(* Title: HOL/MicroJava/BV/BVSpecTypeSafe.thy Author: Cornelia Pusch, Gerwin Klein Copyright 1999 Technische Universitaet Muenchen *) section ‹BV Type Safety Proof \label{sec:BVSpecTypeSafe}› theory BVSpecTypeSafe imports BVConform begin text ‹ This theory contains proof that the specification of the bytecode verifier only admits type safe programs. › subsection ‹Preliminaries› text ‹ Simp and intro setup for the type safety proof: › lemmas defs1 = correct_state_def conf_f_def wt_instr_def eff_def norm_eff_def app_def xcpt_app_def lemmas widen_rules [intro] = conf_widen confT_widen confs_widens confTs_widen subsection ‹Exception Handling› text ‹ For the ‹Invoke› instruction the BV has checked all handlers that guard the current ‹pc›. › lemma Invoke_handlers: "match_ex_table P C pc xt = Some (pc',d') ⟹ ∃(f,t,D,h,d) ∈ set (relevant_entries P (Invoke n M) pc xt). P ⊢ C ≼⇧^{*}D ∧ pc ∈ {f..<t} ∧ pc' = h ∧ d' = d" by (induct xt) (auto simp add: relevant_entries_def matches_ex_entry_def is_relevant_entry_def split: if_split_asm) text ‹ We can prove separately that the recursive search for exception handlers (‹find_handler›) in the frame stack results in a conforming state (if there was no matching exception handler in the current frame). We require that the exception is a valid heap address, and that the state before the exception occurred conforms. › term find_handler lemma uncaught_xcpt_correct: assumes wt: "wf_jvm_prog⇘Φ⇙ P" assumes h: "h xcp = Some obj" shows "⋀f. P,Φ ⊢ (None, h, f#frs)√ ⟹ P,Φ ⊢ (find_handler P xcp h frs) √" (is "⋀f. ?correct (None, h, f#frs) ⟹ ?correct (?find frs)") (*<*) proof (induct frs) ― ‹the base case is trivial as it should be› show "?correct (?find [])" by (simp add: correct_state_def) next ― ‹we will need both forms @{text wf_jvm_prog} and @{text wf_prog} later› from wt obtain mb where wf: "wf_prog mb P" by (simp add: wf_jvm_prog_phi_def) ― ‹the assumption for the cons case:› fix f f' frs' assume cr: "?correct (None, h, f#f'#frs')" ― ‹the induction hypothesis:› assume IH: "⋀f. ?correct (None, h, f#frs') ⟹ ?correct (?find frs')" from cr have cr': "?correct (None, h, f'#frs')" by (fastforce simp add: correct_state_def) obtain stk loc C M pc where [simp]: "f' = (stk,loc,C,M,pc)" by (cases f') from cr obtain Ts T mxs mxl⇩_{0}ins xt where meth: "P ⊢ C sees M:Ts → T = (mxs,mxl⇩_{0},ins,xt) in C" by (simp add: correct_state_def, blast) hence [simp]: "ex_table_of P C M = xt" by simp show "?correct (?find (f'#frs'))" proof (cases "match_ex_table P (cname_of h xcp) pc xt") case None with cr' IH [of f'] show ?thesis by fastforce next fix pc_d assume "match_ex_table P (cname_of h xcp) pc xt = Some pc_d" then obtain pc' d' where match: "match_ex_table P (cname_of h xcp) pc xt = Some (pc',d')" (is "?match (cname_of h xcp) = _") by (cases pc_d) auto from wt meth cr' [simplified] have wti: "P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M" by (fastforce simp add: correct_state_def conf_f_def dest: sees_method_fun elim!: wt_jvm_prog_impl_wt_instr) from cr meth obtain n M' ST LT where ins: "ins!pc = Invoke n M'" (is "_ = ?i") and Φ: "Φ C M ! pc = Some (ST, LT)" by (fastforce dest: sees_method_fun simp add: correct_state_def) from ins match obtain f t D where rel: "(f,t,D,pc',d') ∈ set (relevant_entries P (ins!pc) pc xt)" and D: "P ⊢ cname_of h xcp ≼⇧^{*}D" by (fastforce dest: Invoke_handlers) from rel have "(pc', Some (Class D # drop (size ST - d') ST, LT)) ∈ set (xcpt_eff (ins!pc) P pc (ST,LT) xt)" (is "(_, Some (?ST',_)) ∈ _") by (force simp add: xcpt_eff_def image_def) with wti Φ obtain pc: "pc' < size ins" and "P ⊢ Some (?ST', LT) ≤' Φ C M ! pc'" by (clarsimp simp add: defs1) blast then obtain ST' LT' where Φ': "Φ C M ! pc' = Some (ST',LT')" and less: "P ⊢ (?ST', LT) ≤⇩_{i}(ST',LT')" by (auto simp add: sup_state_opt_any_Some) from cr' Φ meth have "conf_f P h (ST, LT) ins f'" by (unfold correct_state_def) (fastforce dest: sees_method_fun) hence loc: "P,h ⊢ loc [:≤⇩_{⊤}] LT" and stk: "P,h ⊢ stk [:≤] ST" by (unfold conf_f_def) auto hence [simp]: "size stk = size ST" by (simp add: list_all2_lengthD) let ?f = "(Addr