author | schirmer |
Fri, 22 Feb 2002 11:26:44 +0100 | |
changeset 12925 | 99131847fb93 |
parent 12859 | f63315dfffd4 |
child 13337 | f75dfc606ac7 |
permissions | -rw-r--r-- |
12857 | 1 |
(* Title: HOL/Bali/AxSem.thy |
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ID: $Id$ |
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Author: David von Oheimb |
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License: GPL (GNU GENERAL PUBLIC LICENSE) |
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*) |
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header {* Axiomatic semantics of Java expressions and statements |
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(see also Eval.thy) |
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*} |
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theory AxSem = Evaln + TypeSafe: |
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text {* |
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design issues: |
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\begin{itemize} |
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\item a strong version of validity for triples with premises, namely one that |
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takes the recursive depth needed to complete execution, enables |
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correctness proof |
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\item auxiliary variables are handled first-class (-> Thomas Kleymann) |
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\item expressions not flattened to elementary assignments (as usual for |
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axiomatic semantics) but treated first-class => explicit result value |
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handling |
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\item intermediate values not on triple, but on assertion level |
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(with result entry) |
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\item multiple results with semantical substitution mechnism not requiring a |
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stack |
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\item because of dynamic method binding, terms need to be dependent on state. |
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this is also useful for conditional expressions and statements |
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\item result values in triples exactly as in eval relation (also for xcpt |
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states) |
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\item validity: additional assumption of state conformance and well-typedness, |
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which is required for soundness and thus rule hazard required of completeness |
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\end{itemize} |
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restrictions: |
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\begin{itemize} |
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\item all triples in a derivation are of the same type (due to weak |
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polymorphism) |
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\end{itemize} |
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*} |
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types res = vals (* result entry *) |
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syntax |
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Val :: "val \<Rightarrow> res" |
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Var :: "var \<Rightarrow> res" |
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Vals :: "val list \<Rightarrow> res" |
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translations |
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"Val x" => "(In1 x)" |
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"Var x" => "(In2 x)" |
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"Vals x" => "(In3 x)" |
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syntax |
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"Val_" :: "[pttrn] => pttrn" ("Val:_" [951] 950) |
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"Var_" :: "[pttrn] => pttrn" ("Var:_" [951] 950) |
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"Vals_" :: "[pttrn] => pttrn" ("Vals:_" [951] 950) |
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translations |
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"\<lambda>Val:v . b" == "(\<lambda>v. b) \<circ> the_In1" |
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"\<lambda>Var:v . b" == "(\<lambda>v. b) \<circ> the_In2" |
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"\<lambda>Vals:v. b" == "(\<lambda>v. b) \<circ> the_In3" |
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(* relation on result values, state and auxiliary variables *) |
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types 'a assn = "res \<Rightarrow> state \<Rightarrow> 'a \<Rightarrow> bool" |
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translations |
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"res" <= (type) "AxSem.res" |
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"a assn" <= (type) "vals \<Rightarrow> state \<Rightarrow> a \<Rightarrow> bool" |
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constdefs |
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assn_imp :: "'a assn \<Rightarrow> 'a assn \<Rightarrow> bool" (infixr "\<Rightarrow>" 25) |
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"P \<Rightarrow> Q \<equiv> \<forall>Y s Z. P Y s Z \<longrightarrow> Q Y s Z" |
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lemma assn_imp_def2 [iff]: "(P \<Rightarrow> Q) = (\<forall>Y s Z. P Y s Z \<longrightarrow> Q Y s Z)" |
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apply (unfold assn_imp_def) |
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apply (rule HOL.refl) |
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done |
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section "assertion transformers" |
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subsection "peek-and" |
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constdefs |
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peek_and :: "'a assn \<Rightarrow> (state \<Rightarrow> bool) \<Rightarrow> 'a assn" (infixl "\<and>." 13) |
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"P \<and>. p \<equiv> \<lambda>Y s Z. P Y s Z \<and> p s" |
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lemma peek_and_def2 [simp]: "peek_and P p Y s = (\<lambda>Z. (P Y s Z \<and> p s))" |
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apply (unfold peek_and_def) |
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apply (simp (no_asm)) |
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done |
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lemma peek_and_Not [simp]: "(P \<and>. (\<lambda>s. \<not> f s)) = (P \<and>. Not \<circ> f)" |
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apply (rule ext) |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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lemma peek_and_and [simp]: "peek_and (peek_and P p) p = peek_and P p" |
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apply (unfold peek_and_def) |
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apply (simp (no_asm)) |
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done |
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lemma peek_and_commut: "(P \<and>. p \<and>. q) = (P \<and>. q \<and>. p)" |
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apply (rule ext) |
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apply (rule ext) |
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apply (rule ext) |
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apply auto |
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done |
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syntax |
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Normal :: "'a assn \<Rightarrow> 'a assn" |
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translations |
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"Normal P" == "P \<and>. normal" |
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lemma peek_and_Normal [simp]: "peek_and (Normal P) p = Normal (peek_and P p)" |
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apply (rule ext) |
