Lemire.and0: 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 434-442 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=434&end_line=442

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend as C
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/lemmas/proofs/nat_algebra.bend as NA

# x & 0 == 0
def and0(+w: Nat, +x: Nat) -> {SR.and_bits(w, x, 0n) == 0n : Nat}:
  match w:
    case 0n:
      {==}
    case 1n+v:
      %Equal.sym(Nat, Nat.mul(C.bit(x), 0n), 0n, NA.mul_zero(C.bit(x))) : {Nat.add(_, Nat.double(SR.and_bits(v, C.half(x), 0n))) == 0n : Nat}
      %Equal.sym(Nat, SR.and_bits(v, C.half(x), 0n), 0n, and0(v, C.half(x))) : {Nat.add(0n, Nat.double(_)) == 0n : Nat}
      {==}

Uses (9):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  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.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend&name=mul_zero
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=bit
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=half
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/random/rand.bend&name=and_bits