src/Sequents/S4.thy
author wenzelm
Sat Oct 10 20:54:44 2015 +0200 (2015-10-10)
changeset 61386 0a29a984a91b
parent 61385 538100cc4399
permissions -rw-r--r--
more symbols;
     1 (*  Title:      Sequents/S4.thy
     2     Author:     Martin Coen
     3     Copyright   1991  University of Cambridge
     4 *)
     5 
     6 theory S4
     7 imports Modal0
     8 begin
     9 
    10 axiomatization where
    11 (* Definition of the star operation using a set of Horn clauses *)
    12 (* For system S4:  gamma * == {[]P | []P : gamma}               *)
    13 (*                 delta * == {<>P | <>P : delta}               *)
    14 
    15   lstar0:         "|L>" and
    16   lstar1:         "$G |L> $H \<Longrightarrow> []P, $G |L> []P, $H" and
    17   lstar2:         "$G |L> $H \<Longrightarrow>   P, $G |L>      $H" and
    18   rstar0:         "|R>" and
    19   rstar1:         "$G |R> $H \<Longrightarrow> <>P, $G |R> <>P, $H" and
    20   rstar2:         "$G |R> $H \<Longrightarrow>   P, $G |R>      $H" and
    21 
    22 (* Rules for [] and <> *)
    23 
    24   boxR:
    25    "\<lbrakk>$E |L> $E';  $F |R> $F';  $G |R> $G';
    26            $E'         \<turnstile> $F', P, $G'\<rbrakk> \<Longrightarrow> $E          \<turnstile> $F, []P, $G" and
    27   boxL:     "$E,P,$F,[]P \<turnstile>         $G    \<Longrightarrow> $E, []P, $F \<turnstile>          $G" and
    28 
    29   diaR:     "$E          \<turnstile> $F,P,$G,<>P   \<Longrightarrow> $E          \<turnstile> $F, <>P, $G" and
    30   diaL:
    31    "\<lbrakk>$E |L> $E';  $F |L> $F';  $G |R> $G';
    32            $E', P, $F' \<turnstile>         $G'\<rbrakk> \<Longrightarrow> $E, <>P, $F \<turnstile> $G"
    33 
    34 ML \<open>
    35 structure S4_Prover = Modal_ProverFun
    36 (
    37   val rewrite_rls = @{thms rewrite_rls}
    38   val safe_rls = @{thms safe_rls}
    39   val unsafe_rls = @{thms unsafe_rls} @ [@{thm boxR}, @{thm diaL}]
    40   val bound_rls = @{thms bound_rls} @ [@{thm boxL}, @{thm diaR}]
    41   val aside_rls = [@{thm lstar0}, @{thm lstar1}, @{thm lstar2}, @{thm rstar0},
    42     @{thm rstar1}, @{thm rstar2}]
    43 )
    44 \<close>
    45 
    46 method_setup S4_solve =
    47   \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (S4_Prover.solve_tac ctxt 2))\<close>
    48 
    49 
    50 (* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
    51 
    52 lemma "\<turnstile> []P \<longrightarrow> P" by S4_solve
    53 lemma "\<turnstile> [](P \<longrightarrow> Q) \<longrightarrow> ([]P \<longrightarrow> []Q)" by S4_solve   (* normality*)
    54 lemma "\<turnstile> (P --< Q) \<longrightarrow> []P \<longrightarrow> []Q" by S4_solve
    55 lemma "\<turnstile> P \<longrightarrow> <>P" by S4_solve
    56 
    57 lemma "\<turnstile>  [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by S4_solve
    58 lemma "\<turnstile>  <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by S4_solve
    59 lemma "\<turnstile>  [](P \<longleftrightarrow> Q) \<longleftrightarrow> (P >-< Q)" by S4_solve
    60 lemma "\<turnstile>  <>(P \<longrightarrow> Q) \<longleftrightarrow> ([]P \<longrightarrow> <>Q)" by S4_solve
    61 lemma "\<turnstile>        []P \<longleftrightarrow> \<not> <>(\<not> P)" by S4_solve
    62 lemma "\<turnstile>     [](\<not> P) \<longleftrightarrow> \<not> <>P" by S4_solve
