def split

bend-collections-laws-math@1.0.0.0 · proofs/math/random/lemire.bend:373

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

373def split(+w: Nat, +m: Nat, +k: Nat, +N: Nat) -> {cm(w, m, k, Nat.double(N)) == cm2(w, m, k, N) : Nat}:
374  match N:
375    case 0n:
376      {==}
377    case 1n+ +y:
378      +f0 = b2n(Nat.is_eq(SR.and_bits(w, Nat.double(y), m), k))
379      +f1 = b2n(Nat.is_eq(SR.and_bits(w, 1n+Nat.double(y), m), k))
380      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)),
381        N.add_assoc(cm(w, m, k, Nat.double(y)), f0, f1),
382        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)))