Quelle  S4.thy

(*  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' ⊨ $G'] ==> $E, <>P, $F ⊨ $G"

ML ‹
 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}]
 )
 ›

method_setup S4_solve =
  ‹Scan.succeed (fn ctxt => SIMPLE_METHOD (S4_Prover.solve_tac ctxt 2))›


(* 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