Below.and_le: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/math/random/below.bend as Below
Lines 19-32 of bend-collections-laws-math@1.0.0.0/proofs/math/random/below.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/random/below.bend&start_line=19&end_line=32

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
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend as WI
import bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend as WA

# x & m <= m
def and_le(+w: Nat, +a: Nat, +b: Nat) -> {Nat.is_le(SR.and_bits(w, a, b), b) == True{} : Bool}:
  match w:
    case 0n:
      N.zero_le(b)
    case 1n+p:
      +hb = WI.hb(b)
      +h1 = WA.mul_le1(C.bit(a), C.bit(b), WI.bit_le1(a))
      +h2 = N.double_le(SR.and_bits(p, C.half(a), C.half(b)), C.half(b), and_le(p, C.half(a), C.half(b)))
      +h3 = N.le_trans(Nat.add(Nat.mul(C.bit(a), C.bit(b)), Nat.double(SR.and_bits(p, C.half(a), C.half(b)))), Nat.add(C.bit(b), Nat.double(SR.and_bits(p, C.half(a), C.half(b)))), Nat.add(C.bit(b), Nat.double(C.half(b))),
        WI.le_add_r(Nat.mul(C.bit(a), C.bit(b)), C.bit(b), Nat.double(SR.and_bits(p, C.half(a), C.half(b))), h1),
        N.le_add_left(Nat.double(SR.and_bits(p, C.half(a), C.half(b))), Nat.double(C.half(b)), C.bit(b), h2))
      %Equal.sym(Nat, b, Nat.add(C.bit(b), Nat.double(C.half(b))), hb) : {Nat.is_le(Nat.add(Nat.mul(C.bit(a), C.bit(b)), Nat.double(SR.and_bits(p, C.half(a), C.half(b)))), _) == True{} : Bool}
      h3

Uses (18):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.mul
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=double_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_add_left
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_trans
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=zero_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/w64add.bend&name=mul_le1
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend&name=bit_le1
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend&name=hb
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/math/typed/width.bend&name=le_add_r
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/spec/lib/common.bend&name=bit
  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