(* Title: Modal/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'=>seq', seq'=>seq', seq'=>seq',
seq'=>seq', seq'=>seq', seq'=>seq'] => prop"
syntax
"_S43pi" :: "[seq, seq, seq, seq, seq, seq] => prop"
("S43pi((_);(_);(_);(_);(_);(_))" [] 5)
parse_translation {*
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"}, s43pi_tr)] end
*}
print_translation {*
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}, s43pi_tr')] end
*}
axioms
(* Definition of the star operation using a set of Horn clauses *)
(* For system S43: gamma * == {[]P | []P : gamma} *)
(* delta * == {<>P | <>P : delta} *)
lstar0: "|L>"
lstar1: "$G |L> $H ==> []P, $G |L> []P, $H"
lstar2: "$G |L> $H ==> P, $G |L> $H"
rstar0: "|R>"
rstar1: "$G |R> $H ==> <>P, $G |R> <>P, $H"
rstar2: "$G |R> $H ==> P, $G |R> $H"
(* Set of Horn clauses to generate the antecedents for the S43 pi rule *)
(* ie *)
(* S1...Sk,Sk+1...Sk+m *)
(* ---------------------------------- *)
(* <>P1...<>Pk, $G |- $H, []Q1...[]Qm *)
(* *)
(* where Si == <>P1...<>Pi-1,<>Pi+1,..<>Pk,Pi, $G * |- $H *, []Q1...[]Qm *)
(* and Sj == <>P1...<>Pk, $G * |- $H *, []Q1...[]Qj-1,[]Qj+1...[]Qm,Qj *)
(* and 1<=i<=k and k<j<=k+m *)
S43pi0: "S43pi $L;; $R;; $Lbox; $Rdia"
S43pi1:
"[| (S43pi <>P,$L'; $L;; $R; $Lbox;$Rdia); $L',P,$L,$Lbox |- $R,$Rdia |] ==>
S43pi $L'; <>P,$L;; $R; $Lbox;$Rdia"
S43pi2:
"[| (S43pi $L';; []P,$R'; $R; $Lbox;$Rdia); $L',$Lbox |- $R',P,$R,$Rdia |] ==>
S43pi $L';; $R'; []P,$R; $Lbox;$Rdia"
(* Rules for [] and <> for S43 *)
boxL: "$E, P, $F, []P |- $G ==> $E, []P, $F |- $G"
diaR: "$E |- $F, P, $G, <>P ==> $E |- $F, <>P, $G"
pi1:
"[| $L1,<>P,$L2 |L> $Lbox; $L1,<>P,$L2 |R> $Ldia; $R |L> $Rbox; $R |R> $Rdia;
S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==>
$L1, <>P, $L2 |- $R"
pi2:
"[| $L |L> $Lbox; $L |R> $Ldia; $R1,[]P,$R2 |L> $Rbox; $R1,[]P,$R2 |R> $Rdia;
S43pi ; $Ldia;; $Rbox; $Lbox; $Rdia |] ==>
$L |- $R1, []P, $R2"
ML {*
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"]
)
*}
method_setup S43_solve = {*
Scan.succeed (K (SIMPLE_METHOD
(S43_Prover.solve_tac 2 ORELSE S43_Prover.solve_tac 3)))
*} "S4 solver"
(* Theorems of system T from Hughes and Cresswell and Hailpern, LNCS 129 *)
lemma "|- []P --> P" by S43_solve
lemma "|- [](P-->Q) --> ([]P-->[]Q)" by S43_solve (* normality*)
lemma "|- (P--<Q) --> []P --> []Q" by S43_solve
lemma "|- P --> <>P" by S43_solve
lemma "|- [](P & Q) <-> []P & []Q" by S43_solve
lemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solve
lemma "|- [](P<->Q) <-> (P>-<Q)" by S43_solve
lemma "|- <>(P-->Q) <-> ([]P--><>Q)" by S43_solve
lemma "|- []P <-> ~<>(~P)" by S43_solve
lemma "|- [](~P) <-> ~<>P" by S43_solve
lemma "|- ~[]P <-> <>(~P)" by S43_solve
lemma "|- [][]P <-> ~<><>(~P)" by S43_solve
lemma "|- ~<>(P | Q) <-> ~<>P & ~<>Q" by S43_solve
lemma "|- []P | []Q --> [](P | Q)" by S43_solve
lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve
lemma "|- [](P | Q) --> []P | <>Q" by S43_solve
lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve
lemma "|- [](P | Q) --> <>P | []Q" by S43_solve
lemma "|- <>(P-->(Q & R)) --> ([]P --> <>Q) & ([]P--><>R)" by S43_solve
lemma "|- (P--<Q) & (Q--<R) --> (P--<R)" by S43_solve
lemma "|- []P --> <>Q --> <>(P & Q)" by S43_solve
