From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sun, 26 Aug 2007 16:36:34 +0000 (+0000)
Subject: refactoring
X-Git-Tag: make_still_working~6103
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=4f1bd2790a4448a8ebfbe67eb8baa481c124745c;p=helm.git

refactoring
---

diff --git a/helm/software/matita/contribs/LOGIC/NTrack/defs.ma b/helm/software/matita/contribs/LOGIC/NTrack/defs.ma
new file mode 100644
index 000000000..a9c4479ac
--- /dev/null
+++ b/helm/software/matita/contribs/LOGIC/NTrack/defs.ma
@@ -0,0 +1,38 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/LOGIC/Track/defs".
+
+(* PROOF TREE TRACKS
+*)
+
+include "datatypes/Proof.ma".
+include "Insert/defs.ma".
+
+inductive Track: Context \to Proof \to Sequent \to Prop \def
+   | track_proj: \forall P,Q,S,i. Insert S i P Q \to Track Q (lref i) S
+   | track_posr: \forall P,h.
+                 Track P (parx h) (pair (posr h) (posr h))
+   | track_impw: \forall P,r,D,a,b. Track P r (pair lleaf D) \to
+                 Track P (impw r) (pair (impl a b) D)
+   | track_impr: \forall P,r. \forall a,b:Formula. 
+                 Track P r (pair a b) \to 
+                 Track P (impr r) (pair lleaf (impl a b))
+   | track_impi: \forall P,Q,p,q,r,A,B,D,i. \forall a,b:Formula.
+                 Track P p (pair A a) \to
+                 Track P q (pair b B) \to
+		 Track Q r (pair lleaf D) \to
+		 Insert (pair A B) i P Q \to
+		 Track P (impi p q r) (pair (impl a b) D)
+.
diff --git a/helm/software/matita/contribs/LOGIC/NTrack/inv.ma b/helm/software/matita/contribs/LOGIC/NTrack/inv.ma
new file mode 100644
index 000000000..347d7fccb
--- /dev/null
+++ b/helm/software/matita/contribs/LOGIC/NTrack/inv.ma
@@ -0,0 +1,65 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/LOGIC/Track/inv".
+
+include "Track/defs.ma".
+
+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.
+qed.
+
+theorem track_inv_parx: \forall P,S,h. Track P (parx h) S \to
+                        S = pair (posr h) (posr h).
+ intros; inversion H; clear H; intros; subst; 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.
+qed.
+
+theorem track_inv_impr: \forall P,p,S. Track P (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.
+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.
+                        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.
+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
+ ].
+qed.
diff --git a/helm/software/matita/contribs/LOGIC/NTrack/order.ma b/helm/software/matita/contribs/LOGIC/NTrack/order.ma
new file mode 100644
index 000000000..de6ebaf71
--- /dev/null
+++ b/helm/software/matita/contribs/LOGIC/NTrack/order.ma
@@ -0,0 +1,103 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/LOGIC/Track/order".
+
+include "Track/props.ma".
+
+(* Order properties *********************************************************)
+
+theorem track_refl: \forall P. \forall c:Formula. \exists r. 
+                    Track P r (pair c c).
+ intros; elim c; clear c;
+ [ autobatch
+ | lapply (insert_total (pair f f1) zero P); [2:autobatch];
+   decompose; autobatch depth = 5 width = 4 size = 8
+ ].
+qed.
+(*
+theorem track_trans: \forall p,q,A,B. \forall c:Formula.
+                     Track leaf p (pair A c) \to Track leaf q (pair c B) \to
+                     \exists r. Track leaf r (pair A B).
