PROOFOrder.nt.lt_le_trans: def of bend-osdev-u32@0.1.0.0
Import: import bend-osdev-u32@0.1.0.0/PROOF_order.bend as PROOFOrder
Lines 425-438 of bend-osdev-u32@0.1.0.0/PROOF_order.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-u32@0.1.0.0/PROOF_order.bend&start_line=425&end_line=438

import Base

def nt.lt_le_trans(x: Nat, y: Nat, z: Nat, h1: {Nat.is_lt(x, y) == True{} : Bool}, h2: {Nat.is_le(y, z) == True{} : Bool}) -> {Nat.is_lt(x, z) == True{} : Bool}:
  match x y z:
    case 0n 0n _:
      Empty.absurd({Nat.is_lt(0n, z) == True{} : Bool}, ab.ft(h1))
    case 0n 1n+yp 0n:
      Empty.absurd({Nat.is_lt(0n, 0n) == True{} : Bool}, ab.ft(h2))
    case 0n 1n+yp 1n+zp:
      {==}
    case 1n+xp 0n _:
      Empty.absurd({Nat.is_lt(1n+xp, z) == True{} : Bool}, ab.ft(h1))
    case 1n+xp 1n+yp 0n:
      Empty.absurd({Nat.is_lt(1n+xp, 0n) == True{} : Bool}, ab.ft(h2))
    case 1n+xp 1n+yp 1n+zp:
      nt.lt_le_trans(xp, yp, zp, h1, h2)

Uses (6):
  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.is_le
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=bend-osdev-u32@0.1.0.0/PROOF_order.bend&name=ab.ft