// -*- C++ -*-
// automatically generated by autodoc

// ========== HEADER FILE src/bits/average.h: ==========

static inline ulong floor_average(ulong x, ulong y);
// Return floor( (x+y)/2 )
// Result is correct even if (x+y) wouldn't fit into a ulong
// Use:      x+y == ((x&y)<<1) + (x^y)
// that is:  sum ==  carries   + sum_without_carries

static inline ulong ceil_average(ulong x, ulong y);
// Return ceil( (x+y)/2 )
// Result is correct even if (x+y) wouldn't fit into a ulong
// Use:      x+y == ((x|y)<<1) - (x^y)
// ceil_average(x,y) == average(x,y) + ((x^y)&1))

// ========== HEADER FILE src/bits/bin-to-sl-gray.h: ==========

static inline ulong bin_to_sl_gray(ulong k, ulong ldn);
// Unranking for binary SL-Gray:
// Convert binary number to corresponding word in SL-Gray order.
// Successive transitions are adjacent (one-close) or three-close.
// See Joerg Arndt, Subset-lex: did we miss an order?, (2014)
//   http://arxiv.org/abs/1405.6503

static inline ulong sl_gray_to_bin(ulong k, ulong ldn);
// Ranking for binary SL-Gray:
// Convert binary word in SL-Gray order to binary number.
// See Joerg Arndt, Subset-lex: did we miss an order?, (2014)
//   http://arxiv.org/abs/1405.6503

// ========== HEADER FILE src/bits/bin2naf.h: ==========

static inline void bin2naf(ulong x, ulong &np, ulong &nm);
// Compute (nonadjacent form, NAF) signed binary representation of x:
// the unique representation of x as
//   x=\sum_{k}{d_k*2^k} where d_j \in {-1,0,+1}
//   and no two adjacent digits d_j, d_{j+1} are both nonzero.
// np has bits j set where d_j==+1
// nm has bits j set where d_j==-1
// We have:  x = np - nm

static inline ulong naf2bin(ulong np, ulong nm);
// Inverse of bin2naf().
// Works also for signed-binary pairs np, nm
// that are not nonadjacent forms.

static inline void bin2sbin(ulong x, ulong &np, ulong &nm);
// Compute a signed binary representation of x:
// a representation of x as
//   x=\sum_{k}{d_k*2^k} where d_j \in {-1,0,+1}
// Inverse function is naf2bin().

// ========== HEADER FILE src/bits/bit-dragon-r13.h: ==========

static inline bool bit_dragon_r13_turn(ulong &x);
// Increment the radix-13 word x and
// return (tr) whether the lowest nonzero digit
// of the incremented word is either 3, 6, 8, 9, 11, or 12.
// tr determines whether to turn left or right (by 90 degrees)
// with the R13-dragon fractal.
//
// Starting with x==0:
//      x      tr
//  .........  0
//  ........1  0
//  .......1.  1
//  .......11  0
//  ......1..  0
//  ......1.1  1
//  ......11.  0
//  ......111  1
//  .....1...  1
//  .....1..1  0
//  .....1.1.  1
//  .....1.11  1
//  .....11..  0
//  ....1....  0
//  ....1...1  0
//  ....1..1.  1
//  ....1..11  0
//  ....1.1..  0
//  ....1.1.1  1
//  ....1.11.  0
//  ....1.111  1
//  ....11...  1
//  ....11..1  0
//  ....11.1.  1
//  ....11.11  1
//  ....111..  0
//  ...1.....  0
//  ...1....1  0
//  ...1...1.  1
//
// The sequence tr is the fixed point
// of the morphism  0 |--> 0010010110110,  1 |--> 0010010110111.
//
// Also fixed point of morphism (identify + with 0 and - with 1)
//  F |--> F+F+F-F+F+F-F+F-F-F+F-F-F,  + |--> +,  - |--> -

// ========== HEADER FILE src/bits/bit-dragon-r4.h: ==========

static inline int bit_dragon_r4_turn(ulong &x);
// Increment the radix-4 word x and
// return (tr) as follows:
// +1 if lowest nonzero digit of the incremented word is 1,
//   0 if it is 2, otherwise -1 (when it is 3).
// tr determines whether to turn left(-1), right(+1), or not at all(0)
// (by 120 degrees) with the R4-dragon fractal.
//
// The sequence tr starts
//  +0-++0-0+0--+0-++0-++0-0+0--+0-0+0-++0-0+0--+0--+0-++0-0+0--+0-
//
// The sequence tr is the fixed point of morphism
//   + |--> +0-+,  0 |--> +0-0,  - |--> +0--
//
// Also fixed point of morphism
//   F |--> F+F0F-F,  0 |--> 0,  - |--> -,  + |--> +
//  (after omitting the symbols F)
//
// The OEIS sequence A035263 gives the absolute values,
// see also A029883 (different signs).

// ========== HEADER FILE src/bits/bit-dragon-r5.h: ==========

static inline bool bit_dragon_r5_turn(ulong &x);
// Increment the radix-5 word x and
// return (tr) whether the lowest nonzero digit
// of the incremented word is > 2.
// tr determines whether to turn left or right (by 90 degrees)
// with the R5-dragon fractal.
//
// Starting with x==0:
//      x      tr
//  .........  0
//  ........1  0
//  .......1.  1
//  .......11  1
//  ......1..  0
//  .....1...  0
//  .....1..1  0
//  .....1.1.  1
//  .....1.11  1
//  .....11..  0
//  ....1....  0
//  ....1...1  0
//  ....1..1.  1
//  ....1..11  1
//  ....1.1..  1
//  ....11...  0
//
// The sequence tr is the fixed point
// of the morphism  0 |--> 00110,  1 |--> 00111.
// Cf. OEIS sequence A175337.
//
// Also fixed point of morphism (identify + with 0 and - with 1)
// F |--> F+F+F-F-F,  + |--> +,  - |--> -

// ========== HEADER FILE src/bits/bit-dragon-r7.h: ==========

static inline bool bit_dragon_r7_turn(ulong &x);
// Increment the radix-7 word x and
// return (tr) whether the lowest nonzero digit
// of the incremented word is either 2, 3, or 6.
// tr determines whether to turn left or right (by 120 degrees)
// with the R7-dragon fractal.
//
// Starting with x==0:
//      x      tr
//  .........  0
//  ........1  1
//  .......1.  1
//  .......11  0
//  ......1..  0
//  ......1.1  1
//  ......11.  0
//  .....1...  0
//  .....1..1  1
//  .....1.1.  1
//  .....1.11  0
//  .....11..  0
//  .....11.1  1
//  .....111.  1
//  ....1....  0
//  ....1...1  1
//  ....1..1.  1
//  ....1..11  0
//  ....1.1..  0
//
// The sequence tr is the fixed point
// of the morphism  0 |--> 0100110,  1 |--> 0110110.
// Cf. OEIS sequence A176405.
//
// Also fixed point of morphism (identify + with 0 and - with 1)
// F |--> F+F-F-F+F+F-F,  + |--> +,  - |--> -

static inline int bit_dragon_r7_2_turn(ulong &x);
// Increment the radix-7 word x and
// return (tr) according to the lowest nonzero digit d
// of the incremented word:
//  d==[1,2,3,4,5,6]  ==>  rt:=[0,+1,+1,-1,-1,0]
// (tr * 120deg) is the turn with the second R7-dragon.
//
// Starting with x==0:
//      x      tr
//  .........  0
//  ........1  +
//  .......1.  +
//  .......11  -
//  ......1..  -
//  ......1.1  0
//  ......11.  0
//  .....1...  0
//  .....1..1  +
//  .....1.1.  +
//  .....1.11  -
//  .....11..  -
//  .....11.1  0
//  .....111.  +
//  ....1....  0
//  ....1...1  +
//  ....1..1.  +
//  ....1..11  -
//  ....1.1..  -
//  ....1.1.1  0
//  ....1.11.  +
//  ....11...  0
//  ....11..1  +
//
// The sequence tr is the fixed point of the morphism
//   0 |--> 0++--00,  + |--> 0++--0+,  - |--> 0++--0-
// Cf. OEIS sequence A176416.
//
// Also fixed point of morphism
// F |--> F0F+F+F-F-F0F,   + |--> +,  - |--> -,  0 |--> 0

// ========== HEADER FILE src/bits/bit-dragon-r9.h: ==========

static inline bool bit_dragon_r9_turn(ulong &x);
// Increment the radix-9 word x and
// return (tr) whether the lowest nonzero digit
// of the incremented word is either 2, 3, 5, or 8.
// tr determines whether to turn left or right (by 120 degrees)
// with the R9-dragon fractal.
//
// Starting with x==0:
//      x      tr
//  .........  0
//  ........1  1
//  .......1.  1
//  .......11  0
//  ......1..  1
//  ......1.1  0
//  ......11.  0
//  ......111  1
//  .....1...  0
//  ....1....  0
//  ....1...1  1
//  ....1..1.  1
//  ....1..11  0
//  ....1.1..  1
//  ....1.1.1  0
//  ....1.11.  0
//  ....1.111  1
//  ....11...  1
//  ...1.....  0
//  ...1....1  1
//  ...1...1.  1
//  ...1...11  0
//  ...1..1..  1
//  ...1..1.1  0
//  ...1..11.  0
//  ...1..111  1
//  ...1.1...  1
//  ...11....  0
//  ...11...1  1
//  ...11..1.  1
//
// The sequence tr is the fixed point
// of the morphism  0 |--> 011010010,  1 |--> 011010011.
// Also fixed point of 0 |--> 011, 1 |--> 010, because
//   0 |--> 011 |--> 011010010
//   1 |--> 010 |--> 011010011
// See OEIS sequence A156595.
//
// Also fixed point of morphism (identify + with 0 and - with 1)
//  F |--> F+F-F-F+F-F+F+F-F,  + |--> +,  - |--> -
//
// Also fixed point of morphism
//  F |--> G+G-G,  G |--> F-F+F,  + |--> +,  - |--> -

// ========== HEADER FILE src/bits/bit-dragon3.h: ==========

static inline bool bit_dragon3_turn(ulong &x);
// Increment the radix-3 word x and return (tr)
//   whether the number of ones in x is decreased.
// tr determines whether to turn left or right (by 120 degrees)
//   with the terdragon fractal.
//
// Starting with x==0:
//  x (binary) tr   x (ternary)
//  ........   0    0 0 0
//  .......1   0    0 0 1
//  ......1.   1    0 0 2
//  .....1..   0    0 1 0
//  .....1.1   0    0 1 1
//  .....11.   1    0 1 2
//  ....1...   1    0 2 0
//  ....1..1   0    0 2 1
//  ....1.1.   1    0 2 2
//  ...1....   0    1 0 0
// The sequence tr (for x>=1) is entry A080846 in the OEIS,
// the fixed point of the morphism  0 |--> 010,  1 |--> 011.
// See also A060236 (== 1,2,1,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,2, ...).
// Also fixed point of morphism (for k>0, identify + with 0 and - with 1)
// F |--> F+F-F,  + |--> +,  - |--> -

