U32half.hlf_upper: def of bend-collections-laws-math@1.0.0.0
Import: import bend-collections-laws-math@1.0.0.0/proofs/lib/u32half.bend as U32half
Lines 90-100 of bend-collections-laws-math@1.0.0.0/proofs/lib/u32half.bend: https://jend.leeeet.dev/read.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/u32half.bend&start_line=90&end_line=100

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

def hlf_upper(+a: Nat, +b: Nat, +h: {Nat.is_lt(a, b) == True{} : Bool}) -> {Nat.is_lt(hlf(Nat.add(a, b)), b) == True{} : Bool}:
  match a b:
    case 0n 0n:
      Empty.absurd({Nat.is_lt(hlf(Nat.add(0n, 0n)), 0n) == True{} : Bool}, L.false_true(h))
    case 0n 1n+ +q:
      N.le_lt_trans(hlf(1n+q), q, 1n+q, hlf_succ_le(q), N.lt_succ(q))
    case 1n+p 0n:
      Empty.absurd({Nat.is_lt(hlf(Nat.add(1n+p, 0n)), 0n) == True{} : Bool}, L.false_true(h))
    case 1n+ +p 1n+ +q:
      %Equal.sym(Nat, Nat.add(p, 1n+q), 1n+Nat.add(p, q), N.add_succ(p, q)) : {Nat.is_lt(hlf(1n+_), 1n+q) == True{} : Bool}
      hlf_upper(p, q, h)

Uses (12):
  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=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.is_lt
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/logic.bend&name=false_true
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=add_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=le_lt_trans
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/nat.bend&name=lt_succ
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/u32half.bend&name=hlf
  https://jend.leeeet.dev/definition.txt?path=bend-collections-laws-math@1.0.0.0/proofs/lib/u32half.bend&name=hlf_succ_le