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[helm.git] / matita / contribs / LOGIC / NTrack / inv.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+
+
+include "NTrack/defs.ma".
+(*
+theorem ntrack_inv_lref: \forall Q,S,i. NTrack Q (lref i) S \to
+                         \exists P. Insert S i P Q.
+ intros; inversion H; clear H; intros; destruct; autobatch.
+qed.
+
+theorem ntrack_inv_parx: \forall P,S,h. NTrack P (parx h) S \to
+                         S = pair (posr h) (posr h).
+ intros; inversion H; clear H; intros; destruct; autobatch.
+qed.
+
+theorem ntrack_inv_impw: \forall P,p,S. NTrack P (impw p) S \to
+                         \exists B,a,b. 
+                         S = pair (impl a b) B \land 
+                         NTrack P p (pair lleaf B).
+ intros; inversion H; clear H; intros; destruct; autobatch depth = 5.
+qed.
+
+theorem ntrack_inv_impr: \forall P,p,S. NTrack P (impr p) S \to
+                         \exists a,b:Formula. 
+                         S = pair lleaf (impl a b) \land
+                         NTrack P p (pair a b).
+ intros; inversion H; clear H; intros; destruct; autobatch depth = 4.
+qed.
+
+theorem ntrack_inv_impi: \forall P,p,q,r,S. NTrack P (impi p q r) S \to
+                         \exists Q,A,B,D,i. \exists a,b:Formula.
+                         S = pair (impl a b) D \land
+                         NTrack P p (pair A a) \land
+                         NTrack P q (pair b B) \land
+                         NTrack Q r (pair lleaf D) \land
+                         Insert (pair A B) i P Q.
+ intros; inversion H; clear H; intros; destruct; autobatch depth = 12 width = 5 size = 16.
+qed.
+
+theorem ntrack_inv_scut: \forall P,q,r,S. NTrack P (scut q r) S \to False.
+ intros; inversion H; clear H; intros; destruct.
+qed.
+
+theorem ntrack_inv_lleaf_impl: 
+   \forall Q,p,a,b. NTrack Q p (pair lleaf (impl a b)) \to
+   (\exists P,i. p = lref i \land Insert (pair lleaf (impl a b)) i P Q) \lor
+   (\exists r. p = impr r \land NTrack Q r (pair a b)).
+ intros; inversion H; clear H; intros; destruct;
+ [ autobatch depth = 5
+ | destruct; autobatch depth = 4
+ ].
+qed.
+*)