// ========== HEADER FILE src/bits/bit-isolate.h: ==========


// Extraction of ones, zeros, or blocks near transitions.
// Here we assume that the outside bits are all zero.
// Therefore we have
//   interior_values(v) != ( interior_ones(v) | interior_ones(~v) )
// Functions ignoring outside values have names ending in _xi

static inline ulong single_ones(ulong x);
// Return word with only the isolated ones of x set.
// Assume outside values are 0.

static inline ulong single_ones_xi(ulong x);
// Return word with only the isolated ones of x set.
// Ignore outside values.

static inline ulong single_zeros(ulong x);
// Return word with only the isolated zeros of x set.
// Assume outside values are 0.

static inline ulong single_zeros_xi(ulong x);
// Return word with only the isolated zeros of x set.
// Ignore outside values.

static inline ulong single_values(ulong x);
// Return word where only the isolated ones and zeros of x are set.
// Assume outside values are 0.

static inline ulong single_values_xi(ulong x);
// Return word where only the isolated ones and zeros of x are set.
// Ignore outside values.

static inline ulong border_ones(ulong x);
// Return word where only those ones of x are set that lie next to a zero.

static inline ulong border_values(ulong x);
// Return word where those bits of x are set that lie on a transition.
// Assume outside values are 0.

static inline ulong high_border_ones(ulong x);
// Return word where only those ones of x are set
//   that lie right to (i.e. in the next lower bin of) a zero.

static inline ulong low_border_ones(ulong x);
// Return word where only those ones of x are set
//   that lie left to (i.e. in the next higher bin of) a zero.

static inline ulong block_border_ones(ulong x);
// Return word where only those ones of x are set
//   that are at the border of a block of at least 2 ones.

static inline ulong low_block_border_ones(ulong x);
// Return word where only those ones of x are set
//   that are at left of a border of a block of at least 2 ones.

static inline ulong high_block_border_ones(ulong x);
// Return word where only those ones of x are set
//   that are at right of a border of a block of at least 2 ones.

static inline ulong block_ones(ulong x);
// Return word where only those ones of x are set
//   that are part of a block of at least 2 ones.

static inline ulong block_values(ulong x);
// Return word where only those bits of x are set
// that do not lie next to an opposite value.

static inline ulong interior_ones(ulong x);
// Return word where only those ones of x are set
//   that do not have a zero to their left or right.

static inline ulong interior_values(ulong x);
// Return word where only those values of x are set
//   that do have a transitions on both sides.

// ========== HEADER FILE src/bits/bit-necklace.h: ==========

class bit_necklace;
// Binary necklaces as binary words.
// The cyclic maximal words are produced.
// Must have 0<n<=BITS_PER_LONG.

// ========== HEADER FILE src/bits/bit-paper-fold.h: ==========

static inline bool bit_paper_fold(ulong k);
// Return element number k of the paper-folding sequence:
// k=      0, 1, 2, 3, 4, 5, ...
// seq(k)= 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, ...
// Sequence (for k>=1) is entry A014577 of the OEIS.
// Also: for k>=1, the dragon curve turns left if seq(k)=1, else right.
// Also fixed point of morphism (for k>0, identify + with 1 and - with 0)
//   L |--> L+R+L-R,  R |--> L+R-L-R,  + |--> +,  - |--> -
// Also fixed point of morphism (for k>0)
//   L |--> L+R,  R |--> L-R,  + |--> +,  - |--> -

static inline ulong bit_dragon_rot(ulong k);
// Return total rotation (as multiple of the right angle)
//   after k steps in the dragon curve corresponding to
//   the paper-folding sequence.
// k=      0, 1, 2, 3, 4, 5, ...
// seq(k)= 0, 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 3, 2, ...
// move =  +  ^  -  ^  -  v  -  ^  -  v  +  v  -  v  -  ...
// (+==right, -==left, ^==up, v==down).
// Sequence is entry A005811 of the OEIS.
// We take result modulo 4 to ignore multiples of 360 degree.
// Algorithm: count the ones in gray_code(k).

static inline bool bit_paper_fold_alt(ulong k);
// Return element number k of the alternate paper-folding sequence:
// k=      0, 1, 2, 3, 4, 5, ...
// seq(k)= 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, ...
// Sequence (for k>=1) is entry A106665 of the OEIS.
// Also: the corresponding curve turns left if seq(k)=1, else right.
// Also fixed point of morphism (for k>0, identify + with 1 and - with 0)
// L |--> R+L+R-L,  R |--> R+L-R-L,  + |--> +,  - |--> -

static inline ulong bit_paper_fold_alt_rot(ulong k);
// Return total rotation (as multiple of the right angle)
//   after k steps in the alternate paper-folding curve.
// k=      0, 1, 2, 3, 4, 5, ...
// seq(k)= 0, 1, 0, 3, 0, 1, 2, 1, 0, 1, 0, 3, 2, 3, 0, ...
// move =  +  ^  +  v  +  ^  -  ^  +  ^  +  v  -  v  +
// (+==right, -==left, ^==up, v==down).
// Algorithm: count the ones in  (w ^ gray_code(k)).

static inline bool bit_paper_fold_general(ulong k, ulong w);
// Return element number k of the general paper-folding sequence:
// bit number x of the words w determines whether
// a left or right fold is made at the step x.
// With w==0 the result is  ! bit_paper_fold(k).
// With w==~0 the result is bit_paper_fold(k).
// The result with ~w is the complement of the result with w.

static inline ulong bit_paper_fold_general_rot(ulong k, ulong w);
// Return total rotation (as multiple of the right angle)
//   after k steps in the general paper-folding curve.
// Algorithm: count the ones in  (w ^ gray_code(k)).
// With w==~0 the result is bit_dragon_rot(k).

// ========== HEADER FILE src/bits/bit-rll2.h: ==========

class bit_rll2;
// Run length limited (RLL) words and Fibonacci words.
// The RLL words are in lexicographic order,
// the Fibonacci words are in a minimal change order (Gray code).

// ========== HEADER FILE src/bits/bit-sl-gray.h: ==========

class bit_sl_gray;
// Binary words in a minimal-change order
//   related so subset-lex order ("SL-Gray" order).
// Successive transitions are mostly adjacent (one-close),
//   and otherwise have distance 3 (three-close).
// Loopless algorithm.

// ========== HEADER FILE src/bits/bit2adic.h: ==========

static inline ulong inv2adic(ulong x);
// Return inverse modulo 2**BITS_PER_LONG
// x must be odd
// The number of correct bits is doubled with each step
// ==> loop is executed prop. log_2(BITS_PER_LONG) times
// precision is 3, 6, 12, 24, 48, 96, ... bits (or better)

static inline ulong invsqrt2adic(ulong d);
// Return inverse square root modulo 2**BITS_PER_LONG
// Must have:  d==1 mod 8
// The number of correct bits is doubled with each step
// ==> loop is executed prop. log_2(BITS_PER_LONG) times
// precision is 4, 8, 16, 32, 64, ... bits (or better)

static inline ulong sqrt2adic(ulong d);
// Return square root modulo 2**BITS_PER_LONG
// Must have: d==1 mod 8  or  d==4 mod 32,  d==16 mod 128
//   ... d==4**k mod 4**(k+3)
// Result undefined if condition does not hold

// ========== HEADER FILE src/bits/bit2pow.h: ==========

static inline ulong ld(ulong x);
// Return floor(log2(x)),
// i.e. return k so that 2^k <= x < 2^(k+1)
// If x==0, then 0 is returned (!)

static inline bool is_pow_of_2(ulong x);
// Return whether x == 0(!) or x == 2**k

static inline bool one_bit_q(ulong x);
// Return whether x \in {1,2,4,8,16,...}

static inline ulong next_pow_of_2(ulong x);
// Return x if x=2**k
// else return 2**ceil(log_2(x))
// Exception: returns 0 for x==0

static inline ulong next_exp_of_2(ulong x);
// Return k if x=2**k else return k+1.
// Exception: returns 0 for x==0.

// ========== HEADER FILE src/bits/bitasm-amd64.h: ==========

static inline ulong asm_bit_count(ulong x);

static inline ulong asm_bsf(ulong x);
// Bit Scan Forward

static inline ulong asm_bsr(ulong x);
// Bit Scan Reverse

static inline ulong asm_bswap(ulong x);
// Byte swap

static inline ulong asm_rol(ulong x, ulong r);
// Rotate Left

static inline ulong asm_ror(ulong x, ulong r);
// Rotate Right

static inline ulong asm_parity(ulong x);
// Return the parity of x (which is the
// _complement_ of AMD64's parity flag).
// As parity is for the low byte only,
// therefore we need to prepend
// x ^= (x>>32);  x ^= (x>>16);  x ^= (x>>8);
// in the code

static inline ulong asm_bt(const ulong *f, ulong i);
// Bit Test

static inline ulong asm_bts(ulong *f, ulong i);
// Bit Test and Set

static inline void asm_b_s(ulong *f, ulong i);
// Bit Set

static inline ulong asm_btr(ulong *f, ulong i);
// Bit Test and Reset

static inline void asm_b_r(ulong *f, ulong i);
// Bit Reset

static inline ulong asm_btc(ulong *f, ulong i);
// Bit Test and Complement

static inline void asm_b_c(ulong *f, ulong i);
// Bit Complement

// ========== HEADER FILE src/bits/bitasm-i386.h: ==========

static inline ulong asm_bsf(ulong x);
// Bit Scan Forward: return index of lowest one.

static inline ulong asm_bsr(ulong x);
// Bit Scan Reverse: return index of highest one.

static inline ulong asm_bswap(ulong x);
// Byte swap

static inline ulong asm_rol(ulong x, ulong r);
// Rotate Left

static inline ulong asm_ror(ulong x, ulong r);
// Rotate Right

static inline ulong asm_parity(ulong x);
// Return the parity of x (which is the
// _complement_ of x86's parity flag).
// As parity is for the low byte only,
// therefore we need to prepend
// x ^= (x>>16);  x ^= (x>>8);
// in the code

static inline ulong asm_bt(const ulong *f, ulong i);
// Bit Test

static inline ulong asm_bts(ulong *f, ulong i);
// Bit Test and Set

static inline void asm_b_s(ulong *f, ulong i);
// Bit Set

static inline ulong asm_btr(ulong *f, ulong i);
// Bit Test and Reset

static inline void asm_b_r(ulong *f, ulong i);
// Bit Reset

static inline ulong asm_btc(ulong *f, ulong i);
// Bit Test and Complement

static inline void asm_b_c(ulong *f, ulong i);
// Bit Complement

// ========== HEADER FILE src/bits/bitasm-sse.h: ==========

static inline ulong sse_byte_zip(ulong x);
// Return word with lower half bytes in even indices
//  and upper half bits in odd indices.
//.
// Slower than the butterfly_8/16() based version!
// This is apparently not what SSE was made for.

