P3104R1
Bit permutations

1. Revision history

1.1. Changes since R0

• Expand § 4.3 Hardware support, taking more instructions into account, including AVX-512.

• Minor editorial fixes.

2. Introduction

The C++ bit manipulation library in <bit> is an invaluable abstraction from hardware operations. Functions like countl_zero help the programmer avoid use of intrinsic functions or inline assembly.

However, there are still a few operations which are non-trivial to implement in software and have widely available hardware support. Therefore, I propose to expand the bit manipulation library with multiple bit permutation functions described below. Even without hardware support, these functions provide great utility and/or help the developer write more expressive code.

3. Proposed features

I propose to add the following functions to the bit manipulation library (<bit>).

Note: All constraints are exposition-only. The function bodies contain a naive implementation that merely illustrates the behavior.

Note: See § 4.3 Hardware support for the corresponding hardware instructions.

3.1. bit_reverse

template<unsigned_integral T>
constexpr T bit_reverse(T x) noexcept
{
    T result = 0;
    for (int i = 0; i < numeric_limits<T>::digits; ++i) {
        result <<= 1;
        result |= x & 1;
        x >>= 1;
    }
    return result;
}

Reverses the bits of x so that the least significant bit becomes the most significant.

3.2. bit_repeat

template<unsigned_integral T>
constexpr T bit_repeat(T x, int length) noexcept(false)
{
    T result = 0;
    for (int i = 0; i < numeric_limits<T>::digits; ++i) {
        result |= ((x >> (i % length)) & 1) << i;
    }
    return result;
}

Repeats a pattern stored in the least significant length bits of x, as many times as fits into x.

3.3. next_bit_permutation and prev_bit_permutation

template<unsigned_integral T>
constexpr T next_bit_permutation(T x) noexcept
{
    const int count = popcount(x);
    while (x != 0 && popcount(++x) != count) {}
    return x;
}

Returns the lowest number y in (x, numeric_limits<T>::max()] for which popcount(y) == popcount(x), or 0 if none exists.

template<unsigned_integral T>
constexpr T prev_bit_permutation(T x) noexcept
{
    const int count = popcount(x);
    while (x != 0 && popcount(--x) != count) {}
    return x;
}

Returns the greatest number y in [0, x) for which popcount(y) == popcount(x), or 0 if none exists.

for (uint32_t x = 0b111; x != 0; x = next_bit_permutation(x)) // ...

dec:    7,   11,   13,   14,    19,    21, ...
bin:  111, 1011, 1101, 1110, 10011, 10101, ...

3.4. bit_compress

template<unsigned_integral T>
constexpr T bit_compressr(T x, T m) noexcept
{
    T result = 0;
    for (int i = 0, j = 0; i < numeric_limits<T>::digits; ++i) {
        bool mask_bit = (m >> i) & 1;
        result |= (mask_bit & (x >> i)) << j;
        j += mask_bit;
    }
    return result;
}

For each one-bit in m, the corresponding bit in x is taken and packed contiguously into the result, starting with the least significant result bit.

template<unsigned_integral T>
constexpr T bit_compressl(T x, T m) noexcept
{
    return bit_reverse(bit_compressr(bit_reverse(x)), bit_reverse(m)));
}

For each one-bit in m, the corresponding bit in x is taken and packed contiguously into the result, starting with the most significant result bit.

x:
+---+---+---+---+
| a | b | c | d |
+---+---+---+---+
m:    |       |        bit_compressl(x, m):  bit_compressr(x, m):
+---+---+---+---+      +---+---+---+---+      +---+---+---+---+
| 0 | 1 | 0 | 1 |      | b | d | 0 | 0 |      | 0 | 0 | b | d |
+---+---+---+---+      +---+---+---+---+      +---+---+---+---+

Note: Intuitively, m is a "mask" that determines which bits of x are packed.

