author | wenzelm |
Fri, 20 Apr 2012 23:15:44 +0200 | |
changeset 47632 | 50f9f699b2d7 |
parent 44037 | 25011c3a5c3d |
child 56073 | 29e308b56d23 |
permissions | -rw-r--r-- |
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(* Title: HOL/MicroJava/J/Eval.thy |
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Author: David von Oheimb |
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Copyright 1999 Technische Universitaet Muenchen |
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*) |
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header {* \isaheader{Operational Evaluation (big step) Semantics} *} |
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theory Eval imports State WellType begin |
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-- "Auxiliary notions" |
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definition fits :: "java_mb prog \<Rightarrow> state \<Rightarrow> val \<Rightarrow> ty \<Rightarrow> bool" ("_,_\<turnstile>_ fits _"[61,61,61,61]60) where |
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"G,s\<turnstile>a' fits T \<equiv> case T of PrimT T' \<Rightarrow> False | RefT T' \<Rightarrow> a'=Null \<or> G\<turnstile>obj_ty(lookup_obj s a')\<preceq>T" |
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definition catch :: "java_mb prog \<Rightarrow> xstate \<Rightarrow> cname \<Rightarrow> bool" ("_,_\<turnstile>catch _"[61,61,61]60) where |
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"G,s\<turnstile>catch C\<equiv> case abrupt s of None \<Rightarrow> False | Some a \<Rightarrow> G,store s\<turnstile> a fits Class C" |
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definition lupd :: "vname \<Rightarrow> val \<Rightarrow> state \<Rightarrow> state" ("lupd'(_\<mapsto>_')"[10,10]1000) where |
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"lupd vn v \<equiv> \<lambda> (hp,loc). (hp, (loc(vn\<mapsto>v)))" |
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definition new_xcpt_var :: "vname \<Rightarrow> xstate \<Rightarrow> xstate" where |
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"new_xcpt_var vn \<equiv> \<lambda>(x,s). Norm (lupd(vn\<mapsto>the x) s)" |
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-- "Evaluation relations" |
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inductive |
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eval :: "[java_mb prog,xstate,expr,val,xstate] => bool " |
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("_ \<turnstile> _ -_\<succ>_-> _" [51,82,60,82,82] 81) |
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and evals :: "[java_mb prog,xstate,expr list, |
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val list,xstate] => bool " |
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("_ \<turnstile> _ -_[\<succ>]_-> _" [51,82,60,51,82] 81) |
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and exec :: "[java_mb prog,xstate,stmt, xstate] => bool " |
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("_ \<turnstile> _ -_-> _" [51,82,60,82] 81) |
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for G :: "java_mb prog" |
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where |
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-- "evaluation of expressions" |
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XcptE:"G\<turnstile>(Some xc,s) -e\<succ>undefined-> (Some xc,s)" -- "cf. 15.5" |
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-- "cf. 15.8.1" |
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| NewC: "[| h = heap s; (a,x) = new_Addr h; |
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h'= h(a\<mapsto>(C,init_vars (fields (G,C)))) |] ==> |
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G\<turnstile>Norm s -NewC C\<succ>Addr a-> c_hupd h' (x,s)" |
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-- "cf. 15.15" |
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| Cast: "[| G\<turnstile>Norm s0 -e\<succ>v-> (x1,s1); |
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x2 = raise_if (\<not> cast_ok G C (heap s1) v) ClassCast x1 |] ==> |
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G\<turnstile>Norm s0 -Cast C e\<succ>v-> (x2,s1)" |
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-- "cf. 15.7.1" |
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| Lit: "G\<turnstile>Norm s -Lit v\<succ>v-> Norm s" |
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| BinOp:"[| G\<turnstile>Norm s -e1\<succ>v1-> s1; |
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G\<turnstile>s1 -e2\<succ>v2-> s2; |
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v = (case bop of Eq => Bool (v1 = v2) |
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| Add => Intg (the_Intg v1 + the_Intg v2)) |] ==> |
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G\<turnstile>Norm s -BinOp bop e1 e2\<succ>v-> s2" |
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-- "cf. 15.13.1, 15.2" |
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| LAcc: "G\<turnstile>Norm s -LAcc v\<succ>the (locals s v)-> Norm s" |
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-- "cf. 15.25.1" |
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| LAss: "[| G\<turnstile>Norm s -e\<succ>v-> (x,(h,l)); |
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l' = (if x = None then l(va\<mapsto>v) else l) |] ==> |
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G\<turnstile>Norm s -va::=e\<succ>v-> (x,(h,l'))" |
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-- "cf. 15.10.1, 15.2" |
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| FAcc: "[| G\<turnstile>Norm s0 -e\<succ>a'-> (x1,s1); |
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v = the (snd (the (heap s1 (the_Addr a'))) (fn,T)) |] ==> |
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G\<turnstile>Norm s0 -{T}e..fn\<succ>v-> (np a' x1,s1)" |
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-- "cf. 15.25.1" |
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| FAss: "[| G\<turnstile> Norm s0 -e1\<succ>a'-> (x1,s1); a = the_Addr a'; |
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G\<turnstile>(np a' x1,s1) -e2\<succ>v -> (x2,s2); |
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h = heap s2; (c,fs) = the (h a); |
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h' = h(a\<mapsto>(c,(fs((fn,T)\<mapsto>v)))) |] ==> |
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G\<turnstile>Norm s0 -{T}e1..fn:=e2\<succ>v-> c_hupd h' (x2,s2)" |