// ========== HEADER FILE src/bits/bitasm.h: ==========

// ========== HEADER FILE src/bits/bitblock.h: ==========

static inline ulong bit_block(ulong p, ulong n);
// Return word with length-n bit block starting at bit p set.
// Both p and n are effectively taken modulo BITS_PER_LONG.

static inline ulong cyclic_bit_block(ulong p, ulong n);
// Return word with length-n bit block starting at bit p set.
// The result is possibly wrapped around the word boundary.
// Both p and n are effectively taken modulo BITS_PER_LONG.

// ========== HEADER FILE src/bits/bitbutterfly.h: ==========

// The butterfly_*() functions are used in bit_zip() and bit_unzip()
static inline ulong butterfly_16(ulong x);
// Swap the two central blocks of 16 bits.

static inline ulong butterfly_8(ulong x);
// Swap in each block of 32 bits the two central blocks of 8 bits.

static inline ulong butterfly_4(ulong x);
// Swap in each block of 16 bits the two central blocks of 4 bits.

static inline ulong butterfly_2(ulong x);
// Swap in each block of 8 bits the two central blocks of 2 bits.

static inline ulong butterfly_1(ulong x);
// Swap in each block of 4 bits the two central bits.

// ========== HEADER FILE src/bits/bitcombcolex.h: ==========

static inline ulong first_comb(ulong k);
// Return the first combination of (i.e. smallest word with) k bits,
// i.e.  00..001111..1 (k low bits set)
// Must have:  0 <= k <= BITS_PER_LONG

static inline ulong last_comb(ulong k, ulong n=BITS_PER_LONG);
// Return the last combination of (biggest n-bit word with) k bits
// i.e.  1111..100..00 (k high bits set)
// Must have:  0 <= k <= n <= BITS_PER_LONG

static inline ulong next_colex_comb(ulong x);
// Return smallest integer greater than x with the same number of bits set.
//
// colex order (5 over 3):
//   set        word    set reversed (sorted!)
//  0  1  2    ..111    2  1  0
//  0  1  3    .1.11    3  1  0
//  0  2  3    .11.1    3  2  0
//  1  2  3    .111.    3  2  1
//  0  1  4    1..11    4  1  0
//  0  2  4    1.1.1    4  2  0
//  1  2  4    1.11.    4  2  1
//  0  3  4    11..1    4  3  0
//  1  3  4    11.1.    4  3  1
//  2  3  4    111..    4  3  2
//
//  Examples:
//    000001 -> 000010 -> 000100 -> 001000 -> 010000 -> 100000
//    000011 -> 000101 -> 000110 -> 001001 -> 001010 -> 001100 -> 010001 -> ...
//    000111 -> 001011 -> 001101 -> 001110 -> 010011 -> 010101 -> 010110 -> ...
//
//  Special cases:
//    0 -> 0
//    all bits on the high side (i.e. last combination) -> 0
//.
// based on code by Doug Moore / Glenn Rhoads
// note: might want to use bitscan near end

static inline ulong next_colex_comb(ulong x);
// Return smallest integer greater than x with the same number of bits set.
//
//  Example: (the bits marked by '.' remain fixed)
//    x         = ....01111100000000
//    z         = 000000000100000000   (the lowest set bit)
//    v=x+z     = ....10000000000000   (first zero beyond burst of ones)
//    v^x       = 000011111100000000
//    v^x/z     = 000000000000111111
//    v^x/z>>2  = 000000000000001111
//    next      = ....10000000001111
//.
//  Code based on code taken from Torsten Sillke's bitmani.h:
//  http://www.mathematik.uni-bielefeld.de/~sillke/ALGORITHMS/
//  The algorithm can be found in hakmem, shortcut by me.

static inline ulong prev_colex_comb(ulong x);
// Inverse of next_colex_comb()

// ========== HEADER FILE src/bits/bitcombminchange.h: ==========

static inline ulong igc_next_minchange_comb(ulong x);
// Return the inverse Gray code of the next combination in minimal-change order.
// Input must be the inverse Gray code of the current combination.
//.
// Original code by by Doug Moore.
// Only cosmetical changes by me.

static inline ulong igc_next_minchange_comb(ulong x);
// Alternative version, faster.
// Constant amortized time (CAT).

static inline ulong igc_next_minchange_comb(ulong x, ulong k);
// Alternative version, uses the fact that the difference
// of two successive x is the smallest possible power of 2.
// Should be fast if the CPU has a bitcount instruction.
// k must be the bit-count of x
// Constant amortized time (CAT).
// Note: this version has 2 arguments.

static inline ulong igc_prev_minchange_comb(ulong x, ulong k);
// Return the inverse Gray code of the previous combination in minimal-change order.
// Input must be the inverse Gray code of the current combination.
// Constant amortized time (CAT).
// With input==first the output is the last for n=BITS_PER_LONG

static inline ulong igc_last_comb(ulong k, ulong n);
// Return the (inverse Gray code of the) last combination
// as in igc_next_minchange_comb().
//
// Example (n=6)   c:=first_comb(n) == 111111
//
//   k:  f=first_seq(k)   c^(f>>1) == return
//   0:     ......        111111 (!) special case:  return 0==......
//   1:     .....1        111111
//   2:     ....1.        11111.
//   3:     ...1.1        1111.1
//   4:     ..1.1.        111.1.
//   5:     .1.1.1        11.1.1
//   6:     1.1.1.        1.1.1.

static inline ulong next_minchange_comb(ulong x, ulong last);
// Not efficient, just to explain the usage of igc_next_minchange_comb()
// Must have: last==igc_last_comb(k, n)
//
// Example with  k==3, n==5:
//      x       inverse_gray_code(x)
//    ..111       ..1.1 == first_sequency(k)
//    .11.1       .1..1
//    .111.       .1.11
//    .1.11       .11.1
//    11..1       1...1
//    11.1.       1..11
//    111..       1.111
//    1.1.1       11..1
//    1.11.       11.11
//    1..11       111.1 == igc_last_comb(k, n)

// ========== HEADER FILE src/bits/bitcombshifts.h: ==========

class bit_comb_shifts;
// Bit combinations in shifts-order.

// ========== HEADER FILE src/bits/bitcopy.h: ==========

static inline ulong copy_bit(ulong a, ulong isrc, ulong idst);
// Copy bit at [isrc] to position [idst].
// Return the modified word.

static inline ulong mask_copy_bit(ulong a, ulong msrc, ulong mdst);
// Copy bit according at src-mask (msrc)
//  to the bit according to the dest-mask (mdst).
// Both msrc and mdst must have exactly one bit set.
// Return the modified word.

static inline ulong mask_or_bit(ulong a, ulong msrc, ulong mdst);
// Or bit according to src-mask (msrc)
//  to the bit according to the dest-mask (mdst).
// Both msrc and mdst must have exactly one bit set.
// If the mdst-position is known to be zero this function
//  is equivalent to mask_copy_bit().
// Return the modified word.

// ========== HEADER FILE src/bits/bitcount.h: ==========

static inline ulong bit_count(ulong x);
// Return number of set bits.
// The sequence of values returned for x = 0, 1, 2, 3, ... is
// 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, ...
// (OEIS sequence A000120).

static inline ulong bit_count_15(ulong x);
// Return number of set bits, must have at most 15 set bits.

static inline ulong bit_count_3(ulong x);
// Return number of set bits, must have at most 3 set bits.

static inline int bit_count_cmp(const ulong &a, const ulong &b);
// Compare bit counts of a and b.

static inline ulong bit_count_sparse(ulong x);
// Return number of bits set.

static inline ulong bit_count_dense(ulong x);
// Return number of bits set.
// The loop (of bit_count_sparse()) will execute once for
//   each unset bit (i.e. zero) of x.

static inline ulong bit_block_count(ulong x);
// Return number of bit blocks.
// E.g.:
// ..1..11111...111.  -> 3
// ...1..11111...111  -> 3
// ......1.....1.1..  -> 3
// .........111.1111  -> 2

static inline ulong bit_block_ge2_count(ulong x);
// Return number of bit blocks with at least 2 bits.
// E.g.:
// ..1..11111...111.  -> 2
// ...1..11111...111  -> 2
// ......1.....1.1..  -> 0
// .........111.1111  -> 2

static inline ulong bit_count_01(ulong x);
// Return number of bits in a word
// for words of the special form 00...0001...11

// ----- SRCFILE=src/bits/bitcount-v.cc: -----
ulong bit_count_v(const ulong *x, ulong n);
// Return sum(j=0, n-1, bit_count(x[j]) )

// ========== HEADER FILE src/bits/bitcyclic-dist.h: ==========

static inline ulong bit_cyclic_dist(ulong a, ulong b);
// Return minimal bitcount of (t ^ b)
// where t runs through the cyclic rotations of a.

static inline ulong bit_cyclic_dist(ulong a, ulong b, ulong n);
// Return minimal bitcount of (t ^ b)
// where t runs through the cyclic rotations of a.
// Using n-bit words.

// ========== HEADER FILE src/bits/bitcyclic-match.h: ==========

static inline ulong bit_cyclic_match(ulong x, ulong y);
// Return  r if x==rotate_right(y, r) else return ~0UL.
// In other words: return
//   how often the right arg must be rotated right (to match the left)
// or, equivalently:
//   how often the left arg must be rotated left (to match the right)

static inline ulong bit_cyclic_match(ulong x, ulong y, ulong n);
// Return  r if x==rotate_right(y, r, n) else return ~0UL
//  (using n-bit words)

static inline ulong bit_cyclic_dist1_match(ulong x, ulong y);
// Return  the minimal r so that
//   c = (x ^ rotate_right(y, r, n)) is a one-bit word.
// Return ~0UL if there is no such r.

static inline ulong bit_cyclic_dist1_match(ulong x, ulong y, ulong n);
// Return  the minimal r so that
//   c = (x ^ rotate_right(y, r, n)) is a one-bit word.
// Return ~0UL if there is no such r.
//  (using n-bit words)

// ========== HEADER FILE src/bits/bitcyclic-minmax.h: ==========

static inline ulong bit_cyclic_min(ulong x);
// Return minimum of all rotations of x

static inline ulong bit_cyclic_min(ulong x, ulong n);
// Return minimum of all rotations of x using n-bit words

static inline ulong bit_cyclic_max(ulong x);
// Return maximum of all rotations of x

static inline ulong bit_cyclic_max(ulong x, ulong n);
// Return maximum of all rotations of x using n-bit words

static inline bool is_bit_cyclic_min(ulong x);
// Return whether x is the minimum of all rotations of x

static inline bool is_bit_cyclic_min(ulong x, ulong n);
// Return whether x is the minimum of all rotations of x as n-bit word

static inline bool is_bit_cyclic_max(ulong x);
// Return whether x is the maximum of all rotations of x

static inline bool is_bit_cyclic_max(ulong x, ulong n);
// Return whether x is the maximum of all rotations of x as n-bit word

