src/Sequents/S43.thy
author wenzelm
Sun, 11 Jan 2009 21:49:59 +0100
changeset 29450 ac7f67be7f1f
parent 21426 87ac12bed1ab
child 30510 4120fc59dd85
permissions -rw-r--r--
tuned categories;
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
17481
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
     1
(*  Title:      Modal/S43.thy
2073
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
     2
    ID:         $Id$
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
     3
    Author:     Martin Coen
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
     4
    Copyright   1991  University of Cambridge
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
     5
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
     6
This implements Rajeev Gore's sequent calculus for S43.
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
     7
*)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
     8
17481
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
     9
theory S43
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    10
imports Modal0
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    11
begin
2073
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    12
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    13
consts
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    14
  S43pi :: "[seq'=>seq', seq'=>seq', seq'=>seq',
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    15
             seq'=>seq', seq'=>seq', seq'=>seq'] => prop"
14765
bafb24c150c1 proper use of 'syntax';
wenzelm
parents: 2073
diff changeset
    16
syntax
2073
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    17
  "@S43pi" :: "[seq, seq, seq, seq, seq, seq] => prop"
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    18
                         ("S43pi((_);(_);(_);(_);(_);(_))" [] 5)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    19
17481
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    20
ML {*
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    21
  val S43pi  = "S43pi";
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    22
  val SS43pi = "@S43pi";
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    23
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    24
  val tr  = seq_tr;
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    25
  val tr' = seq_tr';
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    26
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    27
  fun s43pi_tr[s1,s2,s3,s4,s5,s6]=
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    28
        Const(S43pi,dummyT)$tr s1$tr s2$tr s3$tr s4$tr s5$tr s6;
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    29
  fun s43pi_tr'[s1,s2,s3,s4,s5,s6] =
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    30
        Const(SS43pi,dummyT)$tr' s1$tr' s2$tr' s3$tr' s4$tr' s5$tr' s6;
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    31
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    32
*}
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    33
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    34
parse_translation {* [(SS43pi,s43pi_tr)] *}
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    35
print_translation {* [(S43pi,s43pi_tr')] *}
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    36
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    37
axioms
2073
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    38
(* Definition of the star operation using a set of Horn clauses  *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    39
(* For system S43: gamma * == {[]P | []P : gamma}                *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    40
(*                 delta * == {<>P | <>P : delta}                *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    41
17481
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    42
  lstar0:         "|L>"
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    43
  lstar1:         "$G |L> $H ==> []P, $G |L> []P, $H"
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    44
  lstar2:         "$G |L> $H ==>   P, $G |L>      $H"
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    45
  rstar0:         "|R>"
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    46
  rstar1:         "$G |R> $H ==> <>P, $G |R> <>P, $H"
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    47
  rstar2:         "$G |R> $H ==>   P, $G |R>      $H"
2073
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    48
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    49
(* Set of Horn clauses to generate the antecedents for the S43 pi rule       *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    50
(* ie                                                                        *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    51
(*           S1...Sk,Sk+1...Sk+m                                             *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    52
(*     ----------------------------------                                    *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    53
(*     <>P1...<>Pk, $G |- $H, []Q1...[]Qm                                    *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    54
(*                                                                           *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    55
(*  where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, $G * |- $H *, []Q1...[]Qm    *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    56
(*    and Sj == <>P1...<>Pk, $G * |- $H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj    *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    57
(*    and 1<=i<=k and k<j<=k+m                                               *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    58
17481
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    59
  S43pi0:         "S43pi $L;; $R;; $Lbox; $Rdia"
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    60
  S43pi1:
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    61
   "[| (S43pi <>P,$L';     $L;; $R; $Lbox;$Rdia);   $L',P,$L,$Lbox |- $R,$Rdia |] ==>
2073
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    62
       S43pi     $L'; <>P,$L;; $R; $Lbox;$Rdia"
17481
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    63
  S43pi2:
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    64
   "[| (S43pi $L';; []P,$R';     $R; $Lbox;$Rdia);  $L',$Lbox |- $R',P,$R,$Rdia |] ==>
2073
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    65
       S43pi $L';;     $R'; []P,$R; $Lbox;$Rdia"
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    66
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    67
(* Rules for [] and <> for S43 *)
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    68
17481
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    69
  boxL:           "$E, P, $F, []P |- $G ==> $E, []P, $F |- $G"
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    70
  diaR:           "$E |- $F, P, $G, <>P ==> $E |- $F, <>P, $G"
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    71
  pi1:
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    72
   "[| $L1,<>P,$L2 |L> $Lbox;  $L1,<>P,$L2 |R> $Ldia;  $R |L> $Rbox;  $R |R> $Rdia;
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    73
      S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==>
2073
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    74
   $L1, <>P, $L2 |- $R"
17481
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    75
  pi2:
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    76
   "[| $L |L> $Lbox;  $L |R> $Ldia;  $R1,[]P,$R2 |L> $Rbox;  $R1,[]P,$R2 |R> $Rdia;
75166ebb619b converted to Isar theory format;
wenzelm
parents: 14765
diff changeset
    77
      S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==>
2073
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    78
   $L |- $R1, []P, $R2"
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
    79
21426
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    80
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    81
ML {*
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    82
structure S43_Prover = Modal_ProverFun
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    83
(
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    84
  val rewrite_rls = thms "rewrite_rls"
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    85
  val safe_rls = thms "safe_rls"
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    86
  val unsafe_rls = thms "unsafe_rls" @ [thm "pi1", thm "pi2"]
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    87
  val bound_rls = thms "bound_rls" @ [thm "boxL", thm "diaR"]
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    88
  val aside_rls = [thm "lstar0", thm "lstar1", thm "lstar2", thm "rstar0",
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    89
    thm "rstar1", thm "rstar2", thm "S43pi0", thm "S43pi1", thm "S43pi2"]
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    90
)
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    91
*}
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    92
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    93
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    94
method_setup S43_solve = {*
