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(* Title: HOL/MicroJava/BV/JVM.thy
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ID: $Id$
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Author: Tobias Nipkow, Gerwin Klein
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Copyright 2000 TUM
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*)
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header {* \isaheader{LBV for the JVM}\label{sec:JVM} *}
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17090
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theory LBVJVM
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imports LBVCorrect LBVComplete Typing_Framework_JVM
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begin
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types prog_cert = "cname \<Rightarrow> sig \<Rightarrow> state list"
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constdefs
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check_cert :: "jvm_prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> state list \<Rightarrow> bool"
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"check_cert G mxs mxr n cert \<equiv> check_types G mxs mxr cert \<and> length cert = n+1 \<and>
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(\<forall>i<n. cert!i \<noteq> Err) \<and> cert!n = OK None"
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lbvjvm :: "jvm_prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> ty \<Rightarrow> exception_table \<Rightarrow>
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state list \<Rightarrow> instr list \<Rightarrow> state \<Rightarrow> state"
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"lbvjvm G maxs maxr rT et cert bs \<equiv>
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wtl_inst_list bs cert (JVMType.sup G maxs maxr) (JVMType.le G maxs maxr) Err (OK None) (exec G maxs rT et bs) 0"
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wt_lbv :: "jvm_prog \<Rightarrow> cname \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow>
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exception_table \<Rightarrow> state list \<Rightarrow> instr list \<Rightarrow> bool"
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"wt_lbv G C pTs rT mxs mxl et cert ins \<equiv>
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check_bounded ins et \<and>
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check_cert G mxs (1+size pTs+mxl) (length ins) cert \<and>
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0 < size ins \<and>
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(let start = Some ([],(OK (Class C))#((map OK pTs))@(replicate mxl Err));
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result = lbvjvm G mxs (1+size pTs+mxl) rT et cert ins (OK start)
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in result \<noteq> Err)"
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wt_jvm_prog_lbv :: "jvm_prog \<Rightarrow> prog_cert \<Rightarrow> bool"
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"wt_jvm_prog_lbv G cert \<equiv>
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wf_prog (\<lambda>G C (sig,rT,(maxs,maxl,b,et)). wt_lbv G C (snd sig) rT maxs maxl et (cert C sig) b) G"
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mk_cert :: "jvm_prog \<Rightarrow> nat \<Rightarrow> ty \<Rightarrow> exception_table \<Rightarrow> instr list
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\<Rightarrow> method_type \<Rightarrow> state list"
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"mk_cert G maxs rT et bs phi \<equiv> make_cert (exec G maxs rT et bs) (map OK phi) (OK None)"
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prg_cert :: "jvm_prog \<Rightarrow> prog_type \<Rightarrow> prog_cert"
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"prg_cert G phi C sig \<equiv> let (C,rT,(maxs,maxl,ins,et)) = the (method (G,C) sig) in
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mk_cert G maxs rT et ins (phi C sig)"
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lemma wt_method_def2:
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fixes pTs and mxl and G and mxs and rT and et and bs and phi
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defines [simp]: "mxr \<equiv> 1 + length pTs + mxl"
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defines [simp]: "r \<equiv> sup_state_opt G"
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defines [simp]: "app0 \<equiv> \<lambda>pc. app (bs!pc) G mxs rT pc et"
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defines [simp]: "step0 \<equiv> \<lambda>pc. eff (bs!pc) G pc et"
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shows
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"wt_method G C pTs rT mxs mxl bs et phi =
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(bs \<noteq> [] \<and>
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length phi = length bs \<and>
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check_bounded bs et \<and>
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check_types G mxs mxr (map OK phi) \<and>
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wt_start G C pTs mxl phi \<and>
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wt_app_eff r app0 step0 phi)"
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by (auto simp add: wt_method_def wt_app_eff_def wt_instr_def lesub_def
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dest: check_bounded_is_bounded boundedD)
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lemma check_certD:
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"check_cert G mxs mxr n cert \<Longrightarrow> cert_ok cert n Err (OK None) (states G mxs mxr)"
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apply (unfold cert_ok_def check_cert_def check_types_def)
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17090
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apply (auto simp add: list_all_iff)
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done
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lemma wt_lbv_wt_step:
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assumes wf: "wf_prog wf_mb G"
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assumes lbv: "wt_lbv G C pTs rT mxs mxl et cert ins"
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assumes C: "is_class G C"
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assumes pTs: "set pTs \<subseteq> types G"
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defines [simp]: "mxr \<equiv> 1+length pTs+mxl"
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shows "\<exists>ts \<in> list (size ins) (states G mxs mxr).
