(* Title: Sequents/S43.thy
Author: Martin Coen
Copyright 1991 University of Cambridge
This implements Rajeev Gore's sequent calculus for S43.
*)
theory S43
imports Modal0
begin
consts
S43pi :: "[seq'\<Rightarrow>seq', seq'\<Rightarrow>seq', seq'\<Rightarrow>seq',
seq'\<Rightarrow>seq', seq'\<Rightarrow>seq', seq'\<Rightarrow>seq'] \<Rightarrow> prop"
syntax
"_S43pi" :: "[seq, seq, seq, seq, seq, seq] \<Rightarrow> prop"
("S43pi((_);(_);(_);(_);(_);(_))" [] 5)
parse_translation \<open>
let
val tr = seq_tr;
fun s43pi_tr [s1, s2, s3, s4, s5, s6] =
Const (@{const_syntax S43pi}, dummyT) $ tr s1 $ tr s2 $ tr s3 $ tr s4 $ tr s5 $ tr s6;
in [(@{syntax_const "_S43pi"}, K s43pi_tr)] end
\<close>
print_translation \<open>
let
val tr' = seq_tr';
fun s43pi_tr' [s1, s2, s3, s4, s5, s6] =
Const(@{syntax_const "_S43pi"}, dummyT) $ tr' s1 $ tr' s2 $ tr' s3 $ tr' s4 $ tr' s5 $ tr' s6;
in [(@{const_syntax S43pi}, K s43pi_tr')] end
\<close>
axiomatization where
(* Definition of the star operation using a set of Horn clauses *)
(* For system S43: 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
(* Set of Horn clauses to generate the antecedents for the S43 pi rule *)
(* ie *)
(* S1...Sk,Sk+1...Sk+m *)
(* ---------------------------------- *)
(* <>P1...<>Pk, $G \<turnstile> $H, []Q1...[]Qm *)
(* *)
(* where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, $G * \<turnstile> $H *, []Q1...[]Qm *)
(* and Sj == <>P1...<>Pk, $G * \<turnstile> $H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj *)
(* and 1<=i<=k and k<j<=k+m *)
S43pi0: "S43pi $L;; $R;; $Lbox; $Rdia" and
S43pi1:
"\<lbrakk>(S43pi <>P,$L'; $L;; $R; $Lbox;$Rdia); $L',P,$L,$Lbox \<turnstile> $R,$Rdia\<rbrakk> \<Longrightarrow>
S43pi $L'; <>P,$L;; $R; $Lbox;$Rdia" and
S43pi2:
"\<lbrakk>(S43pi $L';; []P,$R'; $R; $Lbox;$Rdia); $L',$Lbox \<turnstile> $R',P,$R,$Rdia\<rbrakk> \<Longrightarrow>
S43pi $L';; $R'; []P,$R; $Lbox;$Rdia" and
(* Rules for [] and <> for S43 *)
boxL: "$E, P, $F, []P \<turnstile> $G \<Longrightarrow> $E, []P, $F \<turnstile> $G" and
diaR: "$E \<turnstile> $F, P, $G, <>P \<Longrightarrow> $E \<turnstile> $F, <>P, $G" and
pi1:
"\<lbrakk>$L1,<>P,$L2 |L> $Lbox; $L1,<>P,$L2 |R> $Ldia; $R |L> $Rbox; $R |R> $Rdia;
S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia\<rbrakk> \<Longrightarrow>
$L1, <>P, $L2 \<turnstile> $R" and
pi2:
"\<lbrakk>$L |L> $Lbox; $L |R> $Ldia; $R1,[]P,$R2 |L> $Rbox; $R1,[]P,$R2 |R> $Rdia;
S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia\<rbrakk> \<Longrightarrow>
$L \<turnstile> $R1, []P, $R2"
ML \<open>
structure S43_Prover = Modal_ProverFun
(
val rewrite_rls = @{thms rewrite_rls}
val safe_rls = @{thms safe_rls}
val unsafe_rls = @{thms unsafe_rls} @ [@{thm pi1}, @{thm pi2}]
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}, @{thm S43pi0}, @{thm S43pi1}, @{thm S43pi2}]
)
\<close>
method_setup S43_solve = \<open>
Scan.succeed (fn ctxt => SIMPLE_METHOD
(S43_Prover.solve_tac ctxt 2 ORELSE S43_Prover.solve_tac ctxt 3))
\<close>
(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
lemma "\<turnstile> []P \<longrightarrow> P" by S43_solve
lemma "\<turnstile> [](P \<longrightarrow> Q) \<longrightarrow> ([]P \<longrightarrow> []Q)" by S43_solve (* normality*)
lemma "\<turnstile> (P--<Q) \<longrightarrow> []P \<longrightarrow> []Q" by S43_solve
lemma "\<turnstile> P \<longrightarrow> <>P" by S43_solve