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apply (rule ext) |
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apply (rule ext) |
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apply auto |
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done |
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subsection "assn-supd" |
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constdefs |
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assn_supd :: "'a assn \<Rightarrow> (state \<Rightarrow> state) \<Rightarrow> 'a assn" (infixl ";." 13) |
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"P ;. f \<equiv> \<lambda>Y s' Z. \<exists>s. P Y s Z \<and> s' = f s" |
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lemma assn_supd_def2 [simp]: "assn_supd P f Y s' Z = (\<exists>s. P Y s Z \<and> s' = f s)" |
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apply (unfold assn_supd_def) |
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apply (simp (no_asm)) |
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done |
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subsection "supd-assn" |
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constdefs |
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supd_assn :: "(state \<Rightarrow> state) \<Rightarrow> 'a assn \<Rightarrow> 'a assn" (infixr ".;" 13) |
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"f .; P \<equiv> \<lambda>Y s. P Y (f s)" |
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lemma supd_assn_def2 [simp]: "(f .; P) Y s = P Y (f s)" |
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apply (unfold supd_assn_def) |
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apply (simp (no_asm)) |
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done |
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lemma supd_assn_supdD [elim]: "((f .; Q) ;. f) Y s Z \<Longrightarrow> Q Y s Z" |
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apply auto |
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done |
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lemma supd_assn_supdI [elim]: "Q Y s Z \<Longrightarrow> (f .; (Q ;. f)) Y s Z" |
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apply (auto simp del: split_paired_Ex) |
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done |
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subsection "subst-res" |
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constdefs |
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subst_res :: "'a assn \<Rightarrow> res \<Rightarrow> 'a assn" ("_\<leftarrow>_" [60,61] 60) |
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"P\<leftarrow>w \<equiv> \<lambda>Y. P w" |
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lemma subst_res_def2 [simp]: "(P\<leftarrow>w) Y = P w" |
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apply (unfold subst_res_def) |
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apply (simp (no_asm)) |
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done |
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lemma subst_subst_res [simp]: "P\<leftarrow>w\<leftarrow>v = P\<leftarrow>w" |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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lemma peek_and_subst_res [simp]: "(P \<and>. p)\<leftarrow>w = (P\<leftarrow>w \<and>. p)" |
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apply (rule ext) |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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(*###Do not work for some strange (unification?) reason |
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lemma subst_res_Val_beta [simp]: "(\<lambda>Y. P (the_In1 Y))\<leftarrow>Val v = (\<lambda>Y. P v)" |
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apply (rule ext) |
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by simp |
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lemma subst_res_Var_beta [simp]: "(\<lambda>Y. P (the_In2 Y))\<leftarrow>Var vf = (\<lambda>Y. P vf)"; |
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apply (rule ext) |
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by simp |
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lemma subst_res_Vals_beta [simp]: "(\<lambda>Y. P (the_In3 Y))\<leftarrow>Vals vs = (\<lambda>Y. P vs)"; |
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apply (rule ext) |
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by simp |
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*) |
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subsection "subst-Bool" |
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constdefs |
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subst_Bool :: "'a assn \<Rightarrow> bool \<Rightarrow> 'a assn" ("_\<leftarrow>=_" [60,61] 60) |
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"P\<leftarrow>=b \<equiv> \<lambda>Y s Z. \<exists>v. P (Val v) s Z \<and> (normal s \<longrightarrow> the_Bool v=b)" |
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lemma subst_Bool_def2 [simp]: |
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"(P\<leftarrow>=b) Y s Z = (\<exists>v. P (Val v) s Z \<and> (normal s \<longrightarrow> the_Bool v=b))" |
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apply (unfold subst_Bool_def) |
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apply (simp (no_asm)) |
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done |
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lemma subst_Bool_the_BoolI: "P (Val b) s Z \<Longrightarrow> (P\<leftarrow>=the_Bool b) Y s Z" |
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apply auto |
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done |
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subsection "peek-res" |
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constdefs |
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peek_res :: "(res \<Rightarrow> 'a assn) \<Rightarrow> 'a assn" |
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"peek_res Pf \<equiv> \<lambda>Y. Pf Y Y" |
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syntax |
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"@peek_res" :: "pttrn \<Rightarrow> 'a assn \<Rightarrow> 'a assn" ("\<lambda>_:. _" [0,3] 3) |
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translations |
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"\<lambda>w:. P" == "peek_res (\<lambda>w. P)" |
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lemma peek_res_def2 [simp]: "peek_res P Y = P Y Y" |
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apply (unfold peek_res_def) |
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apply (simp (no_asm)) |
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done |
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lemma peek_res_subst_res [simp]: "peek_res P\<leftarrow>w = P w\<leftarrow>w" |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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(* unused *) |
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lemma peek_subst_res_allI: |
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"(\<And>a. T a (P (f a)\<leftarrow>f a)) \<Longrightarrow> \<forall>a. T a (peek_res P\<leftarrow>f a)" |
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apply (rule allI) |
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apply (simp (no_asm)) |
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apply fast |
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done |
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subsection "ign-res" |
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constdefs |
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ign_res :: " 'a assn \<Rightarrow> 'a assn" ("_\<down>" [1000] 1000) |
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"P\<down> \<equiv> \<lambda>Y s Z. \<exists>Y. P Y s Z" |
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lemma ign_res_def2 [simp]: "P\<down> Y s Z = (\<exists>Y. P Y s Z)" |
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apply (unfold ign_res_def) |
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apply (simp (no_asm)) |
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done |
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lemma ign_ign_res [simp]: "P\<down>\<down> = P\<down>" |
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apply (rule ext) |
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apply (rule ext) |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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lemma ign_subst_res [simp]: "P\<down>\<leftarrow>w = P\<down>" |