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wt_step (JVMType.le G mxs mxr) Err (exec G mxs rT et ins) ts
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\<and> OK (Some ([],(OK (Class C))#((map OK pTs))@(replicate mxl Err))) <=_(JVMType.le G mxs mxr) ts!0"
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proof -
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let ?step = "exec G mxs rT et ins"
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let ?r = "JVMType.le G mxs mxr"
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let ?f = "JVMType.sup G mxs mxr"
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let ?A = "states G mxs mxr"
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have "semilat (JVMType.sl G mxs mxr)"
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by (rule semilat_JVM_slI, rule wf_prog_ws_prog)
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hence "semilat (?A, ?r, ?f)" by (unfold sl_triple_conv)
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moreover
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have "top ?r Err" by (simp add: JVM_le_unfold)
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moreover
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have "Err \<in> ?A" by (simp add: JVM_states_unfold)
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moreover
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have "bottom ?r (OK None)"
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by (simp add: JVM_le_unfold bottom_def)
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moreover
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have "OK None \<in> ?A" by (simp add: JVM_states_unfold)
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moreover
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from lbv
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have "bounded ?step (length ins)"
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by (clarsimp simp add: wt_lbv_def exec_def)
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(intro bounded_lift check_bounded_is_bounded)
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moreover
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from lbv
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have "cert_ok cert (length ins) Err (OK None) ?A"
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by (unfold wt_lbv_def) (auto dest: check_certD)
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moreover
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have "pres_type ?step (length ins) ?A" by (rule exec_pres_type)
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moreover
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let ?start = "OK (Some ([],(OK (Class C))#(map OK pTs)@(replicate mxl Err)))"
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from lbv
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have "wtl_inst_list ins cert ?f ?r Err (OK None) ?step 0 ?start \<noteq> Err"
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by (simp add: wt_lbv_def lbvjvm_def)
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moreover
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from C pTs have "?start \<in> ?A"
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by (unfold JVM_states_unfold) (auto intro: list_appendI, force)
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moreover
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from lbv have "0 < length ins" by (simp add: wt_lbv_def)
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ultimately
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show ?thesis by (rule lbvs.wtl_sound_strong)
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qed
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lemma wt_lbv_wt_method:
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assumes wf: "wf_prog wf_mb G"
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assumes lbv: "wt_lbv G C pTs rT mxs mxl et cert ins"
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assumes C: "is_class G C"
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assumes pTs: "set pTs \<subseteq> types G"
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shows "\<exists>phi. wt_method G C pTs rT mxs mxl ins et phi"
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proof -
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let ?mxr = "1 + length pTs + mxl"
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let ?step = "exec G mxs rT et ins"
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let ?r = "JVMType.le G mxs ?mxr"
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let ?f = "JVMType.sup G mxs ?mxr"
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let ?A = "states G mxs ?mxr"
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let ?start = "OK (Some ([],(OK (Class C))#(map OK pTs)@(replicate mxl Err)))"
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from lbv have l: "ins \<noteq> []" by (simp add: wt_lbv_def)
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moreover
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from wf lbv C pTs
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obtain phi where
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list: "phi \<in> list (length ins) ?A" and
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step: "wt_step ?r Err ?step phi" and
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start: "?start <=_?r phi!0"
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by (blast dest: wt_lbv_wt_step)
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from list have [simp]: "length phi = length ins" by simp
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have "length (map ok_val phi) = length ins" by simp
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moreover
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from l have 0: "0 < length phi" by simp
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with step obtain phi0 where "phi!0 = OK phi0"
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by (unfold wt_step_def) blast
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with start 0
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have "wt_start G C pTs mxl (map ok_val phi)"
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by (simp add: wt_start_def JVM_le_Err_conv lesub_def)
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moreover
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from lbv have chk_bounded: "check_bounded ins et"
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by (simp add: wt_lbv_def)
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moreover {
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from list
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have "check_types G mxs ?mxr phi"
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by (simp add: check_types_def)
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also from step
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have [symmetric]: "map OK (map ok_val phi) = phi"
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by (auto intro!: map_id simp add: wt_step_def)
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finally have "check_types G mxs ?mxr (map OK (map ok_val phi))" .
