Quelle  Effect.thy

(*  Title:      HOL/MicroJava/BV/Effect.thy
  Author: Gerwin Klein
  Copyright 2000 Technische Universitaet Muenchen
*)

section ‹Effect of Instructions on the State Type›

theory Effect 
imports JVMType "../JVM/JVMExceptions"
begin

type_synonym succ_type = "(p_count × state_type option) list"

text ‹Program counter of successor instructions:›
primrec succs :: "instr ==> p_count ==> p_count list" where
  "succs (Load idx) pc = [pc+1]"
| "succs (Store idx) pc = [pc+1]"
| "succs (LitPush v) pc = [pc+1]"
| "succs (Getfield F C) pc = [pc+1]"
| "succs (Putfield F C) pc = [pc+1]"
| "succs (New C) pc = [pc+1]"
| "succs (Checkcast C) pc = [pc+1]"
| "succs Pop pc = [pc+1]"
| "succs Dup pc = [pc+1]"
| "succs Dup_x1 pc = [pc+1]"
| "succs Dup_x2 pc = [pc+1]"
| "succs Swap pc = [pc+1]"
| "succs IAdd pc = [pc+1]"
| "succs (Ifcmpeq b) pc = [pc+1, nat (int pc + b)]"
| "succs (Goto b) pc = [nat (int pc + b)]"
| "succs Return pc = [pc]"  
| "succs (Invoke C mn fpTs) pc = [pc+1]"
| "succs Throw pc = [pc]"

text "Effect of instruction on the state type:"

fun eff' :: "instr × jvm_prog × state_type ==> state_type"
where
"eff' (Load idx, G, (ST, LT)) = (ok_val (LT ! idx) # ST, LT)" |
"eff' (Store idx, G, (ts#ST, LT)) = (ST, LT[idx:= OK ts])" |
"eff' (LitPush v, G, (ST, LT)) = (the (typeof (λv. None) v) # ST, LT)" |
"eff' (Getfield F C, G, (oT#ST, LT)) = (snd (the (field (G,C) F)) # ST, LT)" |
"eff' (Putfield F C, G, (vT#oT#ST, LT)) = (ST,LT)" |
"eff' (New C, G, (ST,LT)) = (Class C # ST, LT)" |
"eff' (Checkcast C, G, (RefT rt#ST,LT)) = (Class C # ST,LT)" |
"eff' (Pop, G, (ts#ST,LT)) = (ST,LT)" |
"eff' (Dup, G, (ts#ST,LT)) = (ts#ts#ST,LT)" |
"eff' (Dup_x1, G, (ts1#ts2#ST,LT)) = (ts1#ts2#ts1#ST,LT)" |
"eff' (Dup_x2, G, (ts1#ts2#ts3#ST,LT)) = (ts1#ts2#ts3#ts1#ST,LT)" |
"eff' (Swap, G, (ts1#ts2#ST,LT)) = (ts2#ts1#ST,LT)" |
"eff' (IAdd, G, (PrimT Integer#PrimT Integer#ST,LT))
                                         = (PrimT Integer#ST,LT)" |
"eff' (Ifcmpeq b, G, (ts1#ts2#ST,LT)) = (ST,LT)" |
"eff' (Goto b, G, s) = s" |
  🍋 ‹Return has no successor instruction in the same method›
"eff' (Return, G, s) = s" |
  🍋 ‹Throw always terminates abruptly›
"eff' (Throw, G, s) = s" |
"eff' (Invoke C mn fpTs, G, (ST,LT)) = (let ST' = drop (length fpTs) ST
  in (fst (snd (the (method (G,C) (mn,fpTs))))#(tl ST'),LT))" 



primrec match_any :: "jvm_prog ==> p_count ==> exception_table ==> cname list" where
  "match_any G pc [] = []"
| "match_any G pc (e#es) = (let (start_pc, end_pc, handler_pc, catch_type) = e;
                                es' = match_any G pc es
                            in
                            if start_pc <= pc ∧ pc < end_pc then catch_type#es' else es')"

primrec match :: "jvm_prog ==> xcpt ==> p_count ==> exception_table ==> cname list" where
  "match G X pc [] = []"
| "match G X pc (e#es) =
  (if match_exception_entry G (Xcpt X) pc e then [Xcpt X] else match G X pc es)"

