PROOF.bs.below: def of bend-osdev-algo@0.1.0.0
Import: import bend-osdev-algo@0.1.0.0/PROOF.bend as PROOF
Lines 1111-1119 of bend-osdev-algo@0.1.0.0/PROOF.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&start_line=1111&end_line=1119

import Base
import bend-osdev-algo@0.1.0.0/bsearch.bend as S

def bs.below(xs: List<&2, U32>, +key: U32, i: Nat, +x: U32, e: {List.get(&2, U32, xs, i) == Some{x} : Maybe<&2, U32>}, l: {Nat.is_lt(i, S.bsearch.lower_bound(xs, key)) == True{} : Bool}) -> {U32.is_lt(x, key) == True{} : Bool}:
  match xs i:
    case Nil{} _:
      Empty.absurd({U32.is_lt(x, key) == True{} : Bool}, absurd.ns32(x, e))
    case Con{+x0, t} 0n:
      %some_inj32(x0, x, e) : {U32.is_lt(_, key) == True{} : Bool}
      bs.step0(U32.is_lt(x0, key), S.bsearch.lower_bound(t, key), l)
    case Con{+x0, +t} 1n+ +ip:
      bs.below(t, key, ip, x, e, bs.step1(U32.is_lt(x0, key), S.bsearch.lower_bound(t, key), ip, l))

Uses (14):
  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=List
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List.get
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Maybe
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.is_lt
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&name=absurd.ns32
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&name=bs.step0
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&name=bs.step1
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&name=some_inj32
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/bsearch.bend&name=bsearch.lower_bound