src/Sequents/T.thy
 author haftmann Thu, 23 Nov 2017 17:03:27 +0000 changeset 67087 733017b19de9 parent 61386 0a29a984a91b permissions -rw-r--r--
generalized more lemmas

(*  Title:      Sequents/T.thy
Author:     Martin Coen
*)

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