PROOF.rot_lo.double.bnd: def of bend-kit-collections@0.1.4.1
Import: import bend-kit-collections@0.1.4.1/PROOF.bend as PROOF
Lines 328-355 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=328&end_line=355

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

def rot_lo.double.bnd(
  ~K: Data, ~V: Data, ~cmp: K -> K -> Cmp, ~trans: Trans(~K, ~cmp),
  hl: O.OMap<K, V>, +lb: Maybe<&2, K>, +ub: Maybe<&2, K>, +k: K, +v: V,
  +lo: O.OMap<K, V>, +hk: K, +hv: V, +hh: O.OMap<K, V>,
  +ea: {Laws.above(~K, ~cmp, lb, k) == True{} : Bool},
  el: {Laws.bounded(~K, ~V, ~cmp, lo, lb, Some{k}) == True{} : Bool},
  +eh: {Laws.bounded(~K, ~V, ~cmp, O.OBin{0, hk, hv, hl, hh}, Some{k}, ub) == True{} : Bool}
) -> {Laws.bounded(~K, ~V, ~cmp,
       O.OMap.rot_lo.double(K, V, k, v, lo, hk, hv, hl, hh), lb, ub) == True{} : Bool}:
  match hl:
    case O.OTip{}:
      rot_lo.single.bnd(~K, ~V, ~cmp, ~trans, lb, ub, k, v, lo, hk, hv, O.OTip{}, hh,
        ea, el, eh)
    case O.OBin{s, +mk, +mv, +ml, +mh}:
      +hl = {O.OBin{0, mk, mv, ml, mh} : O.OMap<K, V>}
      +h_b = bn.b(~K, ~V, ~cmp, Some{k}, ub, hk, hv, hl, hh, eh)
      +h_l = bn.l(~K, ~V, ~cmp, Some{k}, ub, hk, hv, hl, hh, eh)
      +m_a = bn.a(~K, ~V, ~cmp, Some{k}, Some{hk}, mk, mv, ml, mh, h_l)
      +m_b = bn.b(~K, ~V, ~cmp, Some{k}, Some{hk}, mk, mv, ml, mh, h_l)
      bn.i(~K, ~V, ~cmp, lb, ub, mk, mv, O.OMap.bin(K, V, k, v, lo, ml),
        O.OMap.bin(K, V, hk, hv, mh, hh),
        above_trans(~K, ~cmp, ~trans, lb, k, mk, ea, m_a),
        below_trans(~K, ~cmp, ~trans, ub, mk, hk, m_b, h_b),
        bn.i(~K, ~V, ~cmp, lb, Some{mk}, k, v, lo, ml, ea, m_a, el,
          bn.l(~K, ~V, ~cmp, Some{k}, Some{hk}, mk, mv, ml, mh, h_l)),
        bn.i(~K, ~V, ~cmp, Some{mk}, ub, hk, hv, mh, hh, m_b, h_b,
          bn.h(~K, ~V, ~cmp, Some{k}, Some{hk}, mk, mv, ml, mh, h_l),
          bn.h(~K, ~V, ~cmp, Some{k}, ub, hk, hv, hl, hh, eh)))

Uses (17):
  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=Maybe
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/LAWS.bend&name=above
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/LAWS.bend&name=bounded
  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=above_trans
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=below_trans
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=bn.a
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=bn.b
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=bn.h
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=bn.i
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=bn.l
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=rot_lo.single.bnd
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/omap.bend&name=OMap
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/omap.bend&name=OMap.bin
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/omap.bend&name=OMap.rot_lo.double