author  wenzelm 
Fri, 04 Jan 2019 23:22:53 +0100  
changeset 69605  3dda49e08b9d 
parent 62144  bdab93ccfaf8 
child 69618  a96320074298 
permissions  rwrr 
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(* Title: Sequents/Modal0.thy 
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Author: Martin Coen 
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Copyright 1991 University of Cambridge 
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*) 
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theory Modal0 
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imports LK0 

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begin 

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ML_file "modal.ML" 
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consts 
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box :: "o\<Rightarrow>o" ("[]_" [50] 50) 
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dia :: "o\<Rightarrow>o" ("<>_" [50] 50) 

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Lstar :: "two_seqi" 
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Rstar :: "two_seqi" 

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syntax 

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"_Lstar" :: "two_seqe" ("(_)L>(_)" [6,6] 5) 
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"_Rstar" :: "two_seqe" ("(_)R>(_)" [6,6] 5) 

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ML \<open> 
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fun star_tr c [s1, s2] = Const(c, dummyT) $ seq_tr s1 $ seq_tr s2; 
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fun star_tr' c [s1, s2] = Const(c, dummyT) $ seq_tr' s1 $ seq_tr' s2; 

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\<close> 
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parse_translation \<open> 
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[(\<^syntax_const>\<open>_Lstar\<close>, K (star_tr \<^const_syntax>\<open>Lstar\<close>)), 
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(\<^syntax_const>\<open>_Rstar\<close>, K (star_tr \<^const_syntax>\<open>Rstar\<close>))] 

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\<close> 
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print_translation \<open> 
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[(\<^const_syntax>\<open>Lstar\<close>, K (star_tr' \<^syntax_const>\<open>_Lstar\<close>)), 
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(\<^const_syntax>\<open>Rstar\<close>, K (star_tr' \<^syntax_const>\<open>_Rstar\<close>))] 

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\<close> 
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definition strimp :: "[o,o]\<Rightarrow>o" (infixr "<" 25) 
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where "P < Q == [](P \<longrightarrow> Q)" 

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definition streqv :: "[o,o]\<Rightarrow>o" (infixr "><" 25) 

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where "P >< Q == (P < Q) \<and> (Q < P)" 

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lemmas rewrite_rls = strimp_def streqv_def 

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lemma iffR: "\<lbrakk>$H,P \<turnstile> $E,Q,$F; $H,Q \<turnstile> $E,P,$F\<rbrakk> \<Longrightarrow> $H \<turnstile> $E, P \<longleftrightarrow> Q, $F" 
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apply (unfold iff_def) 
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apply (assumption  rule conjR impR)+ 

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done 

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lemma iffL: "\<lbrakk>$H,$G \<turnstile> $E,P,Q; $H,Q,P,$G \<turnstile> $E \<rbrakk> \<Longrightarrow> $H, P \<longleftrightarrow> Q, $G \<turnstile> $E" 
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apply (unfold iff_def) 
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apply (assumption  rule conjL impL basic)+ 

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done 

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lemmas safe_rls = basic conjL conjR disjL disjR impL impR notL notR iffL iffR 

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and unsafe_rls = allR exL 

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and bound_rls = allL exR 

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end 