Nat.bit_double_lt: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as Nat
Lines 401-411 of bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&start_line=401&end_line=411

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend as S

# bit + 2u < 2v gives u < v.
def bit_double_lt(+b: Bool, +u: Nat, +v: Nat, e: {Nat.is_lt(Nat.add(S.bit_value(b), Nat.double(u)), Nat.double(v)) == True{} : Bool}) -> {Nat.is_lt(u, v) == True{} : Bool}:
  match b u v:
    case _ _ 0n:
      Empty.absurd({Nat.is_lt(u, 0n) == True{} : Bool}, lt_zero_absurd(Nat.add(S.bit_value(b), Nat.double(u)), e))
    case _ 0n 1n+q:
      {==}
    case False{} 1n+p 1n+q:
      bit_double_lt(False{}, p, q, e)
    case True{} 1n+p 1n+q:
      bit_double_lt(True{}, p, q, e)

Uses (8):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Empty.absurd
  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=Nat.double
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/lemmas/spec/numeric.bend&name=bit_value
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=lt_zero_absurd