Theory EffectMono (Isabelle2005: October 2005)

# Theory EffectMono

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theory EffectMono
imports Effect
begin

    (*  Title:      HOL/MicroJava/BV/EffMono.thy
        ID:         $Id: EffectMono.thy,v 1.9 2005/09/22 21:56:40 nipkow Exp $
        Author:     Gerwin Klein
        Copyright   2000 Technische Universitaet Muenchen
    *)

    header {* \isaheader{Monotonicity of eff and app} *}

    theory EffectMono imports Effect begin

    lemma PrimT_PrimT: "(G \<turnstile> xb \<preceq> PrimT p) = (xb = PrimT p)"
      by (auto elim: widen.elims)

    lemma sup_loc_some [rule_format]:
    "∀y n. (G \<turnstile> b <=l y) --> n < length y --> y!n = OK t -->
      (∃t. b!n = OK t ∧ (G \<turnstile> (b!n) <=o (y!n)))" (is "?P b")
    proof (induct (open) ?P b)
      show "?P []" by simp

      case Cons
      show "?P (a#list)"
      proof (clarsimp simp add: list_all2_Cons1 sup_loc_def Listn.le_def lesub_def)
        fix z zs n
        assume * :
          "G \<turnstile> a <=o z" "list_all2 (sup_ty_opt G) list zs"
          "n < Suc (length list)" "(z # zs) ! n = OK t"

        show "(∃t. (a # list) ! n = OK t) ∧ G \<turnstile>(a # list) ! n <=o OK t"
        proof (cases n)
          case 0
          with * show ?thesis by (simp add: sup_ty_opt_OK)
        next
          case Suc
          with Cons *
          show ?thesis by (simp add: sup_loc_def Listn.le_def lesub_def)
        qed
      qed
    qed

    lemma all_widen_is_sup_loc:
    "∀b. length a = length b -->
         (∀x∈set (zip a b). x ∈ widen G) = (G \<turnstile> (map OK a) <=l (map OK b))"
     (is "∀b. length a = length b --> ?Q a b" is "?P a")
    proof (induct "a")
      show "?P []" by simp

      fix l ls assume Cons: "?P ls"

      show "?P (l#ls)"
      proof (intro allI impI)
        fix b
        assume "length (l # ls) = length (b::ty list)"
        with Cons
        show "?Q (l # ls) b" by - (cases b, auto)
      qed
    qed

    lemma append_length_n [rule_format]:
    "∀n. n ≤ length x --> (∃a b. x = a@b ∧ length a = n)" (is "?P x")
    proof (induct (open) ?P x)
      show "?P []" by simp

      fix l ls assume Cons: "?P ls"

      show "?P (l#ls)"
      proof (intro allI impI)
        fix n
        assume l: "n ≤ length (l # ls)"

        show "∃a b. l # ls = a @ b ∧ length a = n"
        proof (cases n)
          assume "n=0" thus ?thesis by simp
        next
          fix n' assume s: "n = Suc n'"
          with l have  "n' ≤ length ls" by simp
          hence "∃a b. ls = a @ b ∧ length a = n'" by (rule Cons [rule_format])
          then obtain a b where "ls = a @ b" "length a = n'" by iprover
          with s have "l # ls = (l#a) @ b ∧ length (l#a) = n" by simp
          thus ?thesis by blast
        qed
      qed
    qed

    lemma rev_append_cons:
    "n < length x ==> ∃a b c. x = (rev a) @ b # c ∧ length a = n"
    proof -
      assume n: "n < length x"
      hence "n ≤ length x" by simp
      hence "∃a b. x = a @ b ∧ length a = n" by (rule append_length_n)
      then obtain r d where x: "x = r@d" "length r = n" by iprover
      with n have "∃b c. d = b#c" by (simp add: neq_Nil_conv)
      then obtain b c where "d = b#c" by iprover
      with x have "x = (rev (rev r)) @ b # c ∧ length (rev r) = n" by simp
      thus ?thesis by blast
    qed

    lemma sup_loc_length_map:
      "G \<turnstile> map f a <=l map g b ==> length a = length b"
    proof -
      assume "G \<turnstile> map f a <=l map g b"
      hence "length (map f a) = length (map g b)" by (rule sup_loc_length)
      thus ?thesis by simp
    qed

    lemmas [iff] = not_Err_eq

    lemma app_mono:
    "[|G \<turnstile> s <=' s'; app i G m rT pc et s'|] ==> app i G m rT pc et s"
    proof -

