TwoList.half_le: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/lib/two_list.bend as TwoList
Lines 47-56 of bend-collections-laws-math@1.0.0.0/proofs/lib/two_list.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/two_list.bend&start_line=47&end_line=56

import Base
import bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend as N

# half of a nonempty length is below it
def half_le(+n: Nat) -> {Nat.is_le(Nat.div(1n+n, 2n), n) == True{} : Bool}:
  match n:
    case 0n:
      {==}
    case 1n+0n:
      {==}
    case 1n+1n+k:
      %Equal.sym(Nat, Nat.div(Nat.add(2n, 1n+k), 2n), 1n+Nat.div(1n+k, 2n), N.div2_step(1n+k)) : {Nat.is_le(_, 2n+k) == True{} : Bool}
      N.le_trans(1n+Nat.div(1n+k, 2n), 1n+k, 2n+k, half_le(k), N.le_succ(1n+k))

Uses (9):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  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.div
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_le
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=div2_step
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_trans