lemma "\<turnstile> [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by S43_solve
lemma "\<turnstile> <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by S43_solve
lemma "\<turnstile> [](P \<longleftrightarrow> Q) \<longleftrightarrow> (P>-<Q)" by S43_solve
lemma "\<turnstile> <>(P \<longrightarrow> Q) \<longleftrightarrow> ([]P \<longrightarrow> <>Q)" by S43_solve
lemma "\<turnstile> []P \<longleftrightarrow> \<not> <>(\<not> P)" by S43_solve
lemma "\<turnstile> [](\<not>P) \<longleftrightarrow> \<not> <>P" by S43_solve
lemma "\<turnstile> \<not> []P \<longleftrightarrow> <>(\<not> P)" by S43_solve
lemma "\<turnstile> [][]P \<longleftrightarrow> \<not> <><>(\<not> P)" by S43_solve
lemma "\<turnstile> \<not> <>(P \<or> Q) \<longleftrightarrow> \<not> <>P \<and> \<not> <>Q" by S43_solve
lemma "\<turnstile> []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by S43_solve
lemma "\<turnstile> <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by S43_solve
lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by S43_solve
lemma "\<turnstile> <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by S43_solve
lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by S43_solve
lemma "\<turnstile> <>(P \<longrightarrow> (Q \<and> R)) \<longrightarrow> ([]P \<longrightarrow> <>Q) \<and> ([]P \<longrightarrow> <>R)" by S43_solve
lemma "\<turnstile> (P --< Q) \<and> (Q --<R ) \<longrightarrow> (P --< R)" by S43_solve
lemma "\<turnstile> []P \<longrightarrow> <>Q \<longrightarrow> <>(P \<and> Q)" by S43_solve
(* Theorems of system S4 from Hughes and Cresswell, p.46 *)
lemma "\<turnstile> []A \<longrightarrow> A" by S43_solve (* refexivity *)
lemma "\<turnstile> []A \<longrightarrow> [][]A" by S43_solve (* transitivity *)
lemma "\<turnstile> []A \<longrightarrow> <>A" by S43_solve (* seriality *)
lemma "\<turnstile> <>[](<>A \<longrightarrow> []<>A)" by S43_solve
lemma "\<turnstile> <>[](<>[]A \<longrightarrow> []A)" by S43_solve
lemma "\<turnstile> []P \<longleftrightarrow> [][]P" by S43_solve
lemma "\<turnstile> <>P \<longleftrightarrow> <><>P" by S43_solve
lemma "\<turnstile> <>[]<>P \<longrightarrow> <>P" by S43_solve
lemma "\<turnstile> []<>P \<longleftrightarrow> []<>[]<>P" by S43_solve
lemma "\<turnstile> <>[]P \<longleftrightarrow> <>[]<>[]P" by S43_solve
(* Theorems for system S4 from Hughes and Cresswell, p.60 *)
lemma "\<turnstile> []P \<or> []Q \<longleftrightarrow> []([]P \<or> []Q)" by S43_solve
lemma "\<turnstile> ((P >-< Q) --< R) \<longrightarrow> ((P >-< Q) --< []R)" by S43_solve
(* These are from Hailpern, LNCS 129 *)
lemma "\<turnstile> [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by S43_solve
lemma "\<turnstile> <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by S43_solve
lemma "\<turnstile> <>(P \<longrightarrow> Q) \<longleftrightarrow> ([]P \<longrightarrow> <>Q)" by S43_solve
lemma "\<turnstile> [](P \<longrightarrow> Q) \<longrightarrow> (<>P \<longrightarrow> <>Q)" by S43_solve
lemma "\<turnstile> []P \<longrightarrow> []<>P" by S43_solve
lemma "\<turnstile> <>[]P \<longrightarrow> <>P" by S43_solve
lemma "\<turnstile> []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by S43_solve
lemma "\<turnstile> <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by S43_solve
lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by S43_solve
