# Theory S43

```(*  Title:      Sequents/S43.thy
Author:     Martin Coen

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] =
Syntax.const \<^const_syntax>‹S43pi› \$ tr s1 \$ tr s2 \$ tr s3 \$ tr s4 \$ tr s5 \$ tr s6;
in [(\<^syntax_const>‹_S43pi›, K s43pi_tr)] end
›

print_translation ‹
let
val tr' = seq_tr';
fun s43pi_tr' [s1, s2, s3, s4, s5, s6] =
Syntax.const \<^syntax_const>‹_S43pi› \$ tr' s1 \$ tr' s2 \$ tr' s3 \$ tr' s4 \$ tr' s5 \$ tr' s6;
in [(\<^const_syntax>‹S43pi›, K s43pi_tr')] end
›

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 ⟹ []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

(* 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" and
S43pi1:
"⟦(S43pi <>P,\$L';     \$L;; \$R; \$Lbox;\$Rdia);   \$L',P,\$L,\$Lbox ⊢ \$R,\$Rdia⟧ ⟹
S43pi     \$L'; <>P,\$L;; \$R; \$Lbox;\$Rdia" and
S43pi2:
"⟦(S43pi \$L';; []P,\$R';     \$R; \$Lbox;\$Rdia);  \$L',\$Lbox ⊢ \$R',P,\$R,\$Rdia⟧ ⟹
S43pi \$L';;     \$R'; []P,\$R; \$Lbox;\$Rdia" and

(* Rules for [] and <> for S43 *)

boxL:           "\$E, P, \$F, []P ⊢ \$G ⟹ \$E, []P, \$F ⊢ \$G" and
diaR:           "\$E ⊢ \$F, P, \$G, <>P ⟹ \$E ⊢ \$F, <>P, \$G" and
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" and
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 (fn ctxt => SIMPLE_METHOD
(S43_Prover.solve_tac ctxt 2 ORELSE S43_Prover.solve_tac ctxt 3))
›

(* 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
```