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(* Title: HOL/MicroJava/BV/Effect.thy
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ID: $Id$
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Author: Gerwin Klein
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Copyright 2000 Technische Universitaet Muenchen
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*)
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12911
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header {* \isaheader{Effect of Instructions on the State Type} *}
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15481
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theory Effect
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imports JVMType "../JVM/JVMExceptions"
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begin
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types
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succ_type = "(p_count \<times> state_type option) list"
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text {* Program counter of successor instructions: *}
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consts
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succs :: "instr \<Rightarrow> p_count \<Rightarrow> p_count list"
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primrec
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"succs (Load idx) pc = [pc+1]"
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"succs (Store idx) pc = [pc+1]"
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"succs (LitPush v) pc = [pc+1]"
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"succs (Getfield F C) pc = [pc+1]"
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"succs (Putfield F C) pc = [pc+1]"
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"succs (New C) pc = [pc+1]"
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"succs (Checkcast C) pc = [pc+1]"
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"succs Pop pc = [pc+1]"
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"succs Dup pc = [pc+1]"
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"succs Dup_x1 pc = [pc+1]"
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"succs Dup_x2 pc = [pc+1]"
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"succs Swap pc = [pc+1]"
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"succs IAdd pc = [pc+1]"
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"succs (Ifcmpeq b) pc = [pc+1, nat (int pc + b)]"
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"succs (Goto b) pc = [nat (int pc + b)]"
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"succs Return pc = [pc]"
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"succs (Invoke C mn fpTs) pc = [pc+1]"
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"succs Throw pc = [pc]"
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text "Effect of instruction on the state type:"
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consts
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eff' :: "instr \<times> jvm_prog \<times> state_type \<Rightarrow> state_type"
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recdef eff' "{}"
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"eff' (Load idx, G, (ST, LT)) = (ok_val (LT ! idx) # ST, LT)"
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"eff' (Store idx, G, (ts#ST, LT)) = (ST, LT[idx:= OK ts])"
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"eff' (LitPush v, G, (ST, LT)) = (the (typeof (\<lambda>v. None) v) # ST, LT)"
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"eff' (Getfield F C, G, (oT#ST, LT)) = (snd (the (field (G,C) F)) # ST, LT)"
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"eff' (Putfield F C, G, (vT#oT#ST, LT)) = (ST,LT)"
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"eff' (New C, G, (ST,LT)) = (Class C # ST, LT)"
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"eff' (Checkcast C, G, (RefT rt#ST,LT)) = (Class C # ST,LT)"
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"eff' (Pop, G, (ts#ST,LT)) = (ST,LT)"
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"eff' (Dup, G, (ts#ST,LT)) = (ts#ts#ST,LT)"
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"eff' (Dup_x1, G, (ts1#ts2#ST,LT)) = (ts1#ts2#ts1#ST,LT)"
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"eff' (Dup_x2, G, (ts1#ts2#ts3#ST,LT)) = (ts1#ts2#ts3#ts1#ST,LT)"
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"eff' (Swap, G, (ts1#ts2#ST,LT)) = (ts2#ts1#ST,LT)"
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"eff' (IAdd, G, (PrimT Integer#PrimT Integer#ST,LT))
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= (PrimT Integer#ST,LT)"
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"eff' (Ifcmpeq b, G, (ts1#ts2#ST,LT)) = (ST,LT)"
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"eff' (Goto b, G, s) = s"
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-- "Return has no successor instruction in the same method"
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"eff' (Return, G, s) = s"
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-- "Throw always terminates abruptly"
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"eff' (Throw, G, s) = s"
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"eff' (Invoke C mn fpTs, G, (ST,LT)) = (let ST' = drop (length fpTs) ST
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in (fst (snd (the (method (G,C) (mn,fpTs))))#(tl ST'),LT))"
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consts
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match_any :: "jvm_prog \<Rightarrow> p_count \<Rightarrow> exception_table \<Rightarrow> cname list"
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primrec
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"match_any G pc [] = []"
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"match_any G pc (e#es) = (let (start_pc, end_pc, handler_pc, catch_type) = e;
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es' = match_any G pc es