      { fix s1 s2
        assume G:   "G \<turnstile> s2 <=s s1"
        assume app: "app i G m rT pc et (Some s1)"

        note [simp] = sup_loc_length sup_loc_length_map

        have "app i G m rT pc et (Some s2)"
        proof (cases (open) i)
          case Load

          from G Load app
          have "G \<turnstile> snd s2 <=l snd s1" by (auto simp add: sup_state_conv)

          with G Load app show ?thesis
            by (cases s2) (auto simp add: sup_state_conv dest: sup_loc_some)
        next
          case Store
          with G app show ?thesis
            by (cases s2, auto simp add: sup_loc_Cons2 sup_state_conv)
        next
          case LitPush
          with G app show ?thesis by (cases s2, auto simp add: sup_state_conv)
        next
          case New
          with G app show ?thesis by (cases s2, auto simp add: sup_state_conv)
        next
          case Getfield
          with app G show ?thesis
            by (cases s2) (clarsimp simp add: sup_state_Cons2, rule widen_trans)
        next
          case Putfield

          with app
          obtain vT oT ST LT b
            where s1: "s1 = (vT # oT # ST, LT)" and
                      "field (G, cname) vname = Some (cname, b)"
                      "is_class G cname" and
                  oT: "G\<turnstile> oT\<preceq> (Class cname)" and
                  vT: "G\<turnstile> vT\<preceq> b" and
                  xc: "Ball (set (match G NullPointer pc et)) (is_class G)"
            by force
          moreover
          from s1 G
          obtain vT' oT' ST' LT'
            where s2:  "s2 = (vT' # oT' # ST', LT')" and
                  oT': "G\<turnstile> oT' \<preceq> oT" and
                  vT': "G\<turnstile> vT' \<preceq> vT"
            by - (cases s2, simp add: sup_state_Cons2, elim exE conjE, simp, rule that)
          moreover
          from vT' vT
          have "G \<turnstile> vT' \<preceq> b" by (rule widen_trans)
          moreover
          from oT' oT
          have "G\<turnstile> oT' \<preceq> (Class cname)" by (rule widen_trans)
          ultimately
          show ?thesis by (auto simp add: Putfield xc)
        next
          case Checkcast
          with app G show ?thesis
            by (cases s2, auto intro!: widen_RefT2 simp add: sup_state_Cons2)
        next
          case Return
          with app G show ?thesis
            by (cases s2) (auto simp add: sup_state_Cons2, rule widen_trans)
        next
          case Pop
          with app G show ?thesis
            by (cases s2, clarsimp simp add: sup_state_Cons2)
        next
          case Dup
          with app G show ?thesis
            by (cases s2, clarsimp simp add: sup_state_Cons2,
                auto dest: sup_state_length)
        next
          case Dup_x1
          with app G show ?thesis
            by (cases s2, clarsimp simp add: sup_state_Cons2,
                auto dest: sup_state_length)
        next
          case Dup_x2
          with app G show ?thesis
            by (cases s2, clarsimp simp add: sup_state_Cons2,
                auto dest: sup_state_length)
        next
          case Swap
          with app G show ?thesis
            by (cases s2, clarsimp simp add: sup_state_Cons2)
        next
          case IAdd
          with app G show ?thesis
            by (cases s2, auto simp add: sup_state_Cons2 PrimT_PrimT)
        next
          case Goto
          with app show ?thesis by simp
        next
          case Ifcmpeq
          with app G show ?thesis
            by (cases s2, auto simp add: sup_state_Cons2 PrimT_PrimT widen_RefT2)
        next
          case Invoke