xcp # drop (length stk - d') stk, loc, C, M, pc')" have "conf_f P h (ST',LT') ins ?f" proof - from wf less loc have "P,h ⊢ loc [:≤⇩_{⊤}] LT'" by simp blast moreover from D h have "P,h ⊢ Addr xcp :≤ Class D" by (simp add: conf_def obj_ty_def case_prod_unfold) with less stk have "P,h ⊢ Addr xcp # drop (length stk - d') stk [:≤] ST'" by (auto intro!: list_all2_dropI) ultimately show ?thesis using pc by (simp add: conf_f_def) qed with cr' match Φ' meth pc show ?thesis by (unfold correct_state_def) (fastforce dest: sees_method_fun) qed qed (*>*) text ‹ The requirement of lemma ‹uncaught_xcpt_correct› (that the exception is a valid reference on the heap) is always met for welltyped instructions and conformant states: › lemma exec_instr_xcpt_h: "⟦ fst (exec_instr (ins!pc) P h stk vars Cl M pc frs) = Some xcp; P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M; P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√ ⟧ ⟹ ∃obj. h xcp = Some obj" (is "⟦ ?xcpt; ?wt; ?correct ⟧ ⟹ ?thesis") (*<*) proof - note [simp] = split_beta note [split] = if_split_asm option.split_asm assume wt: ?wt ?correct hence pre: "preallocated h" by (simp add: correct_state_def hconf_def) assume xcpt: ?xcpt with pre show ?thesis proof (cases "ins!pc") case Throw with xcpt wt pre show ?thesis by (clarsimp iff: list_all2_Cons2 simp add: defs1) (auto dest: non_npD simp: is_refT_def elim: preallocatedE) qed (auto elim: preallocatedE) qed (*>*) lemma conf_sys_xcpt: "⟦preallocated h; C ∈ sys_xcpts⟧ ⟹ P,h ⊢ Addr (addr_of_sys_xcpt C) :≤ Class C" by (auto elim: preallocatedE) lemma match_ex_table_SomeD: "match_ex_table P C pc xt = Some (pc',d') ⟹ ∃(f,t,D,h,d) ∈ set xt. matches_ex_entry P C pc (f,t,D,h,d) ∧ h = pc' ∧ d=d'" by (induct xt) (auto split: if_split_asm) text ‹ Finally we can state that, whenever an exception occurs, the next state always conforms: › lemma xcpt_correct: fixes σ' :: jvm_state assumes wtp: "wf_jvm_prog⇘Φ⇙ P" assumes meth: "P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C" assumes wt: "P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M" assumes xp: "fst (exec_instr (ins!pc) P h stk loc C M pc frs) = Some xcp" assumes s': "Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs)" assumes correct: "P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√" shows "P,Φ ⊢ σ'√" (*<*) proof - from wtp obtain wfmb where wf: "wf_prog wfmb P" by (simp add: wf_jvm_prog_phi_def) note conf_sys_xcpt [elim!] note xp' = meth s' xp note wtp moreover from xp wt correct obtain obj where h: "h xcp = Some obj" by (blast dest: exec_instr_xcpt_h) moreover note correct ultimately have "P,Φ ⊢ find_handler P xcp h frs √" by (rule uncaught_xcpt_correct) with xp' have "match_ex_table P (cname_of h xcp) pc xt = None ⟹ ?thesis" (is "?m (cname_of h xcp) = _ ⟹ _" is "?match = _ ⟹ _") by (simp add: split_beta) moreover { fix pc_d assume "?match = Some pc_d" then obtain pc' d' where some_handler: "?match = Some (pc',d')" by (cases pc_d) auto from correct meth obtain ST LT where h_ok: "P ⊢ h √" and Φ_pc: "Φ C M ! pc = Some (ST, LT)" and frame: "conf_f P h (ST,LT) ins (stk,loc,C,M,pc)" and frames: "conf_fs P h Φ M (size Ts) T frs" by (unfold correct_state_def) (fastforce dest: sees_method_fun) from h_ok have preh: "preallocated h" by (simp add: hconf_def) from frame obtain stk: "P,h ⊢ stk [:≤] ST" and loc: "P,h ⊢ loc [:≤⇩_{⊤}] LT" and pc: "pc < size ins" by (unfold conf_f_def) auto from stk have [simp]: "size stk = size ST" .. from wt Φ_pc have eff: "∀(pc', s')∈set (xcpt_eff (ins!pc) P pc (ST,LT) xt). pc' < size ins ∧ P ⊢ s' ≤' Φ C M!pc'" by (auto simp add: defs1) let ?stk' = "Addr xcp # drop (length stk - d') stk" let ?f = "(?stk', loc, C, M, pc')" from some_handler xp' have σ': "σ' = (None, h, ?f#frs)" by (simp add: split_beta) from some_handler obtain f t D where xt: "(f,t,D,pc',d') ∈ set xt" and "matches_ex_entry P (cname_of h xcp) pc (f,t,D,pc',d')" by (auto dest: match_ex_table_SomeD) hence match: "P ⊢ cname_of h xcp ≼⇧^{*}D" "pc ∈ {f..