U32.and_low_hy: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/lib/u32.bend as U32
Lines 207-212 of bend-collections-laws-crypto@1.0.0.0/proofs/lib/u32.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/u32.bend&start_line=207&end_line=212

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend as N
import bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend as SC
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend as S

def and_low_hy(+m: Nat, +d: Nat, +yb: Bool, +yt: Word(m), +hy: {Nat.add(S.bit_value(yb), Nat.double(S.unsigned(m, yt))) == Nat.sub(SC.pow2(d), 1n) : Nat}) -> {S.unsigned(m, yt) == Nat.sub(SC.pow2(and_low_prev(d)), 1n) : Nat}:
  match d:
    case 0n:
      N.bit_split_val(yb, False{}, S.unsigned(m, yt), 0n, hy)
    case 1n+k:
      N.bit_split_val(yb, True{}, S.unsigned(m, yt), Nat.sub(SC.pow2(k), 1n), Equal.trans(Nat, Nat.add(S.bit_value(yb), Nat.double(S.unsigned(m, yt))), Nat.sub(Nat.double(SC.pow2(k)), 1n), 1n+Nat.double(Nat.sub(SC.pow2(k), 1n)), hy, N.pred_double(SC.pow2(k), N.pow2_pos(k))))

Uses (14):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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.sub
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend&name=bit_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend&name=unsigned
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=bit_split_val
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=pow2_pos
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/nat.bend&name=pred_double
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/u32.bend&name=and_low_prev
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/spec/lib/common.bend&name=pow2