// ========== HEADER FILE src/bits/bitcyclic-period.h: ==========

static inline ulong bit_cyclic_period(ulong x);
// Return minimal positive bit-rotation that transforms x into itself.
// (same as bit_cyclic_period(x, BITS_PER_LONG) )
//
// The returned value is a divisor of the word length,
//   i.e. 1,2,4,8,...,BITS_PER_LONG.

static inline ulong bit_cyclic_period(ulong x, ulong n);
// Return minimal positive bit-rotation that transforms x into itself.
//  (using n-bit words)
// The returned value is a divisor of n.
//
// Examples for n=6:
//   word   period
//  ......  1
//  .....1  6
//  ....11  6
//  ...1.1  6
//  ...111  6
//  ..1..1  3
//  ..1.11  6
//  ..11.1  6
//  ..1111  6
//  .1.1.1  2
//  .1.111  6
//  .11.11  3
//  .11111  6
//  111111  1

// ========== HEADER FILE src/bits/bitcyclic-xor.h: ==========

static inline ulong bit_cyclic_rxor(ulong x);
// Similar to Gray code, but the bit shifted
// out at the right is moved to the highest bit.
// The returned word has an even number of bits set.

static inline ulong bit_cyclic_rxor(ulong x, ulong n);
// Version for n-bit words

static inline ulong bit_cyclic_inv_rxor(ulong x);
// Return v so that bit_cyclic_rxor(v) == x.
// The same relation holds for ~v.
// x must have an even number of bits.
// Lowest bit of result is zero for valid inputs.
// Also the inverse of bit_cyclic_rxor(x,n) for any n.

static inline ulong bit_cyclic_lxor(ulong x);
// Similar to reversed Gray code, but the bit shifted
// out at the left is moved to the lowest bit.
// The returned word has an even number of bits set.

static inline ulong bit_cyclic_lxor(ulong x, ulong n);
// Version for n-bit words

static inline ulong bit_cyclic_inv_lxor(ulong x);
// Return v so that bit_cyclic_lxor(v) == x.
// The same relation holds for ~v.
// x must have an even number of bits.
// Highest bit of result is zero for valid inputs.
// Also the inverse of bit_cyclic_lxor(x,n) for any n.

// ========== HEADER FILE src/bits/bitfibgray.h: ==========

class bit_fibgray;
// Fibonacci Gray code with binary words.

// ========== HEADER FILE src/bits/bitgather.h: ==========

static inline ulong bit_gather(ulong w, ulong m);
// Return  word with bits of w collected as indicated by m:
// Example:
//  w = ABCDEFGH
//  m = 00101100
//  ==> 00000CEF
// This is the inverse of bit_scatter()

static inline ulong bit_scatter(ulong w, ulong m);
// Return  word with bits of w distributed as indicated by m:
// Example:
//  w = 00000ABC
//  m = 00101100
//  ==> 00A0BC00
// This is the inverse of bit_gather()

// ========== HEADER FILE src/bits/bitgraypermute.h: ==========

static inline ulong left_swap_16(ulong x);
// Return 64-bit word with two leftmost quarters swapped.

static inline ulong left_swap_8(ulong x);
// Return word with two leftmost quarters of each 32-bit block swapped.

static inline ulong left_swap_4(ulong x);
// Return word with two leftmost quarters of each 16-bit block swapped.

static inline ulong left_swap_2(ulong x);
// Return word with two leftmost quarters of each 8-bit block swapped.

static inline ulong left_swap_1(ulong x);
// Return word with two leftmost bits of each 4-bit block swapped.

static inline ulong bit_gray_permute(ulong x);
// Return word with gray-permuted bits.

static inline ulong bit_inverse_gray_permute(ulong x);
// Inverse of bit_gray_permute()

// ========== HEADER FILE src/bits/bithigh-edge.h: ==========

static inline ulong highest_one_01edge(ulong x);
// Return word where all bits from (including) the
//   highest set bit to bit 0 are set.
// Return 0 if no bit is set.
//
// Feed the result into bit_count() to get
//   the index of the highest bit set.

static inline ulong highest_one_10edge(ulong x);
// Return word where all bits from  (including) the
//   highest set bit to most significant bit are set.
// Return 0 if no bit is set.

// ========== HEADER FILE src/bits/bithigh.h: ==========

static inline ulong highest_one(ulong x);
// Return word where only the highest bit in x is set.
// Return 0 if no bit is set.

static inline ulong highest_zero(ulong x);
// Return word where only the highest unset bit in x is set.
// Return 0 if all bits are set.

static inline ulong set_highest_zero(ulong x);
// Return word where the highest unset bit in x is set.
// Return ~0 for input == ~0.

static inline ulong highest_one_idx(ulong x);
// Return index of highest bit set.
// Return 0 if no bit is set.

static inline ulong highest_zero_idx(ulong x);

static inline ulong high_zeros(ulong x);
// Return word where all the (high end) zeros are set.
// e.g.:  00011001 --> 11100000
// Return 0 if highest bit is set:
//        11011001 --> 00000000

static inline ulong high_ones(ulong x);
// Return word where all the (high end) ones are set.
// e.g.  11001011 --> 11000000
// Return 0 if highest bit is zero:
//       01110110 --> 00000000

// ========== HEADER FILE src/bits/bitldeq.h: ==========

static inline bool ld_eq(ulong x, ulong y);
// Return whether floor(log2(x))==floor(log2(y)).
// For arguments 0 and 1 (either order)
// ld_eq(x,y) correctly returns false
// whereas ld(x) == ld(y) gives true
// (because ld(x) returns 0 for both x==1 and x==0).

static inline bool ld_neq(ulong x, ulong y);
// Return whether floor(log2(x))!=floor(log2(y))

// ========== HEADER FILE src/bits/bitlex.h: ==========

static inline ulong next_lexrev(ulong x);
// Return next word in subset-lexrev order.
// Start with a one-bit word at position n-1 to
// generate 2**n subsets of length n.
// E.g., for n==4 the subsets are
//     word   subset of {0,1,2,3}
//     1...   {0}
//     11..   {0, 1}
//     111.   {0, 1, 2}
//     1111   {0, 1, 2, 3}
//     11.1   {0, 1, 3}
//     1.1.   {0, 2}
//     1.11   {0, 2, 3}
//     1..1   {0, 3}
//     .1..   {1}
//     .11.   {1, 2}
//     .111   {1, 2, 3}
//     .1.1   {1, 3}
//     ..1.   {2}
//     ..11   {2, 3}
//     ...1   {3}
//     ....   {}
// Note (1): the first element of the subset corresponds
// to the highest set bit.  When interpreting the binary
// words via "bit(n)==element n" (as usual), the order
// would be:
//     1...           { 3 }
//     11..        { 2, 3 }
//     111.     { 1, 2, 3 }
//     1111  { 0, 1, 2, 3 }
//     11.1     { 0, 2, 3 }
//     1.1.        { 1, 3 }
//     1.11     { 0, 1, 3 }
//     1..1        { 0, 3 }
//     .1..           { 2 }
//     .11.        { 1, 2 }
//     .111     { 0, 1, 2 }
//     .1.1        { 0, 2 }
//     ..1.           { 1 }
//     ..11        { 0, 1 }
//     ...1           { 0 }
// Note (2): the lex order for the delta sets would simply
// be the counting order (of the words or reversed words
// depending on the interpretation as explained above).

static inline ulong prev_lexrev(ulong x);
// Return previous word in subset-lexrev order.
// Start with zero and use 2**n calls
// generate 2**n subsets of length n.
// E.g., for n==4:
//  ....  =  0
//  ...1  =  1
//  ..11  =  3
//  ..1.  =  2
//  .1.1  =  5
//  .111  =  7
//  .11.  =  6
//  .1..  =  4
//  1..1  =  9
//  1.11  = 11
//  1.1.  = 10
//  11.1  = 13
//  1111  = 15
//  111.  = 14
//  11..  = 12
//  1...  =  8
//

static inline ulong negidx2lexrev(ulong k);
//
//   k:  negidx2lexrev(k)
//   0:  .....
//   1:  ....1
//   2:  ...11
//   3:  ...1.
//   4:  ..1.1
//   5:  ..111
//   6:  ..11.
//   7:  ..1..
//   8:  .1..1
//   9:  .1.11
//  10:  .1.1.
//  11:  .11.1
//  12:  .1111
//  13:  .111.
//  14:  .11..
//  15:  .1...
//  16:  1...1
//

static inline ulong lexrev2negidx(ulong x);
// inverse of negidx2lexrev()

static inline bool is_lexrev_fixed_point(ulong x);
// Return whether x is a fixed point in the prev_lexrev() - sequence

// ========== HEADER FILE src/bits/bitlow-edge.h: ==========

static inline ulong lowest_one_10edge(ulong x);
// Return word where all bits from (including) the
//   lowest set bit to most significant bit are set.
// Return 0 if no bit is set.
// Example:  00110100 --> 11111100

static inline ulong lowest_one_01edge(ulong x);
// Return word where all bits from  (including) the
//   lowest set bit to the least significant bit are set.
// Return 0 if no bit is set.
// Example:  00110100 --> 00000111

// ========== HEADER FILE src/bits/bitlow.h: ==========

static inline ulong lowest_one_idx(ulong x);
// Return index of lowest bit set.
// Bit index ranges from zero to BITS_PER_LONG-1.
// Examples:
//    ***1 --> 0
//    **10 --> 1
//    *100 --> 2
// Return 0 (also) if no bit is set.

static inline ulong lowest_one_idx_parity(ulong x);
// Return the parity of the index of the lowest set bit.
// Return zero with x=0.
// Sequence for x>=0 (OEIS sequence A096268):
// 0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,...

static inline ulong lowest_one(ulong x);
// Return word where only the lowest set bit in x is set.
// Return 0 if no bit is set.

static inline ulong lowest_zero(ulong x);
// Return word where only the lowest unset bit in x is set.
// Return 0 if all bits are set.

static inline ulong lowest_block(ulong x);
// Isolate lowest block of ones.

static inline ulong clear_lowest_one(ulong x);
// Return word where the lowest bit set in x is cleared.
// Return 0 for input == 0.

static inline ulong set_lowest_zero(ulong x);
// Return word where the lowest unset bit in x is set.
// Return ~0 for input == ~0.

static inline ulong low_ones(ulong x);
// Return word where all the (low end) ones are set.
// Example:  01011011 --> 00000011
// Return 0 if lowest bit is zero:
//       10110110 --> 0

static inline ulong low_zeros(ulong x);
// Return word where all the (low end) zeros are set.
// Example:  01011000 --> 00000111
// Return 0 if all bits are set.

static inline ulong low_match(ulong x, ulong y);
// Return word that contains all bits at the low end where x and y match.
// If x = *0W and y = *1W, then 00W is returned.