<t}" by (auto simp: matches_ex_entry_def) from xp obtain "(f,t,D,pc',d') ∈ set (relevant_entries P (ins!pc) pc xt)" and conf: "P,h ⊢ Addr xcp :≤ Class D" proof (cases "ins!pc") case Return with xp have False by (auto simp: split_beta split: if_split_asm) thus ?thesis .. next case New with xp have [simp]: "xcp = addr_of_sys_xcpt OutOfMemory" by simp with New match preh have "is_relevant_entry P (ins!pc) pc (f,t,D,pc',d')" by (simp add: is_relevant_entry_def) with match preh xt show ?thesis by (fastforce simp add: relevant_entries_def intro: that) next case Getfield with xp have [simp]: "xcp = addr_of_sys_xcpt NullPointer" by (simp add: split_beta split: if_split_asm) with Getfield match preh have "is_relevant_entry P (ins!pc) pc (f,t,D,pc',d')" by (simp add: is_relevant_entry_def) with match preh xt show ?thesis by (fastforce simp add: relevant_entries_def intro: that) next case Putfield with xp have [simp]: "xcp = addr_of_sys_xcpt NullPointer" by (simp add: split_beta split: if_split_asm) with Putfield match preh have "is_relevant_entry P (ins!pc) pc (f,t,D,pc',d')" by (simp add: is_relevant_entry_def) with match preh xt show ?thesis by (fastforce simp add: relevant_entries_def intro: that) next case Checkcast with xp have [simp]: "xcp = addr_of_sys_xcpt ClassCast" by (simp add: split_beta split: if_split_asm) with Checkcast match preh have "is_relevant_entry P (ins!pc) pc (f,t,D,pc',d')" by (simp add: is_relevant_entry_def) with match preh xt show ?thesis by (fastforce simp add: relevant_entries_def intro: that) next case Invoke with xp match have "is_relevant_entry P (ins!pc) pc (f,t,D,pc',d')" by (simp add: is_relevant_entry_def) moreover from xp wt correct obtain obj where xcp: "h xcp = Some obj" by (blast dest: exec_instr_xcpt_h) ultimately show ?thesis using xt match by (auto simp add: relevant_entries_def conf_def case_prod_unfold intro: that) next case Throw with xp match preh have "is_relevant_entry P (ins!pc) pc (f,t,D,pc',d')" by (simp add: is_relevant_entry_def) moreover from xp wt correct obtain obj where xcp: "h xcp = Some obj" by (blast dest: exec_instr_xcpt_h) ultimately show ?thesis using xt match by (auto simp add: relevant_entries_def conf_def case_prod_unfold intro: that) qed auto with eff obtain ST' LT' where Φ_pc': "Φ C M ! pc' = Some (ST', LT')" and pc': "pc' < size ins" and less: "P ⊢ (Class D # drop (size ST - d') ST, LT) ≤⇩_{i}(ST', LT')" by (fastforce simp add: xcpt_eff_def sup_state_opt_any_Some) with conf loc stk have "conf_f P h (ST',LT') ins ?f" by (auto simp add: defs1 intro: list_all2_dropI) with meth h_ok frames Φ_pc' σ' have ?thesis by (unfold correct_state_def) (fastforce dest: sees_method_fun) } ultimately show ?thesis by (cases "?match") blast+ qed (*>*) subsection ‹Single Instructions› text ‹ In this section we prove for each single (welltyped) instruction that the state after execution of the instruction still conforms. Since we have already handled exceptions above, we can now assume that no exception occurs in this step. › declare defs1 [simp] lemma Invoke_correct: fixes σ' :: jvm_state assumes wtprog: "wf_jvm_prog⇘Φ⇙ P" assumes meth_C: "P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C" assumes ins: "ins ! pc = Invoke M' n" assumes wti: "P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M" assumes σ': "Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs)" assumes approx: "P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√" assumes no_xcp: "fst (exec_instr (ins!pc) P h stk loc C M pc frs) = None" shows "P,Φ ⊢ σ'√" (*<*) proof - note split_paired_Ex [simp del] from wtprog obtain wfmb where wfprog: "wf_prog wfmb P" by (simp add: wf_jvm_prog_phi_def) from ins meth_C approx obtain ST LT where heap_ok: "P⊢ h√" and Φ_pc: "Φ C M!pc = Some (ST,LT)" and frame: "conf_f P h (ST,LT) ins (stk,loc,C,M,pc)" and frames: "conf_fs P h Φ M (size Ts) T frs" by (fastforce dest: sees_method_fun) from ins wti Φ_pc have n: "n < size ST" by simp { assume "stk!n = Null" with ins no_xcp have False by (simp add: split_beta) hence ?thesis .. } moreover { assume "ST!n = NT" moreover from frame have "P,h ⊢ stk [:≤] ST" by simp with n have "P,h ⊢ stk!n :≤ ST!n" by (simp add: list_all2_conv_all_nth) ultimately have "stk!n = Null" by