Lemire.pairs: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/lemire.bend as Lemire
Lines 424-432 of bend-collections-laws-math@1.0.0.0/proofs/math/random/lemire.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/lemire.bend&start_line=424&end_line=432

import Base
import bend-collections-laws-math@1.0.0.0/spec/lib/common.bend as C
import bend-collections-laws-math@1.0.0.0/spec/math/random/rand.bend as SR

# the pairs one width up count what the lower width counts
def pairs(+v: Nat, +h: Nat, +k: Nat, +N: Nat) -> {cm2(1n+v, 1n+Nat.double(h), k, N) == cm(v, h, C.half(k), N) : Nat}:
  match N:
    case 0n:
      {==}
    case 1n+ +y:
      Equal.trans(Nat, Nat.add(cm2(1n+v, 1n+Nat.double(h), k, y), Nat.add(b2n(Nat.is_eq(SR.and_bits(1n+v, Nat.double(y), 1n+Nat.double(h)), k)), b2n(Nat.is_eq(SR.and_bits(1n+v, 1n+Nat.double(y), 1n+Nat.double(h)), k)))), Nat.add(cm(v, h, C.half(k), y), Nat.add(b2n(Nat.is_eq(SR.and_bits(1n+v, Nat.double(y), 1n+Nat.double(h)), k)), b2n(Nat.is_eq(SR.and_bits(1n+v, 1n+Nat.double(y), 1n+Nat.double(h)), k)))), Nat.add(cm(v, h, C.half(k), y), b2n(Nat.is_eq(SR.and_bits(v, y, h), C.half(k)))),
        Equal.cong(Nat, Nat, z => Nat.add(z, Nat.add(b2n(Nat.is_eq(SR.and_bits(1n+v, Nat.double(y), 1n+Nat.double(h)), k)), b2n(Nat.is_eq(SR.and_bits(1n+v, 1n+Nat.double(y), 1n+Nat.double(h)), k)))), cm2(1n+v, 1n+Nat.double(h), k, y), cm(v, h, C.half(k), y), pairs(v, h, k, y)),
        Equal.cong(Nat, Nat, z => Nat.add(cm(v, h, C.half(k), y), z), Nat.add(b2n(Nat.is_eq(SR.and_bits(1n+v, Nat.double(y), 1n+Nat.double(h)), k)), b2n(Nat.is_eq(SR.and_bits(1n+v, 1n+Nat.double(y), 1n+Nat.double(h)), k))), b2n(Nat.is_eq(SR.and_bits(v, y, h), C.half(k))), pair_and(v, h, k, y)))

Uses (12):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.trans
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.double
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_eq
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/lemire.bend&name=b2n
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/lemire.bend&name=cm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/lemire.bend&name=cm2
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/lemire.bend&name=pair_and
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=half
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/math/random/rand.bend&name=and_bits