(* 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'⇒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