simp with ins no_xcp have False by (simp add: split_beta) hence ?thesis .. } moreover { assume NT: "ST!n ≠ NT" and Null: "stk!n ≠ Null" from NT ins wti Φ_pc obtain D D' Ts T m ST' LT' where D: "ST!n = Class D" and pc': "pc+1 < size ins" and m_D: "P ⊢ D sees M': Ts→T = m in D'" and Ts: "P ⊢ rev (take n ST) [≤] Ts" and Φ': "Φ C M ! (pc+1) = Some (ST', LT')" and LT': "P ⊢ LT [≤⇩_{⊤}] LT'" and ST': "P ⊢ (T # drop (n+1) ST) [≤] ST'" by (clarsimp simp add: sup_state_opt_any_Some) from frame obtain stk: "P,h ⊢ stk [:≤] ST" and loc: "P,h ⊢ loc [:≤⇩_{⊤}] LT" by simp from n stk D have "P,h ⊢ stk!n :≤ Class D" by (auto simp add: list_all2_conv_all_nth) with Null obtain a C' fs where Addr: "stk!n = Addr a" and obj: "h a = Some (C',fs)" and C'subD: "P ⊢ C' ≼⇧^{*}D" by (fastforce dest!: conf_ClassD) with wfprog m_D obtain Ts' T' m' D'' mxs' mxl' ins' xt' where m_C': "P ⊢ C' sees M': Ts'→T' = (mxs',mxl',ins',xt') in D''" and T': "P ⊢ T' ≤ T" and Ts': "P ⊢ Ts [≤] Ts'" by (auto dest: sees_method_mono) let ?loc' = "Addr a # rev (take n stk) @ replicate mxl' undefined" let ?f' = "([], ?loc', D'', M', 0)" let ?f = "(stk, loc, C, M, pc)" from Addr obj m_C' ins σ' meth_C have s': "σ' = (None, h, ?f' # ?f # frs)" by simp from Ts n have [simp]: "size Ts = n" by (auto dest: list_all2_lengthD simp: min_def) with Ts' have [simp]: "size Ts' = n" by (auto dest: list_all2_lengthD) from m_C' wfprog obtain mD'': "P ⊢ D'' sees M':Ts'→T'=(mxs',mxl',ins',xt') in D''" by (fast dest: sees_method_idemp) moreover with wtprog obtain start: "wt_start P D'' Ts' mxl' (Φ D'' M')" and ins': "ins' ≠ []" by (auto dest: wt_jvm_prog_impl_wt_start) then obtain LT⇩_{0}where LT⇩_{0}: "Φ D'' M' ! 0 = Some ([], LT⇩_{0})" by (clarsimp simp add: wt_start_def defs1 sup_state_opt_any_Some) moreover have "conf_f P h ([], LT⇩_{0}) ins' ?f'" proof - let ?LT = "OK (Class D'') # (map OK Ts') @ (replicate mxl' Err)" from stk have "P,h ⊢ take n stk [:≤] take n ST" .. hence "P,h ⊢ rev (take n stk) [:≤] rev (take n ST)" by simp also note Ts also note Ts' finally have "P,h ⊢ rev (take n stk) [:≤⇩_{⊤}] map OK Ts'" by simp also have "P,h ⊢ replicate mxl' undefined [:≤⇩_{⊤}] replicate mxl' Err" by simp also from m_C' have "P ⊢ C' ≼⇧^{*}D''" by (rule sees_method_decl_above) with obj have "P,h ⊢ Addr a :≤ Class D''" by (simp add: conf_def) ultimately have "P,h ⊢ ?loc' [:≤⇩_{⊤}] ?LT" by simp also from start LT⇩_{0}have "P ⊢ … [≤⇩_{⊤}] LT⇩_{0}" by (simp add: wt_start_def) finally have "P,h ⊢ ?loc' [:≤⇩_{⊤}] LT⇩_{0}" . thus ?thesis using ins' by simp qed ultimately have ?thesis using s' Φ_pc approx meth_C m_D T' ins D by (fastforce dest: sees_method_fun [of _ C]) } ultimately show ?thesis by blast qed (*>*) declare list_all2_Cons2 [iff] lemma Return_correct: fixes σ' :: jvm_state assumes wt_prog: "wf_jvm_prog⇘Φ⇙ P" assumes meth: "P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C" assumes ins: "ins ! pc = Return" assumes wt: "P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M" assumes s': "Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs)" assumes correct: "P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√" shows "P,Φ ⊢ σ'√" (*<*) proof - from wt_prog obtain wfmb where wf: "wf_prog wfmb P" by (simp add: wf_jvm_prog_phi_def) from meth ins s' have "frs = [] ⟹ ?thesis" by (simp add: correct_state_def) moreover { fix f frs' assume frs': "frs = f#frs'" moreover obtain stk' loc' C' M' pc' where f: "f = (stk',loc',C',M',pc')" by (cases f) moreover note meth ins s' ultimately have σ': "σ' = (None,h,(hd stk#(drop (1+size Ts) stk'),loc',C',M',pc'+1)#frs')" (is "σ' = (None,h,?f'#frs')") by simp from correct meth obtain ST LT where h_ok: "P ⊢ h √" and Φ_pc: "Φ C M ! pc = Some (ST, LT)" and frame: "conf_f P h (ST, LT) ins (stk,loc,C,M,pc)" and frames: "conf_fs P h Φ M (size Ts) T frs" by (auto dest: sees_method_fun simp add: correct_state_def) from Φ_pc ins wt obtain U ST⇩_{0}where "ST = U # ST⇩_{0}" "P ⊢ U ≤ T" by (simp add: wt_instr_def app_def) blast with wf frame have hd_stk: "P,h ⊢ hd stk :≤ T" by (auto simp add: conf_f_def) from f frs' frames obtain ST' LT' Ts'' T'' mxs' mxl⇩_{0}' ins' xt' D