(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/LOGIC/Track/defs".
+set "baseuri" "cic:/matita/LOGIC/NTrack/defs".
-(* PROOF TREE TRACKS
+(* NORMAL 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)
+inductive NTrack: Context \to Proof \to Sequent \to Prop \def
+ | ntrack_proj: \forall P,Q,S,i. Insert S i P Q \to NTrack Q (lref i) S
+ | ntrack_posr: \forall P,h.
+ NTrack P (parx h) (pair (posr h) (posr h))
+ | ntrack_impw: \forall P,r,D,a,b. NTrack P r (pair lleaf D) \to
+ NTrack P (impw r) (pair (impl a b) D)
+ | ntrack_impr: \forall P,r. \forall a,b:Formula.
+ NTrack P r (pair a b) \to
+ NTrack P (impr r) (pair lleaf (impl a b))
+ | ntrack_impi: \forall P,Q,p,q,r,A,B,D,i. \forall a,b:Formula.
+ NTrack P p (pair A a) \to
+ NTrack P q (pair b B) \to
+ NTrack Q r (pair lleaf D) \to
+ Insert (pair A B) i P Q \to
+ NTrack P (impi p q r) (pair (impl a b) D)
.
(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/LOGIC/Track/inv".
+set "baseuri" "cic:/matita/LOGIC/NTrack/inv".
-include "Track/defs.ma".
+include "NTrack/defs.ma".
-theorem track_inv_lref: \forall Q,S,i. Track Q (lref i) S \to
- \exists P. Insert S i P Q.
+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; subst; autobatch.
qed.
-theorem track_inv_parx: \forall P,S,h. Track P (parx h) S \to
- S = pair (posr h) (posr h).
+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; 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).
+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; 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).
+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; 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.
+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; 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.
+theorem ntrack_inv_scut: \forall P,q,r,S. NTrack 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
+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 Track Q r (pair a b)).
+ (\exists r. p = impr r \land NTrack Q r (pair a b)).
intros; inversion H; clear H; intros; subst;
[ autobatch depth = 5
| subst; autobatch depth = 4
(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/LOGIC/Track/order".
+set "baseuri" "cic:/matita/LOGIC/NTrack/order".
-include "Track/props.ma".
+include "Track/order.ma".
+include "NTrack/props.ma".
(* Order properties *********************************************************)
-theorem track_refl: \forall P. \forall c:Formula. \exists r.
- Track P r (pair c c).
+theorem ntrack_refl: \forall P. \forall c:Formula. \exists r.
+ NTrack P r (pair c c).
intros; elim c; clear c;
[ autobatch
| lapply (insert_total (pair f f1) zero P); [2:autobatch];
].
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;
-*)
+theorem ntrack_trans: \forall p,q,A,B. \forall c:Formula.
+ NTrack leaf p (pair A c) \to NTrack leaf q (pair c B) \to
+ \exists r. NTrack leaf r (pair A B).
+ intros;
+ lapply linear ntrack_track to H as H0;
+ lapply linear ntrack_track to H1 as H;
+ lapply linear track_trans to H0, H as H1; decompose;
*)
(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/LOGIC/Track/props".
+set "baseuri" "cic:/matita/LOGIC/NTrack/props".
include "Insert/props.ma".
-include "Track/inv.ma".
+include "Track/defs.ma".
+include "NTrack/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.
+theorem ntrack_weak: \forall R,p,P,Q,S,i.
+ NTrack P p S \to Insert R i P Q \to
+ \exists q. NTrack Q q S.
intros 2; elim p; clear p;
- [ lapply linear track_inv_lref to H as H0; decompose;
+ [ lapply linear ntrack_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 ntrack_inv_parx to H; subst; autobatch
+ | lapply linear ntrack_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 ntrack_inv_impr to H1; decompose; subst;
lapply linear H to H4, H2; decompose; autobatch
- | lapply linear track_inv_impi to H3; decompose; subst;
+ | lapply linear ntrack_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
+ | lapply linear ntrack_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.
+theorem ntrack_comp: \forall R,q,p,P,Q,S,i.
+ NTrack P p R \to NTrack Q q S \to Insert R i P Q \to
+ \exists r. NTrack P r S.
intros 2; elim q; clear q;
- [ lapply linear track_inv_lref to H1 as H0; decompose;
+ [ lapply linear ntrack_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 ntrack_inv_parx to H1; subst; autobatch
+ | lapply linear ntrack_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 ntrack_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 linear ntrack_inv_impi to H4; decompose; subst;
lapply insert_trans to H5, H7; clear H7; decompose;
- lapply track_weak to H3, H6; decompose;
+ lapply ntrack_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
+ | lapply linear ntrack_inv_scut to H3; decompose
].
qed.
+
+theorem ntrack_track: \forall R,p,P. NTrack P p R \to Track P p R.
+ intros; elim H names 0; clear H P p R; intros; autobatch width = 4.
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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)
+ | track_scut: \forall P,p,q,A,B. \forall c:Formula.
+ Track P p (pair A c) \to
+ Track P q (pair c B) \to
+ Track P (scut p q) (pair A B)
+.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "Insert/fun.ma".
+include "Track/defs.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,p,q,A,B. \forall c:Formula.
+ Track P p (pair A c) \to Track P q (pair c B) \to
+ \exists r. Track P r (pair A B).
+ intros; autobatch.
+qed.
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