Word.inc_join: def of bend-collections-laws-crypto@1.0.0.0
Import: import bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend as Word
Lines 196-204 of bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend&start_line=196&end_line=204

import Base
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/src/wide.bend as W

# incrementing a joined word carries into the high part iff the low part is all ones
def inc_join(+n: Nat, +m: Nat, +a: Word(n), +b: Word(m)) -> {Word.inc(Nat.add(n, m), W.join(n, m, a, b)) == W.join(n, m, Word.inc(n, a), incif(m, ones(n, a), b)) : Word(Nat.add(n, m))}:
  match n a:
    case 0n WNil{}:
      {==}
    case 1n+p WCon{False{}, t}:
      {==}
    case 1n+p WCon{True{}, t}:
      Equal.cong(Word(Nat.add(p, m)), Word(1n+Nat.add(p, m)), w => WCon{False{}, w}, Word.inc(Nat.add(p, m), W.join(p, m, t, b)), W.join(p, m, Word.inc(p, t), incif(m, ones(p, t), b)), inc_join(p, m, t, b))

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=Nat.add
  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.inc
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/src/wide.bend&name=join
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend&name=incif
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend&name=ones