Ts' T' m D' where Φ': "Φ C' M' ! pc' = Some (ST', LT')" and meth_C': "P ⊢ C' sees M':Ts''→T''=(mxs',mxl⇩_{0}',ins',xt') in C'" and ins': "ins' ! pc' = Invoke M (size Ts)" and D: "ST' ! (size Ts) = Class D" and meth_D: "P ⊢ D sees M: Ts'→T' = m in D'" and T': "P ⊢ T ≤ T'" and frame': "conf_f P h (ST',LT') ins' f" and conf_fs: "conf_fs P h Φ M' (size Ts'') T'' frs'" by clarsimp from f frame' obtain stk': "P,h ⊢ stk' [:≤] ST'" and loc': "P,h ⊢ loc' [:≤⇩_{⊤}] LT'" and pc': "pc' < size ins'" by (simp add: conf_f_def) from wt_prog meth_C' pc' have "P,T'',mxs',size ins',xt' ⊢ ins'!pc',pc' :: Φ C' M'" by (rule wt_jvm_prog_impl_wt_instr) with ins' Φ' D meth_D obtain aTs ST'' LT'' where Φ_suc: "Φ C' M' ! Suc pc' = Some (ST'', LT'')" and less: "P ⊢ (T' # drop (size Ts+1) ST', LT') ≤⇩_{i}(ST'', LT'')" and suc_pc': "Suc pc' < size ins'" by (clarsimp simp add: sup_state_opt_any_Some) from hd_stk T' have hd_stk': "P,h ⊢ hd stk :≤ T'" .. have frame'': "conf_f P h (ST'',LT'') ins' ?f'" proof - from stk' have "P,h ⊢ drop (1+size Ts) stk' [:≤] drop (1+size Ts) ST'" .. moreover with hd_stk' less have "P,h ⊢ hd stk # drop (1+size Ts) stk' [:≤] ST''" by auto moreover from wf loc' less have "P,h ⊢ loc' [:≤⇩_{⊤}] LT''" by auto moreover note suc_pc' ultimately show ?thesis by (simp add: conf_f_def) qed with σ' frs' f meth h_ok hd_stk Φ_suc frames meth_C' Φ' have ?thesis by (fastforce dest: sees_method_fun [of _ C']) } ultimately show ?thesis by (cases frs) blast+ qed (*>*) declare sup_state_opt_any_Some [iff] declare not_Err_eq [iff] lemma Load_correct: "⟦ wf_prog wt P; P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C; ins!pc = Load idx; P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M; Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs); P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√ ⟧ ⟹ P,Φ ⊢ σ'√" by (fastforce dest: sees_method_fun [of _ C] elim!: confTs_confT_sup) declare [[simproc del: list_to_set_comprehension]] lemma Store_correct: "⟦ wf_prog wt P; P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C; ins!pc = Store idx; P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M; Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs); P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√ ⟧ ⟹ P,Φ ⊢ σ'√" (*<*) apply clarsimp apply (drule (1) sees_method_fun) apply clarsimp apply (blast intro!: list_all2_update_cong) done (*>*) lemma Push_correct: "⟦ wf_prog wt P; P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C; ins!pc = Push v; P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M; Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs); P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√ ⟧ ⟹ P,Φ ⊢ σ'√" (*<*) apply clarsimp apply (drule (1) sees_method_fun) apply clarsimp apply (blast dest: typeof_lit_conf) done (*>*) lemma Cast_conf2: "⟦ wf_prog ok P; P,h ⊢ v :≤ T; is_refT T; cast_ok P C h v; P ⊢ Class C ≤ T'; is_class P C⟧ ⟹ P,h ⊢ v :≤ T'" (*<*) apply (unfold cast_ok_def is_refT_def) apply (frule Class_widen) apply (elim exE disjE) apply simp apply simp apply simp apply (clarsimp simp add: conf_def obj_ty_def) apply (cases v) apply (auto intro: rtrancl_trans) done (*>*) lemma Checkcast_correct: "⟦ wf_jvm_prog⇘Φ⇙ P; P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C; ins!pc = Checkcast D; P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M; Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs) ; P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√; fst (exec_instr (ins!pc) P h stk loc C M pc frs) = None ⟧ ⟹ P,Φ ⊢ σ'√" (*<*) apply (clarsimp simp add: wf_jvm_prog_phi_def split: if_split_asm) apply (drule (1) sees_method_fun) apply (blast intro: Cast_conf2) done (*>*) declare split_paired_All [simp del] lemmas widens_Cons [iff] = list_all2_Cons1 [of "widen P"] for P lemma Getfield_correct: fixes σ' :: jvm_state assumes wf: "wf_prog wt P" assumes mC: "P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C" assumes i: "ins!pc = Getfield F D" assumes wt: "P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M" assumes s': "Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs)" assumes cf: "P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√" assumes xc: "fst (exec_instr (ins!pc) P h stk loc C M pc frs) = None" shows "P,Φ ⊢ σ'√" (*<*) proof - from mC cf obtain ST LT where "h√": "P ⊢ h √" and Φ: "Φ C M ! pc = Some (ST,LT)" and stk: "P,h ⊢ stk [:≤] ST" and loc: "P,h ⊢ loc [:≤⇩_{⊤}] LT" and pc: "pc < size ins" and fs: "conf_fs P h Φ M (size Ts) T frs" by (fastforce dest: sees_method_fun) from i Φ wt obtain oT ST'' vT ST' LT' vT' where oT: "P ⊢ oT ≤ Class D" and ST: "ST = oT # ST''" and F: "P ⊢ D sees F:vT in D" and pc': "pc+1 < size ins" and Φ': "Φ C M ! (pc+1) = Some (vT'#ST', LT')" and ST': "P ⊢ ST'' [≤] ST'" and LT': "P ⊢ LT [≤⇩_{⊤}] LT'" and vT': "P ⊢ vT ≤ vT'" by fastforce from stk ST obtain ref stk' where stk': "stk = ref#stk'" and ref: "P,h ⊢ ref :≤ oT" and ST'': "P,h ⊢ stk' [:≤] ST''" by auto from stk' i mC s' xc have "ref ≠ Null" by (simp add: split_beta split:if_split_asm) moreover from ref oT have "P,h ⊢ ref :≤ Class D" .. ultimately obtain a D' fs where a: "ref = Addr a" and h: "h a = Some (D', fs)" and D': "P ⊢ D' ≼⇧^{*}D" by (blast dest: non_npD) from D' F have has_field: "P ⊢ D' has F:vT in D" by (blast intro: has_field_mono has_visible_field) moreover from "h√" h have "P,h ⊢ (D', fs) √" by (rule hconfD) ultimately obtain v where v: "fs (F, D) = Some v" "P,h ⊢ v :≤ vT" by (clarsimp simp add: oconf_def has_field_def) (blast dest: has_fields_fun) from a h i mC s' stk' v have "σ' = (None, h, (v#stk',loc,C,M,pc+1)#frs)" by simp moreover from ST'' ST' have "P,h ⊢ stk' [:≤] ST'" .. moreover from v vT' have "P,h ⊢ v :≤ vT'" by blast moreover from loc LT' have "P,h ⊢ loc [:≤⇩_{⊤}] LT'" .. moreover note "h√" mC Φ' pc' v fs ultimately show "P,Φ ⊢ σ' √" by fastforce qed (*>*) lemma Putfield_correct: fixes σ' :: jvm_state assumes wf: "wf_prog wt P" assumes mC: "P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C" assumes i: "ins!pc = Putfield F D" assumes wt: "P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M" assumes s': "Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs)" assumes cf: "P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√" assumes xc: "fst (exec_instr (ins!pc) P h stk loc C M pc frs) = None" shows "P,Φ ⊢ σ'√" (*<*) proof - from mC cf obtain ST LT where "h√": "P ⊢ h √" and Φ: "Φ C M ! pc = Some (ST,LT)" and stk: "P,h ⊢ stk [:≤] ST" and loc: "P,h ⊢ loc [:≤⇩_{⊤}] LT" and pc: "pc < size ins" and fs: "conf_fs P h Φ M (size Ts) T frs" by (fastforce dest: sees_method_fun) from i Φ wt obtain vT vT' oT ST'' ST' LT' where ST: "ST = vT # oT # ST''" and field: "P ⊢ D sees F:vT' in D" and oT: "P ⊢ oT ≤ Class D" and vT: "P ⊢ vT ≤ vT'" and pc': "pc+1 < size ins" and Φ': "Φ C M!(pc+1) = Some (ST',LT')" and ST': "P ⊢ ST'' [≤] ST'" and LT': "P ⊢ LT [≤⇩_{⊤}] LT'" by clarsimp from stk ST obtain v ref stk' where stk': "stk = v#ref#stk'" and v: "P,h ⊢ v :≤ vT" and ref: "P,h ⊢ ref :≤ oT" and ST'': "P,h ⊢ stk' [:≤] ST''" by auto from stk' i mC s' xc have "ref ≠ Null" by (auto simp add: split_beta) moreover from ref oT have "P,h ⊢ ref :≤ Class D" .. ultimately obtain a D' fs where a: "ref = Addr a" and h: "h a = Some (D', fs)" and D': "P ⊢ D' ≼⇧^{*}D" by (blast dest: non_npD) from v vT have vT': "P,h ⊢ v :≤ vT'" .. from field D' have has_field: "P ⊢ D' has F:vT' in D" by (blast intro: has_field_mono has_visible_field) let ?h' = "h(a↦(D', fs((F, D)↦v)))" and ?f' = "(stk',loc,C,M,pc+1)" from h have hext: "h ⊴ ?h'" by (rule hext_upd_obj) from a h i mC s' stk' have "σ' = (None, ?h', ?f'#frs)" by simp moreover from "h√" h have "P,h ⊢ (D',fs)√" by (rule hconfD) with has_field vT' have "P,h ⊢ (D',fs((F, D)↦v))√" .. with "h√" h have "P ⊢ ?h'√" by (rule hconf_upd_obj) moreover from ST'' ST' have "P,h ⊢ stk' [:≤] ST'" .. from this hext have "P,?h' ⊢ stk' [:≤] ST'" by (rule confs_hext) moreover from loc LT' have "P,h ⊢ loc [:≤⇩_{⊤}] LT'" .. from this hext have "P,?h' ⊢ loc [:≤⇩_{⊤}] LT'" by (rule confTs_hext) moreover from fs hext have "conf_fs P ?h' Φ M (size Ts) T frs" by (rule conf_fs_hext) moreover note mC Φ' pc' ultimately show "P,Φ ⊢ σ' √" by fastforce qed (*>*) (* FIXME: move *) lemma has_fields_b_fields: "P ⊢ C