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src/Sequents/S4.thy
changeset 61386 0a29a984a91b
parent 61385 538100cc4399 
--- a/src/Sequents/S4.thy	Sat Oct 10 20:51:39 2015 +0200
+++ b/src/Sequents/S4.thy	Sat Oct 10 20:54:44 2015 +0200
@@ -23,13 +23,13 @@
 
   boxR:
    "\<lbrakk>$E |L> $E';  $F |R> $F';  $G |R> $G';
-           $E'         |- $F', P, $G'\<rbrakk> \<Longrightarrow> $E          |- $F, []P, $G" and
-  boxL:     "$E,P,$F,[]P |-         $G    \<Longrightarrow> $E, []P, $F |-          $G" and
+           $E'         \<turnstile> $F', P, $G'\<rbrakk> \<Longrightarrow> $E          \<turnstile> $F, []P, $G" and
+  boxL:     "$E,P,$F,[]P \<turnstile>         $G    \<Longrightarrow> $E, []P, $F \<turnstile>          $G" and
 
-  diaR:     "$E          |- $F,P,$G,<>P   \<Longrightarrow> $E          |- $F, <>P, $G" and
+  diaR:     "$E          \<turnstile> $F,P,$G,<>P   \<Longrightarrow> $E          \<turnstile> $F, <>P, $G" and
   diaL:
    "\<lbrakk>$E |L> $E';  $F |L> $F';  $G |R> $G';
-           $E', P, $F' |-         $G'\<rbrakk> \<Longrightarrow> $E, <>P, $F |- $G"
+           $E', P, $F' \<turnstile>         $G'\<rbrakk> \<Longrightarrow> $E, <>P, $F \<turnstile> $G"
 
 ML \<open>
 structure S4_Prover = Modal_ProverFun
@@ -49,63 +49,63 @@
 
 (* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
 
-lemma "|- []P \<longrightarrow> P" by S4_solve
-lemma "|- [](P \<longrightarrow> Q) \<longrightarrow> ([]P \<longrightarrow> []Q)" by S4_solve   (* normality*)
-lemma "|- (P --< Q) \<longrightarrow> []P \<longrightarrow> []Q" by S4_solve
-lemma "|- P \<longrightarrow> <>P" by S4_solve
+lemma "\<turnstile> []P \<longrightarrow> P" by S4_solve
+lemma "\<turnstile> [](P \<longrightarrow> Q) \<longrightarrow> ([]P \<longrightarrow> []Q)" by S4_solve   (* normality*)
+lemma "\<turnstile> (P --< Q) \<longrightarrow> []P \<longrightarrow> []Q" by S4_solve
+lemma "\<turnstile> P \<longrightarrow> <>P" by S4_solve
 
-lemma "|-  [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by S4_solve
-lemma "|-  <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by S4_solve
-lemma "|-  [](P \<longleftrightarrow> Q) \<longleftrightarrow> (P >-< Q)" by S4_solve
-lemma "|-  <>(P \<longrightarrow> Q) \<longleftrightarrow> ([]P \<longrightarrow> <>Q)" by S4_solve
-lemma "|-        []P \<longleftrightarrow> \<not> <>(\<not> P)" by S4_solve
-lemma "|-     [](\<not> P) \<longleftrightarrow> \<not> <>P" by S4_solve
-lemma "|-       \<not> []P \<longleftrightarrow> <>(\<not> P)" by S4_solve
-lemma "|-      [][]P \<longleftrightarrow> \<not> <><>(\<not> P)" by S4_solve
-lemma "|- \<not> <>(P \<or> Q) \<longleftrightarrow> \<not> <>P \<and> \<not> <>Q" by S4_solve
+lemma "\<turnstile>  [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by S4_solve
+lemma "\<turnstile>  <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by S4_solve
+lemma "\<turnstile>  [](P \<longleftrightarrow> Q) \<longleftrightarrow> (P >-< Q)" by S4_solve
+lemma "\<turnstile>  <>(P \<longrightarrow> Q) \<longleftrightarrow> ([]P \<longrightarrow> <>Q)" by S4_solve
+lemma "\<turnstile>        []P \<longleftrightarrow> \<not> <>(\<not> P)" by S4_solve
+lemma "\<turnstile>     [](\<not> P) \<longleftrightarrow> \<not> <>P" by S4_solve
+lemma "\<turnstile>       \<not> []P \<longleftrightarrow> <>(\<not> P)" by S4_solve
+lemma "\<turnstile>      [][]P \<longleftrightarrow> \<not> <><>(\<not> P)" by S4_solve
+lemma "\<turnstile> \<not> <>(P \<or> Q) \<longleftrightarrow> \<not> <>P \<and> \<not> <>Q" by S4_solve
 