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in
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if start_pc <= pc \<and> pc < end_pc then catch_type#es' else es')"
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consts
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match :: "jvm_prog \<Rightarrow> xcpt \<Rightarrow> p_count \<Rightarrow> exception_table \<Rightarrow> cname list"
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primrec
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"match G X pc [] = []"
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"match G X pc (e#es) =
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(if match_exception_entry G (Xcpt X) pc e then [Xcpt X] else match G X pc es)"
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lemma match_some_entry:
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"match G X pc et = (if \<exists>e \<in> set et. match_exception_entry G (Xcpt X) pc e then [Xcpt X] else [])"
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by (induct et) auto
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consts
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xcpt_names :: "instr \<times> jvm_prog \<times> p_count \<times> exception_table \<Rightarrow> cname list"
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recdef xcpt_names "{}"
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"xcpt_names (Getfield F C, G, pc, et) = match G NullPointer pc et"
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"xcpt_names (Putfield F C, G, pc, et) = match G NullPointer pc et"
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"xcpt_names (New C, G, pc, et) = match G OutOfMemory pc et"
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"xcpt_names (Checkcast C, G, pc, et) = match G ClassCast pc et"
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"xcpt_names (Throw, G, pc, et) = match_any G pc et"
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"xcpt_names (Invoke C m p, G, pc, et) = match_any G pc et"
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"xcpt_names (i, G, pc, et) = []"
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constdefs
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xcpt_eff :: "instr \<Rightarrow> jvm_prog \<Rightarrow> p_count \<Rightarrow> state_type option \<Rightarrow> exception_table \<Rightarrow> succ_type"
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"xcpt_eff i G pc s et ==
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map (\<lambda>C. (the (match_exception_table G C pc et), case s of None \<Rightarrow> None | Some s' \<Rightarrow> Some ([Class C], snd s')))
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(xcpt_names (i,G,pc,et))"
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norm_eff :: "instr \<Rightarrow> jvm_prog \<Rightarrow> state_type option \<Rightarrow> state_type option"
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"norm_eff i G == option_map (\<lambda>s. eff' (i,G,s))"
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eff :: "instr \<Rightarrow> jvm_prog \<Rightarrow> p_count \<Rightarrow> exception_table \<Rightarrow> state_type option \<Rightarrow> succ_type"
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"eff i G pc et s == (map (\<lambda>pc'. (pc',norm_eff i G s)) (succs i pc)) @ (xcpt_eff i G pc s et)"
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constdefs
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isPrimT :: "ty \<Rightarrow> bool"
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"isPrimT T == case T of PrimT T' \<Rightarrow> True | RefT T' \<Rightarrow> False"
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isRefT :: "ty \<Rightarrow> bool"
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"isRefT T == case T of PrimT T' \<Rightarrow> False | RefT T' \<Rightarrow> True"
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lemma isPrimT [simp]:
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"isPrimT T = (\<exists>T'. T = PrimT T')" by (simp add: isPrimT_def split: ty.splits)
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lemma isRefT [simp]:
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"isRefT T = (\<exists>T'. T = RefT T')" by (simp add: isRefT_def split: ty.splits)
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lemma "list_all2 P a b \<Longrightarrow> \<forall>(x,y) \<in> set (zip a b). P x y"
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by (simp add: list_all2_def)
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text "Conditions under which eff is applicable:"
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consts
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app' :: "instr \<times> jvm_prog \<times> p_count \<times> nat \<times> ty \<times> state_type \<Rightarrow> bool"
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recdef app' "{}"
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"app' (Load idx, G, pc, maxs, rT, s) =
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(idx < length (snd s) \<and> (snd s) ! idx \<noteq> Err \<and> length (fst s) < maxs)"
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"app' (Store idx, G, pc, maxs, rT, (ts#ST, LT)) =
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(idx < length LT)"
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"app' (LitPush v, G, pc, maxs, rT, s) =
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(length (fst s) < maxs \<and> typeof (\<lambda>t. None) v \<noteq> None)"
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"app' (Getfield F C, G, pc, maxs, rT, (oT#ST, LT)) =
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(is_class G C \<and> field (G,C) F \<noteq> None \<and> fst (the (field (G,C) F)) = C \<and>
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G \<turnstile> oT \<preceq> (Class C))"
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"app' (Putfield F C, G, pc, maxs, rT, (vT#oT#ST, LT)) =
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(is_class G C \<and> field (G,C) F \<noteq> None \<and> fst (the (field (G,C) F)) = C \<and>