3.5. bit_expand

template<unsigned_integral T>
constexpr T bit_expandr(T x, T m) noexcept
{
    T result = 0;
    for (int i = 0, j = 0; i < numeric_limits<T>::digits; ++i) {
        bool mask_bit = (m >> i) & 1;
        result |= (mask_bit & (x >> j)) << i;
        j += mask_bit;
    }
    return result;
}

For each one-bit in m, a bit from x, starting with the least significant bit is taken and shifted into the corresponding position of the m bit.

template<unsigned_integral T>
constexpr T bit_expandl(T x, T m) noexcept
{
    return bit_reverse(bit_expandr(bit_reverse(x)), bit_reverse(m)));
}

For each one-bit in m, a bit from x, starting with the most significant bit is taken and shifted into the corresponding position of the m bit.

x:
+---+---+---+---+
| a | b | c | d |
+---+---+---+---+
m:                     bit_expandl(x, m):    bit_expandr(x, m):
+---+---+---+---+      +---+---+---+---+      +---+---+---+---+
| 0 | 1 | 0 | 1 |      | 0 | a | 0 | b |      | 0 | c | 0 | d |
+---+---+---+---+      +---+---+---+---+      +---+---+---+---+

Note: Intuitively, m is a "mask" that determines where the bits of x are unpacked into.

4. Motivation and scope

Bit-reversal, repetition, compression, and expansion are fundamental operations that meet multiple criteria which make them suitable for standardization:

1. They are common and useful operations.

2. They can be used to implement numerous other operations.

3. At least on some architectures, they have direct hardware support.

4. They are non-trivial to implement efficiently in software.

5. For known masks, numerous optimization opportunities are available.

next_bit_permutation, is the most niche of these operations and does not meet all criteria. However, I believe it is useful enough to be included in this proposal, and it fits the overall theme well.

4.1. Applications

4.1.1. Applications of bit_reverse

Bit-reversal is a common operation with uses in:

• Cryptography: scrambling bits

• Networking: as part of cyclic redundancy check computation

• Graphics: mirroring of images with one bit per pixel

• Random number generation: reversal can counteract low entropy of low-order bits such as in the case of linear congruential generators

• Digital signal processing: for radix-2 Cooley-Tukey FFT algorithms

• Code obfuscation: security by obscurity

4.1.2. Applications of bit_repeat

The generation of recurring bit patterns is such a fundamental operation that it’s hard to tie to any specific domain, but here are some use cases:

• Debugging: using obvious recurring bit pattenrs like 0xcccc... to identify "garbage memory"

• Bit manipulation: generating alternating sequences of 1 and 0 for various algorithms

• Eliminating integer divison: for x >> (i % N) with very small N, bit_repeat(x, N) >> i can be used instead

• Testing: recurring bit patterns can make for good test cases when implementing numerics

• Error detection: known bit patterns can be introduced to spot failed transmission

4.1.3. Applications of bit_compress and bit_expand

Compression and expansion are also common, with uses in:

• Space-filling curves: Morton/Z-Order and Hilbert curves

• Input/output: especially for variable-length encodings, such as UTF-8 (§ 4.2.3 UTF-8 decoding with bit_compressr)

• Chess engines: for bitboards; see [ChessProgramming1]

• Genomics: according to [ARM1]

A GitHub code search for /(_pdep_u|_pext_u)(32|64)/ AND language:c++ reveals ~1300 files which use the intrinsic wrappers for the x86 instructions.

4.1.4. Applications of next_bit_permutation

next_bit_permutation is useful for iterating over all subsets with a fixed amount of elements (see below for an example).

Unlike algorithms such as next_permutation, it can be used with no modification of the original range. Together with prev_bit_permutation, it is possible to create a bidirectional view of all fixed-size subsets of a range.

4.2. Motivating examples

4.2.1. Implementing countr_zero with bit_repeat

[Anderson1] contains a vast amount of algorithms, many of which involve masks of alternating 0s and 1s.

When written as "magic numbers" in code, these masks can make it quite hard to understand the overall pattern and to generalize these algorithms. bit_repeat allows one to be more expressive here:

unsigned int v;      // 32-bit word input to count zero bits on right
unsigned int c = 32; // c will be the number of zero bits on the right
v &= -v;
if (v) c--;
if (v & 0x0000FFFF bit_repeat((1 << 16) - 1, 32)) c -= 16;
if (v & 0x00FF00FF bit_repeat( (1 << 8) - 1, 16)) c -=  8;
if (v & 0x0F0F0F0F bit_repeat( (1 << 4) - 1,  8)) c -=  4;
if (v & 0x33333333 bit_repeat( (1 << 2) - 1,  4)) c -=  2;
if (v & 0x55555555 bit_repeat( (1 << 1) - 1,  2)) c -=  1;

// ...
for (int i = 16; i != 0; i /= 2) {
    unsigned mask = bit_repeat((1u << i) - 1, i * 2);
    if (v & mask) c -= i;
}

bit_repeat has been an invaluable asset in the implementation of the remaining functions in this proposal (see [Schultke1] for details).

