def and_low_hy

bend-collections-laws-crypto@1.0.0.0 · proofs/lib/u32.bend:207

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

207def 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}:
208  match d:
209    case 0n:
210      N.bit_split_val(yb, False{}, S.unsigned(m, yt), 0n, hy)
211    case 1n+k:
212      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))))