(* Title: HOL/MicroJava/BV/JVM.thy Author: Tobias Nipkow, Gerwin Klein Copyright 2000 TUM *) section ‹Kildall for the JVM \label{sec:JVM}› theory JVM imports Typing_Framework_JVM begin definition kiljvm :: "jvm_prog ⇒ nat ⇒ nat ⇒ ty ⇒ exception_table ⇒ instr list ⇒ JVMType.state list ⇒ JVMType.state list" where "kiljvm G maxs maxr rT et bs == kildall (JVMType.le G maxs maxr) (JVMType.sup G maxs maxr) (exec G maxs rT et bs)" definition wt_kil :: "jvm_prog ⇒ cname ⇒ ty list ⇒ ty ⇒ nat ⇒ nat ⇒ exception_table ⇒ instr list ⇒ bool" where "wt_kil G C pTs rT mxs mxl et ins == check_bounded ins et ∧ 0 < size ins ∧ (let first = Some ([],(OK (Class C))#((map OK pTs))@(replicate mxl Err)); start = OK first#(replicate (size ins - 1) (OK None)); result = kiljvm G mxs (1+size pTs+mxl) rT et ins start in ∀n < size ins. result!n ≠ Err)" definition wt_jvm_prog_kildall :: "jvm_prog ⇒ bool" where "wt_jvm_prog_kildall G == wf_prog (λG C (sig,rT,(maxs,maxl,b,et)). wt_kil G C (snd sig) rT maxs maxl et b) G" theorem is_bcv_kiljvm: "⟦ wf_prog wf_mb G; bounded (exec G maxs rT et bs) (size bs) ⟧ ⟹ is_bcv (JVMType.le G maxs maxr) Err (exec G maxs rT et bs) (size bs) (states G maxs maxr) (kiljvm G maxs maxr rT et bs)" apply (unfold kiljvm_def sl_triple_conv) apply (rule is_bcv_kildall) apply (simp (no_asm) add: sl_triple_conv [symmetric]) apply (force intro!: semilat_JVM_slI dest: wf_acyclic simp add: symmetric sl_triple_conv) apply (simp (no_asm) add: JVM_le_unfold) apply (blast intro!: order_widen wf_converse_subcls1_impl_acc_subtype dest: wf_subcls1 wf_acyclic wf_prog_ws_prog) apply (simp add: JVM_le_unfold) apply (erule exec_pres_type) apply assumption apply (drule wf_prog_ws_prog, erule exec_mono, assumption) done lemma subset_replicate: "set (replicate n x) ⊆ {x}" by (induct n) auto lemma in_set_replicate: "x ∈ set (replicate n y) ⟹ x = y" proof - assume "x ∈ set (replicate n y)" also have "set (replicate n y) ⊆ {y}" by (rule subset_replicate) finally have "x ∈ {y}" . thus ?thesis by simp qed theorem wt_kil_correct: assumes wf: "wf_prog wf_mb G" assumes C: "is_class G C" assumes pTs: "set pTs ⊆ types G" assumes wtk: "wt_kil G C pTs rT maxs mxl et bs" shows "∃phi. wt_method G C pTs rT maxs mxl bs et phi" proof - let ?start = "OK (Some ([],(OK (Class C))#((map OK pTs))@(replicate mxl Err))) #(replicate (size bs - 1) (OK None))" from wtk obtain maxr r where bounded: "check_bounded bs et" and result: "r = kiljvm G maxs maxr rT et bs ?start" and success: "∀n < size bs. r!n ≠ Err" and instrs: "0 < size bs" and maxr: "maxr = Suc (length pTs + mxl)" by (unfold wt_kil_def) simp from bounded have "bounded (exec G maxs rT et bs) (size bs)" by (unfold exec_def) (intro bounded_lift check_bounded_is_bounded) with wf have bcv: "is_bcv (JVMType.le G maxs maxr) Err (exec G maxs rT et bs) (size bs) (states G maxs maxr) (kiljvm G maxs maxr rT et bs)" by (rule is_bcv_kiljvm) from C pTs instrs maxr have "?start ∈ list (length bs) (states G maxs maxr)" apply (unfold JVM_states_unfold) apply (rule listI) apply (auto intro: list_appendI dest!: in_set_replicate) apply force done with bcv success result have "∃ts∈list (length bs) (states G maxs maxr). ?start <=[JVMType.le G maxs maxr] ts ∧ wt_step (JVMType.le G maxs maxr) Err (exec G maxs rT et bs) ts" by (unfold is_bcv_def) auto then obtain phi' where phi': "phi' ∈ list (length bs) (states G maxs maxr)" and s: "?start <=[JVMType.le G maxs maxr] phi'" and w: "wt_step (JVMType.le G maxs maxr) Err (exec G maxs rT et bs) phi'" by blast hence wt_err_step: "wt_err_step (sup_state_opt G) (exec G maxs rT et bs) phi'" by (simp add: wt_err_step_def exec_def JVM_le_Err_conv) from s have le: "JVMType.le G maxs maxr (?start ! 0) (phi'!0)" by (drule_tac p=0 in le_listD) (simp add: lesub_def)+ from phi' have l: "size phi' = size bs" by simp with instrs w have "phi' ! 0 ≠ Err" by (unfold wt_step_def) simp with instrs l have phi0: "OK (map ok_val phi' ! 0) = phi' ! 0" by auto from phi' have "check_types G maxs maxr phi'" by(simp add: check_types_def) also from w have "phi' = map OK (map ok_val phi')" by (auto simp add: wt_step_def intro!: nth_equalityI) finally have check_types: "check_types G maxs maxr (map OK (map ok_val phi'))" . from l bounded have "bounded (λpc. eff (bs!pc) G pc et) (length phi')" by (simp add: exec_def check_bounded_is_bounded) hence bounded': "bounded (exec G maxs rT et bs) (length bs)" by (auto intro: bounded_lift simp add: exec_def l) with wt_err_step have "wt_app_eff (sup_state_opt G) (λpc. app (bs!pc) G maxs rT pc et) (λpc. eff (bs!pc) G pc et) (map ok_val phi')" by (auto intro: wt_err_imp_wt_app_eff simp add: l exec_def) with instrs l le bounded bounded' check_types maxr have "wt_method G C pTs rT maxs mxl bs et (map ok_val phi')" apply (unfold wt_method_def wt_app_eff_def) apply simp apply (rule conjI) apply (unfold wt_start_def) apply (rule JVM_le_convert [THEN iffD1]) apply (simp (no_asm) add: phi0) apply clarify apply (erule allE, erule impE, assumption) apply (elim conjE) apply (clarsimp simp add: lesub_def wt_instr_def) apply (simp add: exec_def) apply (drule bounded_err_stepD, assumption+) apply blast done thus ?thesis by blast qed theorem wt_kil_complete: assumes wf: "wf_prog wf_mb G" assumes C: "is_class G C" assumes pTs: "set pTs ⊆ types G" assumes wtm: "wt_method G C pTs rT maxs mxl bs et phi" shows "wt_kil G C pTs rT maxs mxl et bs" proof - let ?mxr = "1+size pTs+mxl" from wtm obtain instrs: "0 < length bs" and len: "length phi = length bs" and bounded: "check_bounded bs et" and ck_types: "check_types G maxs ?mxr (map OK phi)" and wt_start: "wt_start G C pTs mxl phi" and wt_ins: "∀pc. pc < length bs ⟶ wt_instr (bs ! pc) G rT phi maxs (length bs) et pc" by (unfold wt_method_def) simp from ck_types len have istype_phi: "map OK phi ∈ list (length bs) (states G maxs (1+size pTs+mxl))" by (auto simp add: check_types_def intro!: listI) let ?eff = "λpc. eff (bs!pc) G pc et" let ?app = "λpc. app (bs!pc) G maxs