PROOF.pm.bnd: def of bend-kit-collections@0.1.4.1
Import: import bend-kit-collections@0.1.4.1/PROOF.bend as PROOF
Lines 1100-1115 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=1100&end_line=1115

import Base
import bend-mathlib@0.7.4.0/bool.bend as MBool
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 pm.bnd(
  ~K: Data, ~V: Data, ~cmp: K -> K -> Cmp, ~trans: Trans(~K, ~cmp),
  lo: O.OMap<K, V>, +lb: Maybe<&2, K>, +ub: Maybe<&2, K>, +k: K, +v: V,
  +hi: O.OMap<K, V>,
  +e: {Laws.bounded(~K, ~V, ~cmp, O.OBin{0, k, v, lo, hi}, lb, ub) == True{} : Bool}
) -> {PM(~K, ~V, ~cmp, lb, ub, O.OMap.pop_min(K, V, lo, k, v, hi)) == True{} : Bool}:
  match lo:
    case O.OTip{}:
      MBool.and_eq_true(Bool.and(Laws.above(~K, ~cmp, lb, k), Laws.below(~K, ~cmp, ub, k)), Laws.bounded(~K, ~V, ~cmp, hi, Some{k}, ub), MBool.and_eq_true(Laws.above(~K, ~cmp, lb, k), Laws.below(~K, ~cmp, ub, k), bn.a(~K, ~V, ~cmp, lb, ub, k, v, O.OTip{}, hi, e), bn.b(~K, ~V, ~cmp, lb, ub, k, v, O.OTip{}, hi, e)), bn.h(~K, ~V, ~cmp, lb, ub, k, v, O.OTip{}, hi, e))
    case O.OBin{s, +lk, +lv, +ll, +lh}:
      +lo = {O.OBin{0, lk, lv, ll, lh} : O.OMap<K, V>}
      pm.fin.bnd(~K, ~V, ~cmp, ~trans, O.OMap.pop_min(K, V, ll, lk, lv, lh), lb, ub, k, v, hi,
        pm.bnd(~K, ~V, ~cmp, ~trans, ll, lb, Some{k}, lk, lv, lh,
          bn.l(~K, ~V, ~cmp, lb, ub, k, v, lo, hi, e)),
        bn.b(~K, ~V, ~cmp, lb, ub, k, v, lo, hi, e),
        bn.h(~K, ~V, ~cmp, lb, ub, k, v, lo, hi, e))

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=Bool.and
  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=below
  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=PM
  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=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.l
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=pm.fin.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.pop_min
  https://jend.leeeet.dev/definition.txt?path=bend-mathlib@0.8.1.0/bool.bend&name=and_eq_true