(* Title: Sequents/S4.thy
Author: Martin Coen
Copyright 1991 University of Cambridge
*)
theory S4
imports Modal0
begin
axiomatization where
(* Definition of the star operation using a set of Horn clauses *)
(* For system S4: gamma * == {[]P | []P : gamma} *)
(* delta * == {<>P | <>P : delta} *)
lstar0: "|L>" and
lstar1: "$G |L> $H \ []P, $G |L> []P, $H" and
lstar2: "$G |L> $H \ P, $G |L> $H" and
rstar0: "|R>" and
rstar1: "$G |R> $H \ <>P, $G |R> <>P, $H" and
rstar2: "$G |R> $H \ P, $G |R> $H" and
(* Rules for [] and <> *)
boxR:
"\$E |L> $E'; $F |R> $F'; $G |R> $G';
$E' \ $F', P, $G'\ \ $E \ $F, []P, $G" and
boxL: "$E,P,$F,[]P \ $G \ $E, []P, $F \ $G" and
diaR: "$E \ $F,P,$G,<>P \ $E \ $F, <>P, $G" and
diaL:
"\$E |L> $E'; $F |L> $F'; $G |R> $G';
$E', P, $F' \<turnstile> $G'\<rbrakk> \<Longrightarrow> $E, <>P, $F \<turnstile> $G"
ML \<open>
structure S4_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 S4_solve =
\<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (S4_Prover.solve_tac ctxt 2))\<close>
(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
lemma "\ []P \ P" by S4_solve
lemma "\ [](P \ Q) \ ([]P \ []Q)" by S4_solve (* normality*)
lemma "\ (P --< Q) \ []P \ []Q" by S4_solve
lemma "\ P \ <>P" by S4_solve
lemma "\ [](P \ Q) \ []P \ []Q" by S4_solve
lemma "\ <>(P \ Q) \ <>P \ <>Q" by S4_solve
lemma "\ [](P \ Q) \ (P >-< Q)" by S4_solve
lemma "\ <>(P \ Q) \ ([]P \ <>Q)" by S4_solve
lemma "\ []P \ \ <>(\ P)" by S4_solve
lemma "\ [](\ P) \ \ <>P" by S4_solve
lemma "\ \ []P \ <>(\ P)" by S4_solve
lemma "\ [][]P \ \ <><>(\ P)" by S4_solve
lemma "\ \ <>(P \ Q) \ \ <>P \ \ <>Q" by S4_solve
lemma "\ []P \ []Q \ [](P \ Q)" by S4_solve
lemma "\ <>(P \ Q) \ <>P \ <>Q" by S4_solve
lemma "\ [](P \ Q) \ []P \ <>Q" by S4_solve
lemma "\ <>P \ []Q \ <>(P \ Q)" by S4_solve
lemma "\ [](P \ Q) \ <>P \ []Q" by S4_solve
lemma "\ <>(P \ (Q \ R)) \ ([]P \ <>Q) \ ([]P \ <>R)" by S4_solve
lemma "\ (P --< Q) \ (Q --< R) \ (P --< R)" by S4_solve
lemma "\ []P \ <>Q \ <>(P \ Q)" by S4_solve
(* Theorems of system S4 from Hughes and Cresswell, p.46 *)
lemma "\ []A \ A" by S4_solve (* refexivity *)
lemma "\ []A \ [][]A" by S4_solve (* transitivity *)
lemma "\ []A \ <>A" by S4_solve (* seriality *)
lemma "\ <>[](<>A \ []<>A)" by S4_solve
lemma "\ <>[](<>[]A \ []A)" by S4_solve
lemma "\ []P \ [][]P" by S4_solve
lemma "\ <>P \ <><>P" by S4_solve
lemma "\ <>[]<>P \ <>P" by S4_solve
lemma "\ []<>P \ []<>[]<>P" by S4_solve
lemma "\ <>[]P \ <>[]<>[]P" by S4_solve
(* Theorems for system S4 from Hughes and Cresswell, p.60 *)
lemma "\ []P \ []Q \ []([]P \ []Q)" by S4_solve
lemma "\ ((P >-< Q) --< R) \ ((P >-< Q) --< []R)" by S4_solve
(* These are from Hailpern, LNCS 129 *)
lemma "\ [](P \ Q) \ []P \ []Q" by S4_solve
lemma "\ <>(P \ Q) \ <>P \ <>Q" by S4_solve
lemma "\ <>(P \ Q) \ ([]P \ <>Q)" by S4_solve
lemma "\ [](P \ Q) \ (<>P \ <>Q)" by S4_solve
lemma "\ []P \ []<>P" by S4_solve
lemma "\ <>[]P \ <>P" by S4_solve
lemma "\ []P \ []Q \ [](P \ Q)" by S4_solve
lemma "\ <>(P \ Q) \ <>P \ <>Q" by S4_solve
lemma "\ [](P \ Q) \ []P \ <>Q" by S4_solve
lemma "\ <>P \ []Q \ <>(P \ Q)" by S4_solve
lemma "\ [](P \ Q) \ <>P \ []Q" by S4_solve
end
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