lemma match_some_entry:
  "match G X pc et = (if ∃e ∈ set et. match_exception_entry G (Xcpt X) pc e then [Xcpt X] else [])"
  by (induct et) auto

fun
  xcpt_names :: "instr × jvm_prog × p_count × exception_table ==> cname list" 
where
  "xcpt_names (Getfield F C, G, pc, et) = match G NullPointer pc et" 
| "xcpt_names (Putfield F C, G, pc, et) = match G NullPointer pc et" 
| "xcpt_names (New C, G, pc, et) = match G OutOfMemory pc et"
| "xcpt_names (Checkcast C, G, pc, et) = match G ClassCast pc et"
| "xcpt_names (Throw, G, pc, et) = match_any G pc et"
| "xcpt_names (Invoke C m p, G, pc, et) = match_any G pc et" 
| "xcpt_names (i, G, pc, et) = []" 


definition xcpt_eff :: "instr ==> jvm_prog ==> p_count ==> state_type option ==> exception_table ==> succ_type" where
  "xcpt_eff i G pc s et ==
   map (λC. (the (match_exception_table G C pc et), case s of None ==> None | Some s' ==> Some ([Class C], snd s')))
       (xcpt_names (i,G,pc,et))"

definition norm_eff :: "instr ==> jvm_prog ==> state_type option ==> state_type option" where
  "norm_eff i G == map_option (λs. eff' (i,G,s))"

definition eff :: "instr ==> jvm_prog ==> p_count ==> exception_table ==> state_type option ==> succ_type" where
  "eff i G pc et s == (map (λpc'. (pc',norm_eff i G s)) (succs i pc)) @ (xcpt_eff i G pc s et)"

definition isPrimT :: "ty ==> bool" where
  "isPrimT T == case T of PrimT T' ==> True | RefT T' ==> False"

definition isRefT :: "ty ==> bool" where
  "isRefT T == case T of PrimT T' ==> False | RefT T' ==> True"

lemma isPrimT [simp]:
  "isPrimT T = (∃T'. T = PrimT T')" by (simp add: isPrimT_def split: ty.splits)

lemma isRefT [simp]:
  "isRefT T = (∃T'. T = RefT T')" by (simp add: isRefT_def split: ty.splits)


lemma "list_all2 P a b ==> ∀(x,y) ∈ set (zip a b). P x y"
  by (simp add: list_all2_iff) 


text "Conditions under which eff is applicable:"

fun
app' :: "instr × jvm_prog × p_count × nat × ty × state_type ==> bool"
where
"app' (Load idx, G, pc, maxs, rT, s) =
  (idx < length (snd s) ∧ (snd s) ! idx ≠ Err ∧ length (fst s) < maxs)" |
"app' (Store idx, G, pc, maxs, rT, (ts#ST, LT)) =
  (idx < length LT)" |
"app' (LitPush v, G, pc, maxs, rT, s) =
  (length (fst s) < maxs ∧ typeof (λt. None) v ≠ None)" |
"app' (Getfield F C, G, pc, maxs, rT, (oT#ST, LT)) =
  (is_class G C ∧ field (G,C) F ≠ None ∧ fst (the (field (G,C) F)) = C ∧
  G ⊨ oT ⪯ (Class C))" |
"app' (Putfield F C, G, pc, maxs, rT, (vT#oT#ST, LT)) =
  (is_class G C ∧ field (G,C) F ≠ None ∧ fst (the (field (G,C) F)) = C ∧
  G ⊨ oT ⪯ (Class C) ∧ G ⊨ vT ⪯ (snd (the (field (G,C) F))))" |
"app' (New C, G, pc, maxs, rT, s) =
  (is_class G C ∧ length (fst s) < maxs)" |
"app' (Checkcast C, G, pc, maxs, rT, (RefT rt#ST,LT)) =
  (is_class G C)" |
"app' (Pop, G, pc, maxs, rT, (ts#ST,LT)) =
  True" |
"app' (Dup, G, pc, maxs, rT, (ts#ST,LT)) =
  (1+length ST < maxs)" |
"app' (Dup_x1, G, pc, maxs, rT, (ts1#ts2#ST,LT)) =
  (2+length ST < maxs)" |
"app' (Dup_x2, G, pc, maxs, rT, (ts1#ts2#ts3#ST,LT)) =
  (3+length ST < maxs)" |
"app' (Swap, G, pc, maxs, rT, (ts1#ts2#ST,LT)) =
  True" |
"app' (IAdd, G, pc, maxs, rT, (PrimT Integer#PrimT Integer#ST,LT)) =
  True" |
"app' (Ifcmpeq b, G, pc, maxs, rT, (ts#ts'#ST,LT)) =
  (0 ≤ int pc + b ∧ (isPrimT ts ∧ ts' = ts ∨ isRefT ts ∧ isRefT ts'))" |
"app' (Goto b, G, pc, maxs, rT, s) =
  (0 ≤ int pc + b)" |
"app' (Return, G, pc, maxs, rT, (T#ST,LT)) =
  (G ⊨ T ⪯ rT)" |
"app' (Throw, G, pc, maxs, rT, (T#ST,LT)) =
  isRefT T" |
"app' (Invoke C mn fpTs, G, pc, maxs, rT, s) =
  (length fpTs < length (fst s) ∧
  (let apTs = rev (take (length fpTs) (fst s));
       X = hd (drop (length fpTs) (fst s))
   in
       G ⊨ X ⪯ Class C ∧ is_class G C ∧ method (G,C) (mn,fpTs) ≠ None ∧
       list_all2 (λx y. G ⊨ x ⪯ y) apTs fpTs))" |
  
