1(* Title: Sequents/T.thy 2 Author: Martin Coen 3 Copyright 1991 University of Cambridge 4*) 5 6theory T 7imports Modal0 8begin 9 10axiomatization 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 \<Longrightarrow> []P, $G |L> P, $H" and 17 lstar2: "$G |L> $H \<Longrightarrow> P, $G |L> $H" and 18 rstar0: "|R>" and 19 rstar1: "$G |R> $H \<Longrightarrow> <>P, $G |R> P, $H" and 20 rstar2: "$G |R> $H \<Longrightarrow> P, $G |R> $H" and 21 22(* Rules for [] and <> *) 23 24 boxR: 25 "\<lbrakk>$E |L> $E'; $F |R> $F'; $G |R> $G'; 26 $E' \<turnstile> $F', P, $G'\<rbrakk> \<Longrightarrow> $E \<turnstile> $F, []P, $G" and 27 boxL: "$E, P, $F \<turnstile> $G \<Longrightarrow> $E, []P, $F \<turnstile> $G" and 28 diaR: "$E \<turnstile> $F, P, $G \<Longrightarrow> $E \<turnstile> $F, <>P, $G" and 29 diaL: 30 "\<lbrakk>$E |L> $E'; $F |L> $F'; $G |R> $G'; 31 $E', P, $F'\<turnstile> $G'\<rbrakk> \<Longrightarrow> $E, <>P, $F \<turnstile> $G" 32 33ML \<open> 34structure 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\<close> 44 45method_setup T_solve = \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (T_Prover.solve_tac ctxt 2))\<close> 46 47 48(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *) 49 50lemma "\<turnstile> []P \<longrightarrow> P" by T_solve 51lemma "\<turnstile> [](P \<longrightarrow> Q) \<longrightarrow> ([]P \<longrightarrow> []Q)" by T_solve (* normality*) 52lemma "\<turnstile> (P --< Q) \<longrightarrow> []P \<longrightarrow> []Q" by T_solve 53lemma "\<turnstile> P \<longrightarrow> <>P" by T_solve 54 55lemma "\<turnstile> [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by T_solve 56lemma "\<turnstile> <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by T_solve 57lemma "\<turnstile> [](P \<longleftrightarrow> Q) \<longleftrightarrow> (P >-< Q)" by T_solve 58lemma "\<turnstile> <>(P \<longrightarrow> Q) \<longleftrightarrow> ([]P \<longrightarrow> <>Q)" by T_solve 59lemma "\<turnstile> []P \<longleftrightarrow> \<not> <>(\<not> P)" by T_solve 60lemma "\<turnstile> [](\<not> P) \<longleftrightarrow> \<not> <>P" by T_solve 61lemma "\<turnstile> \<not> []P \<longleftrightarrow> <>(\<not> P)" by T_solve 62lemma "\<turnstile> [][]P \<longleftrightarrow> \<not> <><>(\<not> P)" by T_solve 63lemma "\<turnstile> \<not> <>(P \<or> Q) \<longleftrightarrow> \<not> <>P \<and> \<not> <>Q" by T_solve 64 65lemma "\<turnstile> []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by T_solve 66lemma "\<turnstile> <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by T_solve 67lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by T_solve 68lemma "\<turnstile> <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by T_solve 69lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by T_solve 70lemma "\<turnstile> <>(P \<longrightarrow> (Q \<and> R)) \<longrightarrow> ([]P \<longrightarrow> <>Q) \<and> ([]P \<longrightarrow> <>R)" by T_solve 71lemma "\<turnstile> (P --< Q) \<and> (Q --< R ) \<longrightarrow> (P --< R)" by T_solve 72lemma "\<turnstile> []P \<longrightarrow> <>Q \<longrightarrow> <>(P \<and> Q)" by T_solve 73 74end 75