PROOF.bits.root_z: def of bend-osdev-caps@0.1.0.0
Import: import bend-osdev-caps@0.1.0.0/PROOF.bend as PROOF
Lines 587-602 of bend-osdev-caps@0.1.0.0/PROOF.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-caps@0.1.0.0/PROOF.bend&start_line=587&end_line=602

import Base
import bend-osdev-caps@0.1.0.0/caps.bend as C

def bits.root_z(+n: Nat, +i: Word(n), +b: Word(n)) -> {Word.or(n, Word.or(n, Word.and(n, i, C.CapBits.all(n)), Word.and(n, C.CapBits.all(n), b)), Word.zero(n)) == Word.or(n, i, b) : Word(n)}:
  match n:
    case 0n:
      match i b:
        case WNil{} WNil{}:
          {==}
    case 1n+n1:
      match i b:
        case WCon{True{}, +it} WCon{True{}, +bt}:
          Equal.cong(Word(n1), Word(1n+n1), w => WCon{True{}, w}, Word.or(n1, Word.or(n1, Word.and(n1, it, C.CapBits.all(n1)), Word.and(n1, C.CapBits.all(n1), bt)), Word.zero(n1)), Word.or(n1, it, bt), bits.root_z(n1, it, bt))
        case WCon{True{}, +it} WCon{False{}, +bt}:
          Equal.cong(Word(n1), Word(1n+n1), w => WCon{True{}, w}, Word.or(n1, Word.or(n1, Word.and(n1, it, C.CapBits.all(n1)), Word.and(n1, C.CapBits.all(n1), bt)), Word.zero(n1)), Word.or(n1, it, bt), bits.root_z(n1, it, bt))
        case WCon{False{}, +it} WCon{True{}, +bt}:
          Equal.cong(Word(n1), Word(1n+n1), w => WCon{True{}, w}, Word.or(n1, Word.or(n1, Word.and(n1, it, C.CapBits.all(n1)), Word.and(n1, C.CapBits.all(n1), bt)), Word.zero(n1)), Word.or(n1, it, bt), bits.root_z(n1, it, bt))
        case WCon{False{}, +it} WCon{False{}, +bt}:
          Equal.cong(Word(n1), Word(1n+n1), w => WCon{False{}, w}, Word.or(n1, Word.or(n1, Word.and(n1, it, C.CapBits.all(n1)), Word.and(n1, C.CapBits.all(n1), bt)), Word.zero(n1)), Word.or(n1, it, bt), bits.root_z(n1, it, bt))

Uses (10):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.cong
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.Con
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.Nil
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.and
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.or
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Word.zero
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-caps@0.1.0.0/caps.bend&name=CapBits.all