QueueProof.below_trans: def of bend-kit-scheduler@0.4.1.0
Import: import bend-kit-scheduler@0.4.1.0/queue_proof.bend as QueueProof
Lines 126-133 of bend-kit-scheduler@0.4.1.0/queue_proof.bend: https://jend.leeeet.dev/read.txt?path=bend-kit-scheduler@0.4.1.0/queue_proof.bend&start_line=126&end_line=133

import Base
import bend-kit-collections@0.1.4.1/heap.bend as H
import bend-kit-scheduler@0.4.1.0/queue.bend as Q
import bend-kit-scheduler@0.4.1.0/queue_laws.bend as L

def below_trans(-A: Data, h: H.Heap<Q.Ticket<A>>, +a: Q.Ticket<A>, +b: Q.Ticket<A>, e: {Q.le(A, a, b) == True{} : Bool}, eh: L.below(A, h, b)) -> L.below(A, h, a):
  match h:
    case H.HLeaf{}:
      Unit{}
    case H.HNode{n, t, l, r}:
      L.Both{et, L.Both{el, er}} = eh
      +e = e
      L.Both{le_trans(A, a, b, t, e, et), L.Both{below_trans(A, l, a, b, e, el), below_trans(A, r, a, b, e, er)}}

Uses (8):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Unit
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/heap.bend&name=Heap
  https://jend.leeeet.dev/definition.txt?path=bend-kit-scheduler@0.4.1.0/queue.bend&name=Ticket
  https://jend.leeeet.dev/definition.txt?path=bend-kit-scheduler@0.4.1.0/queue.bend&name=le
  https://jend.leeeet.dev/definition.txt?path=bend-kit-scheduler@0.4.1.0/queue_laws.bend&name=Both
  https://jend.leeeet.dev/definition.txt?path=bend-kit-scheduler@0.4.1.0/queue_laws.bend&name=below
  https://jend.leeeet.dev/definition.txt?path=bend-kit-scheduler@0.4.1.0/queue_proof.bend&name=le_trans