src/HOL/NanoJava/Equivalence.thy
author wenzelm
Sun Nov 02 18:21:45 2014 +0100 (2014-11-02)
changeset 58889 5b7a9633cfa8
parent 58262 85b13d75b2e4
child 58963 26bf09b95dda
permissions -rw-r--r--
modernized header uniformly as section;
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(*  Title:      HOL/NanoJava/Equivalence.thy
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    Author:     David von Oheimb
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    Copyright   2001 Technische Universitaet Muenchen
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*)
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section "Equivalence of Operational and Axiomatic Semantics"
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theory Equivalence imports OpSem AxSem begin
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subsection "Validity"
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definition valid :: "[assn,stmt, assn] => bool" ("|= {(1_)}/ (_)/ {(1_)}" [3,90,3] 60) where
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 "|=  {P} c {Q} \<equiv> \<forall>s   t. P s --> (\<exists>n. s -c  -n\<rightarrow> t) --> Q   t"
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definition evalid   :: "[assn,expr,vassn] => bool" ("|=e {(1_)}/ (_)/ {(1_)}" [3,90,3] 60) where
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 "|=e {P} e {Q} \<equiv> \<forall>s v t. P s --> (\<exists>n. s -e\<succ>v-n\<rightarrow> t) --> Q v t"
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definition nvalid   :: "[nat, triple    ] => bool" ("|=_: _"  [61,61] 60) where
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 "|=n:  t \<equiv> let (P,c,Q) = t in \<forall>s   t. s -c  -n\<rightarrow> t --> P s --> Q   t"
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definition envalid   :: "[nat,etriple    ] => bool" ("|=_:e _" [61,61] 60) where
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 "|=n:e t \<equiv> let (P,e,Q) = t in \<forall>s v t. s -e\<succ>v-n\<rightarrow> t --> P s --> Q v t"
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definition nvalids :: "[nat,       triple set] => bool" ("||=_: _" [61,61] 60) where
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 "||=n: T \<equiv> \<forall>t\<in>T. |=n: t"
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definition cnvalids :: "[triple set,triple set] => bool" ("_ ||=/ _"  [61,61] 60) where
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 "A ||=  C \<equiv> \<forall>n. ||=n: A --> ||=n: C"
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definition cenvalid  :: "[triple set,etriple   ] => bool" ("_ ||=e/ _" [61,61] 60) where
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 "A ||=e t \<equiv> \<forall>n. ||=n: A --> |=n:e t"
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notation (xsymbols)
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  valid  ("\<Turnstile> {(1_)}/ (_)/ {(1_)}" [3,90,3] 60) and
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  evalid  ("\<Turnstile>\<^sub>e {(1_)}/ (_)/ {(1_)}" [3,90,3] 60) and
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  nvalid  ("\<Turnstile>_: _" [61,61] 60) and
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  envalid  ("\<Turnstile>_:\<^sub>e _" [61,61] 60) and
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  nvalids  ("|\<Turnstile>_: _" [61,61] 60) and
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  cnvalids  ("_ |\<Turnstile>/ _" [61,61] 60) and
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  cenvalid  ("_ |\<Turnstile>\<^sub>e/ _"[61,61] 60)
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lemma nvalid_def2: "\<Turnstile>n: (P,c,Q) \<equiv> \<forall>s t. s -c-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q t"
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by (simp add: nvalid_def Let_def)
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lemma valid_def2: "\<Turnstile> {P} c {Q} = (\<forall>n. \<Turnstile>n: (P,c,Q))"
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apply (simp add: valid_def nvalid_def2)
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apply blast
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done
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lemma envalid_def2: "\<Turnstile>n:\<^sub>e (P,e,Q) \<equiv> \<forall>s v t. s -e\<succ>v-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q v t"
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by (simp add: envalid_def Let_def)
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lemma evalid_def2: "\<Turnstile>\<^sub>e {P} e {Q} = (\<forall>n. \<Turnstile>n:\<^sub>e (P,e,Q))"
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apply (simp add: evalid_def envalid_def2)
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apply blast
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done
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lemma cenvalid_def2: 
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  "A|\<Turnstile>\<^sub>e (P,e,Q) = (\<forall>n. |\<Turnstile>n: A \<longrightarrow> (\<forall>s v t. s -e\<succ>v-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q v t))"
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by(simp add: cenvalid_def envalid_def2) 
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subsection "Soundness"
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declare exec_elim_cases [elim!] eval_elim_cases [elim!]