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}
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moreover {
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let ?app = "\<lambda>pc. app (ins!pc) G mxs rT pc et"
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let ?eff = "\<lambda>pc. eff (ins!pc) G pc et"
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from chk_bounded
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have "bounded (err_step (length ins) ?app ?eff) (length ins)"
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by (blast dest: check_bounded_is_bounded boundedD intro: bounded_err_stepI)
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moreover
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from step
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have "wt_err_step (sup_state_opt G) ?step phi"
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by (simp add: wt_err_step_def JVM_le_Err_conv)
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ultimately
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have "wt_app_eff (sup_state_opt G) ?app ?eff (map ok_val phi)"
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by (auto intro: wt_err_imp_wt_app_eff simp add: exec_def)
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}
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ultimately
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have "wt_method G C pTs rT mxs mxl ins et (map ok_val phi)"
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by - (rule wt_method_def2 [THEN iffD2], simp)
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thus ?thesis ..
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qed
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lemma wt_method_wt_lbv:
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assumes wf: "wf_prog wf_mb G"
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assumes wt: "wt_method G C pTs rT mxs mxl ins et phi"
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assumes C: "is_class G C"
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assumes pTs: "set pTs \<subseteq> types G"
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defines [simp]: "cert \<equiv> mk_cert G mxs rT et ins phi"
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shows "wt_lbv G C pTs rT mxs mxl et cert ins"
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proof -
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let ?mxr = "1 + length pTs + mxl"
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let ?step = "exec G mxs rT et ins"
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let ?app = "\<lambda>pc. app (ins!pc) G mxs rT pc et"
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let ?eff = "\<lambda>pc. eff (ins!pc) G pc et"
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let ?r = "JVMType.le G mxs ?mxr"
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let ?f = "JVMType.sup G mxs ?mxr"
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let ?A = "states G mxs ?mxr"
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let ?phi = "map OK phi"
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let ?cert = "make_cert ?step ?phi (OK None)"
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from wt obtain
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0: "0 < length ins" and
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length: "length ins = length ?phi" and
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ck_bounded: "check_bounded ins et" and
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ck_types: "check_types G mxs ?mxr ?phi" and
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wt_start: "wt_start G C pTs mxl phi" and
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app_eff: "wt_app_eff (sup_state_opt G) ?app ?eff phi"
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by (simp (asm_lr) add: wt_method_def2)
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have "semilat (JVMType.sl G mxs ?mxr)"
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by (rule semilat_JVM_slI, rule wf_prog_ws_prog)
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hence "semilat (?A, ?r, ?f)" by (unfold sl_triple_conv)
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moreover
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have "top ?r Err" by (simp add: JVM_le_unfold)
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moreover
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have "Err \<in> ?A" by (simp add: JVM_states_unfold)
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moreover
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have "bottom ?r (OK None)"
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by (simp add: JVM_le_unfold bottom_def)
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moreover
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have "OK None \<in> ?A" by (simp add: JVM_states_unfold)
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moreover
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from ck_bounded
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have bounded: "bounded ?step (length ins)"
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by (clarsimp simp add: exec_def)
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(intro bounded_lift check_bounded_is_bounded)
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with wf
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have "mono ?r ?step (length ins) ?A"
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by (rule wf_prog_ws_prog [THEN exec_mono])
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hence "mono ?r ?step (length ?phi) ?A" by (simp add: length)
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moreover