    63 lemma "\<turnstile>       \<not> []P \<longleftrightarrow> <>(\<not> P)" by S4_solve
    64 lemma "\<turnstile>      [][]P \<longleftrightarrow> \<not> <><>(\<not> P)" by S4_solve
    65 lemma "\<turnstile> \<not> <>(P \<or> Q) \<longleftrightarrow> \<not> <>P \<and> \<not> <>Q" by S4_solve
    66 
    67 lemma "\<turnstile> []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by S4_solve
    68 lemma "\<turnstile> <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by S4_solve
    69 lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by S4_solve
    70 lemma "\<turnstile> <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by S4_solve
    71 lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by S4_solve
    72 lemma "\<turnstile> <>(P \<longrightarrow> (Q \<and> R)) \<longrightarrow> ([]P \<longrightarrow> <>Q) \<and> ([]P \<longrightarrow> <>R)" by S4_solve
    73 lemma "\<turnstile> (P --< Q) \<and> (Q --< R) \<longrightarrow> (P --< R)" by S4_solve
    74 lemma "\<turnstile> []P \<longrightarrow> <>Q \<longrightarrow> <>(P \<and> Q)" by S4_solve
    75 
    76 
    77 (* Theorems of system S4 from Hughes and Cresswell, p.46 *)
    78 
    79 lemma "\<turnstile> []A \<longrightarrow> A" by S4_solve             (* refexivity *)
    80 lemma "\<turnstile> []A \<longrightarrow> [][]A" by S4_solve         (* transitivity *)
    81 lemma "\<turnstile> []A \<longrightarrow> <>A" by S4_solve           (* seriality *)
    82 lemma "\<turnstile> <>[](<>A \<longrightarrow> []<>A)" by S4_solve
    83 lemma "\<turnstile> <>[](<>[]A \<longrightarrow> []A)" by S4_solve
    84 lemma "\<turnstile> []P \<longleftrightarrow> [][]P" by S4_solve
    85 lemma "\<turnstile> <>P \<longleftrightarrow> <><>P" by S4_solve
    86 lemma "\<turnstile> <>[]<>P \<longrightarrow> <>P" by S4_solve
    87 lemma "\<turnstile> []<>P \<longleftrightarrow> []<>[]<>P" by S4_solve
    88 lemma "\<turnstile> <>[]P \<longleftrightarrow> <>[]<>[]P" by S4_solve
    89 
    90 (* Theorems for system S4 from Hughes and Cresswell, p.60 *)
    91 
    92 lemma "\<turnstile> []P \<or> []Q \<longleftrightarrow> []([]P \<or> []Q)" by S4_solve
    93 lemma "\<turnstile> ((P >-< Q) --< R) \<longrightarrow> ((P >-< Q) --< []R)" by S4_solve
    94 
    95 (* These are from Hailpern, LNCS 129 *)
    96 
    97 lemma "\<turnstile> [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by S4_solve
    98 lemma "\<turnstile> <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by S4_solve
    99 lemma "\<turnstile> <>(P \<longrightarrow> Q) \<longleftrightarrow> ([]P \<longrightarrow> <>Q)" by S4_solve
   100 
   101 lemma "\<turnstile> [](P \<longrightarrow> Q) \<longrightarrow> (<>P \<longrightarrow> <>Q)" by S4_solve
   102 lemma "\<turnstile> []P \<longrightarrow> []<>P" by S4_solve
   103 lemma "\<turnstile> <>[]P \<longrightarrow> <>P" by S4_solve
   104 
   105 lemma "\<turnstile> []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by S4_solve
   106 lemma "\<turnstile> <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by S4_solve
   107 lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by S4_solve
   108 lemma "\<turnstile> <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by S4_solve
   109 lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by S4_solve
   110 
   111 end