<R>>>

While the left-hand side also needs convertible_to<invoke_result_t<F&, T, range_reference_t<R>>, U>.

This is a fairly complicated set of requirements.

But it means that our example, fold(v, 1, std::plus()) yields the more likely expected result of 2.0. So this is the version this paper proposes.

4 Other fold algorithms

[P2214R0] proposed a single fold algorithm that takes an initial value and a binary operation and performs a left fold over the range. But there are a couple variants that are also quite valuable and that we should adopt as a family.

4.1 fold1

Sometimes, there is no good choice for the initial value of the fold and you want to use the first element of the range. For instance, if I want to find the smallest string in a range, I can already do that as ranges::min(r) but the only way to express this in terms of fold is to manually pull out the first element, like so:

auto b = ranges::begin(r);
auto e = ranges::end(r);
ranges::fold(ranges::next(b), e, *b, ranges::min);

But this is both tedious to write, and subtly wrong for input ranges anyway since if the next(b) is evaluated before *b, we have a dangling iterator. This comes up enough that this paper proposes a version of fold that uses the first element in the range as the initial value (and thus has a precondition that the range is not empty).

This algorithm exists in Scala and Kotlin (which call the non-initializer version reduce but the initializer version fold), Haskell (under the name fold1), and Rust (in the Itertools crate under the name fold1 and recently finalized under the name reduce to match Scala and Kotlin [iterator_fold_self], although at some point it was fold_first).

In Python, the single algorithm functools.reduce supports both forms (the initializer is an optional argument). In Julia, foldl and foldr both take an optional initial value as well (though it is mandatory in certain cases).

There are two questions to ask about the version of fold that does not take an extra initializer.

4.1.1 Distinct name?

Should we give this algorithm a different name (e.g. fold_first or fold1, since reduce is clearly not an option for us) or provide it as an overload of fold? To answer that question, we have to deal with the question of ambiguity. For two arguments, fold(xs, a) can only be interpreted as a fold with no initial value using a as the binary operator. For four arguments, fold(xs, a, b, c) can only be interpreted as a fold with a as the initial value, b as the binary operation that is the reduction function, and c as a unary projection.

What about fold(xs, a, b)? It could be:

1. Using a as the initial value and b as a binary reduction of the form (A, X) -> A.
2. Using a as a binary reduction of the form (X, Y) -> X and b as a unary projection of the form X -> Y.

Is it possible for these to collide? It would be an uncommon situation, since b would have to be both a unary and a binary function. But it is definitely possible:

inline constexpr auto first = [](auto x, auto... ){ return x; };
auto r = fold(xs, first, first);

This call is ambiguous! This works with either interpretation. It would either just return first (the lambda) in the first case or the first element of the range in the second case, which makes it either completely useless or just mostly useless.

It’s possible to force either the function or projection to ensure that it can only be interpreted one way or the other, but since the algorithm is sufficiently different (see following section), even if such ambiguity is going to be extremely rare (and possible to deal with even if it does arise), we may as well avoid the issue entirely.

As such, this paper proposes a differently named algorithm for the version that takes no initial value rather than adding an overload under the same name.

4.1.2 optional or UB?

The result of ranges::foldl(empty_range, init, f) is just init. That is straightforward. But what would the result of ranges::foldl1(empty_range, f) be? There are two options:

1. a disengaged optional<T>, or
2. T, but this case is undefined behavior

In other words: empty range is either a valid input for the algorithm, whose result is nullopt, or there is a precondition that the range is non-empty.

Users can always recover the undefined behavior case if they want, by writing *foldl1(empty_range, f), and the optional return allows for easy addition of other functionality, such as providing a sentinel value for the empty range case (foldl1(empty_range, f).value_or(sentinel) reads better than not ranges::empty(r) ? foldl1(r, f) : sentinel, at least to me). It’s also much safer to use in the context where you may not know if the range is empty or not, because it’s adapted: foldl1(r | filter(f), op).

However, this would be the very first algorithm in the standard library that meaningful interacts with one of the sum types. And goes against the convention of algorithms simply being undefined for empty ranges (such as max). Although it’s worth pointing out that max_element is not UB for an empty range, it simply returns the end iterator, and the distinction there is likely due to simply not having had an available sentinel to return. But now we do.