4.2.2. Interleaving bits with bit_expand

A common use case for expansion is interleaving bits. This translates Cartesian coordinates to the index on a Z-order curve. Space filling curves are a popular technique in compression.

unsigned x = 3; // 0b011
unsigned y = 5; // 0b101
const auto i = bit_expandr(x, bit_repeat(0b10u, 2)) // i = 0b01'10'11
             | bit_expandr(y, bit_repeat(0b01u, 2));

4.2.3. UTF-8 decoding with bit_compressr

bit_compress and bit_expand are useful in various I/O-related applications. They are particularly helpful for dealing with variable-length encodings, where "data bits" are interrupted by bits which signal continuation of the data.

uint_least32_t x = load32_little_endian(utf8_data_pointer);
switch (countl_one(uint8_t(x))) {
case 0: return bit_compressr(x, 0b01111111);
case 1: /* error */;
case 2: return bit_compressr(x, 0b00111111'00011111);
case 3: return bit_compressr(x, 0b00111111'00111111'00001111);
case 4: return bit_compressr(x, 0b00111111'00111111'00111111'00000111);
}

4.2.4. Other operations based on bit_compress and bit_expand

Many operations can be built on top of bit_compress and bit_expand. However, direct hardware support is often needed for the proposed functions to efficiently implement them. Even without such support, they can be canonalized into a faster form. The point is that bit_compress and bit_expand allow you to express these operations.

// x & 0xf
bit_expandr(x, 0xf)
bit_compressr(x, 0xf)

// (x & 0xf) << 4
bit_expandr(x, 0xf0)
// (x >> 4) & 0xf
bit_compressr(x, 0xf0)

// Clear the least significant one-bit of x.
x ^= bit_expandr(1, x)
// Clear the nth least significant one-bit of x.
x ^= bit_expandr(1 << n, x)
// Clear the n least significant one-bits of x.
x ^= bit_expandr((1 << n) - 1, x)

// (x >> n) & 1
bit_compressr(x, 1 << n)
// Get the least significant bit of x.
bit_compressr(x, x) & 1
// Get the nth least significant bit of x.
(bit_compressr(x, x) >> n) & 1

// popcount(x)
countr_one(bit_compressr(-1u, x))
countr_one(bit_compressr(x, x))

4.2.5. Iterating over subsets using next_bit_permutation

const char set[] {'a', 'b', 'c', 'd'};

for (unsigned bits = 0b11; bits < 16; bits = next_bit_permutation(bits)) {
    if (bits & 1) std::print("{}", set[0]);
    if (bits & 2) std::print("{}", set[1]);
    if (bits & 4) std::print("{}", set[2]);
    if (bits & 8) std::print("{}", set[3]);
    std::print(" ");
}

ab ac bc ad bd cd

4.3. Hardware support

(Parenthesized) entries signal that the instruction does not directly implement the function, but greatly assists in its implementation.

Note: The AVX-512 vpshufbitqmb instruction can implement any bit permutation in the mathematical sense, and more.

Note: The RISC-V brev8 instruction can also be found under the name rev.b. There appears to have been a name change in 2022.

4.3.1. Support for bit_reverse

This operation is directly implemented in ARM through rbit.

Any architecture with support for byteswap (such as x86 with bswap) also supports bit-reversal in part. [Warren1] presents an O(log n) algorithm which operates by swapping lower and upper N / 2, ..., 16, 8, 4, 2, and 1 bits in parallel. Byte-swapping implements these individual swaps up to 8 bits, requiring only three more parallel swaps in software:

// assuming a byte is an octet of bits, and assuming the width of x is a power of two
x = byteswap(x);
x = (x & 0x0F0F0F0F) <<  4 | (x & 0xF0F0F0F0) >>  4; // ... quartets of bits
x = (x & 0x33333333) <<  2 | (x & 0xCCCCCCCC) >>  2; // ... pairs of bits
x = (x & 0x55555555) <<  1 | (x & 0xAAAAAAAA) >>  1; // ... individual bits

It is worth noting that clang provides a cross-platform family of intrinsics. __builtin_bitreverse uses byte-swapping or bit-reversal instructions if possible.