"app' (i,G, pc,maxs,rT,s) = False"

definition xcpt_app :: "instr ==> jvm_prog ==> nat ==> exception_table ==> bool" where
  "xcpt_app i G pc et ≡ ∀C∈set(xcpt_names (i,G,pc,et)). is_class G C"

definition app :: "instr ==> jvm_prog ==> nat ==> ty ==> nat ==> exception_table ==> state_type option ==> bool" where
  "app i G maxs rT pc et s == case s of None ==> True | Some t ==> app' (i,G,pc,maxs,rT,t) ∧ xcpt_app i G pc et"


lemma match_any_match_table:
  "C ∈ set (match_any G pc et) ==> match_exception_table G C pc et ≠ None"
  apply (induct et)
   apply simp
  apply simp
  apply clarify
  apply (simp split: if_split_asm)
  apply (auto simp add: match_exception_entry_def)
  done

lemma match_X_match_table:
  "C ∈ set (match G X pc et) ==> match_exception_table G C pc et ≠ None"
  apply (induct et)
   apply simp
  apply (simp split: if_split_asm)
  done

lemma xcpt_names_in_et:
  "C ∈ set (xcpt_names (i,G,pc,et)) ==>
  ∃e ∈ set et. the (match_exception_table G C pc et) = fst (snd (snd e))"
  apply (cases i)
  apply (auto dest!: match_any_match_table match_X_match_table 
              dest: match_exception_table_in_et)
  done


lemma 1: "2 < length a ==> (∃l l' l'' ls. a = l#l'#l''#ls)"
proof (cases a)
  fix x xs assume "a = x#xs" "2 < length a"
  thus ?thesis by - (cases xs, simp, cases "tl xs", auto)
qed auto

lemma 2: "¬(2 < length a) ==> a = [] ∨ (∃ l. a = [l]) ∨ (∃ l l'. a = [l,l'])"
proof -
  assume "¬(2 < length a)"
  hence "length a < (Suc (Suc (Suc 0)))" by simp
  hence * : "length a = 0 ∨ length a = Suc 0 ∨ length a = Suc (Suc 0)" 
    by (auto simp add: less_Suc_eq)