This paper proposes returning optional<T>. Which is added motivation for a name distinct from the fold algorithm that takes an initializer.

4.2 fold_right

While ranges::fold would be a left-fold, there is also occasionally the need for a right-fold. As with the previous section, we should also provide overloads of fold_right that do not take an initial value.

There are three questions that would need to be asked about fold_right.

First, the order of operations of to the function. Given fold_right([1, 2, 3], z, f), is the evaluation f(f(f(z, 3), 2), 1) or is the evaluation f(1, f(2, f(3, z)))? Note that either way, we’re choosing the 3 then 2 then 1, both are right folds. It’s a question of if the initial element is the left-hand operand (as it is in the left fold) or the right-hand operand (as it would be if consider the right fold as a flip of the left fold).

One advantage of the former - where the initial call is f(z, 3) - is that we can specify fold_right(r, z, op) as precisely fold_left(views::reverse(r), z, op) and leave it at that. Notably with the same op. With the latter - where the initial call is f(3, z) - we would need slightly more specification and would want to avoid saying flip(op) since directly invoking the operation with the arguments in the correct order is a little better in the case of ranges of prvalues.

If we take a look at how other languages handle left-fold and right-fold, and whether the accumulator is on the same side (and, in these cases, the accumulator is always on the right) or opposite side (the accumulator is on the left-hand side for left fold and on the right-hand side for right fold):

This paper chooses what appears to be the more common approach: the accumulator is on the left-hand side for left fold and the right-hand side for right fold. That is, foldr(r, z, op) is equivalent to foldl(reverse(r), z, flip(op)).

Second, supporting bidirectional ranges is straightforward. Supporting forward ranges involves recursion of the size of the range. Supporting input ranges involves recursion and also copying the whole range first. Are either of these worth supporting? The paper simply supports bidirectional ranges.

Third, the naming question.

4.2.1 Naming for left and right folds

There are roughly four different choices that we could make here:

1. Provide the algorithms fold (a left-fold) and fold_right.
2. Provide the algorithms fold_left and fold_right.
3. Provide the algorithms fold_left and fold_right and also provide an alias fold which is also fold_left.
4. Provide the algorithms foldl and foldr.

There’s language precedents for any of these cases. F# and Kotlin both provide fold as a left-fold and suffixed right-fold (foldBack in F#, foldRight in Kotlin). Elm, Haskell, Julia, and OCaml provide symmetrically named algorithms (foldl/foldr for the first three and fold_left/fold_right for the third). Scala provides a foldLeft and foldRight while also providing fold to also mean foldLeft.

In C++, we don’t have precedent in the library at this point for providing an alias for an algorithm, although we do have precedent in the library for providing an alias for a range adapter (keys and values for elements<0> and elements<1>, and [P2321R0] proposes pairwise and pairwise_transform as aliases for adjacent<2> and adjacent_transform<2>). We also have precedent in the library for asymmetric names (sort vs stable_sort vs partial_sort) and symmetric ones (shift_left vs shift_right), even symmetric ones with terse names (rotl and rotr, although the latter are basically instructions).

All of which is to say, I don’t think there’s a clear answer to this question. There are, I think, three sane option banks:

And this paper proposes option C: foldl and foldr. It’s the right mix of having symmetry between the two names, while also not making them too long. There is preference for fold over fold_left (both because it’s more common than right-fold and thus having it shorter matters), and foldl is only a single character longer.

4.3 Short-circuiting folds

The folds discussed up until now have always evaluated the entirety of the range. That’s very useful in of itself, and several other algorithms that we have in the standard library can be implemented in terms of such a fold (e.g. min or count_if).

But for some algorithms, we really want to short circuit. For instance, we don’t want to define all_of(r, pred) as fold(r, true, logical_and(), pred). This formulation would give the correct answer, but we really don’t want to keep evaluating pred once we got our first false. To do this correctly, we really need short circuiting.

