Lemire.and_self: 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 484-491 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=484&end_line=491

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/math/typed/width.bend as WI

# x & x is the low w bits of x
def and_self(+w: Nat, +x: Nat) -> {SR.and_bits(w, x, x) == C.low(w, x) : Nat}:
  match w:
    case 0n:
      {==}
    case 1n+v:
      %Equal.sym(Nat, Nat.mul(C.bit(x), C.bit(x)), C.bit(x), bitsq_b(C.bit(x), WI.bit_le1(x))) : {Nat.add(_, Nat.double(SR.and_bits(v, C.half(x), C.half(x)))) == Nat.add(C.bit(x), Nat.double(C.low(v, C.half(x)))) : Nat}
      Equal.cong(Nat, Nat, z => Nat.add(C.bit(x), Nat.double(z)), SR.and_bits(v, C.half(x), C.half(x)), C.low(v, C.half(x)), and_self(v, C.half(x)))

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.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/math/random/lemire.bend&name=bitsq_b
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/math/typed/width.bend&name=bit_le1
  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/lib/common.bend&name=low
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/math/random/rand.bend&name=and_bits