--- /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)
+.
--- /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/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.
--- /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 "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;
+*)
+*)
--- /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/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.
+++ /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)
-.
+++ /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/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.
+++ /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 "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;
-*)
-*)
+++ /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/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.
projects / helm.git / commitdiff