Lemire.split: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend as Lemire
Lines 373-382 of bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend&start_line=373&end_line=382

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/math/random/rand.bend as SR
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend as N

def split(+w: Nat, +m: Nat, +k: Nat, +N: Nat) -> {cm(w, m, k, Nat.double(N)) == cm2(w, m, k, N) : Nat}:
  match N:
    case 0n:
      {==}
    case 1n+ +y:
      +f0 = b2n(Nat.is_eq(SR.and_bits(w, Nat.double(y), m), k))
      +f1 = b2n(Nat.is_eq(SR.and_bits(w, 1n+Nat.double(y), m), k))
      Equal.trans(Nat, Nat.add(Nat.add(cm(w, m, k, Nat.double(y)), f0), f1), Nat.add(cm(w, m, k, Nat.double(y)), Nat.add(f0, f1)), Nat.add(cm2(w, m, k, y), Nat.add(f0, f1)),
        N.add_assoc(cm(w, m, k, Nat.double(y)), f0, f1),
        Equal.cong(Nat, Nat, z => Nat.add(z, Nat.add(f0, f1)), cm(w, m, k, Nat.double(y)), cm2(w, m, k, y), split(w, m, k, y)))

Uses (11):
  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-crypto@1.0.0.0/proofs/lib/nat.bend&name=add_assoc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend&name=b2n
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend&name=cm
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/random/lemire.bend&name=cm2
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/random/rand.bend&name=and_bits