Such an intrinsic has been requested from GCC users a number of times in [GNU1].

4.3.2. Support for bit_repeat

Firstly, note that for the pattern length, there are only up to N relevant cases, where N is the operand width in bits. It is feasible to switch between these cases, where the length is constant in each case.

While the AVX-512 instruction vpshufbitqmb can be used for all cases, this is not the ideal solution for most cases. For very low or very great lengths, a naive solution is sufficient (and even optimal), where we simply use << and | to duplicate the pattern. What actually matters is how often the pattern is repeated, i.e. N / length.

Specific cases like bit_repeat(x, 8), bit_repeat(x, 16) can be implemented using permutation/duplication/gather/broadcast instructions.

However, note that the primary use of bit_repeat is to express repeating bit patterns without magic numbers, i.e. to improve code quality. Often, both the pattern and the length are known at compile-time, making hardware support less relevant. Even without hardware support, the reference implementation [Schultke1] requires only O(log N) fundamental bitwise operations.

4.3.3. Support for next_bit_permutation

The next permutation can be efficiently obtained if the platform has a "count trailing zeros" instruction. Integer division can also be used as a fallback, although software computation of trailing zeros may be faster than integer division.

Both options are shown in [Anderson1], and one is implemented in [Schultke1].

prev_bit_permutation can be accelerated using the same operation.

4.3.4. Support for bit_compress and bit_expand

Starting with Haswell (2013), Intel CPUs directly implement compression and expansion with with pext and pdep respectively. AMD CPUs starting with Zen 3 implement pext and pdep with 3 cycles latency, like Intel. Zen 2 and older implement pext/pdep in microcode, with 18 cycles latency.

ARM also supports these operations directly with bext, bdep, and bgrp in the SVE2 instruction set. [Warren1] mentions other older architectures with direct support.

Overall, only recent instruction set extensions offer this functionality directly. However, when the mask is a constant, many different strategies for hardware acceleration open up. For example

• interleaving bits can be assisted (though not fully implemented) using ARM zip1/zip2

• other permutations can be assisted by ARM tbl and tbx

As [Warren1] explains, the cost of computing bit_compress and bit_expand in software is dramatically lower for a constant mask. For specific known masks (such as a mask with a single one-bit), the cost is extremely low.

All in all, there are multiple factors that strongly suggest a standard library implementation:

1. The strategy for computing bit_compress and bit_expand depends greatly on the architecture and on information about the mask, even if the exact mask isn’t known.

   • tzcnt, clmul (see [Schultke1] or [Zp7] for specifics), and popcnt are helpful.

2. ISO C++ does not offer a mechanism through which all of this information can be utilized. Namely, it is not possible to change strategy based on information that only becomes available during optimization passes. Compiler extensions such as __builtin_constant_p offer a workaround.

3. ISO C++ does not offer a mechanism through which function implementations can be chosen based on the surrounding context. In a situation where multiple bit_compress calls with the same mask m are performed, it is significantly faster to pre-compute information based on the mask once, and utilize it in subsequent calls. The same technique can be used to accelerate integer division for multiple divisions with the same divisor.

Bullets 2. and 3. suggest that bit_compress and bit_expand benefit from being implemented directly in the compiler via intrinsic, even if hardware does not directly implement these operations.

Even with a complete lack of hardware support, a software implementation of compress_bitsr in [Schultke1] emits essentially optimal code if the mask is known.

unsigned bit_compress_known_mask(unsigned x) {
    return cxx26bp::bit_compressr(x, 0xf0f0u);
}

bit_compress_known_mask(unsigned int): # bit_compress_known_mask(unsigned int edi)
        mov     eax, edi               # {   unsigned int eax = edi;
        shr     eax, 4                 #     eax >>= 4;
        and     eax, 15                #     eax &= 0xf;
        shr     edi, 8                 #     edi >>= 8;
        and     edi, 240               #     edi &= 0xf0;
        or      eax, edi               #     eax |= edi;
        ret                            #     return eax; }

5. Impact on existing code

This proposal is purely a standard library expansion. No existing code is affected.

6. Design considerations

The design choices in this paper are based on [P0553R4], wherever applicable.