There are (at least) three different approaches for how to have a short-circuiting fold. Here are different approaches to implementing any_of in terms of a short-circuiting fold:

1. You could provide a function that mutates the accumulator and returns true to continue and false to break. That is, all_of(r, pred) would look like

   return fold_while(r, true, [&](bool& state, auto&& elem){
       state = pred(elem);
       return not state;
   });

   and the main loop of the fold_while algorithm would look like:

   for (; first != last; ++first) {
       if (not f(init, *first)) { // NB: not moving init into "f"
           break;                 // since we're mutating in-place
       }
   }
   return init;

2. You could provide a function that returns a variant<continue_<T>, done<T>>. Rust’s Itertools crate provides this under the name fold_while:

   template <typename T> struct continue_ { T value; };
   template <typename T> struct done { T value; };
   template <typename T> using fold_while_t = variant<continue_<T>, done<T>>;
   
   return fold_while(r, true, [&](bool, auto&& elem) -> fold_while_t<bool> {
       if (pred(elem)) {
           return continue{true};
       } else {
           return done{false};
       }
   });

   and the main loop of the fold_while algorithm would look like:

   for (; first != last; ++first) {
       auto next_state = f(move(init), *first);
       if (holds_alternative<continue_<T>>(next_state)) {
           init = get<continue_<T>>(move(next_state)).value;
       } else {
           return get<done<T>>(move(next_state)).value;
       }
   }
   return init;

3. You could provide a function that returns an expected<T, E>, which then the algorithm would return an expected<T, E> (rather than a T). Rust Iterator trait provides this under the name try_fold:

   return fold_while(r, true, [&](bool, auto&& elem) -> expected<bool, bool> {
       if (pred(FWD(elem))) {
           return true;
       } else {
           return unexpected(false);
       }
   }).has_value();

   and the main loop of the fold_while algorithm would look like:

   for (; first != last; ++first) {
       auto next = f(move(init), *first);
       if (not next) {
           return next;
       }
       init = move(*next);
   }
   return init;

Option (1) is a questionable option because of mutating state (note that we cannot use predicate as the constraint on the type, because predicates are not allowed to mutate their arguments), but this approach is probably the most efficient due to not moving the accumulator at all.

Option (2) is an awkward option for C++ because of general ergonomics. The provided lambda couldn’t just return continue_{x} in one case and done{y} in another since those have different types, so you’d basically always have to provide -> fold_while_t<T> as a trailing-return-type. This would also be the first (or second, see above) algorithm which actually meaningfully uses one of the standard library’s sum types.

Option (3) isn’t a great option for C++ because we don’t even have expected<T, E> in the standard library yet, and we’d also want to generalize this approach to any “truthy” type which would require coming up with a way to conceptualize (in the concept sense) “truthy” (since optional<T> would be a valid type as well, as well as any other the various user-defined versions out there).

Note that while the expected<T, E> version does convey failure semantically, more so than the fold_result_t<T> version, the latter can still be used to do so by simply returning a fold_result_t<expected<T, E>>.

4.4 Iterator-returning folds

Up until this point, this paper has only discussed returning a value from fold: whatever we get as the result of f(f(f(f(init, e0), e1), e2), e3). But there is another value that we compute along the way that is thrown out: the end iterator.

An alternative formulation of fold would preserve that information. Rather than returning R, we could do something like this:

template <input_iterator I, typename R>
struct fold_result {
    I in;
    R value;
};

template <input_iterator I, sentinel_for<I> S, class T, class Proj = identity,
    indirectly-binary-left-foldable<T, projected<I, Proj>> F,
    typename R = invoke_result_t<F&, T, indirect_result_t<Proj&, I>>>
constexpr auto foldl(I first, S last, T init, F f, Proj proj = {})
    -> fold_result<I, R>;

But the problem with that direction is, quoting from [P2214R0]:

int total = ranges::sum(numbers);

auto [_, total] = ranges::sum(numbers);

int total = ranges::sum(numbers, 0).value;

Moreover, even if we added a version of this algorithm that returned an iterator (let’s call it fold_with_iterator), we wouldn’t want fold(first, last, init, f) to be defined as

return fold_with_iterator(first, last, init, f).value;

since this would have to incur an extra move of the accumulated result, due to lack of copy elision (we have different return types). So we’d want need this algorithm to be specified separately (or, perhaps, the “Effects: equivalent to” formulation is sufficiently permissive as to allow implementations to do the right thing?)

