Inductive in_ngamma_del_ai_rev_spec (i : Int) (bs : nf_list)
(c : normal_form) : Set :=
    In_NGamma_Del_AI_Rev_Spec_Intro :
      forall (b : normal_form) (n : nat),
      my_nth normal_form n bs b ->
      c = AImp i b -> in_ngamma_del_ai_rev_spec i bs c.Inductive in_ngamma_del_ai_rev_spec (i : Int) (bs : nf_list)
(c : normal_form) : Set :=
    In_NGamma_Del_AI_Rev_Spec_Intro :
      forall (b : normal_form) (n : nat),
      my_nth normal_form n bs b ->
      c = AImp i b -> in_ngamma_del_ai_rev_spec i bs c.

Lemma in_ngamma_shift_work_ai_old : forall (work : nf_list) (ds : disjs) (ni : nested_imps) (i : Int) (b : normal_form) (bs : nf_list) (ai ai' : atomic_imps) (a : atoms) (c : normal_form), LOOKUP nf_list i ai bs -> EQUIV_INS nf_list i (cons b) nf_nil ai ai' -> in_ngamma (AImp i b :: work) ds ni ai a c -> in_ngamma work ds ni ai' a c.