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apply (rule ext) |
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apply (rule ext) |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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lemma peek_and_ign_res [simp]: "(P \<and>. p)\<down> = (P\<down> \<and>. p)" |
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apply (rule ext) |
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apply (rule ext) |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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subsection "peek-st" |
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constdefs |
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peek_st :: "(st \<Rightarrow> 'a assn) \<Rightarrow> 'a assn" |
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"peek_st P \<equiv> \<lambda>Y s. P (store s) Y s" |
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syntax |
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"@peek_st" :: "pttrn \<Rightarrow> 'a assn \<Rightarrow> 'a assn" ("\<lambda>_.. _" [0,3] 3) |
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translations |
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"\<lambda>s.. P" == "peek_st (\<lambda>s. P)" |
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lemma peek_st_def2 [simp]: "(\<lambda>s.. Pf s) Y s = Pf (store s) Y s" |
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apply (unfold peek_st_def) |
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apply (simp (no_asm)) |
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done |
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lemma peek_st_triv [simp]: "(\<lambda>s.. P) = P" |
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apply (rule ext) |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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lemma peek_st_st [simp]: "(\<lambda>s.. \<lambda>s'.. P s s') = (\<lambda>s.. P s s)" |
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apply (rule ext) |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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lemma peek_st_split [simp]: "(\<lambda>s.. \<lambda>Y s'. P s Y s') = (\<lambda>Y s. P (store s) Y s)" |
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apply (rule ext) |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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lemma peek_st_subst_res [simp]: "(\<lambda>s.. P s)\<leftarrow>w = (\<lambda>s.. P s\<leftarrow>w)" |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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lemma peek_st_Normal [simp]: "(\<lambda>s..(Normal (P s))) = Normal (\<lambda>s.. P s)" |
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apply (rule ext) |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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subsection "ign-res-eq" |
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constdefs |
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ign_res_eq :: "'a assn \<Rightarrow> res \<Rightarrow> 'a assn" ("_\<down>=_" [60,61] 60) |
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"P\<down>=w \<equiv> \<lambda>Y:. P\<down> \<and>. (\<lambda>s. Y=w)" |
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lemma ign_res_eq_def2 [simp]: "(P\<down>=w) Y s Z = ((\<exists>Y. P Y s Z) \<and> Y=w)" |
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apply (unfold ign_res_eq_def) |
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apply auto |
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done |
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lemma ign_ign_res_eq [simp]: "(P\<down>=w)\<down> = P\<down>" |
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apply (rule ext) |
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apply (rule ext) |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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(* unused *) |
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lemma ign_res_eq_subst_res: "P\<down>=w\<leftarrow>w = P\<down>" |
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apply (rule ext) |
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apply (rule ext) |
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apply (rule ext) |
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apply (simp (no_asm)) |
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done |
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(* unused *) |
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lemma subst_Bool_ign_res_eq: "((P\<leftarrow>=b)\<down>=x) Y s Z = ((P\<leftarrow>=b) Y s Z \<and> Y=x)" |
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apply (simp (no_asm)) |
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done |
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subsection "RefVar" |
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constdefs |
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RefVar :: "(state \<Rightarrow> vvar \<times> state) \<Rightarrow> 'a assn \<Rightarrow> 'a assn"(infixr "..;" 13) |
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"vf ..; P \<equiv> \<lambda>Y s. let (v,s') = vf s in P (Var v) s'" |
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lemma RefVar_def2 [simp]: "(vf ..; P) Y s = |
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P (Var (fst (vf s))) (snd (vf s))" |
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apply (unfold RefVar_def Let_def) |
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apply (simp (no_asm) add: split_beta) |
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done |
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subsection "allocation" |
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constdefs |
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Alloc :: "prog \<Rightarrow> obj_tag \<Rightarrow> 'a assn \<Rightarrow> 'a assn" |
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"Alloc G otag P \<equiv> \<lambda>Y s Z. |
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\<forall>s' a. G\<turnstile>s \<midarrow>halloc otag\<succ>a\<rightarrow> s'\<longrightarrow> P (Val (Addr a)) s' Z" |
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SXAlloc :: "prog \<Rightarrow> 'a assn \<Rightarrow> 'a assn" |
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"SXAlloc G P \<equiv> \<lambda>Y s Z. \<forall>s'. G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s' \<longrightarrow> P Y s' Z" |
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lemma Alloc_def2 [simp]: "Alloc G otag P Y s Z = |
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(\<forall>s' a. G\<turnstile>s \<midarrow>halloc otag\<succ>a\<rightarrow> s'\<longrightarrow> P (Val (Addr a)) s' Z)" |
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apply (unfold Alloc_def) |
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apply (simp (no_asm)) |
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done |
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lemma SXAlloc_def2 [simp]: |
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"SXAlloc G P Y s Z = (\<forall>s'. G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s' \<longrightarrow> P Y s' Z)" |
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apply (unfold SXAlloc_def) |
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apply (simp (no_asm)) |
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done |
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section "validity" |
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constdefs |
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type_ok :: "prog \<Rightarrow> term \<Rightarrow> state \<Rightarrow> bool" |
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"type_ok G t s \<equiv> \<exists>L T C. (normal s \<longrightarrow> \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T) \<and> s\<Colon>\<preceq>(G,L)" |
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datatype 'a triple = triple "('a assn)" "term" "('a assn)" (** should be |
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something like triple = \<forall>'a. triple ('a assn) term ('a assn) **) |
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("{(1_)}/ _>/ {(1_)}" [3,65,3]75) |
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types 'a triples = "'a triple set" |
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syntax |
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var_triple :: "['a assn, var ,'a assn] \<Rightarrow> 'a triple" |