From a usability perspective, I think it’s important that fold just return the value.

The problem going past that is that we end up with this combinatorial explosion of algorithms based on a lot of orthogonal choices:

1. iterator pair or range
2. left or right fold
3. initial value or no initial value
4. short-circuit or not short-circuit
5. return T or (iterator, T)

Which would be… 32 distinct functions (under 16 different names) if we go all out. And these really are basically orthogonal choices. Indeed, a short-circuiting fold seems even more likely to want the iterator that the algorithm stopped at! Do we need to provide all of them? Maybe we do!

This brings with it its own naming problem. That’s a lot of names. One approach there could be a suffix system:

• foldl is a non-short-circuiting left-fold with an initial value that returns T
• foldl1 is a non-short-circuiting left-fold with no initial value that returns T
• foldl1_while is a short-circuiting left-fold with no initial value that returns T
• foldr_with_iter is a non-short-circuiting right-fold with an initial value that returns (iterator, T)
• foldr1_while_with_iter is a short-circuiting right-fold with no initial value that returns (iterator, T)

with_iter is not the best suffix, but the rest seem to work out ok.

4.5 Short-circuiting and iterator-returning

One solution to the combinatorial explosion problem, as suggested by Tristan Brindle, is to simply not consider all of these options as being necessarily orthogonal. That is, providing a left-fold that returns a value is very valuable and highly desired. But having both a left- and right-fold that do short circuiting and return an iterator? Do we even need a short-circuiting fold that doesn’t return an iterator?

In other words, if you want the common and convenient thing, we’ll provide that: foldl and foldl1 will exist and just return a value. But if you want a more complex tool, it’ll be a little more complicated to use. In other words, what this paper is proposal is six algorithms (with two overloads each):

1. foldl (a left-fold with an initial value that returns T)
2. foldl1 (a left-fold with no initial value that returns T)
3. foldr (a right-fold with an initial value that returns T)
4. foldr1 (a right-fold with no initial value that returns T)
5. foldl_while (a short-circuiting left-fold with an initial value that returns (iterator, T))
6. foldl1_while (a short-circuiting left-fold with no initial value that returns (iterator, T))

There is no short-circuiting right-fold, since you can use views::reverse. This is a little harder to use since the transition from a fold that just returns T to a fold that actually produces an iterator as well isn’t as easy as going from:

auto value = ranges::foldl(r, init, op);

to:

auto [iterator, value] = ranges::foldl_while(r, init, op);

Instead, you have to wrap the binary operation. Though, in general, any binary operation suitable for foldl could be turned into a non-short-circuiting binary operation suitable for foldl_while via (although for specific operations, it could be more efficient to write it by hand):

inline constexpr auto wrap = [](auto op){
    return [=](auto& state, auto&& elem){
        state = op(move(state), FWD(elem)); 
        return true;
    };
};

auto [iterator, value] = ranges::foldl_while(r, init, wrap(op));

Note that for foldl_while and foldl1_while, we don’t have the same kind of return type shenanigans that we have with foldl and foldr - foldl_while takes a T as the initial value and returns that same type (since we have to pass a mutable reference to it into the binary operation).

5 Implementation Experience

Part of this paper (containing the algorithms foldl, foldl1, foldr, and foldr1, where the *1 algorithms return an optional) has been implemented in range-v3 [range-v3-fold]. The short-circuiting alternatives will be added later.

6 Wording

Append to 25.4 [algorithm.syn], first a new result type:

#include <initializer_list>

namespace std {
  namespace ranges {
    // [algorithms.results], algorithm result types
    template<class I, class F>
      struct in_fun_result;

    template<class I1, class I2>
      struct in_in_result;

    template<class I, class O>
      struct in_out_result;

    template<class I1, class I2, class O>
      struct in_in_out_result;

    template<class I, class O1, class O2>
      struct in_out_out_result;

    template<class T>
      struct min_max_result;

    template<class I>
      struct in_found_result;
      
+   template<class I, class T>
+     struct in_value_result;      
  }

  // ...
}

and then also a bunch of fold algorithms:

namespace std {
  // ...
    
