src/Sequents/S43.thy
 changeset 17481 75166ebb619b parent 14765 bafb24c150c1 child 21426 87ac12bed1ab
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17480:fd19f77dcf60 17481:75166ebb619b
`     1 (*  Title:      Modal/S43`
`     1 (*  Title:      Modal/S43.thy`
`     2     ID:         \$Id\$`
`     2     ID:         \$Id\$`
`     3     Author:     Martin Coen`
`     3     Author:     Martin Coen`
`     4     Copyright   1991  University of Cambridge`
`     4     Copyright   1991  University of Cambridge`
`     5 `
`     5 `
`     6 This implements Rajeev Gore's sequent calculus for S43.`
`     6 This implements Rajeev Gore's sequent calculus for S43.`
`     7 *)`
`     7 *)`
`     8 `
`     8 `
`     9 S43 = Modal0 +`
`     9 theory S43`
`       `
`    10 imports Modal0`
`       `
`    11 begin`
`    10 `
`    12 `
`    11 consts`
`    13 consts`
`    12   S43pi :: "[seq'=>seq', seq'=>seq', seq'=>seq',`
`    14   S43pi :: "[seq'=>seq', seq'=>seq', seq'=>seq',`
`    13              seq'=>seq', seq'=>seq', seq'=>seq'] => prop"`
`    15              seq'=>seq', seq'=>seq', seq'=>seq'] => prop"`
`    14 syntax`
`    16 syntax`
`    15   "@S43pi" :: "[seq, seq, seq, seq, seq, seq] => prop"`
`    17   "@S43pi" :: "[seq, seq, seq, seq, seq, seq] => prop"`
`    16                          ("S43pi((_);(_);(_);(_);(_);(_))" [] 5)`
`    18                          ("S43pi((_);(_);(_);(_);(_);(_))" [] 5)`
`    17 `
`    19 `
`    18 rules`
`    20 ML {*`
`       `
`    21   val S43pi  = "S43pi";`
`       `
`    22   val SS43pi = "@S43pi";`
`       `
`    23 `
`       `
`    24   val tr  = seq_tr;`
`       `
`    25   val tr' = seq_tr';`
`       `
`    26 `
`       `
`    27   fun s43pi_tr[s1,s2,s3,s4,s5,s6]=`
`       `
`    28         Const(S43pi,dummyT)\$tr s1\$tr s2\$tr s3\$tr s4\$tr s5\$tr s6;`
`       `
`    29   fun s43pi_tr'[s1,s2,s3,s4,s5,s6] =`
`       `
`    30         Const(SS43pi,dummyT)\$tr' s1\$tr' s2\$tr' s3\$tr' s4\$tr' s5\$tr' s6;`
`       `
`    31 `
`       `
`    32 *}`
`       `
`    33 `
`       `
`    34 parse_translation {* [(SS43pi,s43pi_tr)] *}`
`       `
`    35 print_translation {* [(S43pi,s43pi_tr')] *}`
`       `
`    36 `
`       `
`    37 axioms`
`    19 (* Definition of the star operation using a set of Horn clauses  *)`
`    38 (* Definition of the star operation using a set of Horn clauses  *)`
`    20 (* For system S43: gamma * == {[]P | []P : gamma}                *)`
`    39 (* For system S43: gamma * == {[]P | []P : gamma}                *)`
`    21 (*                 delta * == {<>P | <>P : delta}                *)`
`    40 (*                 delta * == {<>P | <>P : delta}                *)`
`    22 `
`    41 `
`    23   lstar0         "|L>"`
`    42   lstar0:         "|L>"`
`    24   lstar1         "\$G |L> \$H ==> []P, \$G |L> []P, \$H"`
`    43   lstar1:         "\$G |L> \$H ==> []P, \$G |L> []P, \$H"`
`    25   lstar2         "\$G |L> \$H ==>   P, \$G |L>      \$H"`
`    44   lstar2:         "\$G |L> \$H ==>   P, \$G |L>      \$H"`
`    26   rstar0         "|R>"`
`    45   rstar0:         "|R>"`
`    27   rstar1         "\$G |R> \$H ==> <>P, \$G |R> <>P, \$H"`
`    46   rstar1:         "\$G |R> \$H ==> <>P, \$G |R> <>P, \$H"`
`    28   rstar2         "\$G |R> \$H ==>   P, \$G |R>      \$H"`
`    47   rstar2:         "\$G |R> \$H ==>   P, \$G |R>      \$H"`
`    29 `
`    48 `
`    30 (* Set of Horn clauses to generate the antecedents for the S43 pi rule       *)`
`    49 (* Set of Horn clauses to generate the antecedents for the S43 pi rule       *)`
`    31 (* ie                                                                        *)`
