PROOF.bs.none: def of bend-osdev-algo@0.1.0.0
Import: import bend-osdev-algo@0.1.0.0/PROOF.bend as PROOF
Lines 1200-1210 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=1200&end_line=1210

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

def bs.none(xs: List<&2, U32>, +key: U32, i: Nat, +x: U32, s: {S.bsearch.sorted(xs) == True{} : Bool}, e: {S.bsearch.find(xs, key) == None{} : Maybe<&2, Nat>}, g: {List.get(&2, U32, xs, i) == Some{x} : Maybe<&2, U32>}) -> {U32.is_eq(x, key) == False{} : Bool}:
  match xs i:
    case Nil{} _:
      Empty.absurd({U32.is_eq(x, key) == False{} : Bool}, absurd.ns32(x, g))
    case Con{+x0, +t} 0n:
      %some_inj32(x0, x, g) : {U32.is_eq(_, key) == False{} : Bool}
      bs.none0(U32.is_lt(x0, key), x0, key, {==}, t, S.bsearch.lower_bound(t, key), e)
    case Con{+x0, +t} 1n+ +ip:
      +s2 = s
      +g2 = g
      bs.none1(U32.is_lt(x0, key), x0, x, key, {==}, t, S.bsearch.lower_bound(t, key), e, h => bs.none(t, key, ip, x, bs.sorted_tail(t, x0, s2), h, g2), bs.head_le(t, x0, s2, ip, x, g2))

Uses (18):
  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=U32
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32.is_eq
  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.head_le
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&name=bs.none0
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&name=bs.none1
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/PROOF.bend&name=bs.sorted_tail
  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.find
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/bsearch.bend&name=bsearch.lower_bound
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-algo@0.1.0.0/bsearch.bend&name=bsearch.sorted