has_fields FDTs ⟹ fields P C = FDTs" (*<*) apply (unfold fields_def) apply (blast intro: the_equality has_fields_fun) done (*>*) (* FIXME: move *) lemma oconf_blank [intro, simp]: "⟦is_class P C; wf_prog wt P⟧ ⟹ P,h ⊢ blank P C √" (*<*) by (fastforce simp add: blank_def has_fields_b_fields oconf_init_fields dest: wf_Fields_Ex) (*>*) lemma obj_ty_blank [iff]: "obj_ty (blank P C) = Class C" by (simp add: blank_def) lemma New_correct: fixes σ' :: jvm_state assumes wf: "wf_prog wt P" assumes meth: "P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C" assumes ins: "ins!pc = New X" assumes wt: "P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M" assumes exec: "Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs)" assumes conf: "P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√" assumes no_x: "fst (exec_instr (ins!pc) P h stk loc C M pc frs) = None" shows "P,Φ ⊢ σ'√" (*<*) proof - from ins conf meth obtain ST LT where heap_ok: "P⊢ h√" and Φ_pc: "Φ C M!pc = Some (ST,LT)" and frame: "conf_f P h (ST,LT) ins (stk,loc,C,M,pc)" and frames: "conf_fs P h Φ M (size Ts) T frs" by (auto dest: sees_method_fun) from Φ_pc ins wt obtain ST' LT' where is_class_X: "is_class P X" and mxs: "size ST < mxs" and suc_pc: "pc+1 < size ins" and Φ_suc: "Φ C M!(pc+1) = Some (ST', LT')" and less: "P ⊢ (Class X # ST, LT) ≤⇩_{i}(ST', LT')" by auto from ins no_x obtain oref where new_Addr: "new_Addr h = Some oref" by auto hence h: "h oref = None" by (rule new_Addr_SomeD) with exec ins meth new_Addr have σ': "σ' = (None, h(oref ↦ blank P X), (Addr oref#stk,loc,C,M,pc+1)#frs)" (is "σ' = (None, ?h', ?f # frs)") by simp moreover from wf h heap_ok is_class_X have h': "P ⊢ ?h' √" by (auto intro: hconf_new) moreover from h frame less suc_pc wf have "conf_f P ?h' (ST', LT') ins ?f" apply (clarsimp simp add: fun_upd_apply conf_def blank_def split_beta) apply (auto intro: confs_hext confTs_hext) done moreover from h have "h ⊴ ?h'" by simp with frames have "conf_fs P ?h' Φ M (size Ts) T frs" by (rule conf_fs_hext) ultimately show ?thesis using meth Φ_suc by fastforce qed (*>*) lemma Goto_correct: "⟦ wf_prog wt P; P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C; ins ! pc = Goto branch; P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M; Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs) ; P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√ ⟧ ⟹ P,Φ ⊢ σ'√" (*<*) apply clarsimp apply (drule (1) sees_method_fun) apply fastforce done (*>*) lemma IfFalse_correct: "⟦ wf_prog wt P; P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C; ins ! pc = IfFalse branch; P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M; Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs) ; P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√ ⟧ ⟹ P,Φ ⊢ σ'√" (*<*) apply clarsimp apply (drule (1) sees_method_fun) apply fastforce done (*>*) lemma CmpEq_correct: "⟦ wf_prog wt P; P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C; ins ! pc = CmpEq; P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M; Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs) ; P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√ ⟧ ⟹ P,Φ ⊢ σ'√" (*<*) apply clarsimp apply (drule (1) sees_method_fun) apply fastforce done (*>*) lemma Pop_correct: "⟦ wf_prog wt P; P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C; ins ! pc = Pop; P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M; Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs) ; P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√ ⟧ ⟹ P,Φ ⊢ σ'√" (*<*) apply clarsimp apply (drule (1) sees_method_fun) apply fastforce done (*>*) lemma IAdd_correct: "⟦ wf_prog wt P; P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C; ins ! pc = IAdd; P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M; Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs) ; P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√ ⟧ ⟹ P,Φ ⊢ σ'√" (*<*) apply (clarsimp simp add: conf_def) apply (drule (1) sees_method_fun) apply fastforce done (*>*) lemma Throw_correct: "⟦ wf_prog wt P; P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C; ins ! pc = Throw; Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs) ; P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√; fst (exec_instr (ins!pc) P h stk loc C M pc frs) = None ⟧ ⟹ P,Φ ⊢ σ'√" by simp text ‹ The next theorem collects the results of the sections above, i.e.