Definition a_ai_disj (a : atoms) (ai : atomic_imps) :=
  forall i : Int,
  LOOKUP unit i a tt -> forall bs : nf_list, LOOKUP nf_list i ai bs -> False.Definition a_goal_disj (a : atoms) (goal : Int) :=
  LOOKUP unit goal a tt -> False.

Lemma disjs_insert_a : forall (i : Int) (a a' : atoms) (ai : atomic_imps), a_ai_disj a ai -> EQUIV_INS unit i (fun _ : unit => tt) tt a a' -> (forall bs : nf_list, ~ LOOKUP nf_list i ai bs) -> a_ai_disj a' ai.