PROOF.h.absent_put: def of bend-kit-collections@0.1.4.1
Import: import bend-kit-collections@0.1.4.1/PROOF.bend as PROOF
Lines 2394-2410 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=2394&end_line=2410

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/hmap.bend as M

def h.absent_put(
  ~K: Data, ~V: Data, ~eq: K -> K -> Bool, ~refl: @x: K -> {eq(x, x) == True{} : Bool},
  ~sound: @x: K -> @y: K -> {eq(x, y) == True{} : Bool} -> {x == y : K},
  xs: List<&2, M.Entry<K, V>>, +key: K, +k: K, +v: V,
  ab: {Laws.hmap_absent(~K, ~V, ~eq, xs, key) == True{} : Bool},
  +nk: {eq(k, key) == False{} : Bool}
) -> {Laws.hmap_absent(~K, ~V, ~eq, M.HMap.entries.put(~K, ~V, ~eq, xs, k, v), key) == True{} : Bool}:
  match xs:
    case Nil{}:
      %h.sb(eq(key, k), False{}, h.eq_sym_f(~K, ~eq, ~refl, ~sound, k, key, nk))
        : {Bool.and(Bool.not(_), True{}) == True{} : Bool}
      {==}
    case Con{M.Entry{+key2, +val}, +rest}:
      +ab = ab
      h.absent_put.c(~K, ~V, ~eq, key2, val, rest, key, k, v, eq(k, key2), ab,
        h.absent_put(~K, ~V, ~eq, ~refl, ~sound, rest, key, k, v,
          MBool.eq_true_of_and_right(Bool.not(eq(key, key2)), Laws.hmap_absent(~K, ~V, ~eq, rest, key), ab), nk))

Uses (11):
  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=Bool.not
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/LAWS.bend&name=hmap_absent
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=h.absent_put.c
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=h.eq_sym_f
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/PROOF.bend&name=h.sb
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/hmap.bend&name=Entry
  https://jend.leeeet.dev/definition.txt?path=bend-kit-collections@0.1.4.1/hmap.bend&name=HMap.entries.put
  https://jend.leeeet.dev/definition.txt?path=bend-mathlib@0.8.1.0/bool.bend&name=eq_true_of_and_right