  { 
    fix x 
    assume "length x = Suc 0"
    hence "∃ l. x = [l]"  by (cases x) auto
  } note 0 = this

  have "length a = Suc (Suc 0) ==> ∃l l'. a = [l,l']" by (cases a) (auto dest: 0)
  with * show ?thesis by (auto dest: 0)
qed

lemmas [simp] = app_def xcpt_app_def

text ‹
 \medskip
 simp rules for 🍋‹app›
 ›
lemma appNone[simp]: "app i G maxs rT pc et None = True" by simp


lemma appLoad[simp]:
"(app (Load idx) G maxs rT pc et (Some s)) = (∃ST LT. s = (ST,LT) ∧ idx < length LT ∧ LT!idx ≠ Err ∧ length ST < maxs)"
  by (cases s, simp)

lemma appStore[simp]:
"(app (Store idx) G maxs rT pc et (Some s)) = (∃ts ST LT. s = (ts#ST,LT) ∧ idx < length LT)"
  by (cases s, cases "2 < length (fst s)", auto dest: 1 2)

lemma appLitPush[simp]:
"(app (LitPush v) G maxs rT pc et (Some s)) = (∃ST LT. s = (ST,LT) ∧ length ST < maxs ∧ typeof (λv. None) v ≠ None)"
  by (cases s, simp)

lemma appGetField[simp]:
"(app (Getfield F C) G maxs rT pc et (Some s)) =
 (∃ oT vT ST LT. s = (oT#ST, LT) ∧ is_class G C ∧
  field (G,C) F = Some (C,vT) ∧ G ⊨ oT ⪯ (Class C) ∧ (∀x ∈ set (match G NullPointer pc et). is_class G x))"
  by (cases s, cases "2 , auto dest!: 1 2)

lemma appPutField[simp]:
"(app (Putfield F C) G maxs rT pc et (Some s)) =
 (∃ vT vT' oT ST LT. s = (vT#oT#ST, LT) ∧ is_class G C ∧
  field (G,C) F = Some (C, vT') ∧ G ⊨ oT ⪯ (Class C) ∧ G ⊨ vT ⪯ vT' ∧
  (∀x ∈ set (match G NullPointer pc et). is_class G x))"
  by (cases s, cases "2 , auto dest!: 1 2)

lemma appNew[simp]:
  "(app (New C) G maxs rT pc et (Some s)) =
  (∃ST LT. s=(ST,LT) ∧ is_class G C ∧ length ST < maxs ∧
  (∀x ∈ set (match G OutOfMemory pc et). is_class G x))"
  by (cases s, simp)

lemma appCheckcast[simp]: 
  "(app (Checkcast C) G maxs rT pc et (Some s)) =
  (∃rT ST LT. s = (RefT rT#ST,LT) ∧ is_class G C ∧
  (∀x ∈ set (match G ClassCast pc et). is_class G x))"
  by (cases s, cases "fst s", simp) (cases "hd (fst s)", auto)

lemma appPop[simp]: 
"(app Pop G maxs rT pc et (Some s)) = (∃ts ST LT. s = (ts#ST,LT))"
  by (cases s, cases "2 , auto dest: 1 2)


lemma appDup[simp]:
"(app Dup G maxs rT pc et (Some s)) = (∃ts ST LT. s = (ts#ST,LT) ∧ 1+length ST < maxs)" 
  by (cases s, cases "2 , auto dest: 1 2)


lemma appDup_x1[simp]:
"(app Dup_x1 G maxs rT pc et (Some s)) = (∃ts1 ts2 ST LT. s = (ts1#ts2#ST,LT) ∧ 2+length ST < maxs)" 
  by (cases s, cases "2 , auto dest: 1 2)


lemma appDup_x2[simp]:
"(app Dup_x2 G maxs rT pc et (Some s)) = (∃ts1 ts2 ts3 ST LT. s = (ts1#ts2#ts3#ST,LT) ∧ 3+length ST < maxs)"
  by (cases s, cases "2 , auto dest: 1 2)


lemma appSwap[simp]:
"app Swap G maxs rT pc et (Some s) = (∃ts1 ts2 ST LT. s = (ts1#ts2#ST,LT))" 
  by (cases s, cases "2 ) (auto dest: 1 2)


lemma appIAdd[simp]:
"app IAdd G maxs rT pc et (Some s) = (∃ ST LT. s = (PrimT Integer#PrimT Integer#ST,LT))"
  (is "?app s = ?P s")
proof (cases s)
  case (Pair a b)
  have "?app (a,b) = ?P (a,b)"
  proof (cases a)
    fix t ts assume a: "a = t#ts"
    show ?thesis
    proof (cases t)
      fix p assume p: "t = PrimT p"
      show ?thesis
      proof (cases p)
        assume ip: "p = Integer"
        show ?thesis
        proof (cases ts)
          fix t' ts' assume t': "ts = t' # ts'"
          show ?thesis
          proof (cases t')
            fix p' assume "t' = PrimT p'"
            with t' ip p a
            show ?thesis by (cases p') auto
          qed (auto simp add: a p ip t')
        qed (auto simp add: a p ip)
      qed (auto simp add: a p)
    qed (auto simp add: a)
  qed auto
  with Pair show ?thesis by simp
qed