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-- "cf. 15.11.4.1, 15.11.4.2, 15.11.4.4, 15.11.4.5, 14.15" |
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| Call: "[| G\<turnstile>Norm s0 -e\<succ>a'-> s1; a = the_Addr a'; |
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G\<turnstile>s1 -ps[\<succ>]pvs-> (x,(h,l)); dynT = fst (the (h a)); |
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(md,rT,pns,lvars,blk,res) = the (method (G,dynT) (mn,pTs)); |
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G\<turnstile>(np a' x,(h,(init_vars lvars)(pns[\<mapsto>]pvs)(This\<mapsto>a'))) -blk-> s3; |
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G\<turnstile> s3 -res\<succ>v -> (x4,s4) |] ==> |
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G\<turnstile>Norm s0 -{C}e..mn({pTs}ps)\<succ>v-> (x4,(heap s4,l))" |
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-- "evaluation of expression lists" |
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-- "cf. 15.5" |
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| XcptEs:"G\<turnstile>(Some xc,s) -e[\<succ>]undefined-> (Some xc,s)" |
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-- "cf. 15.11.???" |
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| Nil: "G\<turnstile>Norm s0 -[][\<succ>][]-> Norm s0" |
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-- "cf. 15.6.4" |
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| Cons: "[| G\<turnstile>Norm s0 -e \<succ> v -> s1; |
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G\<turnstile> s1 -es[\<succ>]vs-> s2 |] ==> |
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G\<turnstile>Norm s0 -e#es[\<succ>]v#vs-> s2" |
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-- "execution of statements" |
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-- "cf. 14.1" |
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| XcptS:"G\<turnstile>(Some xc,s) -c-> (Some xc,s)" |
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-- "cf. 14.5" |
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| Skip: "G\<turnstile>Norm s -Skip-> Norm s" |
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-- "cf. 14.7" |
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| Expr: "[| G\<turnstile>Norm s0 -e\<succ>v-> s1 |] ==> |
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G\<turnstile>Norm s0 -Expr e-> s1" |
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-- "cf. 14.2" |
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| Comp: "[| G\<turnstile>Norm s0 -c1-> s1; |
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G\<turnstile> s1 -c2-> s2|] ==> |
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G\<turnstile>Norm s0 -c1;; c2-> s2" |
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-- "cf. 14.8.2" |
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| Cond: "[| G\<turnstile>Norm s0 -e\<succ>v-> s1; |
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G\<turnstile> s1 -(if the_Bool v then c1 else c2)-> s2|] ==> |
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G\<turnstile>Norm s0 -If(e) c1 Else c2-> s2" |
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-- "cf. 14.10, 14.10.1" |
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| LoopF:"[| G\<turnstile>Norm s0 -e\<succ>v-> s1; \<not>the_Bool v |] ==> |
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G\<turnstile>Norm s0 -While(e) c-> s1" |
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| LoopT:"[| G\<turnstile>Norm s0 -e\<succ>v-> s1; the_Bool v; |
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G\<turnstile>s1 -c-> s2; G\<turnstile>s2 -While(e) c-> s3 |] ==> |
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G\<turnstile>Norm s0 -While(e) c-> s3" |
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lemmas eval_evals_exec_induct = eval_evals_exec.induct [split_format (complete)] |
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lemma NewCI: "[|new_Addr (heap s) = (a,x); |
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s' = c_hupd (heap s(a\<mapsto>(C,init_vars (fields (G,C))))) (x,s)|] ==> |
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G\<turnstile>Norm s -NewC C\<succ>Addr a-> s'" |
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apply simp |
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apply (rule eval_evals_exec.NewC) |
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apply auto |
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done |
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lemma eval_evals_exec_no_xcpt: |
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"!!s s'. (G\<turnstile>(x,s) -e \<succ> v -> (x',s') --> x'=None --> x=None) \<and> |
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(G\<turnstile>(x,s) -es[\<succ>]vs-> (x',s') --> x'=None --> x=None) \<and> |
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(G\<turnstile>(x,s) -c -> (x',s') --> x'=None --> x=None)" |
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apply(simp only: split_tupled_all) |
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apply(rule eval_evals_exec_induct) |
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apply(unfold c_hupd_def) |
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apply(simp_all) |
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done |
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lemma eval_no_xcpt: "G\<turnstile>(x,s) -e\<succ>v-> (None,s') ==> x=None" |
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apply (drule eval_evals_exec_no_xcpt [THEN conjunct1, THEN mp]) |
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apply (fast) |
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done |
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lemma evals_no_xcpt: "G\<turnstile>(x,s) -e[\<succ>]v-> (None,s') ==> x=None" |
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apply (drule eval_evals_exec_no_xcpt [THEN conjunct2, THEN conjunct1, THEN mp]) |
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apply (fast) |
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done |
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lemma exec_no_xcpt: "G \<turnstile> (x, s) -c-> (None, s') |
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\<Longrightarrow> x = None" |
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apply (drule eval_evals_exec_no_xcpt [THEN conjunct2 [THEN conjunct2], rule_format]) |
