# Theory Effect

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theory Effect
imports JVMType JVMExceptions
begin

    (*  Title:      HOL/MicroJava/BV/Effect.thy
        ID:         $Id: Effect.thy,v 1.11 2005/08/17 09:44:10 nipkow Exp $
        Author:     Gerwin Klein
        Copyright   2000 Technische Universitaet Muenchen
    *)

    header {* \isaheader{Effect of Instructions on the State Type} *}

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

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

    text {* Program counter of successor instructions: *}
    consts
      succs :: "instr => p_count => p_count list"
    primrec
      "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:"
    consts
    eff' :: "instr × jvm_prog × state_type => state_type"

    recdef eff' "{}"
    "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))"

    consts
      match_any :: "jvm_prog => p_count => exception_table => cname list"
    primrec
      "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')"

    consts
      match :: "jvm_prog => xcpt => p_count => exception_table => cname list"
    primrec
      "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

    consts
      xcpt_names :: "instr × jvm_prog × p_count × exception_table => cname list"
    recdef xcpt_names "{}"
      "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)            = []"

    constdefs
      xcpt_eff :: "instr => jvm_prog => p_count => state_type option => exception_table => succ_type"
      "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))"

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

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

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

      isRefT :: "ty => bool"
      "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_def)

    text "Conditions under which eff is applicable:"
    consts
    app' :: "instr × jvm_prog × p_count × nat × ty × state_type => bool"

    recdef app' "{}"
    "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 \<turnstile> oT \<preceq> (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 \<turnstile> oT \<preceq> (Class C) ∧ G \<turnstile> vT \<preceq> (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 \<turnstile> T \<preceq> 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 \<turnstile> X \<preceq> Class C ∧ is_class G C ∧ method (G,C) (mn,fpTs) ≠ None ∧
           list_all2 (λx y. G \<turnstile> x \<preceq> y) apTs fpTs))"

    "app' (i,G, pc,maxs,rT,s) = False"

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

      app :: "instr => jvm_prog => nat => ty => nat => exception_table => state_type option => bool"
      "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: split_if_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: split_if_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 @{term 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 \<turnstile> oT \<preceq> (Class C) ∧ (∀x ∈ set (match G NullPointer pc et). is_class G x))"
      by (cases s, cases "2 <length (fst s)", 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 \<turnstile> oT \<preceq> (Class C) ∧ G \<turnstile> vT \<preceq> vT' ∧
      (∀x ∈ set (match G NullPointer pc et). is_class G x))"
      by (cases s, cases "2 <length (fst s)", 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 add: app_def) (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 <length (fst s)", 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 <length (fst s)", 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 <length (fst s)", 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 <length (fst s)", 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 <length (fst s)", 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 (open) s)
      case Pair
      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 <length (fst s)", auto dest!: 1 2)

    lemma appReturn[simp]:
    "app Return G maxs rT pc et (Some s) = (∃T ST LT. s = (T#ST,LT) ∧ (G \<turnstile> T \<preceq> rT))"
      by (cases s, cases "2 <length (fst s)", 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 \<turnstile> X \<preceq> Class C ∧ (∀(aT,fT)∈set(zip apTs fpTs). G \<turnstile> aT \<preceq> 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 (open) s)
      note list_all2_def [simp]
      case Pair
      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 simp add: app_def)
        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 simp add: min_def)
        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 (unfold app_def, clarsimp) blast
      qed
      with Pair
      have "?app s ==> ?P s" by (simp only:)
      moreover
      have "?P s ==> ?app s" by (unfold app_def) (clarsimp simp add: min_def)
      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

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 [])

lemma isPrimT:

      isPrimT T = (∃T'. T = PrimT T')

lemma isRefT:

      isRefT T = (∃T'. T = RefT T')

lemma

      list_all2 P a b ==> ∀(x, y)∈set (zip a b). P x y

lemma match_any_match_table:

      C ∈ set (match_any G pc et) ==> match_exception_table G C pc et ≠ None

lemma match_X_match_table:

      C ∈ set (match G X pc et) ==> match_exception_table G C pc et ≠ None

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))

lemma 1:

      2 < length a ==> ∃l l' l'' ls. a = l # l' # l'' # ls

lemma 2:

      ¬ 2 < length a ==> a = [] ∨ (∃l. a = [l]) ∨ (∃l l'. a = [l, l'])

lemmas

      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

      xcpt_app i G pc et == ∀C∈set (xcpt_names (i, G, pc, et)). is_class G C

lemmas

      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

      xcpt_app i G pc et == ∀C∈set (xcpt_names (i, G, pc, et)). is_class G C

lemma appNone:

      app i G maxs rT pc et None = True

lemma appLoad:

      app (Load idx) G maxs rT pc et (Some s) =
      (∃ST LT. s = (ST, LT) ∧ idx < length LT ∧ LT ! idx ≠ Err ∧ length ST < maxs)

lemma appStore:

      app (Store idx) G maxs rT pc et (Some s) =
      (∃ts ST LT. s = (ts # ST, LT) ∧ idx < length LT)

lemma appLitPush:

      app (LitPush v) G maxs rT pc et (Some s) =
      (∃ST LT. s = (ST, LT) ∧ length ST < maxs ∧ typeof empty v ≠ None)

lemma appGetField:

      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))

lemma appPutField:

      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))

lemma appNew:

      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))

lemma appCheckcast:

      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))

lemma appPop:

      app Pop G maxs rT pc et (Some s) = (∃ts ST LT. s = (ts # ST, LT))

lemma appDup:

      app Dup G maxs rT pc et (Some s) =
      (∃ts ST LT. s = (ts # ST, LT) ∧ 1 + length ST < maxs)

lemma appDup_x1:

      app Dup_x1 G maxs rT pc et (Some s) =
      (∃ts1 ts2 ST LT. s = (ts1 # ts2 # ST, LT) ∧ 2 + length ST < maxs)

lemma appDup_x2:

      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)

lemma appSwap:

      app Swap G maxs rT pc et (Some s) = (∃ts1 ts2 ST LT. s = (ts1 # ts2 # ST, LT))

lemma appIAdd:

      app IAdd G maxs rT pc et (Some s) =
      (∃ST LT. s = (PrimT Integer # PrimT Integer # ST, LT))

lemma appIfcmpeq:

      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')))

lemma appReturn:

      app Return G maxs rT pc et (Some s) =
      (∃T ST LT. s = (T # ST, LT) ∧ G |- T <= rT)

lemma appGoto:

      app (Goto b) G maxs rT pc et (Some s) = (0 ≤ int pc + b)

lemma appThrow:

      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))

lemma appInvoke:

      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))

lemma effNone:

      (pc', s') ∈ set (eff i G pc et None) ==> s' = None

lemma xcpt_app_lemma:

      xcpt_app i G pc et = list_all (is_class G) (xcpt_names (i, G, pc, et))

lemmas

      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

      xcpt_app i G pc et == ∀C∈set (xcpt_names (i, G, pc, et)). is_class G C

lemmas

      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

      xcpt_app i G pc et == ∀C∈set (xcpt_names (i, G, pc, et)). is_class G C