author  wenzelm 
Mon, 07 Oct 2013 21:24:44 +0200  
changeset 54313  da2e6282a4f5 
parent 52143  36ffe23b25f8 
child 54742  7a86358a3c0b 
permissions  rwrr 
41959  1 
(* Title: Sequents/S43.thy 
2073
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

2 
Author: Martin Coen 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

3 
Copyright 1991 University of Cambridge 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

4 

fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

5 
This implements Rajeev Gore's sequent calculus for S43. 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

6 
*) 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

7 

17481  8 
theory S43 
9 
imports Modal0 

10 
begin 

2073
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

11 

fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

12 
consts 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

13 
S43pi :: "[seq'=>seq', seq'=>seq', seq'=>seq', 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

14 
seq'=>seq', seq'=>seq', seq'=>seq'] => prop" 
14765  15 
syntax 
35113  16 
"_S43pi" :: "[seq, seq, seq, seq, seq, seq] => prop" 
2073
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

17 
("S43pi((_);(_);(_);(_);(_);(_))" [] 5) 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

18 

35113  19 
parse_translation {* 
20 
let 

21 
val tr = seq_tr; 

22 
fun s43pi_tr [s1, s2, s3, s4, s5, s6] = 

23 
Const (@{const_syntax S43pi}, dummyT) $ tr s1 $ tr s2 $ tr s3 $ tr s4 $ tr s5 $ tr s6; 

52143  24 
in [(@{syntax_const "_S43pi"}, K s43pi_tr)] end 
17481  25 
*} 
26 

35113  27 
print_translation {* 
28 
let 

29 
val tr' = seq_tr'; 

30 
fun s43pi_tr' [s1, s2, s3, s4, s5, s6] = 

31 
Const(@{syntax_const "_S43pi"}, dummyT) $ tr' s1 $ tr' s2 $ tr' s3 $ tr' s4 $ tr' s5 $ tr' s6; 

52143  32 
in [(@{const_syntax S43pi}, K s43pi_tr')] end 
35113  33 
*} 
17481  34 

51309  35 
axiomatization where 
2073
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

36 
(* 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

37 
(* For system S43: gamma * == {[]P  []P : gamma} *) 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

38 
(* delta * == {<>P  <>P : delta} *) 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

39 

51309  40 
lstar0: "L>" and 
41 
lstar1: "$G L> $H ==> []P, $G L> []P, $H" and 

42 
lstar2: "$G L> $H ==> P, $G L> $H" and 

43 
rstar0: "R>" and 

44 
rstar1: "$G R> $H ==> <>P, $G R> <>P, $H" and 

45 
rstar2: "$G R> $H ==> P, $G R> $H" and 

2073
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

46 

fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

47 
(* 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

48 
(* ie *) 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

49 
(* S1...Sk,Sk+1...Sk+m *) 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

50 
(*  *) 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

51 
(* <>P1...<>Pk, $G  $H, []Q1...[]Qm *) 
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 
(* where Si == <>P1...<>Pi1,<>Pi+1,..<>Pk,Pi, $G *  $H *, []Q1...[]Qm *) 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

54 
(* and Sj == <>P1...<>Pk, $G *  $H *, []Q1...[]Qj1,[]Qj+1...[]Qm,Qj *) 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

55 
(* and 1<=i<=k and k<j<=k+m *) 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

56 

51309  57 
S43pi0: "S43pi $L;; $R;; $Lbox; $Rdia" and 
17481  58 
S43pi1: 
59 
"[ (S43pi <>P,$L'; $L;; $R; $Lbox;$Rdia); $L',P,$L,$Lbox  $R,$Rdia ] ==> 

51309  60 
S43pi $L'; <>P,$L;; $R; $Lbox;$Rdia" and 
17481  61 
S43pi2: 
62 
"[ (S43pi $L';; []P,$R'; $R; $Lbox;$Rdia); $L',$Lbox  $R',P,$R,$Rdia ] ==> 

51309  63 
S43pi $L';; $R'; []P,$R; $Lbox;$Rdia" and 
2073
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

64 

fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

65 
(* Rules for [] and <> for S43 *) 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

66 

51309  67 
boxL: "$E, P, $F, []P  $G ==> $E, []P, $F  $G" and 
68 
diaR: "$E  $F, P, $G, <>P ==> $E  $F, <>P, $G" and 