(* Theorems of system S4 from Hughes and Cresswell, p.46 *)
lemma "|- []A --> A" by S43_solve (* refexivity *)
lemma "|- []A --> [][]A" by S43_solve (* transitivity *)
lemma "|- []A --> <>A" by S43_solve (* seriality *)
lemma "|- <>[](<>A --> []<>A)" by S43_solve
lemma "|- <>[](<>[]A --> []A)" by S43_solve
lemma "|- []P <-> [][]P" by S43_solve
lemma "|- <>P <-> <><>P" by S43_solve
lemma "|- <>[]<>P --> <>P" by S43_solve
lemma "|- []<>P <-> []<>[]<>P" by S43_solve
lemma "|- <>[]P <-> <>[]<>[]P" by S43_solve
(* Theorems for system S4 from Hughes and Cresswell, p.60 *)
lemma "|- []P | []Q <-> []([]P | []Q)" by S43_solve
lemma "|- ((P>-<Q) --< R) --> ((P>-<Q) --< []R)" by S43_solve
(* These are from Hailpern, LNCS 129 *)
lemma "|- [](P & Q) <-> []P & []Q" by S43_solve
lemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solve
lemma "|- <>(P --> Q) <-> ([]P --> <>Q)" by S43_solve
lemma "|- [](P --> Q) --> (<>P --> <>Q)" by S43_solve
lemma "|- []P --> []<>P" by S43_solve
lemma "|- <>[]P --> <>P" by S43_solve
lemma "|- []P | []Q --> [](P | Q)" by S43_solve
lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve
lemma "|- [](P | Q) --> []P | <>Q" by S43_solve
lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve
lemma "|- [](P | Q) --> <>P | []Q" by S43_solve
(* Theorems of system S43 *)
lemma "|- <>[]P --> []<>P" by S43_solve
lemma "|- <>[]P --> [][]<>P" by S43_solve
lemma "|- [](<>P | <>Q) --> []<>P | []<>Q" by S43_solve
lemma "|- <>[]P & <>[]Q --> <>([]P & []Q)" by S43_solve
lemma "|- []([]P --> []Q) | []([]Q --> []P)" by S43_solve
lemma "|- [](<>P --> <>Q) | [](<>Q --> <>P)" by S43_solve
lemma "|- []([]P --> Q) | []([]Q --> P)" by S43_solve
lemma "|- [](P --> <>Q) | [](Q --> <>P)" by S43_solve
lemma "|- [](P --> []Q-->R) | [](P | ([]R --> Q))" by S43_solve
lemma "|- [](P | (Q --> <>C)) | [](P --> C --> <>Q)" by S43_solve
lemma "|- []([]P | Q) & [](P | []Q) --> []P | []Q" by S43_solve
lemma "|- <>P & <>Q --> <>(<>P & Q) | <>(P & <>Q)" by S43_solve
lemma "|- [](P | Q) & []([]P | Q) & [](P | []Q) --> []P | []Q" by S43_solve
lemma "|- <>P & <>Q --> <>(P & Q) | <>(<>P & Q) | <>(P & <>Q)" by S43_solve
lemma "|- <>[]<>P <-> []<>P" by S43_solve
lemma "|- []<>[]P <-> <>[]P" by S43_solve
(* These are from Hailpern, LNCS 129 *)
lemma "|- [](P & Q) <-> []P & []Q" by S43_solve
lemma "|- <>(P | Q) <-> <>P | <>Q" by S43_solve
lemma "|- <>(P --> Q) <-> []P --> <>Q" by S43_solve
lemma "|- [](P --> Q) --> <>P --> <>Q" by S43_solve
lemma "|- []P --> []<>P" by S43_solve
lemma "|- <>[]P --> <>P" by S43_solve
lemma "|- []<>[]P --> []<>P" by S43_solve
lemma "|- <>[]P --> <>[]<>P" by S43_solve
lemma "|- <>[]P --> []<>P" by S43_solve
lemma "|- []<>[]P <-> <>[]P" by S43_solve
lemma "|- <>[]<>P <-> []<>P" by S43_solve
lemma "|- []P | []Q --> [](P | Q)" by S43_solve
lemma "|- <>(P & Q) --> <>P & <>Q" by S43_solve
lemma "|- [](P | Q) --> []P | <>Q" by S43_solve
lemma "|- <>P & []Q --> <>(P & Q)" by S43_solve
lemma "|- [](P | Q) --> <>P | []Q" by S43_solve
lemma "|- [](P | Q) --> []<>P | []<>Q" by S43_solve
lemma "|- <>[]P & <>[]Q --> <>(P & Q)" by S43_solve
lemma "|- <>[](P & Q) <-> <>[]P & <>[]Q" by S43_solve
lemma "|- []<>(P | Q) <-> []<>P | []<>Q" by S43_solve
end