Inductive cbv : Type :=
| cbv_constr : list cbv -> cache -> term -> cache -> value -> cbv
| cbv_split : list cbv -> cbv -> cache -> term -> cache -> value -> cbv
| cbv_update : nat -> (variable -> value) -> cbv -> cache -> term -> cache -> value -> cbv
| cbv_read : cache -> term -> value -> cbv.Inductive cbv : Type :=
| cbv_constr : list cbv -> cache -> term -> cache -> value -> cbv
| cbv_split : list cbv -> cbv -> cache -> term -> cache -> value -> cbv
| cbv_update : nat -> (variable -> value) -> cbv -> cache -> term -> cache -> value -> cbv
| cbv_read : cache -> term -> value -> cbv.Inductive cbv : Type :=
| cbv_constr : list cbv -> cache -> term -> cache -> value -> cbv
| cbv_split : list cbv -> cbv -> cache -> term -> cache -> value -> cbv
| cbv_update : nat -> (variable -> value) -> cbv -> cache -> term -> cache -> value -> cbv
| cbv_read : cache -> term -> value -> cbv.Inductive cbv : Type :=
| cbv_constr : list cbv -> cache -> term -> cache -> value -> cbv
| cbv_split : list cbv -> cbv -> cache -> term -> cache -> value -> cbv
| cbv_update : nat -> (variable -> value) -> cbv -> cache -> term -> cache -> value -> cbv
| cbv_read : cache -> term -> value -> cbv.Inductive cbv : Type :=
| cbv_constr : list cbv -> cache -> term -> cache -> value -> cbv
| cbv_split : list cbv -> cbv -> cache -> term -> cache -> value -> cbv
| cbv_update : nat -> (variable -> value) -> cbv -> cache -> term -> cache -> value -> cbv
| cbv_read : cache -> term -> value -> cbv.

(* Specifications: *)
Lemma cbv_ind2_gen : forall (P : cbv -> Prop)(Q : list cbv -> Prop), Q nil -> (forall p lp, P p -> Q lp -> Q (p :: lp)) -> (forall lp c1 t c2 v, Q lp -> P (cbv_constr lp c1 t c2 v)) -> (forall lp p c1 t c2 v, Q lp -> P p -> P (cbv_split lp p c1 t c2 v)) -> (forall n s p c1 t c2 v, P p -> P (cbv_update n s p c1 t c2 v)) -> (forall c t v, P (cbv_read c t v)) -> forall p, P p.
Lemma cbv_ind2 : forall (P : cbv -> Prop), (forall lp c1 t c2 v, (forall p, In p lp -> P p) -> P (cbv_constr lp c1 t c2 v)) -> (forall lp p c1 t c2 v, (forall p, In p lp -> P p) -> P p -> P (cbv_split lp p c1 t c2 v)) -> (forall n s p c1 t c2 v, P p -> P (cbv_update n s p c1 t c2 v)) -> (forall c t v, P (cbv_read c t v)) -> forall p, P p.