PROOF.below_trans: def of bend-kit-collections@0.1.4.1
Import: import bend-kit-collections@0.1.4.1/PROOF.bend as PROOF
Lines 251-260 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=251&end_line=260

import Base
import bend-kit-collections@0.1.4.1/LAWS.bend as Laws

def below_trans(
  ~K: Data, ~cmp: K -> K -> Cmp, ~trans: Trans(~K, ~cmp), ub: Maybe<&2, K>,
  +k: K, +hk: K, e: {Cmp.is_lt(cmp(k, hk)) == True{} : Bool},
  eb: {Laws.below(~K, ~cmp, ub, hk) == True{} : Bool}
) -> {Laws.below(~K, ~cmp, ub, k) == True{} : Bool}:
  match ub:
    case None{}:
      {==}
    case Some{+u}:
      lt_trans(~K, ~cmp, ~trans, k, hk, u, e, eb)

Uses (7):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Cmp
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Cmp.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Maybe
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/LAWS.bend&name=below
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=Trans
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=lt_trans