ProofCut.sp1c: law of lawful-fields@0.1.1.0
Import: import lawful-fields@0.1.1.0/proof_cut.bend as ProofCut
Lines 239-247 of lawful-fields@0.1.1.0/proof_cut.bend: https://jend.leeeet.dev/read.txt?path=lawful-fields@0.1.1.0/proof_cut.bend&start_line=239&end_line=247

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

law sp1c:
  for +x: U32
  for +t: List<&2, U32>
  for +sep: U32
  for +b: Bool
  for +hb: {U32.is_eq(x, sep) == b : Bool}
  for +h: {Nat.is_lt(D.run_len(x <> t, sep), List.length(&2, U32, x <> t)) == True{} : Bool}
  for ih: {Nat.is_lt(D.run_len(t, sep), List.length(&2, U32, t)) == True{} : Bool} -> {F.split(t, sep) == List.take(&2, U32, t, D.run_len(t, sep)) <> F.split(List.drop(&2, U32, t, Nat.add(D.run_len(t, sep), 1n)), sep) : List<&2, List<&2, U32>>}
  {F.split(x <> t, sep) == List.take(&2, U32, x <> t, D.run_len(x <> t, sep)) <> F.split(List.drop(&2, U32, x <> t, Nat.add(D.run_len(x <> t, sep), 1n)), sep) : List<&2, 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=Bool.pick
  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.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=Base/base.bend&name=U32.is_eq
  https://jend.leeeet.dev/definition.txt?path=lawful-arrays@0.1.0.0/proof_base.bend&name=subst
  https://jend.leeeet.dev/definition.txt?path=lawful-bytes@0.1.1.0/defs.bend&name=run_len
  https://jend.leeeet.dev/definition.txt?path=lawful-bytes@0.1.1.0/proof_append.bend&name=drop_zero
  https://jend.leeeet.dev/definition.txt?path=lawful-fields@0.1.1.0/defs.bend&name=split
  https://jend.leeeet.dev/definition.txt?path=lawful-fields@0.1.1.0/defs.bend&name=split.fin
  https://jend.leeeet.dev/definition.txt?path=lawful-fields@0.1.1.0/defs.bend&name=split.push