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("{(1_)}/ _=>/ {(1_)}" [3,80,3] 75) |
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expr_triple :: "['a assn, expr ,'a assn] \<Rightarrow> 'a triple" |
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("{(1_)}/ _->/ {(1_)}" [3,80,3] 75) |
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exprs_triple :: "['a assn, expr list ,'a assn] \<Rightarrow> 'a triple" |
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("{(1_)}/ _#>/ {(1_)}" [3,65,3] 75) |
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stmt_triple :: "['a assn, stmt, 'a assn] \<Rightarrow> 'a triple" |
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("{(1_)}/ ._./ {(1_)}" [3,65,3] 75) |
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syntax (xsymbols) |
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triple :: "['a assn, term ,'a assn] \<Rightarrow> 'a triple" |
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("{(1_)}/ _\<succ>/ {(1_)}" [3,65,3] 75) |
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var_triple :: "['a assn, var ,'a assn] \<Rightarrow> 'a triple" |
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("{(1_)}/ _=\<succ>/ {(1_)}" [3,80,3] 75) |
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expr_triple :: "['a assn, expr ,'a assn] \<Rightarrow> 'a triple" |
|
405 |
("{(1_)}/ _-\<succ>/ {(1_)}" [3,80,3] 75) |
|
406 |
exprs_triple :: "['a assn, expr list ,'a assn] \<Rightarrow> 'a triple" |
|
407 |
("{(1_)}/ _\<doteq>\<succ>/ {(1_)}" [3,65,3] 75) |
|
408 |
||
409 |
translations |
|
410 |
"{P} e-\<succ> {Q}" == "{P} In1l e\<succ> {Q}" |
|
411 |
"{P} e=\<succ> {Q}" == "{P} In2 e\<succ> {Q}" |
|
412 |
"{P} e\<doteq>\<succ> {Q}" == "{P} In3 e\<succ> {Q}" |
|
413 |
"{P} .c. {Q}" == "{P} In1r c\<succ> {Q}" |
|
414 |
||
415 |
lemma inj_triple: "inj (\<lambda>(P,t,Q). {P} t\<succ> {Q})" |
|
416 |
apply (rule injI) |
|
417 |
apply auto |
|
418 |
done |
|
419 |
||
420 |
lemma triple_inj_eq: "({P} t\<succ> {Q} = {P'} t'\<succ> {Q'} ) = (P=P' \<and> t=t' \<and> Q=Q')" |
|
421 |
apply auto |
|
422 |
done |
|
423 |
||
424 |
constdefs |
|
425 |
mtriples :: "('c \<Rightarrow> 'sig \<Rightarrow> 'a assn) \<Rightarrow> ('c \<Rightarrow> 'sig \<Rightarrow> expr) \<Rightarrow> |
|
426 |
('c \<Rightarrow> 'sig \<Rightarrow> 'a assn) \<Rightarrow> ('c \<times> 'sig) set \<Rightarrow> 'a triples" |
|
427 |
("{{(1_)}/ _-\<succ>/ {(1_)} | _}"[3,65,3,65]75) |
|
428 |
"{{P} tf-\<succ> {Q} | ms} \<equiv> (\<lambda>(C,sig). {Normal(P C sig)} tf C sig-\<succ> {Q C sig})`ms" |
|
429 |
||
430 |
consts |
|
431 |
||
432 |
triple_valid :: "prog \<Rightarrow> nat \<Rightarrow> 'a triple \<Rightarrow> bool" |
|
433 |
( "_\<Turnstile>_:_" [61,0, 58] 57) |
|
434 |
ax_valids :: "prog \<Rightarrow> 'b triples \<Rightarrow> 'a triples \<Rightarrow> bool" |
|
435 |
("_,_|\<Turnstile>_" [61,58,58] 57) |
|
436 |
ax_derivs :: "prog \<Rightarrow> ('b triples \<times> 'a triples) set" |
|
437 |
||
438 |
syntax |
|
439 |
||
440 |
triples_valid:: "prog \<Rightarrow> nat \<Rightarrow> 'a triples \<Rightarrow> bool" |
|
441 |
( "_||=_:_" [61,0, 58] 57) |
|
442 |
ax_valid :: "prog \<Rightarrow> 'b triples \<Rightarrow> 'a triple \<Rightarrow> bool" |
|
443 |
( "_,_|=_" [61,58,58] 57) |
|
444 |
ax_Derivs:: "prog \<Rightarrow> 'b triples \<Rightarrow> 'a triples \<Rightarrow> bool" |
|
445 |
("_,_||-_" [61,58,58] 57) |
|
446 |
ax_Deriv :: "prog \<Rightarrow> 'b triples \<Rightarrow> 'a triple \<Rightarrow> bool" |
|
447 |
( "_,_|-_" [61,58,58] 57) |
|
448 |
||
449 |
syntax (xsymbols) |
|
450 |
||
451 |
triples_valid:: "prog \<Rightarrow> nat \<Rightarrow> 'a triples \<Rightarrow> bool" |
|
452 |
( "_|\<Turnstile>_:_" [61,0, 58] 57) |
|
453 |
ax_valid :: "prog \<Rightarrow> 'b triples \<Rightarrow> 'a triple \<Rightarrow> bool" |
|
454 |
( "_,_\<Turnstile>_" [61,58,58] 57) |
|
455 |
ax_Derivs:: "prog \<Rightarrow> 'b triples \<Rightarrow> 'a triples \<Rightarrow> bool" |
|
456 |
("_,_|\<turnstile>_" [61,58,58] 57) |
|
457 |
ax_Deriv :: "prog \<Rightarrow> 'b triples \<Rightarrow> 'a triple \<Rightarrow> bool" |
|
458 |
( "_,_\<turnstile>_" [61,58,58] 57) |
|
459 |
||
460 |
defs triple_valid_def: "G\<Turnstile>n:t \<equiv> case t of {P} t\<succ> {Q} \<Rightarrow> |
|
461 |
\<forall>Y s Z. P Y s Z \<longrightarrow> type_ok G t s \<longrightarrow> |
|
462 |
(\<forall>Y' s'. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y',s') \<longrightarrow> Q Y' s' Z)" |
|
463 |
translations "G|\<Turnstile>n:ts" == "Ball ts (triple_valid G n)" |
|
464 |
defs ax_valids_def:"G,A|\<Turnstile>ts \<equiv> \<forall>n. G|\<Turnstile>n:A \<longrightarrow> G|\<Turnstile>n:ts" |
|
465 |
translations "G,A \<Turnstile>t" == "G,A|\<Turnstile>{t}" |
|
466 |
"G,A|\<turnstile>ts" == "(A,ts) \<in> ax_derivs G" |
|
467 |
"G,A \<turnstile>t" == "G,A|\<turnstile>{t}" |
|
468 |
||
469 |
lemma triple_valid_def2: "G\<Turnstile>n:{P} t\<succ> {Q} = |
|
470 |
(\<forall>Y s Z. P Y s Z |
|
471 |
\<longrightarrow> (\<exists>L. (normal s \<longrightarrow> (\<exists>T C. \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T)) \<and> s\<Colon>\<preceq>(G,L)) \<longrightarrow> |
|
472 |
(\<forall>Y' s'. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y',s')\<longrightarrow> Q Y' s' Z))" |
|
473 |
apply (unfold triple_valid_def type_ok_def) |
|
474 |
apply (simp (no_asm)) |
|
475 |
done |
|
476 |
||
477 |
||
478 |
declare split_paired_All [simp del] split_paired_Ex [simp del] |
|
479 |
declare split_if [split del] split_if_asm [split del] |
|
480 |
option.split [split del] option.split_asm [split del] |
|
481 |
ML_setup {* |
|
482 |
simpset_ref() := simpset() delloop "split_all_tac"; |
|
483 |
claset_ref () := claset () delSWrapper "split_all_tac" |
|
484 |
*} |
|
485 |
||
486 |
||
487 |
inductive "ax_derivs G" intros |
|
488 |
||
489 |
empty: " G,A|\<turnstile>{}" |
|
490 |
insert:"\<lbrakk>G,A\<turnstile>t; G,A|\<turnstile>ts\<rbrakk> \<Longrightarrow> |
|
491 |
G,A|\<turnstile>insert t ts" |
|
492 |
||
493 |
asm: "ts\<subseteq>A \<Longrightarrow> G,A|\<turnstile>ts" |
|
494 |
||
495 |
(* could be added for convenience and efficiency, but is not necessary |
|
496 |
cut: "\<lbrakk>G,A'|\<turnstile>ts; G,A|\<turnstile>A'\<rbrakk> \<Longrightarrow> |
|
497 |
G,A |\<turnstile>ts" |
|
498 |
*) |
|
499 |
weaken:"\<lbrakk>G,A|\<turnstile>ts'; ts \<subseteq> ts'\<rbrakk> \<Longrightarrow> G,A|\<turnstile>ts" |
|
500 |
||
501 |
conseq:"\<forall>Y s Z . P Y s Z \<longrightarrow> (\<exists>P' Q'. G,A\<turnstile>{P'} t\<succ> {Q'} \<and> (\<forall>Y' s'. |
|
502 |
(\<forall>Y Z'. P' Y s Z' \<longrightarrow> Q' Y' s' Z') \<longrightarrow> |
|
503 |
Q Y' s' Z )) |
|
504 |
\<Longrightarrow> G,A\<turnstile>{P } t\<succ> {Q }" |
|
505 |
||
506 |
hazard:"G,A\<turnstile>{P \<and>. Not \<circ> type_ok G t} t\<succ> {Q}" |
|
507 |
||
508 |
Abrupt: "G,A\<turnstile>{P\<leftarrow>(arbitrary3 t) \<and>. Not \<circ> normal} t\<succ> {P}" |
|
509 |
||
510 |
(* variables *) |
|
511 |
LVar: " G,A\<turnstile>{Normal (\<lambda>s.. P\<leftarrow>Var (lvar vn s))} LVar vn=\<succ> {P}" |
|
512 |
||
513 |
FVar: "\<lbrakk>G,A\<turnstile>{Normal P} .Init C. {Q}; |
|
514 |
G,A\<turnstile>{Q} e-\<succ> {\<lambda>Val:a:. fvar C stat fn a ..; R}\<rbrakk> \<Longrightarrow> |
|
12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset
|
515 |
G,A\<turnstile>{Normal P} {accC,C,stat}e..fn=\<succ> {R}" |
12854 | 516 |
|
517 |
AVar: "\<lbrakk>G,A\<turnstile>{Normal P} e1-\<succ> {Q}; |
|
518 |
\<forall>a. G,A\<turnstile>{Q\<leftarrow>Val a} e2-\<succ> {\<lambda>Val:i:. avar G i a ..; R}\<rbrakk> \<Longrightarrow> |
|
519 |
G,A\<turnstile>{Normal P} e1.[e2]=\<succ> {R}" |
|
520 |
(* expressions *) |
|
521 |
||
522 |
NewC: "\<lbrakk>G,A\<turnstile>{Normal P} .Init C. {Alloc G (CInst C) Q}\<rbrakk> \<Longrightarrow> |
|
523 |
G,A\<turnstile>{Normal P} NewC C-\<succ> {Q}" |
|
524 |
||
525 |
NewA: "\<lbrakk>G,A\<turnstile>{Normal P} .init_comp_ty T. {Q}; G,A\<turnstile>{Q} e-\<succ> |
|
526 |
{\<lambda>Val:i:. abupd (check_neg i) .; Alloc G (Arr T (the_Intg i)) R}\<rbrakk> \<Longrightarrow> |
|
527 |
G,A\<turnstile>{Normal P} New T[e]-\<succ> {R}" |
|
528 |
||
529 |
Cast: "\<lbrakk>G,A\<turnstile>{Normal P} e-\<succ> {\<lambda>Val:v:. \<lambda>s.. |
|
530 |
abupd (raise_if (\<not>G,s\<turnstile>v fits T) ClassCast) .; Q\<leftarrow>Val v}\<rbrakk> \<Longrightarrow> |
|
531 |
G,A\<turnstile>{Normal P} Cast T e-\<succ> {Q}" |
|
532 |
||
533 |
Inst: "\<lbrakk>G,A\<turnstile>{Normal P} e-\<succ> {\<lambda>Val:v:. \<lambda>s.. |
|
534 |
Q\<leftarrow>Val (Bool (v\<noteq>Null \<and> G,s\<turnstile>v fits RefT T))}\<rbrakk> \<Longrightarrow> |
|
535 |
G,A\<turnstile>{Normal P} e InstOf T-\<succ> {Q}" |
|
536 |
||
537 |
Lit: "G,A\<turnstile>{Normal (P\<leftarrow>Val v)} Lit v-\<succ> {P}" |
