LAWS.join_at: law of lawful-fields@0.1.1.0
Import: import lawful-fields@0.1.1.0/LAWS.bend as LAWS
Lines 145-152 of lawful-fields@0.1.1.0/LAWS.bend: https://jend.leeeet.dev/read.txt?path=lawful-fields@0.1.1.0/LAWS.bend&start_line=145&end_line=152

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

# Joining a list of pieces splits at piece k: the first k+1 pieces, the
# separator, the rest.
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 (12):
  https://jend.leeeet.dev/definition.txt?path=Base/base.bend&name=Bool
  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-fields@0.1.1.0/defs.bend&name=join
  https://jend.leeeet.dev/definition.txt?path=lawful-fields@0.1.1.0/proof_list.bend&name=join_at