ProofList.ej_step: law of lawful-fields@0.1.1.0
Import: import lawful-fields@0.1.1.0/proof_list.bend as ProofList
Lines 362-371 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=362&end_line=371

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

# One step of ej, with the induction hypothesis for the rest (t is not empty).
law ej_step:
  for +p: List<&2, U32>
  for +t: List<&2, List<&2, U32>>
  for +sep: U32
  for +from: Nat
  for +k2: Nat
  for +h: {Nat.is_lt(1n+k2, List.length(&2, List<&2, U32>, p <> t)) == True{} : Bool}
  for +ih: {List.get(&2, Nat, F.ends(t, Nat.add(Nat.add(from, List.length(&2, U32, p)), 1n)), k2) == Some{Nat.add(Nat.add(Nat.add(from, List.length(&2, U32, p)), 1n), List.length(&2, U32, F.join(List.take(&2, List<&2, U32>, t, 1n+k2), sep)))} : Maybe<&2, Nat>}
  {List.get(&2, Nat, F.ends(p <> t, from), 1n+k2) == Some{Nat.add(from, List.length(&2, U32, F.join(List.take(&2, List<&2, U32>, p <> t, 1n+1n+k2), sep)))} : Maybe<&2, Nat>}

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.get
  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=Maybe
  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=length_append
  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=ends
  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=ends_arith