-lemma "|- []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by S4_solve
-lemma "|- <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by S4_solve
-lemma "|- [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by S4_solve
-lemma "|- <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by S4_solve
-lemma "|- [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by S4_solve
-lemma "|- <>(P \<longrightarrow> (Q \<and> R)) \<longrightarrow> ([]P \<longrightarrow> <>Q) \<and> ([]P \<longrightarrow> <>R)" by S4_solve
-lemma "|- (P --< Q) \<and> (Q --< R) \<longrightarrow> (P --< R)" by S4_solve
-lemma "|- []P \<longrightarrow> <>Q \<longrightarrow> <>(P \<and> Q)" by S4_solve
+lemma "\<turnstile> []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by S4_solve
+lemma "\<turnstile> <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by S4_solve
+lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by S4_solve
+lemma "\<turnstile> <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by S4_solve
+lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by S4_solve
+lemma "\<turnstile> <>(P \<longrightarrow> (Q \<and> R)) \<longrightarrow> ([]P \<longrightarrow> <>Q) \<and> ([]P \<longrightarrow> <>R)" by S4_solve
+lemma "\<turnstile> (P --< Q) \<and> (Q --< R) \<longrightarrow> (P --< R)" by S4_solve
+lemma "\<turnstile> []P \<longrightarrow> <>Q \<longrightarrow> <>(P \<and> Q)" by S4_solve
 
 
 (* Theorems of system S4 from Hughes and Cresswell, p.46 *)
 
-lemma "|- []A \<longrightarrow> A" by S4_solve             (* refexivity *)
-lemma "|- []A \<longrightarrow> [][]A" by S4_solve         (* transitivity *)
-lemma "|- []A \<longrightarrow> <>A" by S4_solve           (* seriality *)
-lemma "|- <>[](<>A \<longrightarrow> []<>A)" by S4_solve
-lemma "|- <>[](<>[]A \<longrightarrow> []A)" by S4_solve
-lemma "|- []P \<longleftrightarrow> [][]P" by S4_solve
-lemma "|- <>P \<longleftrightarrow> <><>P" by S4_solve
-lemma "|- <>[]<>P \<longrightarrow> <>P" by S4_solve
-lemma "|- []<>P \<longleftrightarrow> []<>[]<>P" by S4_solve
-lemma "|- <>[]P \<longleftrightarrow> <>[]<>[]P" by S4_solve
+lemma "\<turnstile> []A \<longrightarrow> A" by S4_solve             (* refexivity *)
+lemma "\<turnstile> []A \<longrightarrow> [][]A" by S4_solve         (* transitivity *)
+lemma "\<turnstile> []A \<longrightarrow> <>A" by S4_solve           (* seriality *)
+lemma "\<turnstile> <>[](<>A \<longrightarrow> []<>A)" by S4_solve
+lemma "\<turnstile> <>[](<>[]A \<longrightarrow> []A)" by S4_solve
+lemma "\<turnstile> []P \<longleftrightarrow> [][]P" by S4_solve
+lemma "\<turnstile> <>P \<longleftrightarrow> <><>P" by S4_solve
+lemma "\<turnstile> <>[]<>P \<longrightarrow> <>P" by S4_solve
+lemma "\<turnstile> []<>P \<longleftrightarrow> []<>[]<>P" by S4_solve
+lemma "\<turnstile> <>[]P \<longleftrightarrow> <>[]<>[]P" by S4_solve
 
 (* Theorems for system S4 from Hughes and Cresswell, p.60 *)
 
-lemma "|- []P \<or> []Q \<longleftrightarrow> []([]P \<or> []Q)" by S4_solve
-lemma "|- ((P >-< Q) --< R) \<longrightarrow> ((P >-< Q) --< []R)" by S4_solve
+lemma "\<turnstile> []P \<or> []Q \<longleftrightarrow> []([]P \<or> []Q)" by S4_solve
+lemma "\<turnstile> ((P >-< Q) --< R) \<longrightarrow> ((P >-< Q) --< []R)" by S4_solve
 
 (* These are from Hailpern, LNCS 129 *)
 
-lemma "|- [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by S4_solve
-lemma "|- <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by S4_solve
-lemma "|- <>(P \<longrightarrow> Q) \<longleftrightarrow> ([]P \<longrightarrow> <>Q)" by S4_solve
+lemma "\<turnstile> [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by S4_solve
+lemma "\<turnstile> <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by S4_solve
+lemma "\<turnstile> <>(P \<longrightarrow> Q) \<longleftrightarrow> ([]P \<longrightarrow> <>Q)" by S4_solve
 
-lemma "|- [](P \<longrightarrow> Q) \<longrightarrow> (<>P \<longrightarrow> <>Q)" by S4_solve
-lemma "|- []P \<longrightarrow> []<>P" by S4_solve
-lemma "|- <>[]P \<longrightarrow> <>P" by S4_solve
+lemma "\<turnstile> [](P \<longrightarrow> Q) \<longrightarrow> (<>P \<longrightarrow> <>Q)" by S4_solve
+lemma "\<turnstile> []P \<longrightarrow> []<>P" by S4_solve
+lemma "\<turnstile> <>[]P \<longrightarrow> <>P" by S4_solve
 
-lemma "|- []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by S4_solve
-lemma "|- <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by S4_solve
-lemma "|- [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by S4_solve
-lemma "|- <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by S4_solve
-lemma "|- [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by S4_solve
+lemma "\<turnstile> []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by S4_solve
+lemma "\<turnstile> <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by S4_solve
+lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by S4_solve
+lemma "\<turnstile> <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by S4_solve
+lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by S4_solve
 
 end
isabelle