          with app
          obtain apTs X ST LT mD' rT' b' where
            s1: "s1 = (rev apTs @ X # ST, LT)" and
            l:  "length apTs = length list" and
            c:  "is_class G cname" and
            C:  "G \<turnstile> X \<preceq> Class cname" and
            w:  "∀x ∈ set (zip apTs list). x ∈ widen G" and
            m:  "method (G, cname) (mname, list) = Some (mD', rT', b')" and
            x:  "∀C ∈ set (match_any G pc et). is_class G C"
            by (simp del: not_None_eq, elim exE conjE) (rule that)

          obtain apTs' X' ST' LT' where
            s2: "s2 = (rev apTs' @ X' # ST', LT')" and
            l': "length apTs' = length list"
          proof -
            from l s1 G
            have "length list < length (fst s2)"
              by simp
            hence "∃a b c. (fst s2) = rev a @ b # c ∧ length a = length list"
              by (rule rev_append_cons [rule_format])
            thus ?thesis
              by -  (cases s2, elim exE conjE, simp, rule that)
          qed

          from l l'
          have "length (rev apTs') = length (rev apTs)" by simp

          from this s1 s2 G
          obtain
            G': "G \<turnstile> (apTs',LT') <=s (apTs,LT)" and
            X : "G \<turnstile>  X' \<preceq> X" and "G \<turnstile> (ST',LT') <=s (ST,LT)"
            by (simp add: sup_state_rev_fst sup_state_append_fst sup_state_Cons1)

          with C
          have C': "G \<turnstile> X' \<preceq> Class cname"
            by - (rule widen_trans, auto)

          from G'
          have "G \<turnstile> map OK apTs' <=l map OK apTs"
            by (simp add: sup_state_conv)
          also
          from l w
          have "G \<turnstile> map OK apTs <=l map OK list"
            by (simp add: all_widen_is_sup_loc)
          finally
          have "G \<turnstile> map OK apTs' <=l map OK list" .

          with l'
          have w': "∀x ∈ set (zip apTs' list). x ∈ widen G"
            by (simp add: all_widen_is_sup_loc)

          from Invoke s2 l' w' C' m c x
          show ?thesis
            by (simp del: split_paired_Ex) blast
        next
          case Throw
          with app G show ?thesis
            by (cases s2, clarsimp simp add: sup_state_Cons2 widen_RefT2)
        qed
      } note this [simp]

      assume "G \<turnstile> s <=' s'" "app i G m rT pc et s'"
      thus ?thesis by (cases s, cases s', auto)
    qed

    lemmas [simp del] = split_paired_Ex

    lemma eff'_mono:
    "[| app i G m rT pc et (Some s2); G \<turnstile> s1 <=s s2 |] ==>
      G \<turnstile> eff' (i,G,s1) <=s eff' (i,G,s2)"
    proof (cases s1, cases s2)
      fix a1 b1 a2 b2
      assume s: "s1 = (a1,b1)" "s2 = (a2,b2)"
      assume app2: "app i G m rT pc et (Some s2)"
      assume G: "G \<turnstile> s1 <=s s2"

      note [simp] = eff_def

      hence "G \<turnstile> (Some s1) <=' (Some s2)" by simp
      from this app2
      have app1: "app i G m rT pc et (Some s1)" by (rule app_mono)

      show ?thesis
      proof (cases (open) i)
        case Load

        with s app1
        obtain y where
           y:  "nat < length b1" "b1 ! nat = OK y" by clarsimp

        from Load s app2
        obtain y' where
           y': "nat < length b2" "b2 ! nat = OK y'" by clarsimp

        from G s
        have "G \<turnstile> b1 <=l b2" by (simp add: sup_state_conv)