6.1. Signature of bit_repeat

bit_repeat follows the "use int if possible" rule mentioned in [P0553R4]. Other functions such as std::rotl and std::rotr also accept an int.

It is also the only function not marked noexcept. It does not throw, but it is not noexcept due to its narrow contract (Lakos rule).

6.2. Why the names compress and expand?

The use of compress and expand is consistent with the mask-based permutations for std::simd proposed in [P2664R6].

Furthermore, there are multiple synonymous sets of terminology:

1. deposit and extract

2. compress and expand

3. gather and scatter

I have decided against deposit and extract because of its ambiguity:

Taking the input 0b10101 and densely packing it to 0b111 could be described as:

Extract each second bit from 0b10101 and densely deposit it into the result.

Similarly, taking the input 0b111 and expanding it into 0b10101 could be described as:

Extract each bit from 0b111 and sparsely deposit it in the result.

Both operations can be described with extract and deposit terminology, making it virtually useless for keeping the operations apart. gather and scatter are simply the least common way to describe these operations, which makes compress and expand the best candidates.

Further design choices are consistent with [P0553R4]. The abbreviations l and r for left/right are consistent with rotl/rotr. The prefix bit_ is consistent with bit_floor and bit_ceil.

6.3. Why the lack of generalization?

6.3.1. No generalized bit_compress and bit_expand

[N3864] originally suggested much more general versions of compression and expansion, which support:

1. performing the operation not just on the whole operand, but on "words" of it, in parallel

2. performing the operation not just on bits, but on arbitrarily sized groups of bits

I don’t propose this generality for the following reasons:

1. The utility functions in <bit> are not meant to provide a full bitwise manipulation library, but fundamental operations, especially those that can be accelerated in hardware while still having reasonable software fallbacks.

2. These more general form can be built on top of the proposed hardware-oriented versions. This can be done with relative ease and with little to no overhead.

3. The generality falsely suggests hardware support for all forms, despite the function only being accelerated for specific inputs. This makes the performance characteristics unpredictable.

4. The proposed functions have wide contracts and can be noexcept (Lakos rule). Adding additional parameters would likely require a narrow contract.

5. Generality adds complexity to the standardization process, to implementation, and from the perspective of language users. It is unclear whether this added complexity is worth it in this case.

6.3.2. No generalized bit_reverse

Bit reversal can also be generalized to work with any group size:

template <typename T>
T bit_reverse(T x, int group_size = 1) noexcept(false);

With this generalization, byteswap(x) on conventional platforms is equivalent to bit_reverse(x, 8).

However, this general version is much less used, not as consistently supported in hardware, and has a narrow contract. group_size must be a nonzero factor of x for this operation to be meaningful.

Therefore, a generalized bit-reversal is not proposed in this paper.

6.4. Why does the signature of bit_compress require two same Ts?

Initially, I went through a number of different signatures.

template<unsigned_integral T, unsigned_integral X>
constexpr T bit_compressr(X x, T m) noexcept;

This signature is quite clever because the result never has more bits than the mask m. However, it is surprising that the mask plays such a significant role here.

Furthermore, I’ve realized that while the result never has more bits than m, bit_compressl must still deposit bits starting with the most significant bits of the result. This suggests the following:

template<unsigned_integral T, unsigned_integral X>
constexpr common_type_t<T, X> bit_compressr(X x, T m) noexcept;

However, it is not trivial to juggle bits between the left and right versions of bit_compress and bit_expand. The behavior is also not intuitive when a zero-extension occurs. For wider x, the mask is always zero-extended to the left, which makes the left and right versions slightly asymmetrical.

Since this proposal includes low-level bit operations, it is reasonable and safe to require the user to be explicit. A call to bit_compress or bit_expand with two different types is likely a design flaw or bug. Therefore, I have settled on the very simple signature:

template<unsigned_integral T>
constexpr T bit_compressr(T x, T m) noexcept;

7. Possible implementation

7.1. Reference implementation

All proposed functions have been implemented in [Schultke1]. This reference implementation is compatible with all three major compilers, and leverages hardware support from ARM and x86_64 where possible.

7.2. Other implementations

[Warren1] presents algorithms which are the basis for [Schultke1].

• An O(log n) bit_reverse

• An O(log² n) bit_compress and bit_expand

  • can be O(log n) with hardware support for carry-less multiplication aka. GF(2) polynomial multiplication

[Zp7] offers fast software implementations for pext and pdep, optimized for x86_64.

