(*****************************************************************)


Inductive eqv_ni : nested_imps -> nested_imps -> Prop :=
  | Eqv_NI_Nil : eqv_ni NNil NNil
  | Eqv_NI_Cons_NN :
      forall (x : nimp) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Undecorated x :: ni1) (Undecorated x :: ni2)
  | Eqv_NI_Cons_DN :
      forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Decorated x k :: ni1) (Undecorated x :: ni2)
  | Eqv_NI_Cons_DD :
      forall (x : nimp) (k k' : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Decorated x k :: ni1) (Decorated x k' :: ni2)
  | Eqv_NI_Cons_ND :
      forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Undecorated x :: ni1) (Decorated x k :: ni2).(*****************************************************************)


Inductive eqv_ni : nested_imps -> nested_imps -> Prop :=
  | Eqv_NI_Nil : eqv_ni NNil NNil
  | Eqv_NI_Cons_NN :
      forall (x : nimp) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Undecorated x :: ni1) (Undecorated x :: ni2)
  | Eqv_NI_Cons_DN :
      forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Decorated x k :: ni1) (Undecorated x :: ni2)
  | Eqv_NI_Cons_DD :
      forall (x : nimp) (k k' : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Decorated x k :: ni1) (Decorated x k' :: ni2)
  | Eqv_NI_Cons_ND :
      forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Undecorated x :: ni1) (Decorated x k :: ni2).(*****************************************************************)


Inductive eqv_ni : nested_imps -> nested_imps -> Prop :=
  | Eqv_NI_Nil : eqv_ni NNil NNil
  | Eqv_NI_Cons_NN :
      forall (x : nimp) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Undecorated x :: ni1) (Undecorated x :: ni2)
  | Eqv_NI_Cons_DN :
      forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Decorated x k :: ni1) (Undecorated x :: ni2)
  | Eqv_NI_Cons_DD :
      forall (x : nimp) (k k' : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Decorated x k :: ni1) (Decorated x k' :: ni2)
  | Eqv_NI_Cons_ND :
      forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Undecorated x :: ni1) (Decorated x k :: ni2).(*****************************************************************)


Inductive eqv_ni : nested_imps -> nested_imps -> Prop :=
  | Eqv_NI_Nil : eqv_ni NNil NNil
  | Eqv_NI_Cons_NN :
      forall (x : nimp) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Undecorated x :: ni1) (Undecorated x :: ni2)
  | Eqv_NI_Cons_DN :
      forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Decorated x k :: ni1) (Undecorated x :: ni2)
  | Eqv_NI_Cons_DD :
      forall (x : nimp) (k k' : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Decorated x k :: ni1) (Decorated x k' :: ni2)
  | Eqv_NI_Cons_ND :
      forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Undecorated x :: ni1) (Decorated x k :: ni2).(*****************************************************************)


Inductive eqv_ni : nested_imps -> nested_imps -> Prop :=
  | Eqv_NI_Nil : eqv_ni NNil NNil
  | Eqv_NI_Cons_NN :
      forall (x : nimp) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Undecorated x :: ni1) (Undecorated x :: ni2)
  | Eqv_NI_Cons_DN :
      forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Decorated x k :: ni1) (Undecorated x :: ni2)
  | Eqv_NI_Cons_DD :
      forall (x : nimp) (k k' : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Decorated x k :: ni1) (Decorated x k' :: ni2)
  | Eqv_NI_Cons_ND :
      forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Undecorated x :: ni1) (Decorated x k :: ni2).(*****************************************************************)


Inductive eqv_ni : nested_imps -> nested_imps -> Prop :=
  | Eqv_NI_Nil : eqv_ni NNil NNil
  | Eqv_NI_Cons_NN :
      forall (x : nimp) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Undecorated x :: ni1) (Undecorated x :: ni2)
  | Eqv_NI_Cons_DN :
      forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Decorated x k :: ni1) (Undecorated x :: ni2)
  | Eqv_NI_Cons_DD :
      forall (x : nimp) (k k' : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Decorated x k :: ni1) (Decorated x k' :: ni2)
  | Eqv_NI_Cons_ND :
      forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps),
      eqv_ni ni1 ni2 -> eqv_ni (Undecorated x :: ni1) (Decorated x k :: ni2).

(* Specifications: *)
Lemma eqv_ni_trans : forall ni1 ni2 ni3 : nested_imps, eqv_ni ni1 ni2 -> eqv_ni ni2 ni3 -> eqv_ni ni1 ni3.
Lemma le_eqv : forall ni1 ni2 : nested_imps, le_ni ni1 ni2 -> eqv_ni ni1 ni2.
Lemma eqv_sym : forall ni1 ni2 : nested_imps, eqv_ni ni1 ni2 -> eqv_ni ni2 ni1.
Lemma ge_eqv : forall ni1 ni2 : nested_imps, le_ni ni2 ni1 -> eqv_ni ni1 ni2.
Lemma eqv_ni_rec : forall P : nested_imps -> nested_imps -> Set, P NNil NNil -> (forall (x : nimp) (ni1 ni2 : nested_imps), eqv_ni ni1 ni2 -> P ni1 ni2 -> P (Undecorated x :: ni1) (Undecorated x :: ni2)) -> (forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps), eqv_ni ni1 ni2 -> P ni1 ni2 -> P (Decorated x k :: ni1) (Undecorated x :: ni2)) -> (forall (x : nimp) (k k' : kripke_tree) (ni1 ni2 : nested_imps), eqv_ni ni1 ni2 -> P ni1 ni2 -> P (Decorated x k :: ni1) (Decorated x k' :: ni2)) -> (forall (x : nimp) (k : kripke_tree) (ni1 ni2 : nested_imps), eqv_ni ni1 ni2 -> P ni1 ni2 -> P (Undecorated x :: ni1) (Decorated x k :: ni2)) -> forall ni1 ni2 : nested_imps, eqv_ni ni1 ni2 -> P ni1 ni2.