  // [alg.fold], folds
  namespace ranges {
    template<class F>
    struct flipped {  // exposition only
      F f;
      
      template<class T, class U>
        requires invocable<F&, U, T>
      invoke_result_t<F&, U, T> operator()(T&&, U&&);
    };
    
    template <class F, class T, class I, class U>
    concept indirectly-binary-left-foldable-impl =  // exposition only
        movable<T> &&
        movable<U> &&
        convertible_to<T, U> &&
        invocable<F&, U, iter_reference_t<I>> &&
        assignable_from<U&, invoke_result_t<F&, U, iter_reference_t<I>>>;    
    
    template <class F, class T, class I>
    concept indirectly-binary-left-foldable =      // exposition only
        copy_constructible<F> &&
        indirectly_readable<I> &&
        invocable<F&, T, iter_reference_t<I>> &&
        convertible_to<invoke_result_t<F&, T, iter_reference_t<I>>,
               decay_t<invoke_result_t<F&, T, iter_reference_t<I>>>> &&
        indirectly-binary-left-foldable-impl<F, T, I, decay_t<invoke_result_t<F&, T, iter_reference_t<I>>>>; 
        
    template <class F, class T, class I>
    concept indirectly-short-circuit-left-foldable = // exposition only
        copy_constructible<F> &&
        movable<T> &&
        indirectly_readable<I> &&
        invocable<F&, T&, iter_reference_t<I>> &&
        boolean-testable<invoke_result_t<F&, T&, iter_reference_t<I>>>;

    template <class F, class T, class I>
    concept indirectly-binary-right-foldable =    // exposition only
        indirectly-binary-left-foldable<flipped<F>, T, I>;    
  
    template<input_iterator I, sentinel_for<I> S, class T, class Proj = identity,
      indirectly-binary-left-foldable<T, projected<I, Proj>> F>
    constexpr auto foldl(I first, S last, T init, F f, Proj proj = {});

    template<input_range R, class T, class Proj = identity,
      indirectly-binary-left-foldable<T, projected<iterator_t<R>, Proj>> F>
    constexpr auto foldl(R&& r, T init, F f, Proj proj = {});

    template <input_iterator I, sentinel_for<I> S, class Proj = identity,
      indirectly-binary-left-foldable<iter_value_t<I>, projected<I, Proj>> F>
      requires constructible_from<iter_value_t<I>, iter_reference_t<I>>
    constexpr auto foldl1(I first, S last, F f, Proj proj = {});

    template <input_range R, class Proj = identity,
      indirectly-binary-left-foldable<range_value_t<R>, projected<iterator_t<R>, Proj>> F>
      requires constructible_from<range_value_t<R>, range_reference_t<R>>
    constexpr auto foldl1(R&& r, F f, Proj proj = {});
    
    template<bidirectional_iterator I, sentinel_for<I> S, class T, class Proj = identity,
      indirectly-binary-right-foldable<T, projected<I, Proj>> F>
    constexpr auto foldr(I first, S last, T init, F f, Proj proj = {});

    template<bidirectional_range R, class T, class Proj = identity,
      indirectly-binary-right-foldable<T, projected<iterator_t<R>, Proj>> F>
    constexpr auto foldr(R&& r, T init, F f, Proj proj = {});

    template <bidirectional_iterator I, sentinel_for<I> S, class Proj = identity,
      indirectly-binary-right-foldable<iter_value_t<I>, projected<I, Proj>> F>
      requires constructible_from<iter_value_t<I>, iter_reference_t<I>>
    constexpr auto foldr1(I first, S last, F f, Proj proj = {});

    template <bidirectional_range R, class Proj = identity,
      indirectly-binary-right-foldable<range_value_t<R>, projected<iterator_t<R>, Proj>> F>
      requires constructible_from<range_value_t<R>, range_reference_t<R>>
    constexpr auto foldr1(R&& r, F f, Proj proj = {});  

