src/Sequents/T.thy
 author wenzelm Sat Dec 14 17:28:05 2013 +0100 (2013-12-14) changeset 54742 7a86358a3c0b parent 51309 473303ef6e34 child 60770 240563fbf41d permissions -rw-r--r--
proper context for basic Simplifier operations: rewrite_rule, rewrite_goals_rule, rewrite_goals_tac etc.;
clarified tool context in some boundary cases;
```     1 (*  Title:      Sequents/T.thy
```
```     2     Author:     Martin Coen
```
```     3     Copyright   1991  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 theory T
```
```     7 imports Modal0
```
```     8 begin
```
```     9
```
```    10 axiomatization where
```
```    11 (* Definition of the star operation using a set of Horn clauses *)
```
```    12 (* For system T:  gamma * == {P | []P : gamma}                  *)
```
```    13 (*                delta * == {P | <>P : delta}                  *)
```
```    14
```
```    15   lstar0:         "|L>" and
```
```    16   lstar1:         "\$G |L> \$H ==> []P, \$G |L> P, \$H" and
```
```    17   lstar2:         "\$G |L> \$H ==>   P, \$G |L>    \$H" and
```
```    18   rstar0:         "|R>" and
```
```    19   rstar1:         "\$G |R> \$H ==> <>P, \$G |R> P, \$H" and
```
```    20   rstar2:         "\$G |R> \$H ==>   P, \$G |R>    \$H" and
```
```    21
```
```    22 (* Rules for [] and <> *)
```
```    23
```
```    24   boxR:
```
```    25    "[| \$E |L> \$E';  \$F |R> \$F';  \$G |R> \$G';
```
```    26                \$E'        |- \$F', P, \$G'|] ==> \$E          |- \$F, []P, \$G" and
```
```    27   boxL:     "\$E, P, \$F  |-         \$G    ==> \$E, []P, \$F |-          \$G" and
```
```    28   diaR:     "\$E         |- \$F, P,  \$G    ==> \$E          |- \$F, <>P, \$G" and
```
```    29   diaL:
```
```    30    "[| \$E |L> \$E';  \$F |L> \$F';  \$G |R> \$G';
```
```    31                \$E', P, \$F'|-         \$G'|] ==> \$E, <>P, \$F |-          \$G"
```
```    32
```
```    33 ML {*
```
```    34 structure T_Prover = Modal_ProverFun
```
```    35 (
```
```    36   val rewrite_rls = @{thms rewrite_rls}
```
```    37   val safe_rls = @{thms safe_rls}
```
```    38   val unsafe_rls = @{thms unsafe_rls} @ [@{thm boxR}, @{thm diaL}]
```
```    39   val bound_rls = @{thms bound_rls} @ [@{thm boxL}, @{thm diaR}]
```
```    40   val aside_rls = [@{thm lstar0}, @{thm lstar1}, @{thm lstar2}, @{thm rstar0},
```
```    41     @{thm rstar1}, @{thm rstar2}]
```
```    42 )
```
```    43 *}
```
```    44
```
```    45 method_setup T_solve = {* Scan.succeed (fn ctxt => SIMPLE_METHOD (T_Prover.solve_tac ctxt 2)) *}
```
```    46
```
```    47
```
```    48 (* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
```
```    49
```
```    50 lemma "|- []P --> P" by T_solve
```
```    51 lemma "|- [](P-->Q) --> ([]P-->[]Q)" by T_solve   (* normality*)
```
```    52 lemma "|- (P--<Q) --> []P --> []Q" by T_solve
```
```    53 lemma "|- P --> <>P" by T_solve
```
```    54
```
```    55 lemma "|-  [](P & Q) <-> []P & []Q" by T_solve
```
```    56 lemma "|-  <>(P | Q) <-> <>P | <>Q" by T_solve
```
```    57 lemma "|-  [](P<->Q) <-> (P>-<Q)" by T_solve
```
```    58 lemma "|-  <>(P-->Q) <-> ([]P--><>Q)" by T_solve
```
```    59 lemma "|-        []P <-> ~<>(~P)" by T_solve
```
```    60 lemma "|-     [](~P) <-> ~<>P" by T_solve
```
```    61 lemma "|-       ~[]P <-> <>(~P)" by T_solve
```
```    62 lemma "|-      [][]P <-> ~<><>(~P)" by T_solve
```
```    63 lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by T_solve
```
```    64
```
```    65 lemma "|- []P | []Q --> [](P | Q)" by T_solve
```
```    66 lemma "|- <>(P & Q) --> <>P & <>Q" by T_solve
```
```    67 lemma "|- [](P | Q) --> []P | <>Q" by T_solve
```
```    68 lemma "|- <>P & []Q --> <>(P & Q)" by T_solve
```
```    69 lemma "|- [](P | Q) --> <>P | []Q" by T_solve
```
```    70 lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by T_solve
```
```    71 lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by T_solve
```
```    72 lemma "|- []P --> <>Q --> <>(P & Q)" by T_solve
```
```    73
```
```    74 end
```