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G \<turnstile> oT \<preceq> (Class C) \<and> G \<turnstile> vT \<preceq> (snd (the (field (G,C) F))))"
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"app' (New C, G, pc, maxs, rT, s) =
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(is_class G C \<and> length (fst s) < maxs)"
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"app' (Checkcast C, G, pc, maxs, rT, (RefT rt#ST,LT)) =
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(is_class G C)"
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"app' (Pop, G, pc, maxs, rT, (ts#ST,LT)) =
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True"
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"app' (Dup, G, pc, maxs, rT, (ts#ST,LT)) =
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(1+length ST < maxs)"
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"app' (Dup_x1, G, pc, maxs, rT, (ts1#ts2#ST,LT)) =
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(2+length ST < maxs)"
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"app' (Dup_x2, G, pc, maxs, rT, (ts1#ts2#ts3#ST,LT)) =
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(3+length ST < maxs)"
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"app' (Swap, G, pc, maxs, rT, (ts1#ts2#ST,LT)) =
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True"
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"app' (IAdd, G, pc, maxs, rT, (PrimT Integer#PrimT Integer#ST,LT)) =
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True"
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"app' (Ifcmpeq b, G, pc, maxs, rT, (ts#ts'#ST,LT)) =
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(0 \<le> int pc + b \<and> (isPrimT ts \<and> ts' = ts \<or> isRefT ts \<and> isRefT ts'))"
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"app' (Goto b, G, pc, maxs, rT, s) =
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(0 \<le> int pc + b)"
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"app' (Return, G, pc, maxs, rT, (T#ST,LT)) =
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(G \<turnstile> T \<preceq> rT)"
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"app' (Throw, G, pc, maxs, rT, (T#ST,LT)) =
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isRefT T"
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"app' (Invoke C mn fpTs, G, pc, maxs, rT, s) =
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(length fpTs < length (fst s) \<and>
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(let apTs = rev (take (length fpTs) (fst s));
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X = hd (drop (length fpTs) (fst s))
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in
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G \<turnstile> X \<preceq> Class C \<and> is_class G C \<and> method (G,C) (mn,fpTs) \<noteq> None \<and>
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list_all2 (\<lambda>x y. G \<turnstile> x \<preceq> y) apTs fpTs))"
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"app' (i,G, pc,maxs,rT,s) = False"
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constdefs
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xcpt_app :: "instr \<Rightarrow> jvm_prog \<Rightarrow> nat \<Rightarrow> exception_table \<Rightarrow> bool"
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"xcpt_app i G pc et \<equiv> \<forall>C\<in>set(xcpt_names (i,G,pc,et)). is_class G C"
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app :: "instr \<Rightarrow> jvm_prog \<Rightarrow> nat \<Rightarrow> ty \<Rightarrow> nat \<Rightarrow> exception_table \<Rightarrow> state_type option \<Rightarrow> bool"
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"app i G maxs rT pc et s == case s of None \<Rightarrow> True | Some t \<Rightarrow> app' (i,G,pc,maxs,rT,t) \<and> xcpt_app i G pc et"
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lemma match_any_match_table:
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"C \<in> set (match_any G pc et) \<Longrightarrow> match_exception_table G C pc et \<noteq> None"
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apply (induct et)
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apply simp
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apply simp
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apply clarify
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apply (simp split: split_if_asm)
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apply (auto simp add: match_exception_entry_def)
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done
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lemma match_X_match_table:
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"C \<in> set (match G X pc et) \<Longrightarrow> match_exception_table G C pc et \<noteq> None"
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apply (induct et)
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apply simp
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apply (simp split: split_if_asm)
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done
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lemma xcpt_names_in_et:
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"C \<in> set (xcpt_names (i,G,pc,et)) \<Longrightarrow>
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\<exists>e \<in> set et. the (match_exception_table G C pc et) = fst (snd (snd e))"
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apply (cases i)
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apply (auto dest!: match_any_match_table match_X_match_table
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dest: match_exception_table_in_et)
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done
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lemma 1: "2 < length a \<Longrightarrow> (\<exists>l l' l'' ls. a = l#l'#l''#ls)"
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proof (cases a)
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fix x xs assume "a = x#xs" "2 < length a"