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lemma Impl_nvalid_0: "\<Turnstile>0: (P,Impl M,Q)"
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by (clarsimp simp add: nvalid_def2)
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lemma Impl_nvalid_Suc: "\<Turnstile>n: (P,body M,Q) \<Longrightarrow> \<Turnstile>Suc n: (P,Impl M,Q)"
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by (clarsimp simp add: nvalid_def2)
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lemma nvalid_SucD: "\<And>t. \<Turnstile>Suc n:t \<Longrightarrow> \<Turnstile>n:t"
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by (force simp add: split_paired_all nvalid_def2 intro: exec_mono)
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lemma nvalids_SucD: "Ball A (nvalid (Suc n)) \<Longrightarrow>  Ball A (nvalid n)"
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by (fast intro: nvalid_SucD)
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lemma Loop_sound_lemma [rule_format (no_asm)]: 
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"\<forall>s t. s -c-n\<rightarrow> t \<longrightarrow> P s \<and> s<x> \<noteq> Null \<longrightarrow> P t \<Longrightarrow> 
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  (s -c0-n0\<rightarrow> t \<longrightarrow> P s \<longrightarrow> c0 = While (x) c \<longrightarrow> n0 = n \<longrightarrow> P t \<and> t<x> = Null)"
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apply (rule_tac ?P2.1="%s e v n t. True" in exec_eval.induct [THEN conjunct1])
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apply clarsimp+
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done
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lemma Impl_sound_lemma: 
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"\<lbrakk>\<forall>z n. Ball (A \<union> B) (nvalid n) \<longrightarrow> Ball (f z ` Ms) (nvalid n); 
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  Cm\<in>Ms; Ball A (nvalid na); Ball B (nvalid na)\<rbrakk> \<Longrightarrow> nvalid na (f z Cm)"
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by blast
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lemma all_conjunct2: "\<forall>l. P' l \<and> P l \<Longrightarrow> \<forall>l. P l"
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by fast
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lemma all3_conjunct2: 
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  "\<forall>a p l. (P' a p l \<and> P a p l) \<Longrightarrow> \<forall>a p l. P a p l"
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by fast
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lemma cnvalid1_eq: 
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  "A |\<Turnstile> {(P,c,Q)} \<equiv> \<forall>n. |\<Turnstile>n: A \<longrightarrow> (\<forall>s t. s -c-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q t)"
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by(simp add: cnvalids_def nvalids_def nvalid_def2)
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lemma hoare_sound_main:"\<And>t. (A |\<turnstile> C \<longrightarrow> A |\<Turnstile> C) \<and> (A |\<turnstile>\<^sub>e t \<longrightarrow> A |\<Turnstile>\<^sub>e t)"
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apply (tactic "split_all_tac @{context} 1", rename_tac P e Q)
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apply (rule hoare_ehoare.induct)
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(*18*)
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apply (tactic {* ALLGOALS (REPEAT o dresolve_tac [@{thm all_conjunct2}, @{thm all3_conjunct2}]) *})
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apply (tactic {* ALLGOALS (REPEAT o thin_tac @{context} "hoare ?x ?y") *})
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apply (tactic {* ALLGOALS (REPEAT o thin_tac @{context} "ehoare ?x ?y") *})
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apply (simp_all only: cnvalid1_eq cenvalid_def2)
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                 apply fast
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                apply fast
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               apply fast
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              apply (clarify,tactic "smp_tac 1 1",erule(2) Loop_sound_lemma,(rule HOL.refl)+)
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             apply fast
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            apply fast
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           apply fast
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          apply fast
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         apply fast
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        apply fast
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       apply (clarsimp del: Meth_elim_cases) (* Call *)
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      apply (force del: Impl_elim_cases)
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     defer
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     prefer 4 apply blast (*  Conseq *)
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    prefer 4 apply blast (* eConseq *)
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   apply (simp_all (no_asm_use) only: cnvalids_def nvalids_def)
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   apply blast
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  apply blast
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 apply blast
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apply (rule allI)
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apply (rule_tac x=Z in spec)
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apply (induct_tac "n")
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 apply  (clarify intro!: Impl_nvalid_0)
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apply (clarify  intro!: Impl_nvalid_Suc)
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apply (drule nvalids_SucD)
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apply (simp only: HOL.all_simps)
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apply (erule (1) impE)
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apply (drule (2) Impl_sound_lemma)
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 apply  blast
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apply assumption
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done