[StackOverflow1] contains discussion of various possible software implementations of bit_compressr and bit_expandr.

8. Proposed wording

The proposed changes are relative to the working draft of the standard as of [N4917].

8.1. Feature-testing

In subclause 17.3.2 [version.syn] paragraph 2, update the synopsis as follows:

#define __cpp_lib_bitops                    201907L20XXXXL // freestanding, also in <bit>

8.2. Header synopsis

In subclause 22.15.2 [bit.syn], update the synopsis as follows:

+ // 22.15.X, permutations
+ template<class T>
+   constexpr T bit_reverse(T x) noexcept;
+ template<class T>
+   constexpr T bit_repeat(T x, int l);
+ template<class T>
+   constexpr T next_bit_permutation(T x) noexcept;
+ template<class T>
+   constexpr T prev_bit_permutation(T x) noexcept;
+ template<class T>
+   constexpr T bit_compressr(T x, T m) noexcept;
+ template<class T>
+   constexpr T bit_expandr(T x, T m) noexcept;
+ template<class T>
+   constexpr T bit_compressl(T x, T m) noexcept;
+ template<class T>
+   constexpr T bit_expandl(T x, T m) noexcept;

8.3. New subclause

In subclause 22.15 [bit], add the following subclause:

22.15.X Permutation [bit.permute]

1 In the following descriptions, let N denote numeric_limits<T>​::​digits. Let α_n denote the n-th least significant bit of α, so that α equals

template<class T>
  constexpr T bit_reverse(T x) noexcept;

2 Constraints: T is an unsigned integer type ([basic.fundamental]).

3 Returns:
[Note: bit_reverse(bit_reverse(x)) equals x. — end note]

template<class T>
  constexpr T bit_repeat(T x, int l);

4 Constraints: T is an unsigned integer type ([basic.fundamental]).

5 Preconditions: l is greater than zero.

6 Returns:

7 Throws: Nothing.

template<class T>
  constexpr T next_bit_permutation(T x) noexcept;

8 Constraints: T is an unsigned integer type ([basic.fundamental]).

9 Returns: The lowest integer y in (x, numeric_limits<T>::max()] for which popcount(x) equals popcount(y), or 0 if none exists.
[Note: If x is a power of two or zero, next_bit_permutation(x) equals x << 1. — end note]

template<class T>
  constexpr T prev_bit_permutation(T x) noexcept;

10 Constraints: T is an unsigned integer type ([basic.fundamental]).

11 Returns: The greatest integer y in [0, x) for which popcount(x) equals popcount(y), or 0 if none exists.
[Note: If x == 0 || next_bit_permutation(x) != 0 is true, prev_bit_permutation(next_bit_permutation(x)) equals x. end note]

template<class T>
  constexpr T bit_compressr(T x, T m) noexcept;

12 Constraints: T is an unsigned integer type ([basic.fundamental]).

13 Returns:

template<class T>
  constexpr T bit_expandr(T x, T m) noexcept;

14 Constraints: T is an unsigned integer type ([basic.fundamental]).

15 Returns:
[Note: If x & ~m equals zero, then bit_expandr(bit_compressr(x, m), m) equals x. If x >> popcount(m) equals zero, then bit_compressr(bit_expandr(x, m), m) equals x. — end note]

template<class T>
  constexpr T bit_compressl(T x, T m) noexcept;

16 Constraints: T is an unsigned integer type ([basic.fundamental]).

17 Returns: bit_reverse(bit_compressr(bit_reverse(x), bit_reverse(m))).

template<class T>
  constexpr T bit_expandl(T x, T m) noexcept;

18 Constraints: T is an unsigned integer type ([basic.fundamental]).

19 Returns: bit_reverse(bit_expandr(bit_reverse(x), bit_reverse(m))).

Note: I would have preferred a less mathematical approach to defining these functions. However, it is too difficult to precisely define bit_compress and bit_expand without visual aids, pseudo-code, or other crutches.

9. Acknowledgements

I greatly appreciate the assistance of Stack Overflow users in assisting me with research for this proposal. I especially thank Peter Cordes for his tireless and selfess dedication to sharing knowledge.

I also thank various Discord users from Together C & C++ and #include<C++> who have reviewed drafts of this proposal and shared their thoughts.