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thus ?thesis by - (cases xs, simp, cases "tl xs", auto)
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qed auto
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lemma 2: "\<not>(2 < length a) \<Longrightarrow> a = [] \<or> (\<exists> l. a = [l]) \<or> (\<exists> l l'. a = [l,l'])"
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proof -;
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assume "\<not>(2 < length a)"
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hence "length a < (Suc (Suc (Suc 0)))" by simp
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hence * : "length a = 0 \<or> length a = Suc 0 \<or> length a = Suc (Suc 0)"
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by (auto simp add: less_Suc_eq)
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{
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fix x
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assume "length x = Suc 0"
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hence "\<exists> l. x = [l]" by - (cases x, auto)
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} note 0 = this
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have "length a = Suc (Suc 0) \<Longrightarrow> \<exists>l l'. a = [l,l']" by (cases a, auto dest: 0)
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with * show ?thesis by (auto dest: 0)
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qed
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lemmas [simp] = app_def xcpt_app_def
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text {*
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\medskip
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simp rules for @{term app}
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*}
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lemma appNone[simp]: "app i G maxs rT pc et None = True" by simp
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lemma appLoad[simp]:
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"(app (Load idx) G maxs rT pc et (Some s)) = (\<exists>ST LT. s = (ST,LT) \<and> idx < length LT \<and> LT!idx \<noteq> Err \<and> length ST < maxs)"
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by (cases s, simp)
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lemma appStore[simp]:
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"(app (Store idx) G maxs rT pc et (Some s)) = (\<exists>ts ST LT. s = (ts#ST,LT) \<and> idx < length LT)"
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by (cases s, cases "2 < length (fst s)", auto dest: 1 2)
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lemma appLitPush[simp]:
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"(app (LitPush v) G maxs rT pc et (Some s)) = (\<exists>ST LT. s = (ST,LT) \<and> length ST < maxs \<and> typeof (\<lambda>v. None) v \<noteq> None)"
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by (cases s, simp)
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lemma appGetField[simp]:
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"(app (Getfield F C) G maxs rT pc et (Some s)) =
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(\<exists> oT vT ST LT. s = (oT#ST, LT) \<and> is_class G C \<and>
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field (G,C) F = Some (C,vT) \<and> G \<turnstile> oT \<preceq> (Class C) \<and> (\<forall>x \<in> set (match G NullPointer pc et). is_class G x))"
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by (cases s, cases "2 <length (fst s)", auto dest!: 1 2)
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lemma appPutField[simp]:
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"(app (Putfield F C) G maxs rT pc et (Some s)) =
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(\<exists> vT vT' oT ST LT. s = (vT#oT#ST, LT) \<and> is_class G C \<and>
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field (G,C) F = Some (C, vT') \<and> G \<turnstile> oT \<preceq> (Class C) \<and> G \<turnstile> vT \<preceq> vT' \<and>
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(\<forall>x \<in> set (match G NullPointer pc et). is_class G x))"
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by (cases s, cases "2 <length (fst s)", auto dest!: 1 2)
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lemma appNew[simp]:
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"(app (New C) G maxs rT pc et (Some s)) =
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(\<exists>ST LT. s=(ST,LT) \<and> is_class G C \<and> length ST < maxs \<and>
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13717
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(\<forall>x \<in> set (match G OutOfMemory pc et). is_class G x))"
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by (cases s, simp)
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lemma appCheckcast[simp]:
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"(app (Checkcast C) G maxs rT pc et (Some s)) =
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12951
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(\<exists>rT ST LT. s = (RefT rT#ST,LT) \<and> is_class G C \<and>
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13717
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(\<forall>x \<in> set (match G ClassCast pc et). is_class G x))"
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by (cases s, cases "fst s", simp add: app_def) (cases "hd (fst s)", auto)
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lemma appPop[simp]:
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"(app Pop G maxs rT pc et (Some s)) = (\<exists>ts ST LT. s = (ts#ST,LT))"
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by (cases s, cases "2 <length (fst s)", auto dest: 1 2)
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lemma appDup[simp]:
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"(app Dup G maxs rT pc et (Some s)) = (\<exists>ts ST LT. s = (ts#ST,LT) \<and> 1+length ST < maxs)"
