inductive CLE: Nat \to Context \to Prop \def
| cle_zero: \forall Q. CLE zero Q
- | cle_succ: \forall Q,R,i. CLE i Q \to CLE (succ i) (abst Q R)
+ | cle_succ: \forall Q,i. CLE i Q \to
+ \forall p1,p2,R. CLE (succ i) (abst Q p1 p2 R)
.
interpretation "order relation between positions and contexts"
(* INSERT RELATION FOR CONTEXTS
*)
+include "Lift/defs.ma".
include "datatypes/Context.ma".
-inductive Insert (S:Sequent): Nat \to Context \to Context \to Prop \def
- | insert_zero: \forall P. Insert S zero P (abst P S)
- | insert_succ: \forall P,Q,R,i.
- Insert S i P Q \to Insert S (succ i) (abst P R) (abst Q R)
+inductive Insert (p,q:Proof) (S:Sequent):
+ Nat \to Context \to Context \to Prop \def
+ | insert_zero: \forall P. Insert p q S zero P (abst P p q S)
+ | insert_succ: \forall P,Q,i. Insert p q S i P Q \to
+ \forall p1,p2. Lift i (succ zero) p1 p2 \to
+ \forall q1,q2. Lift i (succ zero) q1 q2 \to \forall R.
+ Insert p q S (succ i) (abst P p1 q1 R) (abst Q p2 q2 R)
.
include "Insert/inv.ma".
(* Functional properties ****************************************************)
-
+(*
theorem insert_total: \forall S,i,P. i <= P \to \exists Q. Insert S i P Q.
intros 4; elim H; clear H i P;
decompose; autobatch.
| lapply linear insert_inv_succ to H3; decompose; subst; autobatch
].
qed.
+*)
*)
include "Insert/defs.ma".
-
+(*
theorem insert_inv_zero: \forall S,P,Q. Insert S zero P Q \to Q = abst P S.
intros; inversion H; clear H; intros; subst; autobatch.
qed.
| clear H1. lapply linear insert_inv_leaf_2 to H. decompose
].
qed.
+*)
*)
include "Insert/fun.ma".
-
+(*
theorem insert_conf: \forall P,Q1,S1,i1. Insert S1 i1 P Q1 \to
\forall Q2,S2,i2. Insert S2 i2 P Q2 \to
\exists Q,j1,j2.
| autobatch depth = 7 size = 8
].
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/Lift/defs".
+
+(* PROOF RELOCATION
+*)
+
+include "datatypes/Proof.ma".
+
+inductive Lift: Nat \to Nat \to Proof \to Proof \to Prop \def
+ | lift_lref_lt: \forall jd,jh,i. i < jd \to Lift jd jh (lref i) (lref i)
+ | lift_lref_ge: \forall jd,jh,i1,i2. jd <= i1 \to (i1 + jh == i2) \to
+ Lift jd jh (lref i1) (lref i2)
+ | lift_prin : \forall jd,jh,h. Lift jd jh (prin h) (prin h)
+ | lift_impw : \forall jd,jh,p1,p2.
+ Lift jd jh p1 p2 \to Lift jd jh (impw p1) (impw p2)
+ | lift_impr : \forall jd,jh,p1,p2.
+ Lift jd jh p1 p2 \to Lift jd jh (impr p1) (impr p2)
+ | lift_impi : \forall jd,jh,p1,p2. Lift jd jh p1 p2 \to
+ \forall q1,q2. Lift jd jh q1 q2 \to
+ \forall r1,r2. Lift (succ jd) jh r1 r2 \to
+ Lift jd jh (impi p1 q1 r1) (impi p2 q2 r2)
+ | lift_scut : \forall jd,jh,p1,p2. Lift jd jh p1 p2 \to
+ \forall q1,q2. Lift jd jh q1 q2 \to
+ Lift jd jh (scut p1 q1) (scut p2 q2)
+.
(* NORMAL PROOF TREE TRACKS
*)
-include "datatypes/Proof.ma".
include "Insert/defs.ma".
-
+(*
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.
Insert (pair A B) i P Q \to
NTrack P (impi p q r) (pair (impl a b) D)
.