17481  69 
pi1: 
70 
"[ $L1,<>P,$L2 L> $Lbox; $L1,<>P,$L2 R> $Ldia; $R L> $Rbox; $R R> $Rdia; 

71 
S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia ] ==> 

51309  72 
$L1, <>P, $L2  $R" and 
17481  73 
pi2: 
74 
"[ $L L> $Lbox; $L R> $Ldia; $R1,[]P,$R2 L> $Rbox; $R1,[]P,$R2 R> $Rdia; 

75 
S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia ] ==> 

2073
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

76 
$L  $R1, []P, $R2" 
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

77 

21426  78 

79 
ML {* 

80 
structure S43_Prover = Modal_ProverFun 

81 
( 

39159  82 
val rewrite_rls = @{thms rewrite_rls} 
83 
val safe_rls = @{thms safe_rls} 

84 
val unsafe_rls = @{thms unsafe_rls} @ [@{thm pi1}, @{thm pi2}] 

85 
val bound_rls = @{thms bound_rls} @ [@{thm boxL}, @{thm diaR}] 

86 
val aside_rls = [@{thm lstar0}, @{thm lstar1}, @{thm lstar2}, @{thm rstar0}, 

87 
@{thm rstar1}, @{thm rstar2}, @{thm S43pi0}, @{thm S43pi1}, @{thm S43pi2}] 

21426  88 
) 
89 
*} 

90 

91 

92 
method_setup S43_solve = {* 

30549  93 
Scan.succeed (K (SIMPLE_METHOD 
94 
(S43_Prover.solve_tac 2 ORELSE S43_Prover.solve_tac 3))) 

42814  95 
*} 
21426  96 

97 

98 
(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *) 

99 

100 
lemma " []P > P" by S43_solve 

101 
lemma " [](P>Q) > ([]P>[]Q)" by S43_solve (* normality*) 

102 
lemma " (P<Q) > []P > []Q" by S43_solve 

103 
lemma " P > <>P" by S43_solve 

104 

105 
lemma " [](P & Q) <> []P & []Q" by S43_solve 

106 
lemma " <>(P  Q) <> <>P  <>Q" by S43_solve 

107 
lemma " [](P<>Q) <> (P><Q)" by S43_solve 

108 
lemma " <>(P>Q) <> ([]P><>Q)" by S43_solve 

109 
lemma " []P <> ~<>(~P)" by S43_solve 

110 
lemma " [](~P) <> ~<>P" by S43_solve 

111 
lemma " ~[]P <> <>(~P)" by S43_solve 

112 
lemma " [][]P <> ~<><>(~P)" by S43_solve 

113 
lemma " ~<>(P  Q) <> ~<>P & ~<>Q" by S43_solve 

114 

115 
lemma " []P  []Q > [](P  Q)" by S43_solve 

116 
lemma " <>(P & Q) > <>P & <>Q" by S43_solve 

117 
lemma " [](P  Q) > []P  <>Q" by S43_solve 

118 
lemma " <>P & []Q > <>(P & Q)" by S43_solve 

119 
lemma " [](P  Q) > <>P  []Q" by S43_solve 

120 
lemma " <>(P>(Q & R)) > ([]P > <>Q) & ([]P><>R)" by S43_solve 

121 
lemma " (P<Q) & (Q<R) > (P<R)" by S43_solve 

122 
lemma " []P > <>Q > <>(P & Q)" by S43_solve 

123 

124 

125 
(* Theorems of system S4 from Hughes and Cresswell, p.46 *) 

126 

127 
lemma " []A > A" by S43_solve (* refexivity *) 

128 
lemma " []A > [][]A" by S43_solve (* transitivity *) 

129 
lemma " []A > <>A" by S43_solve (* seriality *) 

130 
lemma " <>[](<>A > []<>A)" by S43_solve 

131 
lemma " <>[](<>[]A > []A)" by S43_solve 

132 
lemma " []P <> [][]P" by S43_solve 

133 
lemma " <>P <> <><>P" by S43_solve 

134 
lemma " <>[]<>P > <>P" by S43_solve 

135 
lemma " []<>P <> []<>[]<>P" by S43_solve 

136 
lemma " <>[]P <> <>[]<>[]P" by S43_solve 

137 

138 
(* Theorems for system S4 from Hughes and Cresswell, p.60 *) 