// ========== HEADER FILE src/bits/bitperiodic.h: ==========

static inline ulong bit_copy_periodic(ulong a, ulong p);
// Return word that consists of the lowest p bits of a repeated.
// E.g.: if p==3 and a=*****xyz (8-bit), the return yzxyzxyz.
// Must have p>0.

static inline ulong bit_copy_periodic(ulong a, ulong p, ulong ldn);
// Return word that consists of the lowest p bits of a repeated
// in the lowest ldn bits (upper bits are zero).
// E.g.: if p==3, ldn=7 and a=*****xyz (8-bit), the return 0zxyzxyz.
// Must have p>0 and ldn>0.

// ========== HEADER FILE src/bits/bitrotate.h: ==========

static inline ulong bit_rotate_left(ulong x, ulong r);
// Return word rotated r bits to the left
// (i.e. toward the most significant bit)
//.
// GCC optimizes the function to asm 'roll %cl,%ebx'

static inline ulong bit_rotate_right(ulong x, ulong r);
// Return word rotated r bits to the right
// (i.e. toward the least significant bit)
//.
// GCC optimizes the function to asm 'rorl %cl,%ebx'

static inline ulong bit_rotate_sgn(ulong x, long r);
// Positive r --> shift away from element zero

static inline ulong bit_rotate_left(ulong x, ulong r, ulong n);
// Return n-bit word rotated r bits to the left
// (i.e. toward the most significant bit)
// Must have  0 <= r <= n

static inline ulong bit_rotate_right(ulong x, ulong r, ulong n);
// Return n-bit word rotated r bits to the right
// (i.e. toward the least significant bit)
// Must have  0 <= r <= n

static inline ulong bit_rotate_sgn(ulong x, long r, ulong n);
// Positive r --> shift away from element zero

// ========== HEADER FILE src/bits/bitseparate.h: ==========

static inline ulong bit_separate(ulong w, ulong m, ulong s=BITS_PER_LONG/2);
// Return  word with bits of w separated as indicated by m.
// Bits at positions where m is 0/1 are moved to the low/high end.
// Examples:
//  w = ABCDEFGH
//  m = 00111100
//  ==> CDEFABGH
//
//  w = ABCDEFGH
//  m = 01010101
//  ==> BDFHACEG
//
// s must contain the number of ones set in m.
// For the default s exactly half of the bits must be set in m

static inline ulong bit_separate_subwords(ulong w, ulong m, ulong s);
// Bit-separate all length-s subwords of w.
// s>=2 must divide BITS_PER_LONG.
// The mask must be such that exactly half of the bits
//   in each subword are set.
// Examples:
//  s = 4
//  w = ABCD EFGH
//  m = 0011 1010
//  ==> CDAB EGFH
//
//  s = 2
//  w = AB CD EF GH
//  m = 01 10 01 10
//  ==> BA CD FE GH
//
// For s==BITS_PER_LONG the result is as with bit_separate(w, m)
//  if half of the bits of m are set.

// ========== HEADER FILE src/bits/bitsequency.h: ==========

static inline ulong bit_sequency(ulong x);
// Return the number of zero-one (or one-zero)
//  transitions (sequency) of x.

static inline ulong first_sequency(ulong k);
// Return the first (i.e. smallest) word with sequency k,
// e.g.  00..00010101010 (seq 8)
// e.g.  00..00101010101 (seq 9)
// Must have:  0 <= k <= BITS_PER_LONG

static inline ulong last_sequency(ulong k, ulong n=BITS_PER_LONG);
// Return the last (i.e. biggest) word with sequency k.

static inline ulong next_sequency(ulong x);
// Return next word with the same number
// of zero-one transitions (sequency) as x.
// The value of the lowest bit is conserved.
//
// Zero is returned when there is no further sequence.
//
// e.g.:
//  ...1.1.1 ->
//  ..11.1.1 ->
//  ..1..1.1 ->
//  ..1.11.1 ->
//  ..1.1..1 ->
//  ..1.1.11 ->
//  .111.1.1 ->
//  .11..1.1 ->
//  .11.11.1 ->
//  .11.1..1 ->
//  .11.1.11 -> ...
//

static inline ulong prev_sequency(ulong x);

static inline ulong complement_sequency(ulong x);
// Return word whose sequency is BITS_PER_LONG - s
// where s is the sequency of x

// ========== HEADER FILE src/bits/bitset2set.h: ==========

static inline ulong bitset2set(ulong b, ulong *f, ulong n, ulong off=0);
// b=[1...1.11] ==> f[] = [0,1,3,7], return=4
// with off=1   ==> f[] = [1,2,4,8], return=4

static inline ulong set2bitset(const ulong *f, ulong n, ulong off=0);
// f[] = [0,1,3,7] ==>  return=[1...1.11]
// Same return with f[] = [1,2,4,8] and off=1

static inline ulong bitset2deltaset(ulong b, ulong *f, ulong n);
// b=[1...1.11] ==> f[] = [1,1,0,1,0,0,0,1], return=4

static inline ulong deltaset2bitset(const ulong *f, ulong n);
// f[] = [1,1,0,1,0,0,0,1] ==> return=[1...1.11]

// ========== HEADER FILE src/bits/bitsperlong.h: ==========

// ========== HEADER FILE src/bits/bitsubset-gray.h: ==========

class bit_subset_gray;
// Generate all all subsets of bits of  a given word,
// in Gray code (minimal-change) order.
//
// E.g., for the word ('.' printed for unset bits)
//   ...11.1.
// these words are produced by subsequent next()-calls:
//   ......1.
//   ....1.1.
//   ....1...
//   ...11...
//   ...11.1.
//   ...1..1.
//   ...1....
//   ........

// ========== HEADER FILE src/bits/bitsubset.h: ==========

class bit_subset;
// Generate all all subsets of bits of  a given word.
//
// E.g., for the word ('.' printed for unset bits)
//   ...11.1.
// these words are produced by subsequent next()-calls:
//   ......1.
//   ....1...
//   ....1.1.
//   ...1....
//   ...1..1.
//   ...11...
//   ...11.1.
//   ........
//

// ========== HEADER FILE src/bits/bitsubsetq.h: ==========

static inline bool is_subset(ulong u, ulong e);
// Return whether the set bits of u are a subset of the set bits of e.
// That is, as bitsets, test whether u is a subset of e.

static inline bool is_proper_subset(ulong u, ulong e);
// Return whether u (as bitset) is a proper subset of e.

// ========== HEADER FILE src/bits/bitswap.h: ==========

static inline ulong bit_swap_01(ulong a, ulong k1, ulong k2);
// Return a with bits at positions [k1] and [k2] swapped.
// Bits must have different values (!)
// (i.e. one is zero, the other one)
// k1==k2 is allowed (a is unchanged then)

static inline ulong bit_swap(ulong a, ulong k1, ulong k2);
// Return a with bits at positions [k1] and [k2] swapped.
// k1==k2 is allowed (a is unchanged then)

static inline ulong bit_swap_1(ulong x);
// Return x with neighbor bits swapped.

static inline ulong bit_swap_2(ulong x);
// Return x with groups of 2 bits swapped.

static inline ulong bit_swap_4(ulong x);
// Return x with groups of 4 bits swapped.

static inline ulong bit_swap_8(ulong x);
// Return x with groups of 8 bits swapped.

static inline ulong bit_swap_16(ulong x);
// Return x with groups of 16 bits swapped.
//
// For 32 bit words this is identical to either of
//   rotate_left(x,16) or rotate_right(x,16)

static inline ulong bit_swap_32(ulong x);
// Return x with groups of 32 bits swapped.

// ========== HEADER FILE src/bits/bittest.h: ==========

static inline ulong test_bit(ulong a, ulong i);
// Return zero if bit[i] is zero,
//  else return one-bit word with bit[i] set.

static inline bool test_bit01(ulong a, ulong i);
// Return whether bit[i] is set.

static inline ulong set_bit(ulong a, ulong i);
// Return a with bit[i] set.

static inline ulong clear_bit(ulong a, ulong i);
// Return a with bit[i] cleared.

static inline ulong change_bit(ulong a, ulong i);
// Return a with bit[i] changed.

// ========== HEADER FILE src/bits/bittransforms.h: ==========

static inline ulong blue_code(ulong a);
// Self-inverse.
// range [0..2^k-1] is mapped onto itself
// work \prop log_2(BITS_PER_LONG)
//.
// With  B:=blue,  G:=gray,  I:=inverse_gray
//   G(B(G(B(a)))) == a
//   G(B(G(a)))    == B(a)
//   I(B(I(a)))    == B(a)
//   G(B(a))       == B(I(a))
// With  P:=parity
//   P(B(a)) == a % 2

static inline ulong yellow_code(ulong a);
// Self-inverse.
// work \prop log_2(BITS_PER_LONG)
//.
// With  Y:=yellow,  B:=blue,  r:=revbin
//   B(a) == Y(r(Y(a)))
//   Y(a) == r(B(r(a)))
//   r(a) == Y(B(Y(a))) == B(Y(B(a)))
// With  P:=parity
//   P(Y(a)) == highest_one(a) == "sign(a)"

static inline ulong red_code(ulong a);
// work \prop log_2(BITS_PER_LONG)
// =^=  revbin( blue_code(a) );

static inline ulong green_code(ulong a);
// work \prop log_2(BITS_PER_LONG)
// =^=  revbin( yellow_code(a) );

// ========== HEADER FILE src/bits/bitxtransforms.h: ==========

static inline ulong blue_xcode(ulong a, ulong x);

static inline ulong yellow_xcode(ulong a, ulong x);

static inline ulong red_xcode(ulong a, ulong x);

static inline ulong green_xcode(ulong a, ulong x);

// ========== HEADER FILE src/bits/bitzip-pairs.h: ==========

static inline ulong bit_zip0_pairs(ulong x);
// Return word with lower half bits(-pairs) in even (pair-)indices,
// i.e., indices 4k+0, 4k+1.
// Upper half must be zero.
// 0000abcd --> 0a0b0c0d (a,b,c,d are pairs of bits).

static inline ulong bit_unzip0_pairs(ulong x);
// Gather pairs bits in even positions (4k+0, 4k+1) into lower half.
// Inverse of bit_zip0_pairs().
// Bit pairs at odd (pair-)positions (4k+2, 4k+3) must be zero.
// 0a0b0c0d --> 0000abcd (a,b,c,d are pairs of bits).

