(* File: In_NGamma.v  (last edited on 27/10/2000) (c) Klaus Weich  *)
(*******************************************************************)
(* The left hand side Gamma of a sequent consists of               *)
(*     work : a list of (arbitray) normalforms                     *)
(*            (to be inserted in the following structures)         *)
(*     ds: a list of disjunctions                                  *)
(*     ni: a list of decorated and undecorated nested implications *)
(*     ai: an AVL tree of atomic implications                      *)
(*     a: an AVL tree of atoms                                     *)
(*******************************************************************)
Require Export Kripke_Trees.
Require Export Normal_Forms.
(************************************************************)
(* disjunctions are stored as pairs of Int's                *)
Definition disj := (Int * Int)%type.
Definition disjs := list disj.
Definition DNil := nil (A:=disj).
Definition disj2form (x : disj) :=
match x with
| (i, j) => OrF (Atom i) (Atom j)
end.
Definition disj2nform (x : disj) := match x with
| (i, j) => NDisj i j
end.
Definition disjs2forms (ds : disjs) := map disj2form ds.
Definition disjs2nforms (ds : disjs) := map disj2nform ds.
(*****************************************************************)
(* Nested implications are stored either as a nimp or as a       *)
(* nimp and a counter-model                                      *)
Inductive nested_imp : Set :=
| Undecorated : nimp -> nested_imp
| Decorated : nimp -> kripke_tree -> nested_imp.
Definition nested_imp2nimp (ni : nested_imp) :=
match ni with
| Undecorated ni => ni
| Decorated ni _ => ni
end.
Definition nested_imp2form (x : nested_imp) := nimp2form (nested_imp2nimp x).
Definition nested_imp2nform (x : nested_imp) := NImp_NF (nested_imp2nimp x).
Definition nested_imps := list nested_imp.
Definition NNil := nil (A:=nested_imp).
Definition nested_imps2forms (ni : nested_imps) := map nested_imp2form ni.
Definition nested_imps2nforms (ni : nested_imps) := map nested_imp2nform ni.
Definition nested_imps2nimps (ni : nested_imps) := map nested_imp2nimp ni.
intros ni1 ni2.
elim ni1; clear ni1.
simpl in |- *; trivial.
intros x ni1 ih.
simpl in |- *.
rewrite ih; trivial.
Qed.

Lemma nested_imps2nimps_length :
forall ni : nested_imps, length ni = length (nested_imps2nimps ni).