PROOF.pm.fin.bnd.go: def of bend-kit-collections@0.1.4.1
Import: import bend-kit-collections@0.1.4.1/PROOF.bend as PROOF
Lines 1071-1085 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=1071&end_line=1085

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.fin.bnd.go(
  ~K: Data, ~V: Data, ~cmp: K -> K -> Cmp, ~trans: Trans(~K, ~cmp),
  +mk: K, +mv: V, +t: O.OMap<K, V>, +lb: Maybe<&2, K>, +ub: Maybe<&2, K>, +k: K,
  +v: V, +hi: O.OMap<K, V>,
  +ih: {PM(~K, ~V, ~cmp, lb, Some{k}, {(mk, mv, t) : K & V & O.OMap<K, V>}) == True{} : Bool},
  +eb: {Laws.below(~K, ~cmp, ub, k) == True{} : Bool},
  eh: {Laws.bounded(~K, ~V, ~cmp, hi, Some{k}, ub) == True{} : Bool}
) -> {PM(~K, ~V, ~cmp, lb, ub,
       O.OMap.pop_min.fin(K, V, k, v, hi, {(mk, mv, t) : K & V & O.OMap<K, V>})) == True{} : Bool}:
  +ab = Bool.and(Laws.above(~K, ~cmp, lb, mk), Laws.below(~K, ~cmp, Some{k}, mk))
  +c = Laws.bounded(~K, ~V, ~cmp, t, Some{mk}, Some{k})
  +eab = MBool.eq_true_of_and_left(ab, c, ih)
  +mkk = MBool.eq_true_of_and_right(Laws.above(~K, ~cmp, lb, mk), Laws.below(~K, ~cmp, Some{k}, mk), eab)
  MBool.and_eq_true(Bool.and(Laws.above(~K, ~cmp, lb, mk), Laws.below(~K, ~cmp, ub, mk)), Laws.bounded(~K, ~V, ~cmp, O.OMap.balance(K, V, k, v, t, hi), Some{mk}, ub), MBool.and_eq_true(Laws.above(~K, ~cmp, lb, mk), Laws.below(~K, ~cmp, ub, mk), MBool.eq_true_of_and_left(Laws.above(~K, ~cmp, lb, mk), Laws.below(~K, ~cmp, Some{k}, mk), eab), below_trans(~K, ~cmp, ~trans, ub, mk, k, mkk, eb)), balance.bnd(~K, ~V, ~cmp, ~trans, Some{mk}, ub, k, v, t, hi, mkk, eb,
      MBool.eq_true_of_and_right(ab, c, ih), eh))

Uses (19):
  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=Base/base.bend&name=Pair
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Sigma
  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=balance.bnd
  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/omap.bend&name=OMap
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/omap.bend&name=OMap.balance
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/omap.bend&name=OMap.pop_min.fin
  https://jend.leeeet.dev/definition.txt?path=bend-mathlib@0.8.1.0/bool.bend&name=and_eq_true
  https://jend.leeeet.dev/definition.txt?path=bend-mathlib@0.8.1.0/bool.bend&name=eq_true_of_and_left
  https://jend.leeeet.dev/definition.txt?path=bend-mathlib@0.8.1.0/bool.bend&name=eq_true_of_and_right