src/Sequents/S43.thy
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2072:6ac12b9478d5 2073:fb0655539d05
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`     1 (*  Title:      Modal/S43`
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`     2     ID:         \$Id\$`
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`     3     Author:     Martin Coen`
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`     4     Copyright   1991  University of Cambridge`
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`     5 `
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`     6 This implements Rajeev Gore's sequent calculus for S43.`
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`     7 *)`
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`     8 `
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`     9 S43 = Modal0 +`
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`    10 `
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`    11 consts`
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`    12   S43pi :: "[seq'=>seq', seq'=>seq', seq'=>seq',`
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`    13              seq'=>seq', seq'=>seq', seq'=>seq'] => prop"`
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`    14   "@S43pi" :: "[seq, seq, seq, seq, seq, seq] => prop"`
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`    15                          ("S43pi((_);(_);(_);(_);(_);(_))" [] 5)`
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`    16 `
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`    17 rules`
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`    18 (* Definition of the star operation using a set of Horn clauses  *)`
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`    19 (* For system S43: gamma * == {[]P | []P : gamma}                *)`
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`    20 (*                 delta * == {<>P | <>P : delta}                *)`
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`    21 `
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`    22   lstar0         "|L>"`
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`    23   lstar1         "\$G |L> \$H ==> []P, \$G |L> []P, \$H"`
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`    24   lstar2         "\$G |L> \$H ==>   P, \$G |L>      \$H"`
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`    25   rstar0         "|R>"`
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`    26   rstar1         "\$G |R> \$H ==> <>P, \$G |R> <>P, \$H"`
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`    27   rstar2         "\$G |R> \$H ==>   P, \$G |R>      \$H"`
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`    28 `
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`    29 (* Set of Horn clauses to generate the antecedents for the S43 pi rule       *)`
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`    30 (* ie                                                                        *)`
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`    31 (*           S1...Sk,Sk+1...Sk+m                                             *)`
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`    32 (*     ----------------------------------                                    *)`
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`    33 (*     <>P1...<>Pk, \$G |- \$H, []Q1...[]Qm                                    *)`
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`    34 (*                                                                           *)`
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`    35 (*  where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, \$G * |- \$H *, []Q1...[]Qm    *)`
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`    36 (*    and Sj == <>P1...<>Pk, \$G * |- \$H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj    *)`
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`    37 (*    and 1<=i<=k and k<j<=k+m                                               *)`
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`    38 `
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`    39   S43pi0         "S43pi \$L;; \$R;; \$Lbox; \$Rdia"`
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`    40   S43pi1`
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`    41    "[| (S43pi <>P,\$L';     \$L;; \$R; \$Lbox;\$Rdia);   \$L',P,\$L,\$Lbox |- \$R,\$Rdia |] ==> `
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`    42        S43pi     \$L'; <>P,\$L;; \$R; \$Lbox;\$Rdia"`
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`    43   S43pi2`
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`    44    "[| (S43pi \$L';; []P,\$R';     \$R; \$Lbox;\$Rdia);  \$L',\$Lbox |- \$R',P,\$R,\$Rdia |] ==> `
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`    45        S43pi \$L';;     \$R'; []P,\$R; \$Lbox;\$Rdia"`
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`    46 `
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`    47 (* Rules for [] and <> for S43 *)`
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`    48 `
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`    49   boxL           "\$E, P, \$F, []P |- \$G ==> \$E, []P, \$F |- \$G"`
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`    50   diaR           "\$E |- \$F, P, \$G, <>P ==> \$E |- \$F, <>P, \$G"`
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`    51   pi1`
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`    52    "[| \$L1,<>P,\$L2 |L> \$Lbox;  \$L1,<>P,\$L2 |R> \$Ldia;  \$R |L> \$Rbox;  \$R |R> \$Rdia;  `
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`    53       S43pi ; \$Ldia;; \$Rbox; \$Lbox; \$Rdia |] ==> `
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`    54    \$L1, <>P, \$L2 |- \$R"`
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`    55   pi2`
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`    56    "[| \$L |L> \$Lbox;  \$L |R> \$Ldia;  \$R1,[]P,\$R2 |L> \$Rbox;  \$R1,[]P,\$R2 |R> \$Rdia;  `
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`    57       S43pi ; \$Ldia;; \$Rbox; \$Lbox; \$Rdia |] ==> `
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`    58    \$L |- \$R1, []P, \$R2"`
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`    59 end`
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`    60 `
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`    61 ML`
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`    62 `
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`    63 local`
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`    64 `
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`    65   val S43pi  = "S43pi";`
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`    66   val SS43pi = "@S43pi";`
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`    67 `
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`    68   val tr  = Sequents.seq_tr;`
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`    69   val tr' = Sequents.seq_tr';`
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`    70 `
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`    71   fun s43pi_tr[s1,s2,s3,s4,s5,s6]=`
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`    72         Const(S43pi,dummyT)\$tr s1\$tr s2\$tr s3\$tr s4\$tr s5\$tr s6;`
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`    73   fun s43pi_tr'[s1,s2,s3,s4,s5,s6] =`
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`    74         Const(SS43pi,dummyT)\$tr' s1\$tr' s2\$tr' s3\$tr' s4\$tr' s5\$tr' s6;`
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`    75 in`
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`    76 val parse_translation = [(SS43pi,s43pi_tr)];`
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`    77 val print_translation = [(S43pi,s43pi_tr')]`
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`    78 end`