lemma "\<turnstile> <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by S43_solve
lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by S43_solve
(* Theorems of system S43 *)
lemma "\<turnstile> <>[]P \<longrightarrow> []<>P" by S43_solve
lemma "\<turnstile> <>[]P \<longrightarrow> [][]<>P" by S43_solve
lemma "\<turnstile> [](<>P \<or> <>Q) \<longrightarrow> []<>P \<or> []<>Q" by S43_solve
lemma "\<turnstile> <>[]P \<and> <>[]Q \<longrightarrow> <>([]P \<and> []Q)" by S43_solve
lemma "\<turnstile> []([]P \<longrightarrow> []Q) \<or> []([]Q \<longrightarrow> []P)" by S43_solve
lemma "\<turnstile> [](<>P \<longrightarrow> <>Q) \<or> [](<>Q \<longrightarrow> <>P)" by S43_solve
lemma "\<turnstile> []([]P \<longrightarrow> Q) \<or> []([]Q \<longrightarrow> P)" by S43_solve
lemma "\<turnstile> [](P \<longrightarrow> <>Q) \<or> [](Q \<longrightarrow> <>P)" by S43_solve
lemma "\<turnstile> [](P \<longrightarrow> []Q \<longrightarrow> R) \<or> [](P \<or> ([]R \<longrightarrow> Q))" by S43_solve
lemma "\<turnstile> [](P \<or> (Q \<longrightarrow> <>C)) \<or> [](P \<longrightarrow> C \<longrightarrow> <>Q)" by S43_solve
lemma "\<turnstile> []([]P \<or> Q) \<and> [](P \<or> []Q) \<longrightarrow> []P \<or> []Q" by S43_solve
lemma "\<turnstile> <>P \<and> <>Q \<longrightarrow> <>(<>P \<and> Q) \<or> <>(P \<and> <>Q)" by S43_solve
lemma "\<turnstile> [](P \<or> Q) \<and> []([]P \<or> Q) \<and> [](P \<or> []Q) \<longrightarrow> []P \<or> []Q" by S43_solve
lemma "\<turnstile> <>P \<and> <>Q \<longrightarrow> <>(P \<and> Q) \<or> <>(<>P \<and> Q) \<or> <>(P \<and> <>Q)" by S43_solve
lemma "\<turnstile> <>[]<>P \<longleftrightarrow> []<>P" by S43_solve
lemma "\<turnstile> []<>[]P \<longleftrightarrow> <>[]P" by S43_solve
(* These are from Hailpern, LNCS 129 *)
lemma "\<turnstile> [](P \<and> Q) \<longleftrightarrow> []P \<and> []Q" by S43_solve
lemma "\<turnstile> <>(P \<or> Q) \<longleftrightarrow> <>P \<or> <>Q" by S43_solve
lemma "\<turnstile> <>(P \<longrightarrow> Q) \<longleftrightarrow> []P \<longrightarrow> <>Q" by S43_solve
lemma "\<turnstile> [](P \<longrightarrow> Q) \<longrightarrow> <>P \<longrightarrow> <>Q" by S43_solve
lemma "\<turnstile> []P \<longrightarrow> []<>P" by S43_solve
lemma "\<turnstile> <>[]P \<longrightarrow> <>P" by S43_solve
lemma "\<turnstile> []<>[]P \<longrightarrow> []<>P" by S43_solve
lemma "\<turnstile> <>[]P \<longrightarrow> <>[]<>P" by S43_solve
lemma "\<turnstile> <>[]P \<longrightarrow> []<>P" by S43_solve
lemma "\<turnstile> []<>[]P \<longleftrightarrow> <>[]P" by S43_solve
lemma "\<turnstile> <>[]<>P \<longleftrightarrow> []<>P" by S43_solve
lemma "\<turnstile> []P \<or> []Q \<longrightarrow> [](P \<or> Q)" by S43_solve
lemma "\<turnstile> <>(P \<and> Q) \<longrightarrow> <>P \<and> <>Q" by S43_solve
lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []P \<or> <>Q" by S43_solve
lemma "\<turnstile> <>P \<and> []Q \<longrightarrow> <>(P \<and> Q)" by S43_solve
lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> <>P \<or> []Q" by S43_solve
lemma "\<turnstile> [](P \<or> Q) \<longrightarrow> []<>P \<or> []<>Q" by S43_solve
lemma "\<turnstile> <>[]P \<and> <>[]Q \<longrightarrow> <>(P \<and> Q)" by S43_solve
lemma "\<turnstile> <>[](P \<and> Q) \<longleftrightarrow> <>[]P \<and> <>[]Q" by S43_solve
lemma "\<turnstile> []<>(P \<or> Q) \<longleftrightarrow> []<>P \<or> []<>Q" by S43_solve
end