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apply simp+ |
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done |
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lemma eval_evals_exec_xcpt: |
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"!!s s'. (G\<turnstile>(x,s) -e \<succ> v -> (x',s') --> x=Some xc --> x'=Some xc \<and> s'=s) \<and> |
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(G\<turnstile>(x,s) -es[\<succ>]vs-> (x',s') --> x=Some xc --> x'=Some xc \<and> s'=s) \<and> |
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(G\<turnstile>(x,s) -c -> (x',s') --> x=Some xc --> x'=Some xc \<and> s'=s)" |
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apply (simp only: split_tupled_all) |
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apply (rule eval_evals_exec_induct) |
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apply (unfold c_hupd_def) |
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apply (simp_all) |
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done |
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lemma eval_xcpt: "G\<turnstile>(Some xc,s) -e\<succ>v-> (x',s') ==> x'=Some xc \<and> s'=s" |
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apply (drule eval_evals_exec_xcpt [THEN conjunct1, THEN mp]) |
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apply (fast) |
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done |
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lemma exec_xcpt: "G\<turnstile>(Some xc,s) -s0-> (x',s') ==> x'=Some xc \<and> s'=s" |
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apply (drule eval_evals_exec_xcpt [THEN conjunct2, THEN conjunct2, THEN mp]) |
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apply (fast) |
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done |
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|
44037
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lemma eval_LAcc_code: "G\<turnstile>Norm (h, l) -LAcc v\<succ>the (l v)-> Norm (h, l)" |
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using LAcc[of G "(h, l)" v] by simp |
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|
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lemma eval_Call_code: |
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"[| G\<turnstile>Norm s0 -e\<succ>a'-> s1; a = the_Addr a'; |
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G\<turnstile>s1 -ps[\<succ>]pvs-> (x,(h,l)); dynT = fst (the (h a)); |
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(md,rT,pns,lvars,blk,res) = the (method (G,dynT) (mn,pTs)); |
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G\<turnstile>(np a' x,(h,(init_vars lvars)(pns[\<mapsto>]pvs)(This\<mapsto>a'))) -blk-> s3; |
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G\<turnstile> s3 -res\<succ>v -> (x4,(h4, l4)) |] ==> |
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G\<turnstile>Norm s0 -{C}e..mn({pTs}ps)\<succ>v-> (x4,(h4,l))" |
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using Call[of G s0 e a' s1 a ps pvs x h l dynT md rT pns lvars blk res mn pTs s3 v x4 "(h4, l4)" C] |
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by simp |
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|
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lemmas [code_pred_intro] = XcptE NewC Cast Lit BinOp |
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lemmas [code_pred_intro LAcc_code] = eval_LAcc_code |
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lemmas [code_pred_intro] = LAss FAcc FAss |
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lemmas [code_pred_intro Call_code] = eval_Call_code |
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lemmas [code_pred_intro] = XcptEs Nil Cons XcptS Skip Expr Comp Cond LoopF |
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lemmas [code_pred_intro LoopT_code] = LoopT |
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|
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code_pred |
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(modes: |
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eval: i \<Rightarrow> i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> o \<Rightarrow> bool |
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and |
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evals: i \<Rightarrow> i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> o \<Rightarrow> bool |
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and |
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exec: i \<Rightarrow> i \<Rightarrow> i \<Rightarrow> o \<Rightarrow> bool) |
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eval |
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proof - |
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case eval |
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from eval.prems show thesis |
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proof(cases (no_simp)) |
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case LAcc with LAcc_code show ?thesis by auto |
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next |
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case (Call a b c d e f g h i j k l m n o p q r s t u v s4) |
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with Call_code show ?thesis |
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by(cases "s4")(simp, (erule meta_allE meta_impE|rule conjI refl| assumption)+) |
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qed(erule (3) that[OF refl]|assumption)+ |
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next |
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case evals |
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from evals.prems show thesis |
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by(cases (no_simp))(erule (3) that[OF refl]|assumption)+ |
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next |
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case exec |
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from exec.prems show thesis |
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proof(cases (no_simp)) |
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case LoopT thus ?thesis by(rule LoopT_code[OF refl]) |
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qed(erule (2) that[OF refl]|assumption)+ |
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qed |
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|
11642 | 243 |
end |