        with y y'
        have "G \<turnstile> y \<preceq> y'"
          by - (drule sup_loc_some, simp+)

        with Load G y y' s app1 app2
        show ?thesis by (clarsimp simp add: sup_state_conv)
      next
        case Store
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_conv sup_loc_update)
      next
        case LitPush
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_Cons1)
      next
        case New
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_Cons1)
      next
        case Getfield
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_Cons1)
      next
        case Putfield
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_Cons1)
      next
        case Checkcast
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_Cons1)
      next
        case Invoke

        with s app1
        obtain a X ST where
          s1: "s1 = (a @ X # ST, b1)" and
          l:  "length a = length list"
          by (simp, elim exE conjE, simp (no_asm_simp))

        from Invoke s app2
        obtain a' X' ST' where
          s2: "s2 = (a' @ X' # ST', b2)" and
          l': "length a' = length list"
          by (simp, elim exE conjE, simp (no_asm_simp))

        from l l'
        have lr: "length a = length a'" by simp

        from lr G s1 s2
        have "G \<turnstile> (ST, b1) <=s (ST', b2)"
          by (simp add: sup_state_append_fst sup_state_Cons1)

        moreover

        obtain b1' b2' where eff':
          "b1' = snd (eff' (i,G,s1))"
          "b2' = snd (eff' (i,G,s2))" by simp

        from Invoke G s eff' app1 app2
        obtain "b1 = b1'" "b2 = b2'" by simp

        ultimately

        have "G \<turnstile> (ST, b1') <=s (ST', b2')" by simp

        with Invoke G s app1 app2 eff' s1 s2 l l'
        show ?thesis
          by (clarsimp simp add: sup_state_conv)
      next
        case Return
        with G
        show ?thesis
          by simp
      next
        case Pop
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_Cons1)
      next
        case Dup
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_Cons1)
      next
        case Dup_x1
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_Cons1)
      next
        case Dup_x2
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_Cons1)
      next
        case Swap
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_Cons1)
      next
        case IAdd
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_Cons1)
      next
        case Goto
        with G s app1 app2
        show ?thesis by simp
      next
        case Ifcmpeq
        with G s app1 app2
        show ?thesis
          by (clarsimp simp add: sup_state_Cons1)
      next
        case Throw
        with G
        show ?thesis
          by simp
      qed
    qed

    lemmas [iff del] = not_Err_eq

    end

lemma PrimT_PrimT:

      G |- xb <= PrimT p = (xb = PrimT p)

lemma sup_loc_some:

      [| G |- b <=l y; n < length y; y ! n = OK t |]
      ==> ∃t. b ! n = OK t ∧ G |- b ! n <=o y ! n

lemma all_widen_is_sup_loc:

      ∀b. length a = length b -->
          (∀x∈set (zip a b). x ∈ widen G) = G |- map OK a <=l map OK b

lemma append_length_n:

      n ≤ length x ==> ∃a b. x = a @ b ∧ length a = n

lemma rev_append_cons:

      n < length x ==> ∃a b c. x = rev a @ b # c ∧ length a = n

lemma sup_loc_length_map:

      G |- map f a <=l map g b ==> length a = length b

lemmas

      (x ≠ Err) = (∃a. x = OK a)

lemmas

      (x ≠ Err) = (∃a. x = OK a)

lemma app_mono:

      [| G |- s <=' s'; app i G m rT pc et s' |] ==> app i G m rT pc et s

lemmas

      (∃x. P x) = (∃a b. P (a, b))

lemmas

      (∃x. P x) = (∃a b. P (a, b))

lemma eff'_mono:

      [| app i G m rT pc et (Some s2.0); G |- s1.0 <=s s2.0 |]
      ==> G |- eff' (i, G, s1.0) <=s eff' (i, G, s2.0)

lemmas

      (x ≠ Err) = (∃a. x = OK a)

lemmas

      (x ≠ Err) = (∃a. x = OK a)