    template<class I, class T>
      using fold_while_result = in_value_result<I, T>;

    template <input_iterator I, sentinel_for<I> S, class T, class Proj = identity,
      indirectly-short-circuit-left-foldable<T, projected<I, Proj>> F>
    constexpr fold_while_result<I, T> foldl_while(I first, S last, T init, F f, Proj proj = {});
    
    template <input_range R, class T, class Proj = identity,
      indirectly-short-circuit-left-foldable<T, projected<iterator_t<R>>, Proj>> F>
    constexpr fold_while_result<borrowed_iterator_t<R>, T> foldl_while(R&& r, T init, F f, Proj proj = {});    

    template <input_iterator I, sentinel_for<I> S, class Proj = identity,
      indirectly-short-circuit-left-foldable<iter_value_t<I>, projected<I, Proj>> F>
      requires constructible_from<iter_value_t<I>, iter_reference_t<I>>
    constexpr fold_while_result<I, optional<iter_value_t<I>>> foldl1_while(I first, S last, F f, Proj proj = {});

    template <input_range R, class Proj = identity,
      indirectly-short-circuit-left-foldable<range_value_t<R>, projected<iterator_t<R>>, Proj>> F>
      requires constructible_from<range_value_t<R>, range_reference_t<R>>
    constexpr fold_while_result<borrowed_iterator_t<R>, optional<range_value_t<R>>> foldl1_while(R&& r, F f, Proj proj = {});       
  }
}

Add a new result type to 25.5 [algorithms.results]:

namespace std::ranges {
  // ...
  
  template<class I>
  struct in_found_result {
    [[no_unique_address]] I in;
    bool found;

    template<class I2>
      requires convertible_­to<const I&, I2>
    constexpr operator in_found_result<I2>() const & {
      return {in, found};
    }
    template<class I2>
      requires convertible_­to<I, I2>
    constexpr operator in_found_result<I2>() && {
      return {std::move(in), found};
    }
  };

+ template<class I, class T>
+ struct in_value_result {
+   [[no_unique_address]] I in;
+   [[no_unique_address]] T value;
+
+   template<class I2, class T2>
+     requires convertible_to<const I&, I2> && convertible_to<const T&, T2>
+   constexpr operator in_value_result<I2, T2>() const & {
+     return {in, value};
+   }
+
+   template<class I2, class T2>
+     requires convertible_to<I, I2> && convertible_to<T, T2>
+   constexpr operator in_value_result<I2, T2>() && {
+     return {std::move(in), std::move(value)};
+   }
+ };  
}

And add a new clause, [alg.fold]:

template<input_iterator I, sentinel_for<I> S, class T, class Proj = identity,
  indirectly-binary-left-foldable<T, projected<I, Proj>> F>
constexpr auto ranges::foldl(I first, S last, T init, F f, Proj proj = {});

template<input_range R, class T, class Proj = identity,
  indirectly-binary-left-foldable<T, projected<iterator_t<R>, Proj>> F>
constexpr auto ranges::foldl(R&& r, T init, F f, Proj proj = {});

1 Effects: Equivalent to:

using U = invoke_result_t<F&, T, indirect_result_t<Proj&, I>>;
if (first == last) {
    return U(std::move(init));
}

U accum = invoke(f, std::move(init), invoke(proj, *first));
for (++first; first != last; ++first) {
    accum = invoke(f, std::move(accum), invoke(proj, *first));
}
return accum;

template <input_iterator I, sentinel_for<I> S, class Proj = identity,
  indirectly-binary-left-foldable<iter_value_t<I>, projected<I, Proj>> F>
  requires constructible_from<iter_value_t<I>, iter_reference_t<I>>
constexpr auto ranges::foldl1(I first, S last, F f, Proj proj = {});

template <input_range R, class Proj = identity,
  indirectly-binary-left-foldable<range_value_t<R>, projected<iterator_t<R>, Proj>> F>
  requires constructible_from<range_value_t<R>, range_reference_t<R>>
constexpr auto ranges::foldl1(R&& r, F f, Proj proj = {});