lemma appIfcmpeq[simp]:
"app (Ifcmpeq b) G maxs rT pc et (Some s) =
  (∃ts1 ts2 ST LT. s = (ts1#ts2#ST,LT) ∧ 0 ≤ int pc + b ∧
  ((∃ p. ts1 = PrimT p ∧ ts2 = PrimT p) ∨ (∃r r'. ts1 = RefT r ∧ ts2 = RefT r')))" 
  by (cases s, cases "2 , auto dest!: 1 2)

lemma appReturn[simp]:
"app Return G maxs rT pc et (Some s) = (∃T ST LT. s = (T#ST,LT) ∧ (G ⊨ T ⪯ rT))" 
  by (cases s, cases "2 , auto dest: 1 2)

lemma appGoto[simp]:
"app (Goto b) G maxs rT pc et (Some s) = (0 ≤ int pc + b)"
  by simp

lemma appThrow[simp]:
  "app Throw G maxs rT pc et (Some s) =
  (∃T ST LT r. s=(T#ST,LT) ∧ T = RefT r ∧ (∀C ∈ set (match_any G pc et). is_class G C))"
  by (cases s, cases "2 < length (fst s)", auto dest: 1 2)

lemma appInvoke[simp]:
"app (Invoke C mn fpTs) G maxs rT pc et (Some s) = (∃apTs X ST LT mD' rT' b'.
  s = ((rev apTs) @ (X # ST), LT) ∧ length apTs = length fpTs ∧ is_class G C ∧
  G ⊨ X ⪯ Class C ∧ (∀(aT,fT)∈set(zip apTs fpTs). G ⊨ aT ⪯ fT) ∧
  method (G,C) (mn,fpTs) = Some (mD', rT', b') ∧
  (∀C ∈ set (match_any G pc et). is_class G C))" (is "?app s = ?P s")
proof (cases s)
  note list_all2_iff [simp]
  case (Pair a b)
  have "?app (a,b) ==> ?P (a,b)"
  proof -
    assume app: "?app (a,b)"
    hence "a = (rev (rev (take (length fpTs) a))) @ (drop (length fpTs) a) ∧
           length fpTs < length a" (is "?a ∧ ?l") 
      by auto
    hence "?a ∧ 0 < length (drop (length fpTs) a)" (is "?a ∧ ?l") 
      by auto
    hence "?a ∧ ?l ∧ length (rev (take (length fpTs) a)) = length fpTs" 
      by (auto)
    hence "∃apTs ST. a = rev apTs @ ST ∧ length apTs = length fpTs ∧ 0 < length ST" 
      by blast
    hence "∃apTs ST. a = rev apTs @ ST ∧ length apTs = length fpTs ∧ ST ≠ []" 
      by blast
    hence "∃apTs ST. a = rev apTs @ ST ∧ length apTs = length fpTs ∧
           (∃X ST'. ST = X#ST')" 
      by (simp add: neq_Nil_conv)
    hence "∃apTs X ST. a = rev apTs @ X # ST ∧ length apTs = length fpTs" 
      by blast
    with app
    show ?thesis by clarsimp blast
  qed
  with Pair 
  have "?app s ==> ?P s" by (simp only:)
  moreover
  have "?P s ==> ?app s" by (clarsimp simp add: min.absorb2)
  ultimately
  show ?thesis by (rule iffI) 
qed 

lemma effNone: 
  "(pc', s') ∈ set (eff i G pc et None) ==> s' = None"
  by (auto simp add: eff_def xcpt_eff_def norm_eff_def)


lemma xcpt_app_lemma [code]:
  "xcpt_app i G pc et = list_all (is_class G) (xcpt_names (i, G, pc, et))"
  by (simp add: list_all_iff)

lemmas [simp del] = app_def xcpt_app_def

end