(* Title: Sequents/T.thy
Author: Martin Coen
Copyright 1991 University of Cambridge
*)
theory T
imports Modal0
begin
axiomatization where
(* Definition of the star operation using a set of Horn clauses *)
(* For system T: gamma * == {P | []P : gamma} *)
(* delta * == {P | <>P : delta} *)
lstar0: "|L>" and
lstar1: "$G |L> $H \<Longrightarrow> []P, $G |L> P, $H" and
lstar2: "$G |L> $H \<Longrightarrow> P, $G |L> $H" and
rstar0: "|R>" and
rstar1: "$G |R> $H \<Longrightarrow> <>P, $G |R> P, $H" and
rstar2: "$G |R> $H \<Longrightarrow> P, $G |R> $H" and
(* Rules for [] and <> *)
boxR:
"\<lbrakk>$E |L> $E'; $F |R> $F'; $G |R> $G';
$E' \<turnstile> $F', P, $G'\<rbrakk> \<Longrightarrow> $E \<turnstile> $F, []P, $G" and
boxL: "$E, P, $F \<turnstile> $G \<Longrightarrow> $E, []P, $F \<turnstile> $G" and
diaR: "$E \<turnstile> $F, P, $G \<Longrightarrow> $E \<turnstile> $F, <>P, $G" and
diaL:
"\<lbrakk>$E |L> $E'; $F |L> $F'; $G |R> $G';
$E', P, $F'\<turnstile> $G'\<rbrakk> \<Longrightarrow> $E, <>P, $F \<turnstile> $G"
ML \<open>
structure T_Prover = Modal_ProverFun
(
val rewrite_rls = @{thms rewrite_rls}
val safe_rls = @{thms safe_rls}
val unsafe_rls = @{thms unsafe_rls} @ [@{thm boxR}, @{thm diaL}]
val bound_rls = @{thms bound_rls} @ [@{thm boxL}, @{thm diaR}]
val aside_rls = [@{thm lstar0}, @{thm lstar1}, @{thm lstar2}, @{thm rstar0},
@{thm rstar1}, @{thm rstar2}]
)
\<close>
method_setup T_solve = \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (T_Prover.solve_tac ctxt 2))\<close>
(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
lemma "\<turnstile> []P \<longrightarrow> P" by T_solve
lemma "\<turnstile> [](P \<longrightarrow> Q) \<longrightarrow> ([]P \<longrightarrow> []Q)" by T_solve (* normality*)
lemma "\<turnstile> (P --< Q) \<longrightarrow> []P \<longrightarrow> []Q" by T_solve
lemma "\<turnstile> P \<longrightarrow> <>P" by T_solve
lemma "\<turnstile> [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by T_solve
lemma "\<turnstile> <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by T_solve
lemma "\<turnstile> [](P \<longleftrightarrow> Q) \<longleftrightarrow> (P >-< Q)" by T_solve
lemma "\<turnstile> <>(P \<longrightarrow> Q) \<longleftrightarrow> ([]P \<longrightarrow> <>Q)" by T_solve
lemma "\<turnstile> []P \<longleftrightarrow> \<not> <>(\<not> P)" by T_solve
lemma "\<turnstile> [](\<not> P) \<longleftrightarrow> \<not> <>P" by T_solve
lemma "\<turnstile> \<not> []P \<longleftrightarrow> <>(\<not> P)" by T_solve
lemma "\<turnstile> [][]P \<longleftrightarrow> \<not> <><>(\<not> P)" by T_solve
lemma "\<turnstile> \<not> <>(P \<or> Q) \<longleftrightarrow> \<not> <>P \<and> \<not> <>Q" by T_solve
lemma "\<turnstile> []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by T_solve
lemma "\<turnstile> <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by T_solve
lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by T_solve
lemma "\<turnstile> <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by T_solve
lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by T_solve
lemma "\<turnstile> <>(P \<longrightarrow> (Q \<and> R)) \<longrightarrow> ([]P \<longrightarrow> <>Q) \<and> ([]P \<longrightarrow> <>R)" by T_solve
lemma "\<turnstile> (P --< Q) \<and> (Q --< R ) \<longrightarrow> (P --< R)" by T_solve
lemma "\<turnstile> []P \<longrightarrow> <>Q \<longrightarrow> <>(P \<and> Q)" by T_solve
end