+ intros 1; elim p names 0; clear p;
+ [ intros; clear H1;
+   lapply linear track_inv_lref to H; decompose;
+   lapply insert_inv_leaf_2 to H; decompose
+ | intros;
+   lapply linear track_inv_parx to H; subst; autobatch;
+ | intros;
+   lapply linear track_inv_impw to H1;
+   decompose; subst;
+   lapply linear H to H4, H2; decompose; autobatch
+ | intros 3; elim q; clear q;
+   [ clear H H1;
+     lapply linear track_inv_lref to H2; decompose;
+     lapply insert_inv_leaf_2 to H; decompose
+   | lapply linear track_inv_parx to H2; subst; autobatch
+   | clear H H1;
+     lapply linear track_inv_impr to H2;
+     lapply linear track_inv_impw to H3;
+     decompose; subst; autobatch
+   | clear H H1 H2;
+     lapply linear track_inv_impr to H3;
+     decompose; subst
+   | clear H2 H3;
+     lapply linear track_inv_impr to H4;
+     lapply linear track_inv_impi to H5;
+     decompose; subst;
+     lapply linear H to H5, H8; decompose;
+     
+     
+     lapply linear insert_inv_leaf_1 to H4; decompose; subst;
+     apply ex_intro; [| ]
+     
+     autobatch depth = 100 width = 40 size = 100
+   ]
+ | intros 3; elim q; clear q;
+   [ clear H H1;
+     lapply linear track_inv_lref to H2; decompose;
+     lapply insert_inv_leaf_2 to H; decompose 
+   | clear H;
+     lapply linear track_inv_parx to H2;
+     subst; autobatch
+   | clear H H1;
+     lapply linear track_inv_impe to H2;
+     lapply linear track_inv_impw to H3;
+     decompose; subst; autobatch
+   | clear H H1 H2;
+     lapply linear track_inv_impi to H3; decompose; subst
+   | clear H H1;
+     lapply linear track_inv_impe to H2;
+     lapply linear track_inv_impe to H3;
+     decompose; subst;
+     lapply linear track_inv_lleaf_impl to H5; decompose; subst;
+     [ clear H4 H6;
+       lapply linear insert_inv_leaf_1 to H3; decompose; subst;
+       lapply linear insert_inv_abst_2 to H2; decompose; subst
+     | lapply insert_conf to H3, H4; clear H4; decompose;
+       lapply linear track_weak to H6, H4; decompose;
+       lapply linear track_comp to H2, H, H1; decompose; autobatch
+     ]
+   ]
+ ].
+qed.    
+(*   
+   | lapply linear track_inv_impi to H4;
+     lapply linear track_inv_impe to H5;
+     decompose; subst;
+     destruct H4; destruct H5; clear H4 H5; subst;
+     unfold linj in Hcut; unfold rinj in Hcut3; destruct Hcut; destruct Hcut3; clear Hcut Hcut3; subst;
+     destruct Hcut2; clear Hcut2; subst;
+*)
+*)
diff --git a/helm/software/matita/contribs/LOGIC/NTrack/props.ma b/helm/software/matita/contribs/LOGIC/NTrack/props.ma
new file mode 100644
index 000000000..fc57cf006
--- /dev/null
+++ b/helm/software/matita/contribs/LOGIC/NTrack/props.ma
@@ -0,0 +1,57 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/LOGIC/Track/props".
+
+include "Insert/props.ma".
+include "Track/inv.ma".
+
+theorem track_weak: \forall R,p,P,Q,S,i. 
+                    Track P p S \to Insert R i P Q \to 
+                    \exists q. Track Q q S.
+ intros 2; elim p; clear p;
+ [ lapply linear track_inv_lref to H as H0; decompose;
+   lapply linear insert_trans to H, H1; decompose; autobatch
+ | lapply linear track_inv_parx to H; subst; autobatch
+ | lapply linear track_inv_impw to H1; decompose; subst;
+   lapply linear H to H4, H2; decompose; autobatch
+ | lapply linear track_inv_impr to H1; decompose; subst;
+   lapply linear H to H4, H2; decompose; autobatch
+ | lapply linear track_inv_impi to H3; decompose; subst;
+   lapply insert_conf to H4, H6; clear H6; decompose;
+   lapply H to H9, H4; clear H H9; lapply linear H1 to H8, H4;
+   lapply linear H2 to H7, H6; decompose; autobatch width = 4
+ | lapply linear track_inv_scut to H2; decompose
+ ]
+qed.
+
+theorem track_comp: \forall R,q,p,P,Q,S,i. 
+                    Track P p R \to Track Q q S \to Insert R i P Q \to
+                    \exists r. Track P r S.
+ intros 2; elim q; clear q;
+ [ lapply linear track_inv_lref to H1 as H0; decompose;
+   lapply linear insert_conf_rev_full to H1,H2; decompose; subst; autobatch
+ | lapply linear track_inv_parx to H1; subst; autobatch
+ | lapply linear track_inv_impw to H2; decompose; subst;
+   lapply linear H to H1, H5, H3; decompose; autobatch
+ | lapply linear track_inv_impr to H2; decompose; subst;
+   lapply linear H to H1, H5, H3; decompose; autobatch
+ | lapply linear track_inv_impi to H4; decompose; subst;
+   lapply insert_trans to H5, H7; clear H7; decompose;
+   lapply track_weak to H3, H6; decompose;
+   lapply H to H3, H10, H5; clear H H10; lapply linear H1 to H3, H9, H5;
+   lapply linear H2 to H4, H8, H7; decompose; autobatch width = 4
+ | lapply linear track_inv_scut to H3; decompose
+ ].
