(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/LOGIC/Track/inv".
+
include "Track/defs.ma".
+(* Main inversion lemmas ****************************************************)
+
theorem track_inv_lref: \forall Q,S,i. Track Q (lref i) S \to
- \exists P. Insert S i P Q.
- intros; inversion H; clear H; intros; subst; autobatch.
+ \exists p1,p2,P. Insert p1 p2 S i P Q.
+ intros; inversion H; clear H; intros; destruct; autobatch depth = 4.
qed.
-theorem track_inv_parx: \forall P,S,h. Track P (parx h) S \to
+theorem track_inv_prin: \forall P,S,h. Track P (prin h) S \to
S = pair (posr h) (posr h).
- intros; inversion H; clear H; intros; subst; autobatch.
+ intros; inversion H; clear H; intros; destruct; autobatch.
qed.
theorem track_inv_impw: \forall P,p,S. Track P (impw p) S \to
\exists B,a,b.
S = pair (impl a b) B \land
Track P p (pair lleaf B).
- intros; inversion H; clear H; intros; subst; autobatch depth = 5.
+ intros; inversion H; clear H; intros; destruct; autobatch depth = 5.
qed.
-theorem track_inv_impr: \forall P,p,S. Track P (impr p) S \to
+theorem track_inv_impr: \forall Q,p,S. Track Q (impr p) S \to
\exists a,b:Formula.
S = pair lleaf (impl a b) \land
- Track P p (pair a b).
- intros; inversion H; clear H; intros; subst; autobatch depth = 4.
+ Track Q p (pair a b).
+ intros; inversion H; clear H; intros; destruct; autobatch depth = 4.
qed.
theorem track_inv_impi: \forall P,p,q,r,S. Track P (impi p q r) S \to
- \exists Q,A,B,D,i. \exists a,b:Formula.
+ \exists A,B,D. \exists a,b:Formula.
S = pair (impl a b) D \land
Track P p (pair A a) \land
Track P q (pair b B) \land
- Track Q r (pair lleaf D) \land
- Insert (pair A B) i P Q.
- intros; inversion H; clear H; intros; subst; autobatch depth = 12 width = 5 size = 16.
-qed.
-
-theorem track_inv_scut: \forall P,q,r,S. Track P (scut q r) S \to False.
- intros; inversion H; clear H; intros; subst.
+ Track (abst P p q (pair A B)) r (pair lleaf D).
+ intros; inversion H; clear H; intros; destruct; autobatch depth = 9 width = 4 size = 12.
qed.
-theorem track_inv_lleaf_impl:
- \forall Q,p,a,b. Track 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 Track Q r (pair a b)).
- intros; inversion H; clear H; intros; subst;
- [ autobatch depth = 5
- | subst; autobatch depth = 4
- ].
+theorem track_inv_scut: \forall P,q,r,S. Track P (scut q r) S \to
+ \exists A,B. \exists c:Formula.
+ S = pair A B \land
+ Track P q (pair A c) \land
+ Track P r (pair c B).
+ intros; inversion H; clear H; intros; destruct; autobatch depth = 6 size = 8.
qed.