ProofList.join_at: law of lawful-fields@0.1.1.0
Import: import lawful-fields@0.1.1.0/proof_list.bend as ProofList
Lines 320-325 of lawful-fields@0.1.1.0/proof_list.bend: https://jend.leeeet.dev/read.txt?path=lawful-fields@0.1.1.0/proof_list.bend&start_line=320&end_line=325

import Base
import lawful-fields@0.1.1.0/defs.bend as F

law join_at:
  for +ps: List<&2, List<&2, U32>>
  for +sep: U32
  for +k: Nat
  for +h: {Nat.is_lt(Nat.add(k, 1n), List.length(&2, List<&2, U32>, ps)) == True{} : Bool}
  {F.join(ps, sep) == List.append(&2, U32, F.join(List.take(&2, List<&2, U32>, ps, Nat.add(k, 1n)), sep), Con{sep, F.join(List.drop(&2, List<&2, U32>, ps, Nat.add(k, 1n)), sep)}) : List<&2, U32>}

Uses (18):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Empty.absurd
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Equal.sym
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List.append
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List.drop
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List.length
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=List.take
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.add
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Nat.is_lt
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=U32
  https://jend.leeeet.dev/definition.txt?path=lawful-arrays@0.1.0.0/proof_base.bend&name=lt_zero_absurd
  https://jend.leeeet.dev/definition.txt?path=lawful-fields@0.1.1.0/defs.bend&name=join
  https://jend.leeeet.dev/definition.txt?path=lawful-fields@0.1.1.0/defs.bend&name=join.go
  https://jend.leeeet.dev/definition.txt?path=lawful-fields@0.1.1.0/proof_list.bend&name=append_assoc
  https://jend.leeeet.dev/definition.txt?path=lawful-fields@0.1.1.0/proof_list.bend&name=ja_go
  https://jend.leeeet.dev/definition.txt?path=lawful-fields@0.1.1.0/proof_list.bend&name=jat0