(*
Ltac flatten_emp_in_mpreds RR :=
 match RR with
 | (SEPx (?R)) =>
   match R with context [emp :: ?R'] =>
      let n := constr:(length R - Datatypes.S (length R'))%nat in
      let n' := eval lazy beta zeta iota delta in n in
      erewrite(@flatten_emp_in_mpreds' n' R _ (eq_refl _));
      [ |
        let RR := fresh "RR" in set (RR := R);
        unfold firstn, app, skipn; subst RR; cbv beta iota;
        apply eq_refl
      ]
   end
 end.

Ltac flatten_emp :=
  match goal with
  | |- semax _ ?PQR _ _ => flatten_emp_in_SEP PQR
  | |-  ?PQR |-- _ => first [flatten_emp_in_SEP PQR |
                             flatten_emp_in_mpreds PQR ]
end.*)


Ltac flatten_emp_in_SEP PQR :=
 match PQR with
 | PROPx ?P (LOCALx ?Q (SEPx (?R))) =>
   match R with context [emp :: ?R'] =>
      let n := constr:((length R - Datatypes.S (length R'))%nat) in
      let n' := eval lazy beta zeta iota delta in n in
      erewrite(@flatten_emp_in_SEP' n' P Q R _ (eq_refl _));
      [ |
        let RR := fresh "RR" in set (RR := R);
        unfold firstn, app, skipn; subst RR; cbv beta iota;
        apply eq_refl
      ]
   end
 end.(*Thawing a freezer results in the sepcon product of the frozen items.*)

(*Tactic requires the resulting goal to be normalized manually.*)
Ltac thaw' name :=
  rewrite (FRZL_ax name); unfold name, abbreviate; clear name.(*add simplification of the list operations inside the freezer,
   flatten the sepcon, and eliminate the emp term*)
Ltac thaw name :=
  thaw' name; simpl nat_of_Z; unfold my_delete_nth, my_nth, fold_right_sepcon;
  repeat flatten_sepcon_in_SEP; repeat flatten_emp.(************************ Ramification ************************)

Inductive split_FRZ_in_SEP: list mpred -> list mpred -> list mpred -> Prop :=
| split_FRZ_in_SEP_nil: split_FRZ_in_SEP nil nil nil
| split_FRZ_in_SEP_FRZ: forall R R' RF F, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (FRZ F :: R) R' (FRZ F :: RF)
| split_FRZ_in_SEP_FRZL: forall R R' RF F, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (FRZL F :: R) R' (FRZL F :: RF)
| split_FRZ_in_SEP_FRZR: forall R R' RF L G w, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (@FRZR L G w :: R) R' (@FRZR L G w :: RF)
| split_FRZ_in_SEP_other: forall R R' RF R0, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (R0 :: R) (R0 :: R') RF.(************************ Ramification ************************)

Inductive split_FRZ_in_SEP: list mpred -> list mpred -> list mpred -> Prop :=
| split_FRZ_in_SEP_nil: split_FRZ_in_SEP nil nil nil
| split_FRZ_in_SEP_FRZ: forall R R' RF F, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (FRZ F :: R) R' (FRZ F :: RF)
| split_FRZ_in_SEP_FRZL: forall R R' RF F, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (FRZL F :: R) R' (FRZL F :: RF)
| split_FRZ_in_SEP_FRZR: forall R R' RF L G w, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (@FRZR L G w :: R) R' (@FRZR L G w :: RF)
| split_FRZ_in_SEP_other: forall R R' RF R0, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (R0 :: R) (R0 :: R') RF.(************************ Ramification ************************)

Inductive split_FRZ_in_SEP: list mpred -> list mpred -> list mpred -> Prop :=
| split_FRZ_in_SEP_nil: split_FRZ_in_SEP nil nil nil
| split_FRZ_in_SEP_FRZ: forall R R' RF F, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (FRZ F :: R) R' (FRZ F :: RF)
| split_FRZ_in_SEP_FRZL: forall R R' RF F, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (FRZL F :: R) R' (FRZL F :: RF)
| split_FRZ_in_SEP_FRZR: forall R R' RF L G w, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (@FRZR L G w :: R) R' (@FRZR L G w :: RF)
| split_FRZ_in_SEP_other: forall R R' RF R0, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (R0 :: R) (R0 :: R') RF.(************************ Ramification ************************)

Inductive split_FRZ_in_SEP: list mpred -> list mpred -> list mpred -> Prop :=
| split_FRZ_in_SEP_nil: split_FRZ_in_SEP nil nil nil
| split_FRZ_in_SEP_FRZ: forall R R' RF F, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (FRZ F :: R) R' (FRZ F :: RF)
| split_FRZ_in_SEP_FRZL: forall R R' RF F, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (FRZL F :: R) R' (FRZL F :: RF)
| split_FRZ_in_SEP_FRZR: forall R R' RF L G w, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (@FRZR L G w :: R) R' (@FRZR L G w :: RF)
| split_FRZ_in_SEP_other: forall R R' RF R0, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (R0 :: R) (R0 :: R') RF.(************************ Ramification ************************)

Inductive split_FRZ_in_SEP: list mpred -> list mpred -> list mpred -> Prop :=
| split_FRZ_in_SEP_nil: split_FRZ_in_SEP nil nil nil
| split_FRZ_in_SEP_FRZ: forall R R' RF F, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (FRZ F :: R) R' (FRZ F :: RF)
| split_FRZ_in_SEP_FRZL: forall R R' RF F, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (FRZL F :: R) R' (FRZL F :: RF)
| split_FRZ_in_SEP_FRZR: forall R R' RF L G w, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (@FRZR L G w :: R) R' (@FRZR L G w :: RF)
| split_FRZ_in_SEP_other: forall R R' RF R0, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (R0 :: R) (R0 :: R') RF.(************************ Ramification ************************)

Inductive split_FRZ_in_SEP: list mpred -> list mpred -> list mpred -> Prop :=
| split_FRZ_in_SEP_nil: split_FRZ_in_SEP nil nil nil
| split_FRZ_in_SEP_FRZ: forall R R' RF F, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (FRZ F :: R) R' (FRZ F :: RF)
| split_FRZ_in_SEP_FRZL: forall R R' RF F, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (FRZL F :: R) R' (FRZL F :: RF)
| split_FRZ_in_SEP_FRZR: forall R R' RF L G w, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (@FRZR L G w :: R) R' (@FRZR L G w :: RF)
| split_FRZ_in_SEP_other: forall R R' RF R0, split_FRZ_in_SEP R R' RF -> split_FRZ_in_SEP (R0 :: R) (R0 :: R') RF.Ltac prove_split_FRZ_in_SEP :=
  solve [
    repeat first
    [ simple apply split_FRZ_in_SEP_nil
    | simple apply split_FRZ_in_SEP_FRZ
    | simple apply split_FRZ_in_SEP_FRZL
    | simple apply split_FRZ_in_SEP_FRZR
    | simple apply split_FRZ_in_SEP_other]].

(* Specifications: *)
Lemma split_FRZ_in_SEP_spec: forall R R' RF, split_FRZ_in_SEP R R' RF -> fold_right_sepcon R = fold_right_sepcon R' * fold_right_sepcon RF.