ProofBsearch.bq.lt_le_trans: def of bend-osdev-algo@0.1.0.0
Import: import bend-osdev-algo@0.1.0.0/proof_bsearch.bend as ProofBsearch
Lines 190-203 of bend-osdev-algo@0.1.0.0/proof_bsearch.bend: https://jend.leeeet.dev/read.txt?path=bend-osdev-algo@0.1.0.0/proof_bsearch.bend&start_line=190&end_line=203

import Base

def bq.lt_le_trans(a: Nat, b: Nat, c: Nat, e: {Nat.is_lt(a, b) == True{} : Bool}, f: {Nat.is_le(b, c) == True{} : Bool}) -> {Nat.is_lt(a, c) == True{} : Bool}:
  match a b c:
    case 0n 0n _:
      Empty.absurd({Nat.is_lt(0n, c) == True{} : Bool}, bq.ft(e))
    case 0n 1n+bp 0n:
      Empty.absurd({Nat.is_lt(0n, 0n) == True{} : Bool}, bq.ft(f))
    case 0n 1n+bp 1n+cp:
      {==}
    case 1n+ap 0n _:
      Empty.absurd({Nat.is_lt(1n+ap, c) == True{} : Bool}, bq.ft(e))
    case 1n+ap 1n+bp 0n:
      Empty.absurd({Nat.is_lt(1n+ap, 0n) == True{} : Bool}, bq.ft(f))
    case 1n+ap 1n+bp 1n+cp:
      bq.lt_le_trans(ap, bp, cp, e, f)

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-algo@0.1.0.0/proof_bsearch.bend&name=bq.ft