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theorem hoare_sound: "{} \<turnstile> {P} c {Q} \<Longrightarrow> \<Turnstile> {P} c {Q}"
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apply (simp only: valid_def2)
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apply (drule hoare_sound_main [THEN conjunct1, rule_format])
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apply (unfold cnvalids_def nvalids_def)
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apply fast
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done
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theorem ehoare_sound: "{} \<turnstile>\<^sub>e {P} e {Q} \<Longrightarrow> \<Turnstile>\<^sub>e {P} e {Q}"
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apply (simp only: evalid_def2)
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apply (drule hoare_sound_main [THEN conjunct2, rule_format])
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apply (unfold cenvalid_def nvalids_def)
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apply fast
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done
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subsection "(Relative) Completeness"
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definition MGT :: "stmt => state => triple" where
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         "MGT  c Z \<equiv> (\<lambda>s. Z = s, c, \<lambda>  t. \<exists>n. Z -c-  n\<rightarrow> t)"
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definition MGTe   :: "expr => state => etriple" where
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         "MGTe e Z \<equiv> (\<lambda>s. Z = s, e, \<lambda>v t. \<exists>n. Z -e\<succ>v-n\<rightarrow> t)"
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notation (xsymbols)
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  MGTe  ("MGT\<^sub>e")
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notation (HTML output)
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  MGTe  ("MGT\<^sub>e")
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lemma MGF_implies_complete:
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 "\<forall>Z. {} |\<turnstile> { MGT c Z} \<Longrightarrow> \<Turnstile>  {P} c {Q} \<Longrightarrow> {} \<turnstile>  {P} c {Q}"
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apply (simp only: valid_def2)
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apply (unfold MGT_def)
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apply (erule hoare_ehoare.Conseq)
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apply (clarsimp simp add: nvalid_def2)
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done
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lemma eMGF_implies_complete:
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 "\<forall>Z. {} |\<turnstile>\<^sub>e MGT\<^sub>e e Z \<Longrightarrow> \<Turnstile>\<^sub>e {P} e {Q} \<Longrightarrow> {} \<turnstile>\<^sub>e {P} e {Q}"
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apply (simp only: evalid_def2)
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apply (unfold MGTe_def)
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apply (erule hoare_ehoare.eConseq)
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apply (clarsimp simp add: envalid_def2)
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done
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declare exec_eval.intros[intro!]
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lemma MGF_Loop: "\<forall>Z. A \<turnstile> {op = Z} c {\<lambda>t. \<exists>n. Z -c-n\<rightarrow> t} \<Longrightarrow> 
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  A \<turnstile> {op = Z} While (x) c {\<lambda>t. \<exists>n. Z -While (x) c-n\<rightarrow> t}"
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apply (rule_tac P' = "\<lambda>Z s. (Z,s) \<in> ({(s,t). \<exists>n. s<x> \<noteq> Null \<and> s -c-n\<rightarrow> t})^*"
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       in hoare_ehoare.Conseq)
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apply  (rule allI)
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apply  (rule hoare_ehoare.Loop)
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apply  (erule hoare_ehoare.Conseq)
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apply  clarsimp
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apply  (blast intro:rtrancl_into_rtrancl)
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apply (erule thin_rl)
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apply clarsimp
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apply (erule_tac x = Z in allE)
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apply clarsimp
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apply (erule converse_rtrancl_induct)
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apply  blast
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apply clarsimp
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apply (drule (1) exec_exec_max)
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apply (blast del: exec_elim_cases)
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done
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lemma MGF_lemma: "\<forall>M Z. A |\<turnstile> {MGT (Impl M) Z} \<Longrightarrow> 
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 (\<forall>Z. A |\<turnstile> {MGT c Z}) \<and> (\<forall>Z. A |\<turnstile>\<^sub>e MGT\<^sub>e e Z)"
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apply (simp add: MGT_def MGTe_def)
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apply (rule stmt_expr.induct)
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apply (rule_tac [!] allI)
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apply (rule Conseq1 [OF hoare_ehoare.Skip])
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apply blast
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apply (rule hoare_ehoare.Comp)
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apply  (erule spec)
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apply (erule hoare_ehoare.Conseq)
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apply clarsimp
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apply (drule (1) exec_exec_max)
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apply blast
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apply (erule thin_rl)
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apply (rule hoare_ehoare.Cond)
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apply  (erule spec)
