Inductive Forallz {A} (P: Z -> A->Prop) : Z -> list A -> Prop :=
 | Forallz_nil : forall i, Forallz P i nil
 | Forallz_cons : forall i x l, P i x -> Forallz P (Z.succ i) l -> Forallz P i (x::l).Inductive Forallz {A} (P: Z -> A->Prop) : Z -> list A -> Prop :=
 | Forallz_nil : forall i, Forallz P i nil
 | Forallz_cons : forall i x l, P i x -> Forallz P (Z.succ i) l -> Forallz P i (x::l).Inductive Forallz {A} (P: Z -> A->Prop) : Z -> list A -> Prop :=
 | Forallz_nil : forall i, Forallz P i nil
 | Forallz_cons : forall i x l, P i x -> Forallz P (Z.succ i) l -> Forallz P i (x::l).Definition struct_pred (m: members) {A: ident * type -> Type} (P: forall it, A it -> val -> mpred) (v: compact_prod (map A m)) (p: val): mpred.(* when unfold, do cbv [struct_pred list_rect]. *)

Definition union_pred (m: members) {A: ident * type -> Type} (P: forall it, A it -> val -> mpred) (v: compact_sum (map A m)) (p: val): mpred.Definition array_Prop {A: Type} (d:A) (lo hi: Z) (P: Z -> A -> Prop) (v: list A) : Prop :=
   Zlength v = hi-lo /\ Forallz P 0 v.Definition struct_Prop (m: members) {A: ident * type -> Type}
                             (P: forall it, A it -> Prop) (v: compact_prod (map A m)) : Prop.Definition union_Prop (m: members) {A: ident * type -> Type}
               (P: forall it, A it -> Prop) (v: compact_sum (map A m)): Prop.

Lemma array_pred_len_1: forall {A}{d: Inhabitant A} i P v p, array_pred i (i + 1) P (v :: nil) p = P i v p.