2 Let U be decltype(ranges::foldl(std::move(first), last, iter_value_t<I>(*first), f, proj)).

3 Effects: Equivalent to:

if (first == last) {
    return optional<U>();
}

iter_value_t<I> init(*first);
++first;
return optional<U>(in_place,
    ranges::foldl(std::move(first), std::move(last), std::move(init), std::move(f), std::move(proj)));

template<bidirectional_iterator I, sentinel_for<I> S, class T, class Proj = identity,
  indirectly-binary-right-foldable<T, projected<I, Proj>> F>
constexpr auto ranges::foldr(I first, S last, T init, F f, Proj proj = {});

template<bidirectional_range R, class T, class Proj = identity,
  indirectly-binary-right-foldable<T, projected<iterator_t<R>, Proj>> F>
constexpr auto ranges::foldr(R&& r, T init, F f, Proj proj = {});

4 Effects: Equivalent to:

using U = invoke_result_t<F&, indirect_result_t<Proj&, I>, T>;

if (first == last) {
    return U(std::move(init));
}

I tail = ranges::next(first, last);
U accum = invoke(f, invoke(proj, *--tail), std::move(init));
while (first != tail) {
    accum = invoke(f, invoke(proj, *--tail), std::move(accum));
}
return accum;

template <bidirectional_iterator I, sentinel_for<I> S, class Proj = identity,
  indirectly-binary-right-foldable<iter_value_t<I>, projected<I, Proj>> F>
  requires constructible_from<iter_value_t<I>, iter_reference_t<I>>
constexpr auto ranges::foldr1(I first, S last, F f, Proj proj = {});

template <bidirectional_range R, class Proj = identity,
  indirectly-binary-right-foldable<range_value_t<R>, projected<iterator_t<R>, Proj>> F>
  requires constructible_from<range_value_t<R>, range_reference_t<R>>
constexpr auto ranges::foldr1(R&& r, F f, Proj proj = {});  

5 Let U be decltype(ranges::foldr(first, last, iter_value_t<I>(*first), f, proj)).

6 Effects: Equivalent to:

if (first == last) {
    return optional<U>();
}

I tail = ranges::prev(ranges::next(first, std::move(last)));
return optional<U>(in_place,
    ranges::foldr(std::move(first), tail, iter_value_t<I>(*tail), std::move(f), std::move(proj)));

template <input_iterator I, sentinel_for<I> S, class T, class Proj = identity,
  indirectly-short-circuit-left-foldable<T, projected<I, Proj>> F>
constexpr ranges::fold_while_result<I, T> ranges::foldl_while(I first, S last, T init, F f, Proj proj = {});

template <input_range R, class T, class Proj = identity,
  indirectly-short-circuit-left-foldable<T, projected<iterator_t<R>>, Proj>> F>
constexpr ranges::fold_while_result<borrowed_iterator_t<R>, T> ranges::foldl_while(R&& r, T init, F f, Proj proj = {}); 

7 Effects: Equivalent to:

for (; first != last; ++first) {
    if (!invoke(f, init, invoke(proj, *first))) {
        break;
    }
}

return {std::move(first), std::move(init)};

template <input_iterator I, sentinel_for<I> S, class Proj = identity,
  indirectly-short-circuit-left-foldable<iter_value_t<I>, projected<I, Proj>> F>
  requires constructible_from<iter_value_t<I>, iter_reference_t<I>>
constexpr ranges::fold_while_result<I, optional<iter_value_t<I>>>
  ranges::foldl1_while(I first, S last, F f, Proj proj = {});

template <input_range R, class Proj = identity,
  indirectly-short-circuit-left-foldable<range_value_t<R>, projected<iterator_t<R>>, Proj>> F>
  requires constructible_from<range_value_t<R>, range_reference_t<R>>
constexpr ranges::fold_while_result<borrowed_iterator_t<R>, optional<range_value_t<R>>>
  ranges::foldl1_while(R&& r, F f, Proj proj = {});   

8 Effects: Equivalent to:

if (first == last) {
    return {std::move(first), nullopt};
}

iter_value_t<I> init(*first);
++first;
return ranges::foldl_while(std::move(first), std::move(last), std::move(init), std::move(proj));

7 References