PROOF.rot_lo.double.get2: def of bend-kit-collections@0.1.4.1
Import: import bend-kit-collections@0.1.4.1/PROOF.bend as PROOF
Lines 653-693 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=653&end_line=693

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

def rot_lo.double.get2(
  ~K: Data, ~V: Data, ~cmp: K -> K -> Cmp, ~refl: Refl(~K, ~cmp), ~eq: Eq(~K, ~cmp),
  ~flip: Flip(~K, ~cmp), ~trans: Trans(~K, ~cmp),
  +x: K, +k: K, +v: V, +lo: O.OMap<K, V>, +hk: K, +hv: V, +mk: K, +mv: V,
  +ml: O.OMap<K, V>, +mh: O.OMap<K, V>, +hh: O.OMap<K, V>,
  +khm: {cmp(k, mk) == LT{} : Cmp}, +mhk: {cmp(mk, hk) == LT{} : Cmp}
) -> {g(~K, ~V, ~cmp, O.OMap.bin(K, V, mk, mv, O.OMap.bin(K, V, k, v, lo, ml),
        O.OMap.bin(K, V, hk, hv, mh, hh)), x)
      == g(~K, ~V, ~cmp, O.OBin{0, k, v, lo, O.OBin{0, hk, hv, O.OBin{0, mk, mv, ml, mh}, hh}}, x)
      : Maybe<&2, V>}:
  +a = {O.OBin{0, k, v, lo, ml} : O.OMap<K, V>}
  +b = {O.OBin{0, hk, hv, mh, hh} : O.OMap<K, V>}
  +l = {O.OBin{0, mk, mv, a, b} : O.OMap<K, V>}
  +m = {O.OBin{0, mk, mv, ml, mh} : O.OMap<K, V>}
  +r1 = {O.OBin{0, hk, hv, m, hh} : O.OMap<K, V>}
  split(c => {g(~K, ~V, ~cmp, l, x) == getc(~K, ~V, ~cmp, c, x, v, lo, r1) : Maybe<&2, V>},
    cmp(x, k),
    +e => mtr(V, g(~K, ~V, ~cmp, l, x), g(~K, ~V, ~cmp, a, x), g(~K, ~V, ~cmp, lo, x),
      get.lt(~K, ~V, ~cmp, x, mk, mv, a, b, trans(x, k, mk, e, khm)),
      get.lt(~K, ~V, ~cmp, x, k, v, lo, ml, e)),
    e => rw(K, y => {g(~K, ~V, ~cmp, l, y) == Some{v} : Maybe<&2, V>}, x, k, eq(x, k, e),
      mtr(V, g(~K, ~V, ~cmp, l, k), g(~K, ~V, ~cmp, a, k), Some{v},
        get.lt(~K, ~V, ~cmp, k, mk, mv, a, b, khm),
        get.here(~K, ~V, ~cmp, ~refl, k, v, lo, ml))),
    +e => split(c => {g(~K, ~V, ~cmp, l, x) == getc(~K, ~V, ~cmp, c, x, hv, m, hh) : Maybe<&2, V>},
      cmp(x, hk),
      +e2 => split(c => {getc(~K, ~V, ~cmp, c, x, mv, a, b)
          == getc(~K, ~V, ~cmp, c, x, mv, ml, mh) : Maybe<&2, V>},
        cmp(x, mk),
        e3 => get.gt(~K, ~V, ~cmp, x, k, v, lo, ml, e),
        e3 => {==},
        e3 => get.lt(~K, ~V, ~cmp, x, hk, hv, mh, hh, e2)),
      e2 => rw(K, y => {g(~K, ~V, ~cmp, l, y) == Some{hv} : Maybe<&2, V>}, x, hk, eq(x, hk, e2),
        mtr(V, g(~K, ~V, ~cmp, l, hk), g(~K, ~V, ~cmp, b, hk), Some{hv},
          get.gt(~K, ~V, ~cmp, hk, mk, mv, a, b, gt_of_lt(~K, ~cmp, ~flip, mk, hk, mhk)),
          get.here(~K, ~V, ~cmp, ~refl, hk, hv, mh, hh))),
      +e2 => mtr(V, g(~K, ~V, ~cmp, l, x), g(~K, ~V, ~cmp, b, x), g(~K, ~V, ~cmp, hh, x),
        get.gt(~K, ~V, ~cmp, x, mk, mv, a, b,
          gt_of_lt(~K, ~cmp, ~flip, mk, x,
            trans(mk, hk, x, mhk, gt_flip(~K, ~cmp, ~flip, x, hk, e2)))),
        get.gt(~K, ~V, ~cmp, x, hk, hv, mh, hh, e2))))

Uses (18):
  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/PROOF.bend&name=Eq
  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=Refl
  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=g
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=get.gt
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=get.here
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=get.lt
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=getc
  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=gt_of_lt
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=mtr
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=rw
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=split
  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