// ========== HEADER FILE src/bits/bitzip.h: ==========

static inline ulong bit_zip(ulong x);
// Return word with lower half bits in even indices
//  and upper half bits in odd indices.

static inline ulong bit_unzip(ulong x);
// Return word with even indexed bits in lower half
//  and odd indexed bits in upper half.
// Inverse of bit_zip()

static inline ulong bit_zip0(ulong x);
// Return word with lower half bits in even indices.
// upper half must be zero.
// Same effect as bit_zip() but faster.
// 0000abcd --> 0a0b0c0d (a,b,c,d are bits).

static inline ulong bit_unzip0(ulong x);
// Gather bits in even positions into lower half.
// Inverse of bit_zip0().
// Bits at odd positions must be zero.
// 0a0b0c0d --> 0000abcd (a,b,c,d are bits).

static inline void bit_zip2(ulong x, ulong &lo, ulong &hi);
// Bits of lower half word spread out into even positions of lo,
// bits of upper half word spread out into even positions of hi.

static inline ulong bit_unzip2(ulong lo, ulong hi);
// Inverse of bit_zip2(x, lo, hi).

static inline ulong bit_zip2(ulong x, ulong y);
// 2-word version:
// only the lower half of x and y are merged

static inline void bit_unzip2(ulong t, ulong &x, ulong &y);
// 2-word version:
// only the lower half of x and y are filled

// ========== HEADER FILE src/bits/blue-fixed-points.h: ==========

static inline ulong blue_fixed_point(ulong s);
// Return unique fixed point of blue_code() for each argument.
//.
// The first few fixed points (bfp) are:
//  arg = arg_2 :     bfp_2     =      bfp
//   0 = ...... :    .......... =        0
//   1 = .....1 :    .........1 =        1
//   2 = ....1. :    .......11. =        6
//   3 = ....11 :    .......111 =        7
//   4 = ...1.. :    .....1.1.. =       20
//   5 = ...1.1 :    .....1..1. =       18
//   6 = ...11. :    .....1.1.1 =       21
//   7 = ...111 :    .....1..11 =       19
//   8 = ..1... :    ...1111... =      120
//   9 = ..1..1 :    ...11.11.. =      108
//  10 = ..1.1. :    ...111111. =      126
//  11 = ..1.11 :    ...11.1.1. =      106
//  12 = ..11.. :    ...1111..1 =      121
//  13 = ..11.1 :    ...11.11.1 =      109
//  14 = ..111. :    ...1111111 =      127
//  15 = ..1111 :    ...11.1.11 =      107
//  16 = .1.... :    .1...1.... =      272
//  17 = .1...1 :    .1.11.1... =      360

static inline ulong blue_fixed_point_idx(ulong f);
// Inverse of blue_fixed_point()

// ========== HEADER FILE src/bits/branchless.h: ==========


// Some functions that avoid if-statements.
// The upos_*() functions only work for a limited range
// in order to have the highest bit emulate the carry flag.
// The trick is to use signed shift right to get a mask
// which == 0xfff... if a carry ocurred or zero else.
//
// With CPUs that have conditional-move instructions
// the compiler should produce branch-free machine code.
// GCC does that (in many situations; check to be sure).

static inline ulong upos_max(ulong a, ulong b);
// Return maximum(a, b)
// Both a and b must not have the most significant bit set

static inline ulong upos_min(ulong a, ulong b);
// Return minimum(a, b)
// Both a and b must not have the most significant bit set

static inline ulong upos_abs_diff(ulong a, ulong b);
// Return abs(a-b)
// Both a and b must not have the most significant bit set

static inline void upos_sort2(ulong &a, ulong &b);
// Set {a, b} := {min(a, b), max(a,b)}
// Both a and b must not have the most significant bit set

static inline ulong upos_add_sat(ulong a, ulong b);
// Return a + b
// If result overflows, return 0xfff...
// Both a and b must not have the most significant bit set

static inline ulong upos_sub_sat(ulong a, ulong b);
// Return a - b
// if result underflows, return zero
// Both a and b must not have the most significant bit set

static inline ushort add_sat16(ushort a, ushort b);
// Add two 16-bit numbers
// if result overflows, return 0xffff.
// Must have: sizeof(int)==4

static inline ushort sub_sat16(ushort a, ushort b);
// Add two 16-bit numbers
// if result underflows, return zero.
// Must have: sizeof(int)==4

static inline long max0(long x);
// Return max(0, x), i.e. return zero for negative input
// No restriction on input range

static inline long min0(long x);
// Return min(0, x), i.e. return zero for positive input
// No restriction on input range

static inline long clip_range0(long x, long m);
// Code equivalent (for m>0) to:
//   if ( x<0 )  x = 0;
//   else if ( x>m )  x = m;
//   return  x;

static inline long clip_range(long x, long mi, long ma);
// Code equivalent to (for mi<=ma):
//   if ( x<mi )  x = mi;
//   else if ( x>ma )  x = ma;

// ========== HEADER FILE src/bits/colormix-fl.h: ==========


// Various operations on 32-bit integers where 3 color channels
// of 8 bit are expected to be in the least significant 24 bits.
// The order (e.g. 0x00RRGGBB vs. 0x00BBGGRR) does not matter.
// This file contains the scaling operations with floating point
// numbers.  Note that the conversion float<-->int is expensive.

static inline uint fl_color01(uint c, double v);
//  0.0   ...  +1.0
//   0          c2

static inline uint fl_ccolor11(uint c, double v);
//      -1.0       ... 0.0 ... +1.0
//  complement(c)       0        c

static inline uint fl_color_mix_11(uint c1, uint c2, double v);
// -1.0  ...   0.0      ... +1.0
//   c1     (c1+c2)/2        c2

static inline uint fl_color_mix_1b1(uint c1, uint c2, double v);
// -1.0  ...   0.0   ...  +1.0
//   c1         0          c2

// ========== HEADER FILE src/bits/colormix.h: ==========


// operations on 32-bit integers where 3 color channels
// of 8 bit are expected to be in the least significant 24 bits.
// The order (e.g. 0x00RRGGBB vs. 0x00BBGGRR) does not matter.

static inline uint color01(uint c, ulong v);
// Return color with each channel scaled by v
// 0 <= v <= (1<<16)  corresponding to 0.0 ... 1.0

static inline uint color_mix(uint c1, uint c2, ulong v);
// Return channel-wise average of colors:  (1.0-v)*c1 + v*c2
// 0 <= v <= (1<<16)  corresponding to 0.0 ... 1.0
// c1   ...    c2

static inline uint color_mix_50(uint c1, uint c2);
// Return channel-wise average of colors c1 and c2
// Shortcut for the special case (50% transparency)
//   of color_mix(c1, c2, "0.5")
// The least significant bits are ignored
// (in all functions color_mix_NN() )

static inline uint color_mix_25(uint c1, uint c2);

static inline uint color_mix_75(uint c1, uint c2);

static inline uint color_mix_25(uint c1, uint c2, uint c3);

static inline uint color_mix_50(uint c1, uint c2, uint c3);

static inline uint color_mix_75(uint c1, uint c2, uint c3);

static inline uint color_sum_adjust(uint s);
// Set color channel to max (0xff) iff an overflow occurred
//  (that is, leftmost bit in channel is set)

static inline uint color_sum(uint c1, uint c2);
// Return channel-wise saturated sum of colors c1 and c2.
// The least significant bits are ignored.

static inline uint color_sum(uint c0, uint c1, uint c2);
// Return channel-wise saturated sum of colors c0, c1 and c2.
// The least significant bits are ignored.

static inline uint color_mult(uint c1, uint c2);
// Return channel-wise product of colors c1 and c2.
// Corresponding to an object of color c1
//  illuminated by a light of color c2

// ========== HEADER FILE src/bits/colormixp.h: ==========


// Versions of some functionss from colormix.h that
// do not ignore the least significant bit
// (and are more expensive).
//
// The non-'perfect' versions should be ok for
//   the majority of applications.

static inline uint perfect_color_mix_50(uint c1, uint c2);
// Return channel-wise average of colors c1 and c2

static inline uint perfect_color_mix_75(uint c1, uint c2);

static inline uint perfect_color_sum(uint c1, uint c2);

static inline uint perfect_color_sum(uint c0, uint c1, uint c2);

// ========== HEADER FILE src/bits/crc32.h: ==========

class crc32;
// 32-bit CRC (cyclic redundancy check)

// ========== HEADER FILE src/bits/crc64.h: ==========

class crc64;
// 64-bit CRC (cyclic redundancy check)

// ========== HEADER FILE src/bits/cswap.h: ==========

static inline void cswap_lt(ulong &a, ulong &b);
// Branchless equivalent to:
// if ( a<b )   { ulong t=a; a=b; b=t; }  // swap if a < b
// NOTE: Whether this routine is faster depends both
// on the machine used and the surrounding code.

static inline void cswap_gt(ulong &a, ulong &b);
// Branchless equivalent to:
// if ( a>b )   { ulong t=a; a=b; b=t; }  // swap if a > b

// ========== HEADER FILE src/bits/evenodd.h: ==========

// ========== HEADER FILE src/bits/fibrep-subset-lexrev.h: ==========

static inline ulong next_subset_lexrev_fib(ulong x);
// Return next Fibonacci word in subset-lex order.
// Start with a one-bit word at position n-1 to
// generate all Fibonacci words of length n
// E.g., for n==6 the words (and subsets) are
//       word     subset of {0,1,2,3,4,5}
//   1:  1.....  =  { 0 }
//   2:  1.1...  =  { 0, 2 }
//   3:  1.1.1.  =  { 0, 2, 4 }
//   4:  1.1..1  =  { 0, 2, 5 }
//   5:  1..1..  =  { 0, 3 }
//   6:  1..1.1  =  { 0, 3, 5 }
//   7:  1...1.  =  { 0, 4 }
//   8:  1....1  =  { 0, 5 }
//   9:  .1....  =  { 1 }
//  10:  .1.1..  =  { 1, 3 }
//  11:  .1.1.1  =  { 1, 3, 5 }
//  12:  .1..1.  =  { 1, 4 }
//  13:  .1...1  =  { 1, 5 }
//  14:  ..1...  =  { 2 }
//  15:  ..1.1.  =  { 2, 4 }
//  16:  ..1..1  =  { 2, 5 }
//  17:  ...1..  =  { 3 }
//  18:  ...1.1  =  { 3, 5 }
//  19:  ....1.  =  { 4 }
//  20:  .....1  =  { 5 }
//  21:  ......  =  { }
//
// Note (1): the first element of the subset corresponds
// to the highest set bit.
// Note (2): the lex order for the delta sets would simply
// be the counting order.

static inline ulong prev_subset_lexrev_fib(ulong x);
// Return previous Fibonacci word in subset-lex order.
// Start with zero to generate all words of length n.
// E.g., for n==6:
//       word     subset of {0,1,2,3,4,5}
//   0:  ......  =   { }
//   1:  .....1  =   { 5 }
//   2:  ....1.  =   { 4 }
//   3:  ...1.1  =   { 3, 5 }
//   4:  ...1..  =   { 3 }
//   5:  ..1..1  =   { 2, 5 }
//   6:  ..1.1.  =   { 2, 4 }
//   7:  ..1...  =   { 2 }
//   8:  .1...1  =   { 1, 5 }
//   9:  .1..1.  =   { 1, 4 }
//  10:  .1.1.1  =   { 1, 3, 5 }
//  11:  .1.1..  =   { 1, 3 }
//  12:  .1....  =   { 1 }
//  13:  1....1  =   { 0, 5 }
//  14:  1...1.  =   { 0, 4 }
//  15:  1..1.1  =   { 0, 3, 5 }
//  16:  1..1..  =   { 0, 3 }
//  17:  1.1..1  =   { 0, 2, 5 }
//  18:  1.1.1.  =   { 0, 2, 4 }
//  19:  1.1...  =   { 0, 2 }
//  20:  1.....  =   { 0 }
//

// ========== HEADER FILE src/bits/fibrep.h: ==========

static inline ulong bin2fibrep(ulong b);
// Return Fibonacci representation of b
// Limitation: the first Fibonacci number greater
//  than b must be representable as ulong.
// 32 bit:  b < 2971215073=F(47) [F(48)=4807526976 > 2^32]
// 64 bit:  b < 12200160415121876738=F(93) [F(94) > 2^64]

static inline ulong fibrep2bin(ulong f);
// Return binary representation of f
// Inverse of bin2fibrep().

static inline ulong next_fibrep(ulong x);
// With x the Fibonacci representation of n
// return Fibonacci representation of n+1.

static inline ulong prev_fibrep(ulong x);
// With x the Fibonacci representation of n
// return Fibonacci representation of n-1.

static inline bool is_fibrep(ulong f);
// Return whether f is a valid Fibonacci representation,
// that is, whether it does not contain two adjacent ones.