+qed.
diff --git a/helm/software/matita/contribs/LOGIC/Track/defs.ma b/helm/software/matita/contribs/LOGIC/Track/defs.ma
deleted file mode 100644
index a9c4479ac..000000000
--- a/helm/software/matita/contribs/LOGIC/Track/defs.ma
+++ /dev/null
@@ -1,38 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/LOGIC/Track/defs".
-
-(* PROOF TREE TRACKS
-*)
-
-include "datatypes/Proof.ma".
-include "Insert/defs.ma".
-
-inductive Track: Context \to Proof \to Sequent \to Prop \def
-   | track_proj: \forall P,Q,S,i. Insert S i P Q \to Track Q (lref i) S
-   | track_posr: \forall P,h.
-                 Track P (parx h) (pair (posr h) (posr h))
-   | track_impw: \forall P,r,D,a,b. Track P r (pair lleaf D) \to
-                 Track P (impw r) (pair (impl a b) D)
-   | track_impr: \forall P,r. \forall a,b:Formula. 
-                 Track P r (pair a b) \to 
-                 Track P (impr r) (pair lleaf (impl a b))
-   | track_impi: \forall P,Q,p,q,r,A,B,D,i. \forall a,b:Formula.
-                 Track P p (pair A a) \to
-                 Track P q (pair b B) \to
-		 Track Q r (pair lleaf D) \to
-		 Insert (pair A B) i P Q \to
-		 Track P (impi p q r) (pair (impl a b) D)
-.
diff --git a/helm/software/matita/contribs/LOGIC/Track/inv.ma b/helm/software/matita/contribs/LOGIC/Track/inv.ma
deleted file mode 100644
index 347d7fccb..000000000
--- a/helm/software/matita/contribs/LOGIC/Track/inv.ma
+++ /dev/null
@@ -1,65 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/LOGIC/Track/inv".
-
-include "Track/defs.ma".
-
-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.
-qed.
-
-theorem track_inv_parx: \forall P,S,h. Track P (parx h) S \to
-                        S = pair (posr h) (posr h).
- intros; inversion H; clear H; intros; subst; 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.
-qed.
-
-theorem track_inv_impr: \forall P,p,S. Track P (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.
-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.
-                        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.
-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
- ].
-qed.
diff --git a/helm/software/matita/contribs/LOGIC/Track/order.ma b/helm/software/matita/contribs/LOGIC/Track/order.ma
deleted file mode 100644
index de6ebaf71..000000000
--- a/helm/software/matita/contribs/LOGIC/Track/order.ma
+++ /dev/null
@@ -1,103 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/LOGIC/Track/order".
-
-include "Track/props.ma".
-
-(* Order properties *********************************************************)
-
-theorem track_refl: \forall P. \forall c:Formula. \exists r. 
-                    Track P r (pair c c).
- intros; elim c; clear c;
- [ autobatch
- | lapply (insert_total (pair f f1) zero P); [2:autobatch];
-   decompose; autobatch depth = 5 width = 4 size = 8
- ].
-qed.
-(*
-theorem track_trans: \forall p,q,A,B. \forall c:Formula.
-                     Track leaf p (pair A c) \to Track leaf q (pair c B) \to
-                     \exists r. Track leaf r (pair A B).