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apply (rule allI)
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apply (simp)
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apply (rule conjI)
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apply  (rule impI, erule hoare_ehoare.Conseq, clarsimp, drule (1) eval_exec_max,
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        erule thin_rl, erule thin_rl, force)+
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apply (erule MGF_Loop)
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apply (erule hoare_ehoare.eConseq [THEN hoare_ehoare.LAss])
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apply fast
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apply (erule thin_rl)
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apply (rename_tac expr1 u v Z, rule_tac Q = "\<lambda>a s. \<exists>n. Z -expr1\<succ>Addr a-n\<rightarrow> s" in hoare_ehoare.FAss)
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apply  (drule spec)
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apply  (erule eConseq2)
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apply  fast
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apply (rule allI)
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apply (erule hoare_ehoare.eConseq)
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apply clarsimp
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apply (drule (1) eval_eval_max)
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apply blast
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apply (simp only: split_paired_all)
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apply (rule hoare_ehoare.Meth)
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apply (rule allI)
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apply (drule spec, drule spec, erule hoare_ehoare.Conseq)
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apply blast
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apply (simp add: split_paired_all)
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apply (rule eConseq1 [OF hoare_ehoare.NewC])
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apply blast
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apply (erule hoare_ehoare.eConseq [THEN hoare_ehoare.Cast])
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apply fast
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apply (rule eConseq1 [OF hoare_ehoare.LAcc])
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apply blast
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apply (erule hoare_ehoare.eConseq [THEN hoare_ehoare.FAcc])
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apply fast
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apply (rename_tac expr1 u expr2 Z)
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apply (rule_tac R = "\<lambda>a v s. \<exists>n1 n2 t. Z -expr1\<succ>a-n1\<rightarrow> t \<and> t -expr2\<succ>v-n2\<rightarrow> s" in
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                hoare_ehoare.Call)
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apply   (erule spec)
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apply  (rule allI)
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apply  (erule hoare_ehoare.eConseq)
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apply  clarsimp
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apply  blast
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apply (rule allI)+
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apply (rule hoare_ehoare.Meth)
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apply (rule allI)
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apply (drule spec, drule spec, erule hoare_ehoare.Conseq)
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apply (erule thin_rl, erule thin_rl)
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apply (clarsimp del: Impl_elim_cases)
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apply (drule (2) eval_eval_exec_max)
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apply (force del: Impl_elim_cases)
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done
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lemma MGF_Impl: "{} |\<turnstile> {MGT (Impl M) Z}"
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apply (unfold MGT_def)
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apply (rule Impl1')
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apply  (rule_tac [2] UNIV_I)
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apply clarsimp
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apply (rule hoare_ehoare.ConjI)
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apply clarsimp
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   295
apply (rule ssubst [OF Impl_body_eq])
oheimb@11376
   296
apply (fold MGT_def)
oheimb@11476
   297
apply (rule MGF_lemma [THEN conjunct1, rule_format])
oheimb@11476
   298
apply (rule hoare_ehoare.Asm)
oheimb@11376
   299
apply force
oheimb@11376
   300
done
oheimb@11376
   301
oheimb@11376
   302
theorem hoare_relative_complete: "\<Turnstile> {P} c {Q} \<Longrightarrow> {} \<turnstile> {P} c {Q}"
oheimb@11376
   303
apply (rule MGF_implies_complete)
oheimb@11376
   304
apply  (erule_tac [2] asm_rl)
oheimb@11376
   305
apply (rule allI)
oheimb@11476
   306
apply (rule MGF_lemma [THEN conjunct1, rule_format])
oheimb@11476
   307
apply (rule MGF_Impl)
oheimb@11476
   308
done
oheimb@11476
   309
oheimb@11486
   310
theorem ehoare_relative_complete: "\<Turnstile>\<^sub>e {P} e {Q} \<Longrightarrow> {} \<turnstile>\<^sub>e {P} e {Q}"
oheimb@11476
   311
apply (rule eMGF_implies_complete)
oheimb@11476
   312
apply  (erule_tac [2] asm_rl)
oheimb@11476
   313
apply (rule allI)
oheimb@11476
   314
apply (rule MGF_lemma [THEN conjunct2, rule_format])
oheimb@11376
   315
apply (rule MGF_Impl)
oheimb@11376
   316
done
oheimb@11376
   317
oheimb@11565
   318
lemma cFalse: "A \<turnstile> {\<lambda>s. False} c {Q}"
oheimb@11565
   319
apply (rule cThin)
oheimb@11565
   320
apply (rule hoare_relative_complete)
oheimb@11565
   321
apply (auto simp add: valid_def)
oheimb@11565
   322
done
oheimb@11565
   323
oheimb@11565
   324
lemma eFalse: "A \<turnstile>\<^sub>e {\<lambda>s. False} e {Q}"
oheimb@11565
   325
apply (rule eThin)
oheimb@11565
   326
apply (rule ehoare_relative_complete)
oheimb@11565
   327
apply (auto simp add: evalid_def)
oheimb@11565
   328
done
oheimb@11565
   329
oheimb@11376
   330
end