PROOF.ncmp_trans: def of bend-kit-collections@0.1.4.1
Import: import bend-kit-collections@0.1.4.1/PROOF.bend as PROOF
Lines 3975-3988 of bend-kit-collections@0.1.4.1/PROOF.bend: https://jend.leeeet.dev/read.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&start_line=3975&end_line=3988

import Base

def ncmp_trans(
  a: Nat, b: Nat, c: Nat, e1: {Nat.cmp(a, b) == LT{} : Cmp}, e2: {Nat.cmp(b, c) == LT{} : Cmp}
) -> {Nat.cmp(a, c) == LT{} : Cmp}:
  match a b c:
    case 0n 0n _:
      lt_absurd({Nat.cmp(0n, c) == LT{} : Cmp}, EQ{}, {==}, e1)
    case 1n+p 0n _:
      lt_absurd({Nat.cmp(1n+p, c) == LT{} : Cmp}, GT{}, {==}, e1)
    case _ 1n+q 0n:
      lt_absurd({Nat.cmp(a, 0n) == LT{} : Cmp}, GT{}, {==}, e2)
    case 0n 1n+q 1n+r:
      {==}
    case 1n+p 1n+q 1n+r:
      ncmp_trans(p, q, r, e1, e2)

Uses (4):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Cmp
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.cmp
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=lt_absurd