PROOF.bits.root_a: def of bend-osdev-caps@0.1.0.0
Import: import bend-osdev-caps@0.1.0.0/PROOF.bend as PROOF
Lines 562-585 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=562&end_line=585

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

def bits.root_a(+n: Nat, +i: Word(n), +b: Word(n), +a: Word(n)) -> {C.CapBits.subset(n, a, i) == True{} : Bool} -> {Word.or(n, Word.or(n, Word.and(n, i, C.CapBits.all(n)), Word.and(n, C.CapBits.all(n), b)), a) == Word.or(n, i, b) : Word(n)}:
  match n:
    case 0n:
      match i b a:
        case WNil{} WNil{} WNil{}:
          h0 => {==}
    case 1n+n1:
      match i b a:
        case WCon{True{}, +it} WCon{True{}, +bt} WCon{True{}, +at}:
          h0 => 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)), at), Word.or(n1, it, bt), bits.root_a(n1, it, bt, at)(h0))
        case WCon{True{}, +it} WCon{True{}, +bt} WCon{False{}, +at}:
          h0 => 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)), at), Word.or(n1, it, bt), bits.root_a(n1, it, bt, at)(h0))
        case WCon{True{}, +it} WCon{False{}, +bt} WCon{True{}, +at}:
          h0 => 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)), at), Word.or(n1, it, bt), bits.root_a(n1, it, bt, at)(h0))
        case WCon{True{}, +it} WCon{False{}, +bt} WCon{False{}, +at}:
          h0 => 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)), at), Word.or(n1, it, bt), bits.root_a(n1, it, bt, at)(h0))
        case WCon{False{}, +it} WCon{True{}, +bt} WCon{True{}, +at}:
          h0 => bits.absurd_eq(Word(1n+n1), Word.or(1n+n1, Word.or(1n+n1, Word.and(1n+n1, WCon{False{}, it}, C.CapBits.all(1n+n1)), Word.and(1n+n1, C.CapBits.all(1n+n1), WCon{True{}, bt})), WCon{True{}, at}), Word.or(1n+n1, WCon{False{}, it}, WCon{True{}, bt}), h0)
        case WCon{False{}, +it} WCon{True{}, +bt} WCon{False{}, +at}:
          h0 => 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)), at), Word.or(n1, it, bt), bits.root_a(n1, it, bt, at)(h0))
        case WCon{False{}, +it} WCon{False{}, +bt} WCon{True{}, +at}:
          h0 => bits.absurd_eq(Word(1n+n1), Word.or(1n+n1, Word.or(1n+n1, Word.and(1n+n1, WCon{False{}, it}, C.CapBits.all(1n+n1)), Word.and(1n+n1, C.CapBits.all(1n+n1), WCon{False{}, bt})), WCon{True{}, at}), Word.or(1n+n1, WCon{False{}, it}, WCon{False{}, bt}), h0)
        case WCon{False{}, +it} WCon{False{}, +bt} WCon{False{}, +at}:
          h0 => 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)), at), Word.or(n1, it, bt), bits.root_a(n1, it, bt, at)(h0))

Uses (11):
  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=bend-osdev-caps@0.1.0.0/PROOF.bend&name=bits.absurd_eq
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-caps@0.1.0.0/caps.bend&name=CapBits.all
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-caps@0.1.0.0/caps.bend&name=CapBits.subset