def bits.root_a

bend-osdev-caps@0.1.0.0 · PROOF.bend:562

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

562def 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)}:
563  match n:
564    case 0n:
565      match i b a:
566        case WNil{} WNil{} WNil{}:
567          h0 => {==}
568    case 1n+n1:
569      match i b a:
570        case WCon{True{}, +it} WCon{True{}, +bt} WCon{True{}, +at}:
571          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))
572        case WCon{True{}, +it} WCon{True{}, +bt} WCon{False{}, +at}:
573          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))
574        case WCon{True{}, +it} WCon{False{}, +bt} WCon{True{}, +at}:
575          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))
576        case WCon{True{}, +it} WCon{False{}, +bt} WCon{False{}, +at}:
577          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))
578        case WCon{False{}, +it} WCon{True{}, +bt} WCon{True{}, +at}:
579          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)
580        case WCon{False{}, +it} WCon{True{}, +bt} WCon{False{}, +at}:
581          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))
582        case WCon{False{}, +it} WCon{False{}, +bt} WCon{True{}, +at}:
583          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)
584        case WCon{False{}, +it} WCon{False{}, +bt} WCon{False{}, +at}:
585          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))