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    95
  Method.no_args (Method.SIMPLE_METHOD
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    96
    (S43_Prover.solve_tac 2 ORELSE S43_Prover.solve_tac 3))
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    97
*} "S4 solver"
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    98
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
    99
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   100
(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   101
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   102
lemma "|- []P --> P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   103
lemma "|- [](P-->Q) --> ([]P-->[]Q)" by S43_solve   (* normality*)
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   104
lemma "|- (P--<Q) --> []P --> []Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   105
lemma "|- P --> <>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   106
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   107
lemma "|-  [](P & Q) <-> []P & []Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   108
lemma "|-  <>(P | Q) <-> <>P | <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   109
lemma "|-  [](P<->Q) <-> (P>-<Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   110
lemma "|-  <>(P-->Q) <-> ([]P--><>Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   111
lemma "|-        []P <-> ~<>(~P)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   112
lemma "|-     [](~P) <-> ~<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   113
lemma "|-       ~[]P <-> <>(~P)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   114
lemma "|-      [][]P <-> ~<><>(~P)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   115
lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   116
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   117
lemma "|- []P | []Q --> [](P | Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   118
lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   119
lemma "|- [](P | Q) --> []P | <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   120
lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   121
lemma "|- [](P | Q) --> <>P | []Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   122
lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   123
lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   124
lemma "|- []P --> <>Q --> <>(P & Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   125
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   126
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   127
(* Theorems of system S4 from Hughes and Cresswell, p.46 *)
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   128
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   129
lemma "|- []A --> A" by S43_solve             (* refexivity *)
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   130
lemma "|- []A --> [][]A" by S43_solve         (* transitivity *)
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   131
lemma "|- []A --> <>A" by S43_solve           (* seriality *)
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   132
lemma "|- <>[](<>A --> []<>A)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   133
lemma "|- <>[](<>[]A --> []A)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   134
lemma "|- []P <-> [][]P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   135
lemma "|- <>P <-> <><>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   136
lemma "|- <>[]<>P --> <>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   137
lemma "|- []<>P <-> []<>[]<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   138
lemma "|- <>[]P <-> <>[]<>[]P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   139
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   140
(* Theorems for system S4 from Hughes and Cresswell, p.60 *)
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   141
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   142
lemma "|- []P | []Q <-> []([]P | []Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   143
lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   144
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   145
(* These are from Hailpern, LNCS 129 *)
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   146
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   147
lemma "|- [](P & Q) <-> []P & []Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   148
lemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   149
lemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   150
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   151
lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   152
lemma "|- []P --> []<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   153
lemma "|- <>[]P --> <>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   154
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   155
lemma "|- []P | []Q --> [](P | Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   156
lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   157
lemma "|- [](P | Q) --> []P | <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   158
lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   159
lemma "|- [](P | Q) --> <>P | []Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   160
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   161
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   162
(* Theorems of system S43 *)
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   163
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   164
lemma "|- <>[]P --> []<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   165
lemma "|- <>[]P --> [][]<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   166
lemma "|- [](<>P | <>Q) --> []<>P | []<>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   167
lemma "|- <>[]P & <>[]Q --> <>([]P & []Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   168
lemma "|- []([]P --> []Q) | []([]Q --> []P)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   169
lemma "|- [](<>P --> <>Q) | [](<>Q --> <>P)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   170
lemma "|- []([]P --> Q) | []([]Q --> P)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   171
lemma "|- [](P --> <>Q) | [](Q --> <>P)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   172
lemma "|- [](P --> []Q-->R) | [](P | ([]R --> Q))" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   173
lemma "|- [](P | (Q --> <>C)) | [](P --> C --> <>Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   174
lemma "|- []([]P | Q) & [](P | []Q) --> []P | []Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   175
lemma "|- <>P & <>Q --> <>(<>P & Q) | <>(P & <>Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   176
lemma "|- [](P | Q) & []([]P | Q) & [](P | []Q) --> []P | []Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   177
lemma "|- <>P & <>Q --> <>(P & Q) | <>(<>P & Q) | <>(P & <>Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   178
lemma "|- <>[]<>P <-> []<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   179
lemma "|- []<>[]P <-> <>[]P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   180
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   181
(* These are from Hailpern, LNCS 129 *)
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   182
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   183
lemma "|- [](P & Q) <-> []P & []Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   184
lemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   185
lemma "|- <>(P --> Q) <-> []P --> <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   186
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   187
lemma "|- [](P --> Q) --> <>P --> <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   188
lemma "|- []P --> []<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   189
lemma "|- <>[]P --> <>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   190
lemma "|- []<>[]P --> []<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   191
lemma "|- <>[]P --> <>[]<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   192
lemma "|- <>[]P --> []<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   193
lemma "|- []<>[]P <-> <>[]P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   194
lemma "|- <>[]<>P <-> []<>P" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   195
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   196
lemma "|- []P | []Q --> [](P | Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   197
lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   198
lemma "|- [](P | Q) --> []P | <>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   199
lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   200
lemma "|- [](P | Q) --> <>P | []Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   201
lemma "|- [](P | Q) --> []<>P | []<>Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   202
lemma "|- <>[]P & <>[]Q --> <>(P & Q)" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   203
lemma "|- <>[](P & Q) <-> <>[]P & <>[]Q" by S43_solve
87ac12bed1ab converted legacy ML scripts;
wenzelm
parents: 17481
diff changeset
   204
lemma "|- []<>(P | Q) <-> []<>P | []<>Q" by S43_solve
2073
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
   205
fb0655539d05 New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff changeset
   206
end