// ========== HEADER FILE src/bits/graycode.h: ==========

static inline ulong gray_code(ulong x);
// Return the Gray code of x
// ('bit-wise derivative modulo 2')

static inline ulong inverse_gray_code(ulong x);
// inverse of gray_code()
// note: the returned value contains at each bit position
// the parity of all bits of the input left from it (incl. itself)
//

static inline ulong byte_gray_code(ulong x);
// Return the Gray code of bytes in parallel

static inline ulong byte_inverse_gray_code(ulong x);
// Return the inverse Gray code of bytes in parallel

// ========== HEADER FILE src/bits/graypower.h: ==========

static inline ulong gray_pow(ulong x, ulong e);
// Return (gray_code**e)(x)
// gray_pow(x, 1) == gray_code(x)
// gray_pow(x, BITS_PER_LONG-1) == inverse_gray_code(x)

static inline ulong inverse_gray_pow(ulong x, ulong e);
// Return (inverse_gray_code**(e))(x)
//   == (gray_code**(-e))(x)
// inverse_gray_pow(x, 1) == inverse_gray_code(x)
// inverse_gray_pow(x, BITS_PER_LONG-1) == gray_code(x)

static inline ulong rev_gray_pow(ulong x, ulong e);
// Return (rev_gray_code**e)(x)
// rev_gray_pow(x, 1) == rev_gray_code(x)
// rev_gray_pow(x, BITS_PER_LONG-1) == inverse_rev_gray_code(x)

static inline ulong inverse_rev_gray_pow(ulong x, ulong e);
// Return (inverse_rev_gray_code**(e))(x)
//   == (rev_gray_code**(-e))(x)
// inverse_rev_gray_pow(x, 1) == inverse_rev_gray_code(x)
// inverse_rev_gray_pow(x, BITS_PER_LONG-1) == rev_gray_code(x)

// ========== HEADER FILE src/bits/grsnegative.h: ==========

static inline ulong grs_negative_q(ulong x);
// Return whether the Golay-Rudin-Shapiro sequence
// (OEIS sequence A020985) is negative for index x
// Return 1 for x =
//  3,6,11,12,13,15,19,22,24,25,26,30,35,38,43,44,45,47,48,49,
//  50,52,53,55,59,60,61,63,67,70,75,76,77,79,83,86,88,89,90,94,
//  96,97,98,100,101,103,104,105,106,110,115,118,120,121,122,
//  126,131,134,139,140, ...
//
// Algorithm: count bit pairs modulo 2
//

static inline ulong grs_next(ulong k, ulong g);
// With g == grs_negative_q(k), compute grs_negative_q(k+1).

// ========== HEADER FILE src/bits/hilbert.h: ==========

// ----- SRCFILE=src/bits/lin2hilbert.cc: -----
void lin2hilbert(ulong t, ulong &x, ulong &y);
// Transform linear coordinate t to Hilbert x and y

ulong hilbert2lin(ulong x, ulong y);
// Transform Hilbert x and y to linear coordinate t

ulong hilbert_gray_code(ulong t);
// A Gray code based on the function lin2hilbert().
// Equivalent to the following sequence of statements:
//  { lin2hilbert(t, x, y);
//    x=gray_code(x);  y=gray_code(y);
//    return bitzip2(x, y); }

ulong inverse_hilbert_gray_code(ulong g);
// Inverse of hilbert_gray_code()

static inline ulong hilbert_p(ulong t);
// Let (dx,dy) be the (horizontal,vertical) move
//   with step t of the Hilbert curve.
// Return  zero if (dx+dy)==-1, else one (then: (dx+dy)==+1).
// Used in hilbert_dir()
// Algorithm: count number of threes in radix 4 expansion of t.

static inline ulong hilbert_m(ulong t);
// Let (dx,dy) be the (horizontal,vertical) move
//   with step t of the Hilbert curve.
// Return  zero if (dx-dy)==-1, else one (then: (dx-dy)==+1).
// Used in hilbert_dir()

static inline ulong hilbert_dir(ulong t);
// Return encoding of the the next move with the Hilbert curve.
//
// d \in {0,1,2,3} as follows:
//   d : direction
//   0 : right (+x:  dx=+1, dy= 0)
//   1 : down  (-y:  dx= 0, dy=-1)
//   2 : up    (+y:  dx= 0, dy=+1)
//   3 : left  (-x:  dx=-1, dy= 0)
//
// To print symbollically:  cout << (">v^<")[ d ];
//
// dx+dy: ++-+++-+++----++++-+++-+++----++++-+++-+++----+---+---+---++++-
// dx-dy: +----+++-+++-+++-++++---+---+----++++---+---+----++++---+---+--
//   dir: >^<^^>v>^>vv<v>>^>v>>^<^>^<<v<^^^>v>>^<^>^<<v<^<<v>vv<^<v<^^>^<

static inline int hilbert_turn(ulong t);
// Return the turn (left or right) with the steps
//   t and t-1 of the Hilbert curve.
// Returned value is
//   0 for no turn
//  +1 for right turn
//  -1 for left turn
//
// To print symbollically:  cout << ("-0+")[ u ];
//   dir: >^<^^>v>^>vv<v>>^>v>>^<^>^<<v<^^^>v>>^<^>^<<v<^<<v>vv<^<v<^^>^<
//  turn: 0--+0++--++0+--0-++-0--++--0-++00++-0--++--0-++-0--+0++--++0+--

// ========== HEADER FILE src/bits/ith-one-idx.h: ==========

static inline ulong ith_one_idx(ulong x, ulong i);
// Return index of the i-th set bit of x
// where 0 <= i < bit_count(x).

// ========== HEADER FILE src/bits/kolakoski-seq.h: ==========

class kolakoski_seq;
// Generator for the Oldenburger-Kolakoski sequence.
// See OEIS sequence A000002.
// Cf. https://en.wikipedia.org/wiki/Kolakoski_sequence
// Algorithm by David Eppstein, see
//   https://11011110.github.io/blog/2016/10/14/kolakoski-sequence-via.html

// ========== HEADER FILE src/bits/negbin.h: ==========

// Conversion to and from radix(-2).
static inline ulong bin2neg(ulong x);
// binary --> radix(-2)
//.
// HAKMEM, item 128

static inline ulong neg2bin(ulong x);
// radix(-2) --> binary
// inverse of bin2neg()

static inline ulong negbin_fixed_point(ulong k);
// Return the k-th fixed point of bin2neg()
// With k=000000ABCDEF (binary) the returned value is 0A0B0C0D0E0F.

static inline ulong next_negbin(ulong x);
// With x the radix(-2) representation of n
// return radix(-2) representation of n+1.

static inline ulong prev_negbin(ulong x);
// With x the radix(-2) representation of n
// return radix(-2) representation of n-1.

static inline ulong negbin_add(ulong a, ulong b);
// Addition of two numbers in radix(-2) representation.

// ========== HEADER FILE src/bits/nextgray.h: ==========

static inline ulong next_gray2(ulong x);
// With input x==gray_code(2*k) the return is gray_code(2*k+2).
// Let x1 be the word x shifted right once
// and i1 its inverse Gray code.
// Let r1 be the return r shifted right once.
// Then r1 = gray_code(i1+1).
// That is, we have a Gray code counter.
// The argument must have an even number of bits.
//
//   k:     g(k)      g(2*k)     g(k) p
//   0:   .......    .......   ...... .   ...... .
//   1:   ......1    .....11   .....1 1   .....+ 1
//   2:   .....11    ....11.   ....11 .   ....+1 .
//   3:   .....1.    ....1.1   ....1. 1   ....1- 1
//   4:   ....11.    ...11..   ...11. .   ...+1. .
//   5:   ....111    ...1111   ...111 1   ...11+ 1
//   6:   ....1.1    ...1.1.   ...1.1 .   ...1-1 .
//   7:   ....1..    ...1..1   ...1.. 1   ...1.- 1
//   8:   ...11..    ..11...   ..11.. .   ..+1.. .
//   9:   ...11.1    ..11.11   ..11.1 1   ..11.+ 1
//  10:   ...1111    ..1111.   ..1111 .   ..11+1 .
//  11:   ...111.    ..111.1   ..111. 1   ..111- 1
//  12:   ...1.1.    ..1.1..   ..1.1. .   ..1-1. .
//  13:   ...1.11    ..1.111   ..1.11 1   ..1.1+ 1
//  14:   ...1..1    ..1..1.   ..1..1 .   ..1.-1 .
//  15:   ...1...    ..1...1   ..1... 1   ..1..- 1
//  16:   ..11...    .11....   .11... .   .+1... .
//  17:   ..11..1    .11..11   .11..1 1   .11..+ 1
//
// Note that the changes with increment always
// happen one position left of the rightmost bit.
//
// Convert an arbitrary (Gray code) word g to
//   x = (g<<1) ^ parity(g)
// in order to use this routine.