|
538 |
||
539 |
Super:" G,A\<turnstile>{Normal (\<lambda>s.. P\<leftarrow>Val (val_this s))} Super-\<succ> {P}" |
|
540 |
||
541 |
Acc: "\<lbrakk>G,A\<turnstile>{Normal P} va=\<succ> {\<lambda>Var:(v,f):. Q\<leftarrow>Val v}\<rbrakk> \<Longrightarrow> |
|
542 |
G,A\<turnstile>{Normal P} Acc va-\<succ> {Q}" |
|
543 |
||
544 |
Ass: "\<lbrakk>G,A\<turnstile>{Normal P} va=\<succ> {Q}; |
|
545 |
\<forall>vf. G,A\<turnstile>{Q\<leftarrow>Var vf} e-\<succ> {\<lambda>Val:v:. assign (snd vf) v .; R}\<rbrakk> \<Longrightarrow> |
|
546 |
G,A\<turnstile>{Normal P} va:=e-\<succ> {R}" |
|
547 |
||
548 |
Cond: "\<lbrakk>G,A \<turnstile>{Normal P} e0-\<succ> {P'}; |
|
549 |
\<forall>b. G,A\<turnstile>{P'\<leftarrow>=b} (if b then e1 else e2)-\<succ> {Q}\<rbrakk> \<Longrightarrow> |
|
550 |
G,A\<turnstile>{Normal P} e0 ? e1 : e2-\<succ> {Q}" |
|
551 |
||
552 |
Call: |
|
553 |
"\<lbrakk>G,A\<turnstile>{Normal P} e-\<succ> {Q}; \<forall>a. G,A\<turnstile>{Q\<leftarrow>Val a} args\<doteq>\<succ> {R a}; |
|
554 |
\<forall>a vs invC declC l. G,A\<turnstile>{(R a\<leftarrow>Vals vs \<and>. |
|
555 |
(\<lambda>s. declC=invocation_declclass G mode (store s) a statT \<lparr>name=mn,parTs=pTs\<rparr> \<and> |
|
556 |
invC = invocation_class mode (store s) a statT \<and> |
|
557 |
l = locals (store s)) ;. |
|
558 |
init_lvars G declC \<lparr>name=mn,parTs=pTs\<rparr> mode a vs) \<and>. |
|
559 |
(\<lambda>s. normal s \<longrightarrow> G\<turnstile>mode\<rightarrow>invC\<preceq>statT)} |
|
560 |
Methd declC \<lparr>name=mn,parTs=pTs\<rparr>-\<succ> {set_lvars l .; S}\<rbrakk> \<Longrightarrow> |
|
12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset
|
561 |
G,A\<turnstile>{Normal P} {accC,statT,mode}e\<cdot>mn({pTs}args)-\<succ> {S}" |
12854 | 562 |
|
563 |
Methd:"\<lbrakk>G,A\<union> {{P} Methd-\<succ> {Q} | ms} |\<turnstile> {{P} body G-\<succ> {Q} | ms}\<rbrakk> \<Longrightarrow> |
|
564 |
G,A|\<turnstile>{{P} Methd-\<succ> {Q} | ms}" |
|
565 |
||
566 |
Body: "\<lbrakk>G,A\<turnstile>{Normal P} .Init D. {Q}; |
|
567 |
G,A\<turnstile>{Q} .c. {\<lambda>s.. abupd (absorb Ret) .; R\<leftarrow>(In1 (the (locals s Result)))}\<rbrakk> |
|
568 |
\<Longrightarrow> |
|
569 |
G,A\<turnstile>{Normal P} Body D c-\<succ> {R}" |
|
570 |
||
571 |
(* expression lists *) |
|
572 |
||
573 |
Nil: "G,A\<turnstile>{Normal (P\<leftarrow>Vals [])} []\<doteq>\<succ> {P}" |
|
574 |
||
575 |
Cons: "\<lbrakk>G,A\<turnstile>{Normal P} e-\<succ> {Q}; |
|
576 |
\<forall>v. G,A\<turnstile>{Q\<leftarrow>Val v} es\<doteq>\<succ> {\<lambda>Vals:vs:. R\<leftarrow>Vals (v#vs)}\<rbrakk> \<Longrightarrow> |
|
577 |
G,A\<turnstile>{Normal P} e#es\<doteq>\<succ> {R}" |
|
578 |
||
579 |
(* statements *) |
|
580 |
||
581 |
Skip: "G,A\<turnstile>{Normal (P\<leftarrow>\<diamondsuit>)} .Skip. {P}" |
|
582 |
||
583 |
Expr: "\<lbrakk>G,A\<turnstile>{Normal P} e-\<succ> {Q\<leftarrow>\<diamondsuit>}\<rbrakk> \<Longrightarrow> |
|
584 |
G,A\<turnstile>{Normal P} .Expr e. {Q}" |
|
585 |
||
586 |
Lab: "\<lbrakk>G,A\<turnstile>{Normal P} .c. {abupd (absorb (Break l)) .; Q}\<rbrakk> \<Longrightarrow> |
|
587 |
G,A\<turnstile>{Normal P} .l\<bullet> c. {Q}" |
|
588 |
||
589 |
Comp: "\<lbrakk>G,A\<turnstile>{Normal P} .c1. {Q}; |
|
590 |
G,A\<turnstile>{Q} .c2. {R}\<rbrakk> \<Longrightarrow> |
|
591 |
G,A\<turnstile>{Normal P} .c1;;c2. {R}" |
|
592 |
||
593 |
If: "\<lbrakk>G,A \<turnstile>{Normal P} e-\<succ> {P'}; |
|
594 |
\<forall>b. G,A\<turnstile>{P'\<leftarrow>=b} .(if b then c1 else c2). {Q}\<rbrakk> \<Longrightarrow> |
|
595 |
G,A\<turnstile>{Normal P} .If(e) c1 Else c2. {Q}" |
|
596 |
(* unfolding variant of Loop, not needed here |
|
597 |
LoopU:"\<lbrakk>G,A \<turnstile>{Normal P} e-\<succ> {P'}; |
|
598 |
\<forall>b. G,A\<turnstile>{P'\<leftarrow>=b} .(if b then c;;While(e) c else Skip).{Q}\<rbrakk> |
|
599 |
\<Longrightarrow> G,A\<turnstile>{Normal P} .While(e) c. {Q}" |
|
600 |
*) |
|
601 |
Loop: "\<lbrakk>G,A\<turnstile>{P} e-\<succ> {P'}; |
|
602 |
G,A\<turnstile>{Normal (P'\<leftarrow>=True)} .c. {abupd (absorb (Cont l)) .; P}\<rbrakk> \<Longrightarrow> |
|
603 |
G,A\<turnstile>{P} .l\<bullet> While(e) c. {(P'\<leftarrow>=False)\<down>=\<diamondsuit>}" |
|
604 |
(** Beware of polymorphic_Loop below: should be identical terms **) |
|
605 |
||
606 |
Do: "G,A\<turnstile>{Normal (abupd (\<lambda>a. (Some (Jump j))) .; P\<leftarrow>\<diamondsuit>)} .Do j. {P}" |
|
607 |
||
608 |
Throw:"\<lbrakk>G,A\<turnstile>{Normal P} e-\<succ> {\<lambda>Val:a:. abupd (throw a) .; Q\<leftarrow>\<diamondsuit>}\<rbrakk> \<Longrightarrow> |
|
609 |
G,A\<turnstile>{Normal P} .Throw e. {Q}" |
|
610 |
||
611 |
Try: "\<lbrakk>G,A\<turnstile>{Normal P} .c1. {SXAlloc G Q}; |
|
612 |
G,A\<turnstile>{Q \<and>. (\<lambda>s. G,s\<turnstile>catch C) ;. new_xcpt_var vn} .c2. {R}; |
|
613 |
(Q \<and>. (\<lambda>s. \<not>G,s\<turnstile>catch C)) \<Rightarrow> R\<rbrakk> \<Longrightarrow> |
|
614 |
G,A\<turnstile>{Normal P} .Try c1 Catch(C vn) c2. {R}" |
|
615 |
||
616 |
Fin: "\<lbrakk>G,A\<turnstile>{Normal P} .c1. {Q}; |
|
617 |
\<forall>x. G,A\<turnstile>{Q \<and>. (\<lambda>s. x = fst s) ;. abupd (\<lambda>x. None)} |
|
618 |
.c2. {abupd (abrupt_if (x\<noteq>None) x) .; R}\<rbrakk> \<Longrightarrow> |
|
619 |
G,A\<turnstile>{Normal P} .c1 Finally c2. {R}" |
|
620 |
||
621 |
Done: "G,A\<turnstile>{Normal (P\<leftarrow>\<diamondsuit> \<and>. initd C)} .Init C. {P}" |
|
622 |
||
623 |
Init: "\<lbrakk>the (class G C) = c; |
|
624 |
G,A\<turnstile>{Normal ((P \<and>. Not \<circ> initd C) ;. supd (init_class_obj G C))} |
|
625 |
.(if C = Object then Skip else Init (super c)). {Q}; |
|
626 |
\<forall>l. G,A\<turnstile>{Q \<and>. (\<lambda>s. l = locals (store s)) ;. set_lvars empty} |
|
627 |
.init c. {set_lvars l .; R}\<rbrakk> \<Longrightarrow> |
|
628 |
G,A\<turnstile>{Normal (P \<and>. Not \<circ> initd C)} .Init C. {R}" |
|
629 |
||
630 |
axioms (** these terms are the same as above, but with generalized typing **) |
|
631 |
polymorphic_conseq: |
|
632 |
"\<forall>Y s Z . P Y s Z \<longrightarrow> (\<exists>P' Q'. G,A\<turnstile>{P'} t\<succ> {Q'} \<and> (\<forall>Y' s'. |
|
633 |
(\<forall>Y Z'. P' Y s Z' \<longrightarrow> Q' Y' s' Z') \<longrightarrow> |
|
634 |
Q Y' s' Z )) |
|
635 |
\<Longrightarrow> G,A\<turnstile>{P } t\<succ> {Q }" |
|
636 |
||
637 |
polymorphic_Loop: |
|
638 |
"\<lbrakk>G,A\<turnstile>{P} e-\<succ> {P'}; |
|
639 |
G,A\<turnstile>{Normal (P'\<leftarrow>=True)} .c. {abupd (absorb (Cont l)) .; P}\<rbrakk> \<Longrightarrow> |
|
640 |
G,A\<turnstile>{P} .l\<bullet> While(e) c. {(P'\<leftarrow>=False)\<down>=\<diamondsuit>}" |
|
641 |
||
642 |
constdefs |
|
643 |
adapt_pre :: "'a assn \<Rightarrow> 'a assn \<Rightarrow> 'a assn \<Rightarrow> 'a assn" |
|
644 |
"adapt_pre P Q Q'\<equiv>\<lambda>Y s Z. \<forall>Y' s'. \<exists>Z'. P Y s Z' \<and> (Q Y' s' Z' \<longrightarrow> Q' Y' s' Z)" |
|
645 |
||
646 |
||
647 |
section "rules derived by induction" |
|
648 |
||
649 |
lemma cut_valid: "\<lbrakk>G,A'|\<Turnstile>ts; G,A|\<Turnstile>A'\<rbrakk> \<Longrightarrow> G,A|\<Turnstile>ts" |
|
650 |
apply (unfold ax_valids_def) |
|
651 |
apply fast |
|
652 |
done |
|
653 |
||
654 |
(*if cut is available |
|
655 |
Goal "\<lbrakk>G,A'|\<turnstile>ts; A' \<subseteq> A; \<forall>P Q t. {P} t\<succ> {Q} \<in> A' \<longrightarrow> (\<exists>T. (G,L)\<turnstile>t\<Colon>T) \<rbrakk> \<Longrightarrow> |
|
656 |
G,A|\<turnstile>ts" |
|
657 |
b y etac ax_derivs.cut 1; |
|
658 |
b y eatac ax_derivs.asm 1 1; |
|
659 |
qed "ax_thin"; |
|
660 |
*) |
|
661 |
lemma ax_thin [rule_format (no_asm)]: |
|
662 |
"G,(A'::'a triple set)|\<turnstile>(ts::'a triple set) \<Longrightarrow> \<forall>A. A' \<subseteq> A \<longrightarrow> G,A|\<turnstile>ts" |
|
663 |
apply (erule ax_derivs.induct) |
|
664 |
apply (tactic "ALLGOALS(EVERY'[Clarify_tac,REPEAT o smp_tac 1])") |
|
665 |
apply (rule ax_derivs.empty) |
|
666 |
apply (erule (1) ax_derivs.insert) |
|
667 |
apply (fast intro: ax_derivs.asm) |
|
668 |
(*apply (fast intro: ax_derivs.cut) *) |
|
669 |
apply (fast intro: ax_derivs.weaken) |
|
670 |
apply (rule ax_derivs.conseq, intro strip, tactic "smp_tac 3 1",clarify, tactic "smp_tac 1 1",rule exI, rule exI, erule (1) conjI) |
|
671 |
(* 31 subgoals *) |
|
672 |
prefer 16 (* Methd *) |
|
673 |
apply (rule ax_derivs.Methd, drule spec, erule mp, fast) |
|
674 |
apply (tactic {* TRYALL (resolve_tac ((funpow 5 tl) (thms "ax_derivs.intros")) |
|
675 |
THEN_ALL_NEW Blast_tac) *}) |
|
676 |
apply (erule ax_derivs.Call) |
|
677 |
apply clarify |
|
678 |
apply blast |
|
679 |
||
680 |
apply (rule allI)+ |
|
681 |
apply (drule spec)+ |
|
682 |
apply blast |
|
683 |
done |
|
684 |
||
685 |