- intros 1; elim p names 0; clear p;
- [ intros; clear H1;
-   lapply linear track_inv_lref to H; decompose;
-   lapply insert_inv_leaf_2 to H; decompose
- | intros;
-   lapply linear track_inv_parx to H; subst; autobatch;
- | intros;
-   lapply linear track_inv_impw to H1;
-   decompose; subst;
-   lapply linear H to H4, H2; decompose; autobatch
- | intros 3; elim q; clear q;
-   [ clear H H1;
-     lapply linear track_inv_lref to H2; decompose;
-     lapply insert_inv_leaf_2 to H; decompose
-   | lapply linear track_inv_parx to H2; subst; autobatch
-   | clear H H1;
-     lapply linear track_inv_impr to H2;
-     lapply linear track_inv_impw to H3;
-     decompose; subst; autobatch
-   | clear H H1 H2;
-     lapply linear track_inv_impr to H3;
-     decompose; subst
-   | clear H2 H3;
-     lapply linear track_inv_impr to H4;
-     lapply linear track_inv_impi to H5;
-     decompose; subst;
-     lapply linear H to H5, H8; decompose;
-     
-     
-     lapply linear insert_inv_leaf_1 to H4; decompose; subst;
-     apply ex_intro; [| ]
-     
-     autobatch depth = 100 width = 40 size = 100
-   ]
- | intros 3; elim q; clear q;
-   [ clear H H1;
-     lapply linear track_inv_lref to H2; decompose;
-     lapply insert_inv_leaf_2 to H; decompose 
-   | clear H;
-     lapply linear track_inv_parx to H2;
-     subst; autobatch
-   | clear H H1;
-     lapply linear track_inv_impe to H2;
-     lapply linear track_inv_impw to H3;
-     decompose; subst; autobatch
-   | clear H H1 H2;
-     lapply linear track_inv_impi to H3; decompose; subst
-   | clear H H1;
-     lapply linear track_inv_impe to H2;
-     lapply linear track_inv_impe to H3;
-     decompose; subst;
-     lapply linear track_inv_lleaf_impl to H5; decompose; subst;
-     [ clear H4 H6;
-       lapply linear insert_inv_leaf_1 to H3; decompose; subst;
-       lapply linear insert_inv_abst_2 to H2; decompose; subst
-     | lapply insert_conf to H3, H4; clear H4; decompose;
-       lapply linear track_weak to H6, H4; decompose;
-       lapply linear track_comp to H2, H, H1; decompose; autobatch
-     ]
-   ]
- ].
-qed.    
-(*   
-   | lapply linear track_inv_impi to H4;
-     lapply linear track_inv_impe to H5;
-     decompose; subst;
-     destruct H4; destruct H5; clear H4 H5; subst;
-     unfold linj in Hcut; unfold rinj in Hcut3; destruct Hcut; destruct Hcut3; clear Hcut Hcut3; subst;
-     destruct Hcut2; clear Hcut2; subst;
-*)
-*)
diff --git a/helm/software/matita/contribs/LOGIC/Track/props.ma b/helm/software/matita/contribs/LOGIC/Track/props.ma
deleted file mode 100644
index fc57cf006..000000000
--- a/helm/software/matita/contribs/LOGIC/Track/props.ma
+++ /dev/null
@@ -1,57 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/LOGIC/Track/props".
-
-include "Insert/props.ma".
-include "Track/inv.ma".
-
-theorem track_weak: \forall R,p,P,Q,S,i. 
-                    Track P p S \to Insert R i P Q \to 
-                    \exists q. Track Q q S.
- intros 2; elim p; clear p;
- [ lapply linear track_inv_lref to H as H0; decompose;
-   lapply linear insert_trans to H, H1; decompose; autobatch
- | lapply linear track_inv_parx to H; subst; autobatch
- | lapply linear track_inv_impw to H1; decompose; subst;
-   lapply linear H to H4, H2; decompose; autobatch
- | lapply linear track_inv_impr to H1; decompose; subst;
-   lapply linear H to H4, H2; decompose; autobatch
- | lapply linear track_inv_impi to H3; decompose; subst;
-   lapply insert_conf to H4, H6; clear H6; decompose;
-   lapply H to H9, H4; clear H H9; lapply linear H1 to H8, H4;
-   lapply linear H2 to H7, H6; decompose; autobatch width = 4
- | lapply linear track_inv_scut to H2; decompose
- ]
-qed.
-
-theorem track_comp: \forall R,q,p,P,Q,S,i. 
-                    Track P p R \to Track Q q S \to Insert R i P Q \to
-                    \exists r. Track P r S.
- intros 2; elim q; clear q;
- [ lapply linear track_inv_lref to H1 as H0; decompose;
-   lapply linear insert_conf_rev_full to H1,H2; decompose; subst; autobatch
- | lapply linear track_inv_parx to H1; subst; autobatch
- | lapply linear track_inv_impw to H2; decompose; subst;
-   lapply linear H to H1, H5, H3; decompose; autobatch
- | lapply linear track_inv_impr to H2; decompose; subst;
-   lapply linear H to H1, H5, H3; decompose; autobatch
- | lapply linear track_inv_impi to H4; decompose; subst;
-   lapply insert_trans to H5, H7; clear H7; decompose;
-   lapply track_weak to H3, H6; decompose;
-   lapply H to H3, H10, H5; clear H H10; lapply linear H1 to H3, H9, H5;
-   lapply linear H2 to H4, H8, H7; decompose; autobatch width = 4
- | lapply linear track_inv_scut to H3; decompose
- ].
-qed.