rT pc et" from bounded have bounded_exec: "bounded (exec G maxs rT et bs) (size bs)" by (unfold exec_def) (intro bounded_lift check_bounded_is_bounded) from wt_ins have "wt_app_eff (sup_state_opt G) ?app ?eff phi" apply (unfold wt_app_eff_def wt_instr_def lesub_def) apply (simp (no_asm) only: len) apply blast done with bounded_exec have "wt_err_step (sup_state_opt G) (err_step (size phi) ?app ?eff) (map OK phi)" by - (erule wt_app_eff_imp_wt_err,simp add: exec_def len) hence wt_err: "wt_err_step (sup_state_opt G) (exec G maxs rT et bs) (map OK phi)" by (unfold exec_def) (simp add: len) from wf bounded_exec have is_bcv: "is_bcv (JVMType.le G maxs ?mxr) Err (exec G maxs rT et bs) (size bs) (states G maxs ?mxr) (kiljvm G maxs ?mxr rT et bs)" by (rule is_bcv_kiljvm) let ?start = "OK (Some ([],(OK (Class C))#((map OK pTs))@(replicate mxl Err))) #(replicate (size bs - 1) (OK None))" from C pTs instrs have start: "?start ∈ list (length bs) (states G maxs ?mxr)" apply (unfold JVM_states_unfold) apply (rule listI) apply (auto intro!: list_appendI dest!: in_set_replicate) apply force done let ?phi = "map OK phi" have less_phi: "?start <=[JVMType.le G maxs ?mxr] ?phi" proof - from len instrs have "length ?start = length (map OK phi)" by simp moreover { fix n from wt_start have "G ⊢ ok_val (?start!0) <=' phi!0" by (simp add: wt_start_def) moreover from instrs len have "0 < length phi" by simp ultimately have "JVMType.le G maxs ?mxr (?start!0) (?phi!0)" by (simp add: JVM_le_Err_conv Err.le_def lesub_def) moreover { fix n' have "JVMType.le G maxs ?mxr (OK None) (?phi!n)" by (auto simp add: JVM_le_Err_conv Err.le_def lesub_def split: err.splits) hence "⟦ n = Suc n'; n < length ?start ⟧ ⟹ JVMType.le G maxs ?mxr (?start!n) (?phi!n)" by simp } ultimately have "n < length ?start ⟹ (?start!n) <=_(JVMType.le G maxs ?mxr) (?phi!n)" by (unfold lesub_def) (cases n, blast+) } ultimately show ?thesis by (rule le_listI) qed from wt_err have "wt_step (JVMType.le G maxs ?mxr) Err (exec G maxs rT et bs) ?phi" by (simp add: wt_err_step_def JVM_le_Err_conv) with start istype_phi less_phi is_bcv have "∀p. p < length bs ⟶ kiljvm G maxs ?mxr rT et bs ?start ! p ≠ Err" by (unfold is_bcv_def) auto with bounded instrs show "wt_kil G C pTs rT maxs mxl et bs" by (unfold wt_kil_def) simp qed theorem jvm_kildall_sound_complete: "wt_jvm_prog_kildall G = (∃Phi. wt_jvm_prog G Phi)" proof let ?Phi = "λC sig. let (C,rT,(maxs,maxl,ins,et)) = the (method (G,C) sig) in SOME phi. wt_method G C (snd sig) rT maxs maxl ins et phi" assume "wt_jvm_prog_kildall G" hence "wt_jvm_prog G ?Phi" apply (unfold wt_jvm_prog_def wt_jvm_prog_kildall_def) apply (erule jvm_prog_lift) apply (auto dest!: wt_kil_correct intro: someI) done thus "∃Phi. wt_jvm_prog G Phi" by fast next assume "∃Phi. wt_jvm_prog G Phi" thus "wt_jvm_prog_kildall G" apply (clarify) apply (unfold wt_jvm_prog_def wt_jvm_prog_kildall_def) apply (erule jvm_prog_lift) apply (auto intro: wt_kil_complete) done qed end