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by (cases s, cases "2 <length (fst s)", auto dest: 1 2)
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lemma appDup_x1[simp]:
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296 |
"(app Dup_x1 G maxs rT pc et (Some s)) = (\<exists>ts1 ts2 ST LT. s = (ts1#ts2#ST,LT) \<and> 2+length ST < maxs)"
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297 |
by (cases s, cases "2 <length (fst s)", auto dest: 1 2)
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298 |
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299 |
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300 |
lemma appDup_x2[simp]:
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301 |
"(app Dup_x2 G maxs rT pc et (Some s)) = (\<exists>ts1 ts2 ts3 ST LT. s = (ts1#ts2#ts3#ST,LT) \<and> 3+length ST < maxs)"
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302 |
by (cases s, cases "2 <length (fst s)", auto dest: 1 2)
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303 |
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304 |
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305 |
lemma appSwap[simp]:
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306 |
"app Swap G maxs rT pc et (Some s) = (\<exists>ts1 ts2 ST LT. s = (ts1#ts2#ST,LT))"
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307 |
by (cases s, cases "2 <length (fst s)", auto dest: 1 2)
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308 |
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309 |
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310 |
lemma appIAdd[simp]:
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311 |
"app IAdd G maxs rT pc et (Some s) = (\<exists> ST LT. s = (PrimT Integer#PrimT Integer#ST,LT))"
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312 |
(is "?app s = ?P s")
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313 |
proof (cases (open) s)
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314 |
case Pair
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315 |
have "?app (a,b) = ?P (a,b)"
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316 |
proof (cases "a")
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317 |
fix t ts assume a: "a = t#ts"
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318 |
show ?thesis
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319 |
proof (cases t)
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320 |
fix p assume p: "t = PrimT p"
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321 |
show ?thesis
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322 |
proof (cases p)
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323 |
assume ip: "p = Integer"
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324 |
show ?thesis
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325 |
proof (cases ts)
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326 |
fix t' ts' assume t': "ts = t' # ts'"
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327 |
show ?thesis
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328 |
proof (cases t')
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329 |
fix p' assume "t' = PrimT p'"
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330 |
with t' ip p a
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331 |
show ?thesis by - (cases p', auto)
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332 |
qed (auto simp add: a p ip t')
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333 |
qed (auto simp add: a p ip)
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334 |
qed (auto simp add: a p)
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335 |
qed (auto simp add: a)
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336 |
qed auto
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337 |
with Pair show ?thesis by simp
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|
338 |
qed
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339 |
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340 |
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341 |
lemma appIfcmpeq[simp]:
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12974
|
342 |
"app (Ifcmpeq b) G maxs rT pc et (Some s) =
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343 |
(\<exists>ts1 ts2 ST LT. s = (ts1#ts2#ST,LT) \<and> 0 \<le> int pc + b \<and>
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344 |
((\<exists> p. ts1 = PrimT p \<and> ts2 = PrimT p) \<or> (\<exists>r r'. ts1 = RefT r \<and> ts2 = RefT r')))"
|
12772
|
345 |
by (cases s, cases "2 <length (fst s)", auto dest!: 1 2)
|
12516
|
346 |
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|
347 |
lemma appReturn[simp]:
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348 |
"app Return G maxs rT pc et (Some s) = (\<exists>T ST LT. s = (T#ST,LT) \<and> (G \<turnstile> T \<preceq> rT))"
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|
349 |
by (cases s, cases "2 <length (fst s)", auto dest: 1 2)
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|
350 |
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|
351 |
lemma appGoto[simp]:
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12974
|
352 |
"app (Goto b) G maxs rT pc et (Some s) = (0 \<le> int pc + b)"
|
12516
|
353 |
by simp
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|
354 |
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|
355 |
lemma appThrow[simp]:
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|
356 |
"app Throw G maxs rT pc et (Some s) =
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|
357 |
(\<exists>T ST LT r. s=(T#ST,LT) \<and> T = RefT r \<and> (\<forall>C \<in> set (match_any G pc et). is_class G C))"
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|
358 |
by (cases s, cases "2 < length (fst s)", auto dest: 1 2)
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|
359 |
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|
360 |
lemma appInvoke[simp]:
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|
361 |
"app (Invoke C mn fpTs) G maxs rT pc et (Some s) = (\<exists>apTs X ST LT mD' rT' b'.