lemma ax_thin_insert: "G,(A::'a triple set)\<turnstile>(t::'a triple) \<Longrightarrow> G,insert x A\<turnstile>t" |
|
686 |
apply (erule ax_thin) |
|
687 |
apply fast |
|
688 |
done |
|
689 |
||
690 |
lemma subset_mtriples_iff: |
|
691 |
"ts \<subseteq> {{P} mb-\<succ> {Q} | ms} = (\<exists>ms'. ms'\<subseteq>ms \<and> ts = {{P} mb-\<succ> {Q} | ms'})" |
|
692 |
apply (unfold mtriples_def) |
|
693 |
apply (rule subset_image_iff) |
|
694 |
done |
|
695 |
||
696 |
lemma weaken: |
|
697 |
"G,(A::'a triple set)|\<turnstile>(ts'::'a triple set) \<Longrightarrow> !ts. ts \<subseteq> ts' \<longrightarrow> G,A|\<turnstile>ts" |
|
698 |
apply (erule ax_derivs.induct) |
|
699 |
(*36 subgoals*) |
|
700 |
apply (tactic "ALLGOALS strip_tac") |
|
701 |
apply (tactic {* ALLGOALS(REPEAT o (EVERY'[dtac (thm "subset_singletonD"), |
|
702 |
etac disjE, fast_tac (claset() addSIs [thm "ax_derivs.empty"])]))*}) |
|
703 |
apply (tactic "TRYALL hyp_subst_tac") |
|
704 |
apply (simp, rule ax_derivs.empty) |
|
705 |
apply (drule subset_insertD) |
|
706 |
apply (blast intro: ax_derivs.insert) |
|
707 |
apply (fast intro: ax_derivs.asm) |
|
708 |
(*apply (blast intro: ax_derivs.cut) *) |
|
709 |
apply (fast intro: ax_derivs.weaken) |
|
710 |
apply (rule ax_derivs.conseq, clarify, tactic "smp_tac 3 1", blast(* unused *)) |
|
711 |
(*31 subgoals*) |
|
712 |
apply (tactic {* TRYALL (resolve_tac ((funpow 5 tl) (thms "ax_derivs.intros")) |
|
713 |
THEN_ALL_NEW Fast_tac) *}) |
|
714 |
(*1 subgoal*) |
|
715 |
apply (clarsimp simp add: subset_mtriples_iff) |
|
716 |
apply (rule ax_derivs.Methd) |
|
717 |
apply (drule spec) |
|
718 |
apply (erule impE) |
|
719 |
apply (rule exI) |
|
720 |
apply (erule conjI) |
|
721 |
apply (rule HOL.refl) |
|
722 |
oops (* dead end, Methd is to blame *) |
|
723 |
||
724 |
||
725 |
section "rules derived from conseq" |
|
726 |
||
727 |
lemma conseq12: "\<lbrakk>G,A\<turnstile>{P'} t\<succ> {Q'}; |
|
728 |
\<forall>Y s Z. P Y s Z \<longrightarrow> (\<forall>Y' s'. (\<forall>Y Z'. P' Y s Z' \<longrightarrow> Q' Y' s' Z') \<longrightarrow> |
|
729 |
Q Y' s' Z)\<rbrakk> |
|
730 |
\<Longrightarrow> G,A\<turnstile>{P ::'a assn} t\<succ> {Q }" |
|
731 |
apply (rule polymorphic_conseq) |
|
732 |
apply clarsimp |
|
733 |
apply blast |
|
734 |
done |
|
735 |
||
736 |
(*unused, but nice variant*) |
|
737 |
lemma conseq12': "\<lbrakk>G,A\<turnstile>{P'} t\<succ> {Q'}; \<forall>s Y' s'. |
|
738 |
(\<forall>Y Z. P' Y s Z \<longrightarrow> Q' Y' s' Z) \<longrightarrow> |
|
739 |
(\<forall>Y Z. P Y s Z \<longrightarrow> Q Y' s' Z)\<rbrakk> |
|
740 |
\<Longrightarrow> G,A\<turnstile>{P } t\<succ> {Q }" |
|
741 |
apply (erule conseq12) |
|
742 |
apply fast |
|
743 |
done |
|
744 |
||
745 |
lemma conseq12_from_conseq12': "\<lbrakk>G,A\<turnstile>{P'} t\<succ> {Q'}; |
|
746 |
\<forall>Y s Z. P Y s Z \<longrightarrow> (\<forall>Y' s'. (\<forall>Y Z'. P' Y s Z' \<longrightarrow> Q' Y' s' Z') \<longrightarrow> |
|
747 |
Q Y' s' Z)\<rbrakk> |
|
748 |
\<Longrightarrow> G,A\<turnstile>{P } t\<succ> {Q }" |
|
749 |
apply (erule conseq12') |
|
750 |
apply blast |
|
751 |
done |
|
752 |
||
753 |
lemma conseq1: "\<lbrakk>G,A\<turnstile>{P'} t\<succ> {Q}; P \<Rightarrow> P'\<rbrakk> \<Longrightarrow> G,A\<turnstile>{P } t\<succ> {Q}" |
|
754 |
apply (erule conseq12) |
|
755 |
apply blast |
|
756 |
done |
|
757 |
||
758 |
lemma conseq2: "\<lbrakk>G,A\<turnstile>{P} t\<succ> {Q'}; Q' \<Rightarrow> Q\<rbrakk> \<Longrightarrow> G,A\<turnstile>{P} t\<succ> {Q}" |
|
759 |
apply (erule conseq12) |
|
760 |
apply blast |
|
761 |
done |
|
762 |
||
763 |
lemma ax_escape: "\<lbrakk>\<forall>Y s Z. P Y s Z \<longrightarrow> G,A\<turnstile>{\<lambda>Y' s' Z'. (Y',s') = (Y,s)} t\<succ> {\<lambda>Y s Z'. Q Y s Z}\<rbrakk> \<Longrightarrow> |
|
764 |
G,A\<turnstile>{P} t\<succ> {Q}" |
|
765 |
apply (rule polymorphic_conseq) |
|
766 |
apply force |
|
767 |
done |
|
768 |
||
769 |
(* unused *) |
|
770 |
lemma ax_constant: "\<lbrakk> C \<Longrightarrow> G,A\<turnstile>{P} t\<succ> {Q}\<rbrakk> \<Longrightarrow> G,A\<turnstile>{\<lambda>Y s Z. C \<and> P Y s Z} t\<succ> {Q}" |
|
771 |
apply (rule ax_escape (* unused *)) |
|
772 |
apply clarify |
|
773 |
apply (rule conseq12) |
|
774 |
apply fast |
|
775 |
apply auto |
|
776 |
done |
|
777 |
(*alternative (more direct) proof: |
|
778 |
apply (rule ax_derivs.conseq) *)(* unused *)(* |
|
779 |
apply (fast) |
|
780 |
*) |
|
781 |
||
782 |
||
783 |
lemma ax_impossible [intro]: "G,A\<turnstile>{\<lambda>Y s Z. False} t\<succ> {Q}" |
|
784 |
apply (rule ax_escape) |
|
785 |
apply clarify |
|
786 |
done |
|
787 |
||
788 |
(* unused *) |
|
789 |
lemma ax_nochange_lemma: "\<lbrakk>P Y s; All (op = w)\<rbrakk> \<Longrightarrow> P w s" |
|
790 |
apply auto |
|
791 |
done |
|
792 |
lemma ax_nochange:"G,A\<turnstile>{\<lambda>Y s Z. (Y,s)=Z} t\<succ> {\<lambda>Y s Z. (Y,s)=Z} \<Longrightarrow> G,A\<turnstile>{P} t\<succ> {P}" |
|
793 |
apply (erule conseq12) |
|
794 |
apply auto |
|
795 |
apply (erule (1) ax_nochange_lemma) |
|
796 |
done |
|
797 |
||
798 |
(* unused *) |
|
799 |
lemma ax_trivial: "G,A\<turnstile>{P} t\<succ> {\<lambda>Y s Z. True}" |
|
800 |
apply (rule polymorphic_conseq(* unused *)) |
|
801 |
apply auto |
|
802 |
done |
|
803 |
||
804 |
(* unused *) |
|
805 |
lemma ax_disj: "\<lbrakk>G,A\<turnstile>{P1} t\<succ> {Q1}; G,A\<turnstile>{P2} t\<succ> {Q2}\<rbrakk> \<Longrightarrow> |
|
806 |
G,A\<turnstile>{\<lambda>Y s Z. P1 Y s Z \<or> P2 Y s Z} t\<succ> {\<lambda>Y s Z. Q1 Y s Z \<or> Q2 Y s Z}" |
|
807 |
apply (rule ax_escape (* unused *)) |
|
808 |
apply safe |
|
809 |
apply (erule conseq12, fast)+ |
|
810 |
done |
|
811 |
||
812 |
(* unused *) |
|
813 |
lemma ax_supd_shuffle: "(\<exists>Q. G,A\<turnstile>{P} .c1. {Q} \<and> G,A\<turnstile>{Q ;. f} .c2. {R}) = |
|
814 |
(\<exists>Q'. G,A\<turnstile>{P} .c1. {f .; Q'} \<and> G,A\<turnstile>{Q'} .c2. {R})" |
|
815 |
apply (best elim!: conseq1 conseq2) |
|
816 |
done |
|
817 |
||
818 |
lemma ax_cases: "\<lbrakk>G,A\<turnstile>{P \<and>. C} t\<succ> {Q}; |
|
819 |
G,A\<turnstile>{P \<and>. Not \<circ> C} t\<succ> {Q}\<rbrakk> \<Longrightarrow> G,A\<turnstile>{P} t\<succ> {Q}" |
|
820 |
apply (unfold peek_and_def) |
|
821 |
apply (rule ax_escape) |
|
822 |
apply clarify |
|
823 |
apply (case_tac "C s") |
|
824 |
apply (erule conseq12, force)+ |
|
825 |
done |
|
826 |
(*alternative (more direct) proof: |
|
827 |
apply (rule rtac ax_derivs.conseq) *)(* unused *)(* |
|
828 |
apply clarify |
|
829 |
apply (case_tac "C s") |
|
830 |
apply force+ |
|
831 |
*) |
|
832 |
||
833 |
lemma ax_adapt: "G,A\<turnstile>{P} t\<succ> {Q} \<Longrightarrow> G,A\<turnstile>{adapt_pre P Q Q'} t\<succ> {Q'}" |
|
834 |
apply (unfold adapt_pre_def) |
|
835 |
apply (erule conseq12) |
|
836 |
apply fast |
|
837 |
done |
|
838 |
||
839 |
lemma adapt_pre_adapts: "G,A\<Turnstile>{P} t\<succ> {Q} \<longrightarrow> G,A\<Turnstile>{adapt_pre P Q Q'} t\<succ> {Q'}" |
|
840 |
apply (unfold adapt_pre_def) |
|
841 |
apply (simp add: ax_valids_def triple_valid_def2) |
|
842 |
apply fast |
|
843 |
done |
|
844 |
||
845 |
||
846 |
lemma adapt_pre_weakest: |
|
847 |
"\<forall>G (A::'a triple set) t. G,A\<Turnstile>{P} t\<succ> {Q} \<longrightarrow> G,A\<Turnstile>{P'} t\<succ> {Q'} \<Longrightarrow> |
|
848 |
P' \<Rightarrow> adapt_pre P Q (Q'::'a assn)" |
|
849 |
apply (unfold adapt_pre_def) |
|
850 |
apply (drule spec) |
|
851 |
apply (drule_tac x = "{}" in spec) |
|
852 |
apply (drule_tac x = "In1r Skip" in spec) |
|
853 |
apply (simp add: ax_valids_def triple_valid_def2) |
|
854 |
oops |
|
855 |
||
856 |
(* |
|
857 |
Goal "\<forall>(A::'a triple set) t. G,A\<Turnstile>{P} t\<succ> {Q} \<longrightarrow> G,A\<Turnstile>{P'} t\<succ> {Q'} \<Longrightarrow> |
|
858 |
wf_prog G \<Longrightarrow> G,(A::'a triple set)\<turnstile>{P} t\<succ> {Q::'a assn} \<Longrightarrow> G,A\<turnstile>{P'} t\<succ> {Q'::'a assn}" |
|
859 |
b y fatac ax_sound 1 1; |
|
860 |
b y asm_full_simp_tac (simpset() addsimps [ax_valids_def,triple_valid_def2]) 1; |
|
861 |
b y rtac ax_no_hazard 1; |
|
862 |
b y etac conseq12 1; |
|
863 |
b y Clarify_tac 1; |
|
864 |
b y case_tac "\<forall>Z. \<not>P Y s Z" 1; |
|
865 |
b y smp_tac 2 1; |
|
866 |
b y etac thin_rl 1; |
|
867 |
b y etac thin_rl 1; |
|
868 |
b y clarsimp_tac (claset(), simpset() addsimps [type_ok_def]) 1; |
|
869 |
b y subgoal_tac "G|\<Turnstile>n:A" 1; |
|
870 |
b y smp_tac 1 1; |
|
871 |
b y smp_tac 3 1; |
|
872 |
b y etac impE 1; |
|
873 |
back(); |
|
874 |
b y Fast_tac 1; |
|
875 |
b y |
|
876 |
b y rotate_tac 2 1; |
|
877 |
b y etac thin_rl 1; |
|
878 |
b y etac thin_rl 2; |
|
879 |
b y etac thin_rl 2; |
|
880 |
b y Clarify_tac 2; |
|
881 |
b y dtac spec 2; |
|
882 |
b y EVERY'[dtac spec, mp_tac] 2; |
|
883 |