~exception handling and the execution step for each instruction. It states type safety for single step execution: in welltyped programs, a conforming state is transformed into another conforming state when one instruction is executed. › theorem instr_correct: "⟦ wf_jvm_prog⇘Φ⇙ P; P ⊢ C sees M:Ts→T=(mxs,mxl⇩_{0},ins,xt) in C; Some σ' = exec (P, None, h, (stk,loc,C,M,pc)#frs); P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√ ⟧ ⟹ P,Φ ⊢ σ'√" (*<*) apply (subgoal_tac "P,T,mxs,size ins,xt ⊢ ins!pc,pc :: Φ C M") prefer 2 apply (erule wt_jvm_prog_impl_wt_instr, assumption) apply clarsimp apply (drule (1) sees_method_fun) apply simp apply (cases "fst (exec_instr (ins!pc) P h stk loc C M pc frs)") prefer 2 apply (erule xcpt_correct, assumption+) apply (frule wt_jvm_progD, erule exE) apply (cases "ins!pc") apply (rule Load_correct, assumption+) apply (rule Store_correct, assumption+) apply (rule Push_correct, assumption+) apply (rule New_correct, assumption+) apply (rule Getfield_correct, assumption+) apply (rule Putfield_correct, assumption+) apply (rule Checkcast_correct, assumption+) apply (rule Invoke_correct, assumption+) apply (rule Return_correct, assumption+) apply (rule Pop_correct, assumption+) apply (rule IAdd_correct, assumption+) apply (rule Goto_correct, assumption+) apply (rule CmpEq_correct, assumption+) apply (rule IfFalse_correct, assumption+) apply (rule Throw_correct, assumption+) done (*>*) subsection ‹Main› lemma correct_state_impl_Some_method: "P,Φ ⊢ (None, h, (stk,loc,C,M,pc)#frs)√ ⟹ ∃m Ts T. P ⊢ C sees M:Ts→T = m in C" by fastforce lemma BV_correct_1 [rule_format]: "⋀σ. ⟦ wf_jvm_prog⇘Φ⇙ P; P,Φ ⊢ σ√⟧ ⟹ P ⊢ σ -jvm→⇩_{1}σ' ⟶ P,Φ ⊢ σ'√" (*<*) apply (simp only: split_tupled_all exec_1_iff) apply (rename_tac xp h frs) apply (case_tac xp) apply (case_tac frs) apply simp apply (simp only: split_tupled_all) apply hypsubst apply (frule correct_state_impl_Some_method) apply clarify apply (rule instr_correct) apply assumption+ apply (rule sym) apply assumption+ apply (case_tac frs) apply simp_all done (*>*) theorem progress: "⟦ xp=None; frs≠[] ⟧ ⟹ ∃σ'. P ⊢ (xp,h,frs) -jvm→⇩_{1}σ'" by (clarsimp simp add: exec_1_iff neq_Nil_conv split_beta simp del: split_paired_Ex) lemma progress_conform: "⟦wf_jvm_prog⇘Φ⇙ P; P,Φ ⊢ (xp,h,frs)√; xp=None; frs≠[]⟧ ⟹ ∃σ'. P ⊢ (xp,h,frs) -jvm→⇩_{1}σ' ∧ P,Φ ⊢ σ'√" (*<*) apply (drule progress) apply assumption apply (fast intro: BV_correct_1) done (*>*) theorem BV_correct [rule_format]: "⟦ wf_jvm_prog⇘Φ⇙ P; P ⊢ σ -jvm→ σ' ⟧ ⟹ P,Φ ⊢ σ√ ⟶ P,Φ ⊢ σ'√" (*<*) apply (simp only: exec_all_def1) apply (erule rtrancl_induct) apply simp apply clarify apply (erule (2) BV_correct_1) done (*>*) lemma hconf_start: assumes wf: "wf_prog wf_mb P" shows "P ⊢ (start_heap P) √" (*<*) apply (unfold hconf_def) apply (simp add: preallocated_start) apply (clarify) apply (drule sym) apply (unfold start_heap_def) apply (insert wf) apply (auto simp add: fun_upd_apply is_class_xcpt split: if_split_asm) done (*>*) lemma BV_correct_initial: shows "⟦ wf_jvm_prog⇘Φ⇙ P; P ⊢ C sees M:[]→T = m in C ⟧ ⟹ P,Φ ⊢ start_state P C M √" (*<*) apply (cases m) apply (unfold start_state_def) apply (unfold correct_state_def) apply (simp del: defs1) apply (rule conjI) apply (simp add: wf_jvm_prog_phi_def hconf_start) apply (drule wt_jvm_prog_impl_wt_start, assumption+) apply (unfold conf_f_def wt_start_def) apply fastforce done declare [[simproc add: list_to_set_comprehension]] (*>*) theorem typesafe: assumes welltyped: "wf_jvm_prog⇘Φ⇙ P" assumes main_method: "P ⊢ C sees M:[]→T = m in C" shows "P ⊢ start_state P C M -jvm→ σ ⟹ P,Φ ⊢ σ √" (*<*) proof - from welltyped main_method have "P,Φ ⊢ start_state P C M √" by (rule BV_correct_initial) moreover assume "P ⊢ start_state P C M -jvm→ σ" ultimately show "P,Φ ⊢ σ √" using welltyped by - (rule BV_correct) qed (*>*) end