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have "pres_type ?step (length ins) ?A" by (rule exec_pres_type)
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hence "pres_type ?step (length ?phi) ?A" by (simp add: length)
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moreover
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from ck_types
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have "set ?phi \<subseteq> ?A" by (simp add: check_types_def)
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hence "\<forall>pc. pc < length ?phi \<longrightarrow> ?phi!pc \<in> ?A \<and> ?phi!pc \<noteq> Err" by auto
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moreover
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from bounded
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have "bounded (exec G mxs rT et ins) (length ?phi)" by (simp add: length)
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moreover
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have "OK None \<noteq> Err" by simp
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moreover
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from bounded length app_eff
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have "wt_err_step (sup_state_opt G) ?step ?phi"
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by (auto intro: wt_app_eff_imp_wt_err simp add: exec_def)
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hence "wt_step ?r Err ?step ?phi"
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by (simp add: wt_err_step_def JVM_le_Err_conv)
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moreover
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let ?start = "OK (Some ([],(OK (Class C))#(map OK pTs)@(replicate mxl Err)))"
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from 0 length have "0 < length phi" by auto
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hence "?phi!0 = OK (phi!0)" by simp
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with wt_start have "?start <=_?r ?phi!0"
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by (clarsimp simp add: wt_start_def lesub_def JVM_le_Err_conv)
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moreover
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from C pTs have "?start \<in> ?A"
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by (unfold JVM_states_unfold) (auto intro: list_appendI, force)
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moreover
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have "?start \<noteq> Err" by simp
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moreover
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note length
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ultimately
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have "wtl_inst_list ins ?cert ?f ?r Err (OK None) ?step 0 ?start \<noteq> Err"
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by (rule lbvc.wtl_complete)
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moreover
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from 0 length have "phi \<noteq> []" by auto
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moreover
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from ck_types
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have "check_types G mxs ?mxr ?cert"
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by (auto simp add: make_cert_def check_types_def JVM_states_unfold)
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moreover
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note ck_bounded 0 length
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ultimately
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show ?thesis
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by (simp add: wt_lbv_def lbvjvm_def mk_cert_def
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check_cert_def make_cert_def nth_append)
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qed
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theorem jvm_lbv_correct:
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"wt_jvm_prog_lbv G Cert \<Longrightarrow> \<exists>Phi. wt_jvm_prog G Phi"
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proof -
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let ?Phi = "\<lambda>C sig. let (C,rT,(maxs,maxl,ins,et)) = the (method (G,C) sig) in
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SOME phi. wt_method G C (snd sig) rT maxs maxl ins et phi"
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assume "wt_jvm_prog_lbv G Cert"
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hence "wt_jvm_prog G ?Phi"
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apply (unfold wt_jvm_prog_def wt_jvm_prog_lbv_def)
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apply (erule jvm_prog_lift)
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apply (auto dest: wt_lbv_wt_method intro: someI)
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done
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thus ?thesis by blast
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qed
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13215
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theorem jvm_lbv_complete:
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"wt_jvm_prog G Phi \<Longrightarrow> wt_jvm_prog_lbv G (prg_cert G Phi)"
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apply (unfold wt_jvm_prog_def wt_jvm_prog_lbv_def)
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apply (erule jvm_prog_lift)
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apply (auto simp add: prg_cert_def intro: wt_method_wt_lbv)
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done
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end
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