b y thin_tac "\<forall>n Y s Z. ?PP n Y s Z" 2; |
|
884 |
b y thin_tac "P' Y s Z" 2; |
|
885 |
b y Blast_tac 2; |
|
886 |
b y smp_tac 3 1; |
|
887 |
b y case_tac "\<forall>Z. \<not>P Y s Z" 1; |
|
888 |
b y dres_inst_tac [("x","In1r Skip")] spec 1; |
|
889 |
b y Full_simp_tac 1; |
|
890 |
*) |
|
891 |
||
892 |
lemma peek_and_forget1_Normal: |
|
893 |
"G,A\<turnstile>{Normal P} t\<succ> {Q} \<Longrightarrow> G,A\<turnstile>{Normal (P \<and>. p)} t\<succ> {Q}" |
|
894 |
apply (erule conseq1) |
|
895 |
apply (simp (no_asm)) |
|
896 |
done |
|
897 |
||
898 |
lemma peek_and_forget1: "G,A\<turnstile>{P} t\<succ> {Q} \<Longrightarrow> G,A\<turnstile>{P \<and>. p} t\<succ> {Q}" |
|
899 |
apply (erule conseq1) |
|
900 |
apply (simp (no_asm)) |
|
901 |
done |
|
902 |
||
903 |
lemmas ax_NormalD = peek_and_forget1 [of _ _ _ _ _ normal] |
|
904 |
||
905 |
lemma peek_and_forget2: "G,A\<turnstile>{P} t\<succ> {Q \<and>. p} \<Longrightarrow> G,A\<turnstile>{P} t\<succ> {Q}" |
|
906 |
apply (erule conseq2) |
|
907 |
apply (simp (no_asm)) |
|
908 |
done |
|
909 |
||
910 |
lemma ax_subst_Val_allI: "\<forall>v. G,A\<turnstile>{(P' v )\<leftarrow>Val v} t\<succ> {Q v} \<Longrightarrow> |
|
911 |
\<forall>v. G,A\<turnstile>{(\<lambda>w:. P' (the_In1 w))\<leftarrow>Val v} t\<succ> {Q v}" |
|
912 |
apply (force elim!: conseq1) |
|
913 |
done |
|
914 |
||
915 |
lemma ax_subst_Var_allI: "\<forall>v. G,A\<turnstile>{(P' v )\<leftarrow>Var v} t\<succ> {Q v} \<Longrightarrow> |
|
916 |
\<forall>v. G,A\<turnstile>{(\<lambda>w:. P' (the_In2 w))\<leftarrow>Var v} t\<succ> {Q v}" |
|
917 |
apply (force elim!: conseq1) |
|
918 |
done |
|
919 |
||
920 |
lemma ax_subst_Vals_allI: "(\<forall>v. G,A\<turnstile>{( P' v )\<leftarrow>Vals v} t\<succ> {Q v}) \<Longrightarrow> |
|
921 |
\<forall>v. G,A\<turnstile>{(\<lambda>w:. P' (the_In3 w))\<leftarrow>Vals v} t\<succ> {Q v}" |
|
922 |
apply (force elim!: conseq1) |
|
923 |
done |
|
924 |
||
925 |
||
926 |
section "alternative axioms" |
|
927 |
||
928 |
lemma ax_Lit2: |
|
929 |
"G,(A::'a triple set)\<turnstile>{Normal P::'a assn} Lit v-\<succ> {Normal (P\<down>=Val v)}" |
|
930 |
apply (rule ax_derivs.Lit [THEN conseq1]) |
|
931 |
apply force |
|
932 |
done |
|
933 |
lemma ax_Lit2_test_complete: |
|
934 |
"G,(A::'a triple set)\<turnstile>{Normal (P\<leftarrow>Val v)::'a assn} Lit v-\<succ> {P}" |
|
935 |
apply (rule ax_Lit2 [THEN conseq2]) |
|
936 |
apply force |
|
937 |
done |
|
938 |
||
939 |
lemma ax_LVar2: "G,(A::'a triple set)\<turnstile>{Normal P::'a assn} LVar vn=\<succ> {Normal (\<lambda>s.. P\<down>=Var (lvar vn s))}" |
|
940 |
apply (rule ax_derivs.LVar [THEN conseq1]) |
|
941 |
apply force |
|
942 |
done |
|
943 |
||
944 |
lemma ax_Super2: "G,(A::'a triple set)\<turnstile> |
|
945 |
{Normal P::'a assn} Super-\<succ> {Normal (\<lambda>s.. P\<down>=Val (val_this s))}" |
|
946 |
apply (rule ax_derivs.Super [THEN conseq1]) |
|
947 |
apply force |
|
948 |
done |
|
949 |
||
950 |
lemma ax_Nil2: |
|
951 |
"G,(A::'a triple set)\<turnstile>{Normal P::'a assn} []\<doteq>\<succ> {Normal (P\<down>=Vals [])}" |
|
952 |
apply (rule ax_derivs.Nil [THEN conseq1]) |
|
953 |
apply force |
|
954 |
done |
|
955 |
||
956 |
||
957 |
section "misc derived structural rules" |
|
958 |
||
959 |
(* unused *) |
|
960 |
lemma ax_finite_mtriples_lemma: "\<lbrakk>F \<subseteq> ms; finite ms; \<forall>(C,sig)\<in>ms. |
|
961 |
G,(A::'a triple set)\<turnstile>{Normal (P C sig)::'a assn} mb C sig-\<succ> {Q C sig}\<rbrakk> \<Longrightarrow> |
|
962 |
G,A|\<turnstile>{{P} mb-\<succ> {Q} | F}" |
|
963 |
apply (frule (1) finite_subset) |
|
964 |
apply (erule make_imp) |
|
965 |
apply (erule thin_rl) |
|
966 |
apply (erule finite_induct) |
|
967 |
apply (unfold mtriples_def) |
|
968 |
apply (clarsimp intro!: ax_derivs.empty ax_derivs.insert)+ |
|
969 |
apply force |
|
970 |
done |
|
971 |
lemmas ax_finite_mtriples = ax_finite_mtriples_lemma [OF subset_refl] |
|
972 |
||
973 |
lemma ax_derivs_insertD: |
|
974 |
"G,(A::'a triple set)|\<turnstile>insert (t::'a triple) ts \<Longrightarrow> G,A\<turnstile>t \<and> G,A|\<turnstile>ts" |
|
975 |
apply (fast intro: ax_derivs.weaken) |
|
976 |
done |
|
977 |
||
978 |
lemma ax_methods_spec: |
|
979 |
"\<lbrakk>G,(A::'a triple set)|\<turnstile>split f ` ms; (C,sig) \<in> ms\<rbrakk>\<Longrightarrow> G,A\<turnstile>((f C sig)::'a triple)" |
|
980 |
apply (erule ax_derivs.weaken) |
|
981 |
apply (force del: image_eqI intro: rev_image_eqI) |
|
982 |
done |
|
983 |
||
984 |
(* this version is used to avoid using the cut rule *) |
|
985 |
lemma ax_finite_pointwise_lemma [rule_format]: "\<lbrakk>F \<subseteq> ms; finite ms\<rbrakk> \<Longrightarrow> |
|
986 |
((\<forall>(C,sig)\<in>F. G,(A::'a triple set)\<turnstile>(f C sig::'a triple)) \<longrightarrow> (\<forall>(C,sig)\<in>ms. G,A\<turnstile>(g C sig::'a triple))) \<longrightarrow> |
|
987 |
G,A|\<turnstile>split f ` F \<longrightarrow> G,A|\<turnstile>split g ` F" |
|
988 |
apply (frule (1) finite_subset) |
|
989 |
apply (erule make_imp) |
|
990 |
apply (erule thin_rl) |
|
991 |
apply (erule finite_induct) |
|
992 |
apply clarsimp+ |
|
993 |
apply (drule ax_derivs_insertD) |
|
994 |
apply (rule ax_derivs.insert) |
|
995 |
apply (simp (no_asm_simp) only: split_tupled_all) |
|
996 |
apply (auto elim: ax_methods_spec) |
|
997 |
done |
|
998 |
lemmas ax_finite_pointwise = ax_finite_pointwise_lemma [OF subset_refl] |
|
999 |
||
1000 |
lemma ax_no_hazard: |
|
1001 |
"G,(A::'a triple set)\<turnstile>{P \<and>. type_ok G t} t\<succ> {Q::'a assn} \<Longrightarrow> G,A\<turnstile>{P} t\<succ> {Q}" |
|
1002 |
apply (erule ax_cases) |
|
1003 |
apply (rule ax_derivs.hazard [THEN conseq1]) |
|
1004 |
apply force |
|
1005 |
done |
|
1006 |
||
1007 |
lemma ax_free_wt: |
|
1008 |
"(\<exists>T L C. \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T) |
|
1009 |
\<longrightarrow> G,(A::'a triple set)\<turnstile>{Normal P} t\<succ> {Q::'a assn} \<Longrightarrow> |
|
1010 |
G,A\<turnstile>{Normal P} t\<succ> {Q}" |
|
1011 |
apply (rule ax_no_hazard) |
|
1012 |
apply (rule ax_escape) |
|
1013 |
apply clarify |
|
1014 |
apply (erule mp [THEN conseq12]) |
|
1015 |
apply (auto simp add: type_ok_def) |
|
1016 |
done |
|
1017 |
||
1018 |
ML {* |
|
1019 |
bind_thms ("ax_Abrupts", sum3_instantiate (thm "ax_derivs.Abrupt")) |
|
1020 |
*} |
|
1021 |
declare ax_Abrupts [intro!] |
|
1022 |
||
1023 |
lemmas ax_Normal_cases = ax_cases [of _ _ normal] |
|
1024 |
||
1025 |
lemma ax_Skip [intro!]: "G,(A::'a triple set)\<turnstile>{P\<leftarrow>\<diamondsuit>} .Skip. {P::'a assn}" |
|
1026 |
apply (rule ax_Normal_cases) |
|
1027 |
apply (rule ax_derivs.Skip) |
|
1028 |
apply fast |
|
1029 |
done |
|
1030 |
lemmas ax_SkipI = ax_Skip [THEN conseq1, standard] |
|
1031 |
||
1032 |
||
1033 |
section "derived rules for methd call" |
|
1034 |
||
1035 |
lemma ax_Call_known_DynT: |
|
1036 |
"\<lbrakk>G\<turnstile>IntVir\<rightarrow>C\<preceq>statT; |
|
1037 |
\<forall>a vs l. G,A\<turnstile>{(R a\<leftarrow>Vals vs \<and>. (\<lambda>s. l = locals (store s)) ;. |
|
1038 |
init_lvars G C \<lparr>name=mn,parTs=pTs\<rparr> IntVir a vs)} |
|
1039 |
Methd C \<lparr>name=mn,parTs=pTs\<rparr>-\<succ> {set_lvars l .; S}; |
|
1040 |
\<forall>a. G,A\<turnstile>{Q\<leftarrow>Val a} args\<doteq>\<succ> |
|
1041 |
{R a \<and>. (\<lambda>s. C = obj_class (the (heap (store s) (the_Addr a))) \<and> |
|
1042 |
C = invocation_declclass |
|
1043 |
G IntVir (store s) a statT \<lparr>name=mn,parTs=pTs\<rparr> )}; |
|
1044 |
G,(A::'a triple set)\<turnstile>{Normal P} e-\<succ> {Q::'a assn}\<rbrakk> |
|
12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset
|
1045 |
\<Longrightarrow> G,A\<turnstile>{Normal P} {accC,statT,IntVir}e\<cdot>mn({pTs}args)-\<succ> {S}" |
12854 | 1046 |
apply (erule ax_derivs.Call) |
1047 |
apply safe |
|
1048 |
apply (erule spec) |
|
1049 |
apply (rule ax_escape, clarsimp) |
|
1050 |
apply (drule spec, drule spec, drule spec,erule conseq12) |
|
1051 |
apply force |
|
1052 |
done |
|
1053 |
||
1054 |
||
1055 |
lemma ax_Call_Static: |
|
1056 |
"\<lbrakk>\<forall>a vs l. G,A\<turnstile>{R a\<leftarrow>Vals vs \<and>. (\<lambda>s. l = locals (store s)) ;. |
|
1057 |
init_lvars G C \<lparr>name=mn,parTs=pTs\<rparr> Static any_Addr vs} |
|
1058 |
Methd C \<lparr>name=mn,parTs=pTs\<rparr>-\<succ> {set_lvars l .; S}; |
|
1059 |
G,A\<turnstile>{Normal P} e-\<succ> {Q}; |
|
1060 |
\<forall> a. G,(A::'a triple set)\<turnstile>{Q\<leftarrow>Val a} args\<doteq>\<succ> {(R::val \<Rightarrow> 'a assn) a |
|
1061 |
\<and>. (\<lambda> s. C=invocation_declclass |
|
1062 |
G Static (store s) a statT \<lparr>name=mn,parTs=pTs\<rparr>)} |
|
12925
99131847fb93
Added check for field/method access to operational semantics and proved the acesses valid.