+*)
set "baseuri" "cic:/matita/LOGIC/NTrack/inv".
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; subst; autobatch.
| subst; autobatch depth = 4
].
qed.
+*)
include "NTrack/props.ma".
(* Order properties *********************************************************)
-
+(*
theorem ntrack_refl: \forall P. \forall c:Formula. \exists r.
NTrack P r (pair c c).
intros; elim c; clear c;
lapply linear ntrack_track to H1 as H;
lapply linear track_trans to H0, H as H1; decompose;
*)
+*)
include "Insert/props.ma".
include "Track/defs.ma".
include "NTrack/inv.ma".
-
+(*
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.
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.
+*)
*)
include "PRed/defs.ma".
-
+(*
inductive PNF: Proof \to Prop \def
| pnf: \forall p. (\forall q. p => q \to p = q) \to PNF p
.
+*)
For cut elimination
*)
-include "datatypes/Proof.ma".
-
-inductive PRed: Proof \to Proof \to Prop \def
- | pred_lref: \forall i. PRed (lref i) (lref i)
- | pred_parx: \forall h. PRed (parx h) (parx h)
- | pred_impw: \forall p1,p2. PRed p1 p2 \to PRed (impw p1) (impw p2)
- | pred_impr: \forall p1,p2. PRed p1 p2 \to PRed (impr p1) (impr p2)
- | pred_impi: \forall p1,p2. PRed p1 p2 \to \forall q1,q2. PRed q1 q2 \to
- \forall r1,r2. PRed r1 r2 \to
- PRed (impi p1 q1 r1) (impi p2 q2 r2)
- | pred_scut: \forall p1,p2. PRed p1 p2 \to \forall q1,q2. PRed q1 q2 \to
- PRed (scut p1 q1) (scut p2 q2)
+include "Insert/defs.ma".
+
+inductive PRed: Context \to Proof \to Sequent \to
+ Context \to Proof \to Sequent \to Prop \def
+ | pred_lref: \forall P,i,S. PRed P (lref i) S P (lref i) S
+ | pred_prin: \forall P,h,S. PRed P (prin h) S P (prin h) S
+ | pred_impw: \forall Q1,Q2,p1,p2. (\forall S. PRed Q1 p1 S Q2 p2 S) \to
+ \forall S. PRed Q1 (impw p1) S Q2 (impw p2) S
+ | pred_impr: \forall Q1,Q2,p1,p2. (\forall S. PRed Q1 p1 S Q2 p2 S) \to
+ \forall S. PRed Q1 (impr p1) S Q2 (impr p2) S
+ | pred_impi: \forall Q1,Q2,p1,p2. (\forall S. PRed Q1 p1 S Q2 p2 S) \to
+ \forall q1,q2. (\forall S. PRed Q1 q1 S Q2 q2 S) \to
+ \forall r1,r2. (\forall S,R. PRed (abst Q1 p1 q1 R) r1 S (abst Q2 p2 q2 R) r2 S) \to
+ \forall S. PRed Q1 (impi p1 q1 r1) S Q2 (impi p2 q2 r2) S
+ | pred_scut: \forall Q1,Q2,p1,p2. (\forall S. PRed Q1 p1 S Q2 p2 S) \to
+ \forall q1,q2. (\forall S. PRed Q1 q1 S Q2 q2 S) \to
+ \forall S. PRed Q1 (scut p1 q1) S Q2 (scut p2 q2) S
+ | pred_a_sx: \forall p1,p2,R,P,Q1,i. Insert p1 p2 R i P Q1 \to
+ \forall q0,S,Q2. Insert p1 (scut p2 q0) S i P Q2 \to
+ \forall q. Lift zero i q0 q \to
+ PRed Q1 (scut (lref i) q) S Q2 (lref i) S
+ | pred_a_dx: \forall q1,q2,R,P,Q1,i. Insert q1 q2 R i P Q1 \to
+ \forall p0,S,Q2. Insert (scut p0 q1) q2 S i P Q2 \to
+ \forall p. Lift zero i p0 p \to
+ PRed Q1 (scut p (lref i)) S Q2 (lref i) S
+ | pred_b_sx: \forall q1,q2,P1,P2,S. PRed P1 q1 S P2 q2 S \to
+ \forall h. PRed P1 (scut (prin h) q1) S P2 q2 S
+ | pred_b_dx: \forall p1,p2,Q1,Q2,S. PRed Q1 p1 S Q2 p2 S \to
+ \forall h. PRed Q1 (scut p1 (prin h)) S Q2 p2 S
.
