PROOF.put.bnd: def of bend-kit-collections@0.1.4.1
Import: import bend-kit-collections@0.1.4.1/PROOF.bend as PROOF
Lines 517-547 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=517&end_line=547

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 put.bnd(
  ~K: Data, ~V: Data, ~cmp: K -> K -> Cmp, ~flip: Flip(~K, ~cmp),
  ~trans: Trans(~K, ~cmp), t: O.OMap<K, V>, +lb: Maybe<&2, K>,
  +ub: Maybe<&2, K>, +k: K, +v: V,
  +bt: {Laws.bounded(~K, ~V, ~cmp, t, lb, ub) == True{} : Bool},
  +ea: {Laws.above(~K, ~cmp, lb, k) == True{} : Bool},
  +eb: {Laws.below(~K, ~cmp, ub, k) == True{} : Bool}
) -> {Laws.bounded(~K, ~V, ~cmp, O.OMap.put(~K, ~V, ~cmp, t, k, v), lb, ub) == True{} : Bool}:
  match t:
    case O.OTip{}:
      bn.i(~K, ~V, ~cmp, lb, ub, k, v, O.OTip{}, O.OTip{}, ea, eb, {==}, {==})
    case O.OBin{+s, +key, +val, +lo, +hi}:
      +b_a = bn.a(~K, ~V, ~cmp, lb, ub, key, val, lo, hi, bt)
      +b_b = bn.b(~K, ~V, ~cmp, lb, ub, key, val, lo, hi, bt)
      +b_l = bn.l(~K, ~V, ~cmp, lb, ub, key, val, lo, hi, bt)
      +b_h = bn.h(~K, ~V, ~cmp, lb, ub, key, val, lo, hi, bt)
      cmp.split(O.OMap<K, V>, Bool, x => Laws.bounded(~K, ~V, ~cmp, x, lb, ub), True{},
        cmp(k, key),
        u => O.OMap.balance(K, V, key, val, O.OMap.put(~K, ~V, ~cmp, lo, k, v), hi),
        u => O.OBin{s, key, v, lo, hi},
        u => O.OMap.balance(K, V, key, val, lo, O.OMap.put(~K, ~V, ~cmp, hi, k, v)),
        e => balance.bnd(~K, ~V, ~cmp, ~trans, lb, ub, key, val,
          O.OMap.put(~K, ~V, ~cmp, lo, k, v), hi, b_a, b_b,
          put.bnd(~K, ~V, ~cmp, ~flip, ~trans, lo, lb, Some{key}, k, v, b_l, ea,
            lt_true(cmp(k, key), e)),
          b_h),
        e => bt,
        e => balance.bnd(~K, ~V, ~cmp, ~trans, lb, ub, key, val, lo,
          O.OMap.put(~K, ~V, ~cmp, hi, k, v), b_a, b_b, b_l,
          put.bnd(~K, ~V, ~cmp, ~flip, ~trans, hi, Some{key}, ub, k, v, b_h,
            lt_true(cmp(key, k), gt_flip(~K, ~cmp, ~flip, k, key, e)), eb)))

Uses (20):
  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=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=Flip
  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=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=cmp.split
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=gt_flip
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=lt_true
  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.put