schirmer
parents:
12859
diff
changeset
|
1063 |
\<rbrakk> \<Longrightarrow> G,A\<turnstile>{Normal P} {accC,statT,Static}e\<cdot>mn({pTs}args)-\<succ> {S}" |
12854 | 1064 |
apply (erule ax_derivs.Call) |
1065 |
apply safe |
|
1066 |
apply (erule spec) |
|
1067 |
apply (rule ax_escape, clarsimp) |
|
1068 |
apply (erule_tac V = "?P \<longrightarrow> ?Q" in thin_rl) |
|
1069 |
apply (drule spec,drule spec,drule spec, erule conseq12) |
|
1070 |
apply (force simp add: init_lvars_def) |
|
1071 |
done |
|
1072 |
||
1073 |
lemma ax_Methd1: |
|
1074 |
"\<lbrakk>G,A\<union>{{P} Methd-\<succ> {Q} | ms}|\<turnstile> {{P} body G-\<succ> {Q} | ms}; (C,sig)\<in> ms\<rbrakk> \<Longrightarrow> |
|
1075 |
G,A\<turnstile>{Normal (P C sig)} Methd C sig-\<succ> {Q C sig}" |
|
1076 |
apply (drule ax_derivs.Methd) |
|
1077 |
apply (unfold mtriples_def) |
|
1078 |
apply (erule (1) ax_methods_spec) |
|
1079 |
done |
|
1080 |
||
1081 |
lemma ax_MethdN: |
|
1082 |
"G,insert({Normal P} Methd C sig-\<succ> {Q}) A\<turnstile> |
|
1083 |
{Normal P} body G C sig-\<succ> {Q} \<Longrightarrow> |
|
1084 |
G,A\<turnstile>{Normal P} Methd C sig-\<succ> {Q}" |
|
1085 |
apply (rule ax_Methd1) |
|
1086 |
apply (rule_tac [2] singletonI) |
|
1087 |
apply (unfold mtriples_def) |
|
1088 |
apply clarsimp |
|
1089 |
done |
|
1090 |
||
1091 |
lemma ax_StatRef: |
|
1092 |
"G,(A::'a triple set)\<turnstile>{Normal (P\<leftarrow>Val Null)} StatRef rt-\<succ> {P::'a assn}" |
|
1093 |
apply (rule ax_derivs.Cast) |
|
1094 |
apply (rule ax_Lit2 [THEN conseq2]) |
|
1095 |
apply clarsimp |
|
1096 |
done |
|
1097 |
||
1098 |
section "rules derived from Init and Done" |
|
1099 |
||
1100 |
lemma ax_InitS: "\<lbrakk>the (class G C) = c; C \<noteq> Object; |
|
1101 |
\<forall>l. G,A\<turnstile>{Q \<and>. (\<lambda>s. l = locals (store s)) ;. set_lvars empty} |
|
1102 |
.init c. {set_lvars l .; R}; |
|
1103 |
G,A\<turnstile>{Normal ((P \<and>. Not \<circ> initd C) ;. supd (init_class_obj G C))} |
|
1104 |
.Init (super c). {Q}\<rbrakk> \<Longrightarrow> |
|
1105 |
G,(A::'a triple set)\<turnstile>{Normal (P \<and>. Not \<circ> initd C)} .Init C. {R::'a assn}" |
|
1106 |
apply (erule ax_derivs.Init) |
|
1107 |
apply (simp (no_asm_simp)) |
|
1108 |
apply assumption |
|
1109 |
done |
|
1110 |
||
1111 |
lemma ax_Init_Skip_lemma: |
|
1112 |
"\<forall>l. G,(A::'a triple set)\<turnstile>{P\<leftarrow>\<diamondsuit> \<and>. (\<lambda>s. l = locals (store s)) ;. set_lvars l'} |
|
1113 |
.Skip. {(set_lvars l .; P)::'a assn}" |
|
1114 |
apply (rule allI) |
|
1115 |
apply (rule ax_SkipI) |
|
1116 |
apply clarsimp |
|
1117 |
done |
|
1118 |
||
1119 |
lemma ax_triv_InitS: "\<lbrakk>the (class G C) = c;init c = Skip; C \<noteq> Object; |
|
1120 |
P\<leftarrow>\<diamondsuit> \<Rightarrow> (supd (init_class_obj G C) .; P); |
|
1121 |
G,A\<turnstile>{Normal (P \<and>. initd C)} .Init (super c). {(P \<and>. initd C)\<leftarrow>\<diamondsuit>}\<rbrakk> \<Longrightarrow> |
|
1122 |
G,(A::'a triple set)\<turnstile>{Normal P\<leftarrow>\<diamondsuit>} .Init C. {(P \<and>. initd C)::'a assn}" |
|
1123 |
apply (rule_tac C = "initd C" in ax_cases) |
|
1124 |
apply (rule conseq1, rule ax_derivs.Done, clarsimp) |
|
1125 |
apply (simp (no_asm)) |
|
1126 |
apply (erule (1) ax_InitS) |
|
1127 |
apply simp |
|
1128 |
apply (rule ax_Init_Skip_lemma) |
|
1129 |
apply (erule conseq1) |
|
1130 |
apply force |
|
1131 |
done |
|
1132 |
||
1133 |
lemma ax_Init_Object: "wf_prog G \<Longrightarrow> G,(A::'a triple set)\<turnstile> |
|
1134 |
{Normal ((supd (init_class_obj G Object) .; P\<leftarrow>\<diamondsuit>) \<and>. Not \<circ> initd Object)} |
|
1135 |
.Init Object. {(P \<and>. initd Object)::'a assn}" |
|
1136 |
apply (rule ax_derivs.Init) |
|
1137 |
apply (drule class_Object, force) |
|
1138 |
apply (simp_all (no_asm)) |
|
1139 |
apply (rule_tac [2] ax_Init_Skip_lemma) |
|
1140 |
apply (rule ax_SkipI, force) |
|
1141 |
done |
|
1142 |
||
1143 |
lemma ax_triv_Init_Object: "\<lbrakk>wf_prog G; |
|
1144 |
(P::'a assn) \<Rightarrow> (supd (init_class_obj G Object) .; P)\<rbrakk> \<Longrightarrow> |
|
1145 |
G,(A::'a triple set)\<turnstile>{Normal P\<leftarrow>\<diamondsuit>} .Init Object. {P \<and>. initd Object}" |
|
1146 |
apply (rule_tac C = "initd Object" in ax_cases) |
|
1147 |
apply (rule conseq1, rule ax_derivs.Done, clarsimp) |
|
1148 |
apply (erule ax_Init_Object [THEN conseq1]) |
|
1149 |
apply force |
|
1150 |
done |
|
1151 |
||
1152 |
||
1153 |
section "introduction rules for Alloc and SXAlloc" |
|
1154 |
||
1155 |
lemma ax_SXAlloc_Normal: "G,A\<turnstile>{P} .c. {Normal Q} \<Longrightarrow> G,A\<turnstile>{P} .c. {SXAlloc G Q}" |
|
1156 |
apply (erule conseq2) |
|
1157 |
apply (clarsimp elim!: sxalloc_elim_cases simp add: split_tupled_all) |
|
1158 |
done |
|
1159 |
||
1160 |
lemma ax_Alloc: |
|
1161 |
"G,A\<turnstile>{P} t\<succ> {Normal (\<lambda>Y (x,s) Z. (\<forall>a. new_Addr (heap s) = Some a \<longrightarrow> |
|
1162 |
Q (Val (Addr a)) (Norm(init_obj G (CInst C) (Heap a) s)) Z)) \<and>. |
|
1163 |
heap_free (Suc (Suc 0))} |
|
1164 |
\<Longrightarrow> G,A\<turnstile>{P} t\<succ> {Alloc G (CInst C) Q}" |
|
1165 |
apply (erule conseq2) |
|
1166 |
apply (auto elim!: halloc_elim_cases) |
|
1167 |
done |
|
1168 |
||
1169 |
lemma ax_Alloc_Arr: |
|
1170 |
"G,A\<turnstile>{P} t\<succ> {\<lambda>Val:i:. Normal (\<lambda>Y (x,s) Z. \<not>the_Intg i<0 \<and> |
|
1171 |
(\<forall>a. new_Addr (heap s) = Some a \<longrightarrow> |
|
1172 |
Q (Val (Addr a)) (Norm (init_obj G (Arr T (the_Intg i)) (Heap a) s)) Z)) \<and>. |
|
1173 |
heap_free (Suc (Suc 0))} \<Longrightarrow> |
|
1174 |
G,A\<turnstile>{P} t\<succ> {\<lambda>Val:i:. abupd (check_neg i) .; Alloc G (Arr T(the_Intg i)) Q}" |
|
1175 |
apply (erule conseq2) |
|
1176 |
apply (auto elim!: halloc_elim_cases) |
|
1177 |
done |
|
1178 |
||
1179 |
lemma ax_SXAlloc_catch_SXcpt: |
|
1180 |
"\<lbrakk>G,A\<turnstile>{P} t\<succ> {(\<lambda>Y (x,s) Z. x=Some (Xcpt (Std xn)) \<and> |
|
1181 |
(\<forall>a. new_Addr (heap s) = Some a \<longrightarrow> |
|
1182 |
Q Y (Some (Xcpt (Loc a)),init_obj G (CInst (SXcpt xn)) (Heap a) s) Z)) |
|
1183 |
\<and>. heap_free (Suc (Suc 0))}\<rbrakk> \<Longrightarrow> |
|
1184 |
G,A\<turnstile>{P} t\<succ> {SXAlloc G (\<lambda>Y s Z. Q Y s Z \<and> G,s\<turnstile>catch SXcpt xn)}" |
|
1185 |
apply (erule conseq2) |
|
1186 |
apply (auto elim!: sxalloc_elim_cases halloc_elim_cases) |
|
1187 |
done |
|
1188 |
||
1189 |
end |