(*CSC: the URI must disappear: there is a bug now *)
interpretation
"single step parallel reduction in B->"
- 'parred x y = (cic:/matita/LOGIC/PRed/defs/PRed.ind#xpointer(1/1) x y)
+ 'parred3 x1 y1 z1 x2 y2 z2 =
+ (cic:/matita/LOGIC/PRed/defs/PRed.ind#xpointer(1/1) x1 y1 z1 x2 y2 z2)
.
(* 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_proj: \forall P,Q,p1,p2,S,i.
+ Insert p1 p2 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 P (prin 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_impi: \forall P,p,q,r,A,B,D. \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 (abst P p q (pair A B)) r (pair lleaf D) \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
--- /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".
+
+(* Main inversion lemmas ****************************************************)
+
+theorem track_inv_lref: \forall Q,S,i. Track Q (lref i) S \to
+ \exists p1,p2,P. Insert p1 p2 S i P Q.
+ intros; inversion H; clear H; intros; subst; autobatch depth = 4.
+qed.
+
+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.
+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 Q,p,S. Track Q (impr p) S \to
+ \exists a,b:Formula.
+ S = pair lleaf (impl a b) \land
+ Track Q 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 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 (abst P p q (pair A B)) r (pair lleaf D).
+ intros; inversion H; clear H; intros; subst; autobatch depth = 9 width = 4 size = 12.
+qed.
+
+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; subst; autobatch depth = 6 size = 8.
+qed.
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;
\exists r. Track P r (pair A B).
intros; autobatch.
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/pred".
+
+(**)
+
+include "Track/inv.ma".
+include "PRed/defs.ma".
+
+theorem track_pred: \forall Q1,Q2,p1,p2,S1,S2. PRed Q1 p1 S1 Q2 p2 S2 \to
+ Track Q1 p1 S1 \to Track Q2 p2 S2.
+ intros 7; elim H; clear H Q1 Q2 p1 p2 S1 S2;
+ [ autobatch
+ | autobatch
+ | lapply linear track_inv_impw to H3; decompose; subst; autobatch
+ | lapply linear track_inv_impr to H3; decompose; subst; autobatch
+ | lapply linear track_inv_impi to H7; decompose; subst; autobatch size = 7
+ | lapply linear track_inv_scut to H5; decompose; subst; autobatch
+ | lapply linear track_inv_scut to H4; decompose; subst;
+ lapply linear track_inv_lref to H6; decompose; autobatch
+ | lapply linear track_inv_scut to H4; decompose; subst;
+ lapply linear track_inv_lref to H5; decompose; autobatch
+ | lapply linear track_inv_scut to H3; decompose; subst;
+ lapply linear track_inv_prin to H5; subst; autobatch
+ | lapply linear track_inv_scut to H3; decompose; subst;
+ lapply linear track_inv_prin to H4; subst; autobatch
+ ].
+qed.
- contexts: P Q
*)
+include "datatypes/Proof.ma".
include "datatypes/Sequent.ma".
inductive Context: Type \def
| leaf: Context
- | abst: Context \to Sequent \to Context
+ | abst: Context \to Proof \to Proof \to Sequent \to Context
.
-
+(*
definition inj: Sequent \to Context \def abst leaf.
coercion cic:/matita/LOGIC/datatypes/Context/inj.con.
+*)
inductive Proof: Type \def
| lref: Nat \to Proof (* projection *)
- | parx: Nat \to Proof (* *)
+ | prin: Nat \to Proof (* pos rel inhabitant *)
| impw: Proof \to Proof (* weakening *)
| impr: Proof \to Proof (* right introduction *)
| impi: Proof \to Proof \to Proof \to Proof (* left introduction *)
(* PREAMBLE
*)
-include "../RELATIONAL/datatypes/Nat.ma".
+include "../RELATIONAL/NLE/defs.ma".
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