139 

140 
lemma " []P  []Q <> []([]P  []Q)" by S43_solve 

141 
lemma " ((P><Q) < R) > ((P><Q) < []R)" by S43_solve 

142 

143 
(* These are from Hailpern, LNCS 129 *) 

144 

145 
lemma " [](P & Q) <> []P & []Q" by S43_solve 

146 
lemma " <>(P  Q) <> <>P  <>Q" by S43_solve 

147 
lemma " <>(P > Q) <> ([]P > <>Q)" by S43_solve 

148 

149 
lemma " [](P > Q) > (<>P > <>Q)" by S43_solve 

150 
lemma " []P > []<>P" by S43_solve 

151 
lemma " <>[]P > <>P" by S43_solve 

152 

153 
lemma " []P  []Q > [](P  Q)" by S43_solve 

154 
lemma " <>(P & Q) > <>P & <>Q" by S43_solve 

155 
lemma " [](P  Q) > []P  <>Q" by S43_solve 

156 
lemma " <>P & []Q > <>(P & Q)" by S43_solve 

157 
lemma " [](P  Q) > <>P  []Q" by S43_solve 

158 

159 

160 
(* Theorems of system S43 *) 

161 

162 
lemma " <>[]P > []<>P" by S43_solve 

163 
lemma " <>[]P > [][]<>P" by S43_solve 

164 
lemma " [](<>P  <>Q) > []<>P  []<>Q" by S43_solve 

165 
lemma " <>[]P & <>[]Q > <>([]P & []Q)" by S43_solve 

166 
lemma " []([]P > []Q)  []([]Q > []P)" by S43_solve 

167 
lemma " [](<>P > <>Q)  [](<>Q > <>P)" by S43_solve 

168 
lemma " []([]P > Q)  []([]Q > P)" by S43_solve 

169 
lemma " [](P > <>Q)  [](Q > <>P)" by S43_solve 

170 
lemma " [](P > []Q>R)  [](P  ([]R > Q))" by S43_solve 

171 
lemma " [](P  (Q > <>C))  [](P > C > <>Q)" by S43_solve 

172 
lemma " []([]P  Q) & [](P  []Q) > []P  []Q" by S43_solve 

173 
lemma " <>P & <>Q > <>(<>P & Q)  <>(P & <>Q)" by S43_solve 

174 
lemma " [](P  Q) & []([]P  Q) & [](P  []Q) > []P  []Q" by S43_solve 

175 
lemma " <>P & <>Q > <>(P & Q)  <>(<>P & Q)  <>(P & <>Q)" by S43_solve 

176 
lemma " <>[]<>P <> []<>P" by S43_solve 

177 
lemma " []<>[]P <> <>[]P" by S43_solve 

178 

179 
(* These are from Hailpern, LNCS 129 *) 

180 

181 
lemma " [](P & Q) <> []P & []Q" by S43_solve 

182 
lemma " <>(P  Q) <> <>P  <>Q" by S43_solve 

183 
lemma " <>(P > Q) <> []P > <>Q" by S43_solve 

184 

185 
lemma " [](P > Q) > <>P > <>Q" by S43_solve 

186 
lemma " []P > []<>P" by S43_solve 

187 
lemma " <>[]P > <>P" by S43_solve 

188 
lemma " []<>[]P > []<>P" by S43_solve 

189 
lemma " <>[]P > <>[]<>P" by S43_solve 

190 
lemma " <>[]P > []<>P" by S43_solve 

191 
lemma " []<>[]P <> <>[]P" by S43_solve 

192 
lemma " <>[]<>P <> []<>P" by S43_solve 

193 

194 
lemma " []P  []Q > [](P  Q)" by S43_solve 

195 
lemma " <>(P & Q) > <>P & <>Q" by S43_solve 

196 
lemma " [](P  Q) > []P  <>Q" by S43_solve 

197 
lemma " <>P & []Q > <>(P & Q)" by S43_solve 

198 
lemma " [](P  Q) > <>P  []Q" by S43_solve 

199 
lemma " [](P  Q) > []<>P  []<>Q" by S43_solve 

200 
lemma " <>[]P & <>[]Q > <>(P & Q)" by S43_solve 

201 
lemma " <>[](P & Q) <> <>[]P & <>[]Q" by S43_solve 

202 
lemma " []<>(P  Q) <> []<>P  []<>Q" by S43_solve 

2073
fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

203 

fb0655539d05
New unified treatment of sequent calculi by Sara Kalvala
paulson
parents:
diff
changeset

204 
end 