// ========== HEADER FILE src/bits/parenwords.h: ==========

static inline bool is_parenword(ulong x);
// Return whether x is a valid paren word.
//
// Binary words < 16, those that are valid
//  'paren words' are marked with 'P':
//  .... P   [empty string]
//  ...1 P   ()
//  ..1.
//  ..11 P   (())
//  .1..
//  .1.1 P   ()()
//  .11.
//  .111 P   ((()))
//  1...
//  1..1
//  1.1.
//  1.11 P   (()())
//  11..
//  11.1 P   ()(())
//  111.
//  1111 P   (((())))

static inline void parenword2str(ulong x, char *str);
// Fill paren string corresponding to x into str.
//
// Binary words < 32 that are valid
//  'paren words' together with paren-string:
//  .....   [empty string]
//  ....1   ()
//  ...11   (())
//  ..1.1   ()()
//  ..111   ((()))
//  .1.11   (()())
//  .11.1   ()(())
//  .1111   (((())))
//  1..11   (())()
//  1.1.1   ()()()
//  1.111   ((()()))
//  11.11   (()(()))
//  111.1   ()((()))
//  11111   ((((()))))
// Note 1: lower bits in word (right end) correspond
//         to the begin of string (left end).
// Note 2: Word is extended with zeros (to the left) when necessary,
//         so the length of str must be >= 1 + 2*(number of set bits)

static inline ulong first_parenword(ulong n);
// Return least binary word corresponding to n pairs of parens.
// Example, n=5:  .....11111   ((((()))))

static inline ulong last_parenword(ulong n);
// Return biggest binary word corresponding to n pairs of parens.
// Must have: 1 <= n <= BITS_PER_LONG/2.
// Example, n=5:  .1.1.1.1.1   ()()()()()

static inline ulong next_parenword(ulong x);
// Next (colex order) binary word that is a paren word.
// With n=4 and starting with first_parenword(n)
//  one gets the following sequence:
//  .....1111   (((())))
//  ....1.111   ((()()))
//  ....11.11   (()(()))
//  ....111.1   ()((()))
//  ...1..111   ((())())
//  ...1.1.11   (()()())
//  ...1.11.1   ()(()())
//  ...11..11   (())(())
//  ...11.1.1   ()()(())
//  ..1...111   ((()))()
//  ..1..1.11   (()())()
//  ..1..11.1   ()(())()
//  ..1.1..11   (())()()
//  ..1.1.1.1   ()()()()
//  .........    [zero]

// ========== HEADER FILE src/bits/parity.h: ==========

static inline ulong byte_parity(ulong x);
// Return the parities of bytes in parallel

static inline ulong parity(ulong x);
// Return 0 if the number of set bits is even, else 1

// ========== HEADER FILE src/bits/pcrc64.h: ==========

class pcrc64;
// Parallel computation of 64-bit CRCs for each bit of the input words.
// Primitive polynomial used is x^64 + x^4 + x^3 + x^2 + 1

// ========== HEADER FILE src/bits/print-bin.h: ==========

// ----- SRCFILE=src/bits/print-bin.cc: -----
void print_bin(const char *bla, unsigned long long x, ulong pd/*=0*/, const char *c01/*=nullptr*/);
// Print x to radix-2.
// pd: how many bits to print.

void print_bin_vec(const char *bla, unsigned long long x, ulong pd/*=0*/, const char *c01/*=nullptr*/);
// Print x to radix-2, least significant bits first (as Vector).
// pd: how many bits to print.

void print_idx_seq(const char *bla, unsigned long long x, ulong off/*=0*/);
// Print x as index sequence:
//   x=....1..11  ==>  [0, 1, 4]
// With offsets off!=0 start index with off instead of zero.

// ----- SRCFILE=src/bits/print-bindiff.cc: -----
void print_bin_diff(const char*bla,
               unsigned long long x1,
               unsigned long long x2,
               ulong pd/*=0*/,
               const char *c01pm/*=".1+-"*/);
// Binary print difference between two values x1 and x2:
// Bits that agree between x1 and x2 are printed as
//    c01pm[0]=='.' for zero and c01pm[1]='1' for one
// Bits that differ between x1 and x2 are printed as
//    c01pm[2]=='+' if set in x2 (i.e. bit was added)
//    c01pm[3]=='-' if set in x1 (i.e. bit was removed)
// Example:
//         x1==1..11.111
//         x2==1.111...1
//         eq==11.111..1
//   output = "1.+11.--1"

void print_bin_vec_diff(const char*bla,
                   unsigned long long x1,
                   unsigned long long x2,
                   ulong pd/*=0*/,
                   const char *c01pm/*=".1+-"*/);

// ----- SRCFILE=src/bits/print-bitset.cc: -----
void print_bit_set(const char *bla, ulong x, ulong rq/*=0*/);
// Print delta set given as word x as set.
// Example:       x=[0,0,1,0,1,1] ==> { 0, 1, 3 }
// if rq!=0 then print set for rq-bit reversed binary word:
// Example: rq=6, x=[0,0,1,0,1,1] ==> { 3, 1, 0 }

// ========== HEADER FILE src/bits/radix-2i.h: ==========

// Conversion to and from radix(2*i), "quater-imaginary base".
// The radix(2*i) representation for complex integers needs
// one digit (i.e., two bits) after the point.
static inline ulong bin_real_to_rad2i(ulong x);
// binary --> radix(2*i)

static inline ulong bin_imag_to_rad2i(ulong x);
// binary * i --> radix(2*i)

static inline ulong bin_to_rad2i(ulong re, ulong im);
// binary ( re + i * im ) --> radix(2*i)

static inline ulong rad2i_to_bin_real(ulong x);
// radix(2*i) (must be purely real) --> binary

static inline ulong rad2i_to_bin_imag(ulong x);
// radix(2*) (must be purely imaginary) --> binary

static inline void rad2i_to_bin(ulong z, ulong &re, ulong &im);
// radix(2*i) --> binary ( re + i * im )

static inline bool rad2i_is_real(ulong z);
// Return whether z is purely real.

static inline bool rad2i_is_imag(ulong z);
// Return whether z is purely imaginary.

// ========== HEADER FILE src/bits/radix-m1pi.h: ==========

// Conversion to and from (complex) radix(-1+i).
static inline ulong bin_real_to_radm1pi(ulong x);
// binary --> radix(-1+i)

static inline ulong bin_imag_to_radm1pi(ulong x);
// binary * i --> radix(-1+i)

static inline ulong add_radm1pi(ulong x, ulong y);
// Add radix(-1+i) representations x and y.
// Note that x + (x<<1) == x + (-1+i)*x == i*x

static inline ulong bin_to_radm1pi(ulong re, ulong im);
// binary ( re + i * im ) --> radix(-1+I)

static inline ulong radm1pi_to_bin_real(ulong x);
// convert radix(-1+i) number (which must be purely real) to binary.

static inline ulong radm1pi_to_bin_imag(ulong x);
// convert radix(-1+i) number (which must be purely imaginary) to binary.

static inline void radm1pi_to_bin(ulong z, ulong &re, ulong &im);
// radix(-1+i) --> binary ( re + i * im )

static inline bool radm1pi_is_real(ulong z);
// Return whether z is purely real.

static inline bool radm1pi_is_imag(ulong z);
// Return whether z is purely imaginary.

// ========== HEADER FILE src/bits/radix-m4.h: ==========

// Conversion to and from radix(-4).
static inline ulong bin_to_radm4(ulong x);
// binary --> radix(-4)

static inline ulong radm4_to_bin(ulong x);
// radix(-4) --> binary
// inverse of bin_to_radm4()

static inline ulong next_radm4(ulong x);
// With x the radix(-4) representation of n
// return radix(-4) representation of n+1.

static inline ulong prev_radm4(ulong x);
// With x the radix(-4) representation of n
// return radix(-4) representation of n-1.

static inline ulong radm4_add(ulong a, ulong b);
// Addition of two numbers in radix(-4) representation.

// ========== HEADER FILE src/bits/revbin-upd.h: ==========

static inline ulong revbin_upd(ulong r, ulong h);
// Let n=2**ldn and h=n/2.
// Then, with r == revbin(x, ldn) at entry, return revbin(x+1, ldn)
// NOTE: routine will hang if called with r the all-ones word

static inline void make_revbin_upd_tab(ulong h);
// Initialize lookup table used by revbin_tupd()

static inline ulong revbin_tupd(ulong r, ulong k);
// Let r==revbin(k, ldn) then
// return revbin(k+1, ldn).
// NOTE 1: need to call make_revbin_upd_tab(ldn) before usage
//         where ldn=log_2(n)
// NOTE 2: different argument structure than revbin_upd()

// ========== HEADER FILE src/bits/revbin.h: ==========

static inline ulong bswap(ulong x);
// Return word with reversed byte order.

static inline ulong revbin(ulong x);
// Return x with reversed bit order.

static inline ulong revbin_t(ulong x);
// Return x with reversed bit order.
// Table based version.

static inline ulong revbin_t_le32(ulong x);
// Return x with reversed bit order.
// Table based version for lengths less or equal 32 bits.

static inline ulong revbin_t_le16(ulong x);
// Return x with reversed bit order.
// Table based version for lengths less or equal 16 bits.

static inline ulong xrevbin(ulong a, ulong x);
// Symbolic power of revbin.

static inline ulong revbin(ulong x, ulong ldn);
// Return word with the ldn least significant bits
//   (i.e. bit_0 ... bit_{ldn-1})  of x reversed,
//   the other bits are set to zero.

// ========== HEADER FILE src/bits/revgraycode.h: ==========

static inline ulong rev_gray_code(ulong x);
// Return the reversed Gray code of x.
// ('bit-wise derivative modulo 2 towards high bits').
// Also: multiplication by x+1 as binary polynomial.
// Also: return x^2+x in binary normal basis.
// rev_gray_code(x) == revbin( gray_code( revbin(x) ) )

static inline ulong inverse_rev_gray_code(ulong x);
// Inverse of rev_gray_code()
// Note: the returned value contains at each bit position the parity
//   of all bits of the input right from it (incl. itself).
// Also: division by x+1 as powers series over GF(2)
// Also: return solution of z = x^2+x in binary normal basis.
// Note: the statements can be reordered at will.
// inverse_rev_gray_code(x) == revbin( inverse_gray_code( revbin(x) ) )

// ========== HEADER FILE src/bits/tcrc64.h: ==========

class tcrc64;
// 64-bit CRC (cyclic redundancy check) with lookup table

// ========== HEADER FILE src/bits/thue-morse.h: ==========

class thue_morse;
// Thue-Morse sequence

// ========== HEADER FILE src/bits/tinyfactors.h: ==========

static inline bool is_tiny_prime(ulong n);
// For n < BITS_PER_LONG (!)
// return whether n is prime

static inline bool is_tiny_factor(ulong x, ulong d);
// For x,d < BITS_PER_LONG (!)
// return whether d divides x  (1 and x included as divisors)
// No need to check whether d==0
//

// ========== HEADER FILE src/bits/zerobyte.h: ==========

static inline ulong contains_zero_byte(ulong x);
// Determine if any sub-byte of x is zero:
// Return zero when x contains no zero-byte and nonzero when it does.
//
// To scan for other values than zero (e.g. 0xa5) use:
//  contains_zero_byte( x ^ 0xa5a5a5a5UL )

// ========== HEADER FILE src/bits/zorder.h: ==========

// Z-order (or Morton order), a non self-intersecting space filling curve.
static inline void zorder_next(ulong &x, ulong &y);

static inline void zorder_prev(ulong &x, ulong &y);

static inline void zorder3d_next(ulong &x, ulong &y, ulong &z);

static inline void zorder3d_prev(ulong &x, ulong &y, ulong &z);