`    50 (* ie                                                                        *)`
`    32 (*           S1...Sk,Sk+1...Sk+m                                             *)`
`    51 (*           S1...Sk,Sk+1...Sk+m                                             *)`
`    33 (*     ----------------------------------                                    *)`
`    52 (*     ----------------------------------                                    *)`
`    35 (*                                                                           *)`
`    54 (*                                                                           *)`
`    36 (*  where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, \$G * |- \$H *, []Q1...[]Qm    *)`
`    55 (*  where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, \$G * |- \$H *, []Q1...[]Qm    *)`
`    37 (*    and Sj == <>P1...<>Pk, \$G * |- \$H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj    *)`
`    56 (*    and Sj == <>P1...<>Pk, \$G * |- \$H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj    *)`
`    38 (*    and 1<=i<=k and k<j<=k+m                                               *)`
`    57 (*    and 1<=i<=k and k<j<=k+m                                               *)`
`    39 `
`    58 `
`    40   S43pi0         "S43pi \$L;; \$R;; \$Lbox; \$Rdia"`
`    59   S43pi0:         "S43pi \$L;; \$R;; \$Lbox; \$Rdia"`
`    41   S43pi1`
`    60   S43pi1:`
`    42    "[| (S43pi <>P,\$L';     \$L;; \$R; \$Lbox;\$Rdia);   \$L',P,\$L,\$Lbox |- \$R,\$Rdia |] ==> `
`    61    "[| (S43pi <>P,\$L';     \$L;; \$R; \$Lbox;\$Rdia);   \$L',P,\$L,\$Lbox |- \$R,\$Rdia |] ==>`
`    43        S43pi     \$L'; <>P,\$L;; \$R; \$Lbox;\$Rdia"`
`    62        S43pi     \$L'; <>P,\$L;; \$R; \$Lbox;\$Rdia"`
`    44   S43pi2`
`    63   S43pi2:`
`    45    "[| (S43pi \$L';; []P,\$R';     \$R; \$Lbox;\$Rdia);  \$L',\$Lbox |- \$R',P,\$R,\$Rdia |] ==> `
`    64    "[| (S43pi \$L';; []P,\$R';     \$R; \$Lbox;\$Rdia);  \$L',\$Lbox |- \$R',P,\$R,\$Rdia |] ==>`
`    46        S43pi \$L';;     \$R'; []P,\$R; \$Lbox;\$Rdia"`
`    65        S43pi \$L';;     \$R'; []P,\$R; \$Lbox;\$Rdia"`
`    47 `
`    66 `
`    48 (* Rules for [] and <> for S43 *)`
`    67 (* Rules for [] and <> for S43 *)`
`    49 `
`    68 `
`    50   boxL           "\$E, P, \$F, []P |- \$G ==> \$E, []P, \$F |- \$G"`
`    69   boxL:           "\$E, P, \$F, []P |- \$G ==> \$E, []P, \$F |- \$G"`
`    51   diaR           "\$E |- \$F, P, \$G, <>P ==> \$E |- \$F, <>P, \$G"`
`    70   diaR:           "\$E |- \$F, P, \$G, <>P ==> \$E |- \$F, <>P, \$G"`
`    52   pi1`
`    71   pi1:`
`    53    "[| \$L1,<>P,\$L2 |L> \$Lbox;  \$L1,<>P,\$L2 |R> \$Ldia;  \$R |L> \$Rbox;  \$R |R> \$Rdia;  `
`    72    "[| \$L1,<>P,\$L2 |L> \$Lbox;  \$L1,<>P,\$L2 |R> \$Ldia;  \$R |L> \$Rbox;  \$R |R> \$Rdia;`
`    54       S43pi ; \$Ldia;; \$Rbox; \$Lbox; \$Rdia |] ==> `
`    73       S43pi ; \$Ldia;; \$Rbox; \$Lbox; \$Rdia |] ==>`
`    55    \$L1, <>P, \$L2 |- \$R"`
`    74    \$L1, <>P, \$L2 |- \$R"`
`    56   pi2`
`    75   pi2:`
`    57    "[| \$L |L> \$Lbox;  \$L |R> \$Ldia;  \$R1,[]P,\$R2 |L> \$Rbox;  \$R1,[]P,\$R2 |R> \$Rdia;  `
`    76    "[| \$L |L> \$Lbox;  \$L |R> \$Ldia;  \$R1,[]P,\$R2 |L> \$Rbox;  \$R1,[]P,\$R2 |R> \$Rdia;`
`    58       S43pi ; \$Ldia;; \$Rbox; \$Lbox; \$Rdia |] ==> `
`    77       S43pi ; \$Ldia;; \$Rbox; \$Lbox; \$Rdia |] ==>`
`    59    \$L |- \$R1, []P, \$R2"`
`    78    \$L |- \$R1, []P, \$R2"`
`       `
`    79 `
`       `
`    80 ML {* use_legacy_bindings (the_context ()) *}`
`       `
`    81 `
`    60 end`
`    82 end`