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|
362 |
s = ((rev apTs) @ (X # ST), LT) \<and> length apTs = length fpTs \<and> is_class G C \<and>
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|
363 |
G \<turnstile> X \<preceq> Class C \<and> (\<forall>(aT,fT)\<in>set(zip apTs fpTs). G \<turnstile> aT \<preceq> fT) \<and>
|
|
364 |
method (G,C) (mn,fpTs) = Some (mD', rT', b') \<and>
|
|
365 |
(\<forall>C \<in> set (match_any G pc et). is_class G C))" (is "?app s = ?P s")
|
|
366 |
proof (cases (open) s)
|
12772
|
367 |
note list_all2_def [simp]
|
12516
|
368 |
case Pair
|
13006
|
369 |
have "?app (a,b) \<Longrightarrow> ?P (a,b)"
|
12516
|
370 |
proof -
|
|
371 |
assume app: "?app (a,b)"
|
|
372 |
hence "a = (rev (rev (take (length fpTs) a))) @ (drop (length fpTs) a) \<and>
|
|
373 |
length fpTs < length a" (is "?a \<and> ?l")
|
|
374 |
by (auto simp add: app_def)
|
|
375 |
hence "?a \<and> 0 < length (drop (length fpTs) a)" (is "?a \<and> ?l")
|
|
376 |
by auto
|
|
377 |
hence "?a \<and> ?l \<and> length (rev (take (length fpTs) a)) = length fpTs"
|
|
378 |
by (auto simp add: min_def)
|
|
379 |
hence "\<exists>apTs ST. a = rev apTs @ ST \<and> length apTs = length fpTs \<and> 0 < length ST"
|
|
380 |
by blast
|
|
381 |
hence "\<exists>apTs ST. a = rev apTs @ ST \<and> length apTs = length fpTs \<and> ST \<noteq> []"
|
|
382 |
by blast
|
|
383 |
hence "\<exists>apTs ST. a = rev apTs @ ST \<and> length apTs = length fpTs \<and>
|
|
384 |
(\<exists>X ST'. ST = X#ST')"
|
|
385 |
by (simp add: neq_Nil_conv)
|
|
386 |
hence "\<exists>apTs X ST. a = rev apTs @ X # ST \<and> length apTs = length fpTs"
|
|
387 |
by blast
|
|
388 |
with app
|
|
389 |
show ?thesis by (unfold app_def, clarsimp) blast
|
|
390 |
qed
|
|
391 |
with Pair
|
12772
|
392 |
have "?app s \<Longrightarrow> ?P s" by (simp only:)
|
12516
|
393 |
moreover
|
12772
|
394 |
have "?P s \<Longrightarrow> ?app s" by (unfold app_def) (clarsimp simp add: min_def)
|
12516
|
395 |
ultimately
|
12772
|
396 |
show ?thesis by (rule iffI)
|
12516
|
397 |
qed
|
|
398 |
|
|
399 |
lemma effNone:
|
|
400 |
"(pc', s') \<in> set (eff i G pc et None) \<Longrightarrow> s' = None"
|
|
401 |
by (auto simp add: eff_def xcpt_eff_def norm_eff_def)
|
|
402 |
|
12772
|
403 |
|
|
404 |
text {* some helpers to make the specification directly executable: *}
|
|
405 |
declare list_all2_Nil [code]
|
|
406 |
declare list_all2_Cons [code]
|
|
407 |
|
|
408 |
lemma xcpt_app_lemma [code]:
|
|
409 |
"xcpt_app i G pc et = list_all (is_class G) (xcpt_names (i, G, pc, et))"
|
17088
|
410 |
by (simp add: list_all_iff)
|
12772
|
411 |
|
12516
|
412 |
lemmas [simp del] = app_def xcpt_app_def
|
|
413 |
|
|
414 |
end
|