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products/Sources/formale Sprachen/PVS/trig/ (Beweissystem der NASA Version 6.0.9^©) Datei vom 28.9.2014 mit Größe 681 B

Quelle AxSound.thy Sprache: Isabelle

(*  Title:      HOL/Bali/AxSound.thy
    Author:     David von Oheimb and Norbert Schirmer
*)
subsection \<open>Soundness proof for Axiomatic semantics of Java expressions and
          statements
\<close>

theory AxSound imports AxSem begin

subsubsection "validity"

definition
  triple_valid2 :: "prog \ nat \ 'a triple \ bool" (\_\_\_\[61,0, 58] 57)
  where
    "G\n\t =
      (case t of {P} t\<succ> {Q} \<Rightarrow>
        \<forall>Y s Z. P Y s Z \<longrightarrow> (\<forall>L. s\<Colon>\<preceq>(G,L)
          \<longrightarrow> (\<forall>T C A. (normal s \<longrightarrow> (\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T \<and>
            \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>dom (locals (store s))\<guillemotright>t\<guillemotright>A)) \<longrightarrow>
             (\<forall>Y' s'. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y',s') \<longrightarrow> Q Y' s' Z \<and> s'\<Colon>\<preceq>(G,L)))))"

text \<open>This definition differs from the ordinary  \<open>triple_valid_def\<close>
manly in the conclusion: We also ensures conformance of the result state. So
we don't have to apply the type soundness lemma all the time during
induction. This definition is only introduced for the soundness
proof of the axiomatic semantics, in the end we will conclude to
the ordinary definition.
\<close>

definition
  ax_valids2 :: "prog \ 'a triples \ 'a triples \ bool" (\_,_|\\_\ [61,58,58] 57)
  where "G,A|\\ts = (\n. (\t\A. G\n\t) \ (\t\ts. G\n\t))"

lemma triple_valid2_def2: "G\n\{P} t\ {Q} =
(\<forall>Y s Z. P Y s Z \<longrightarrow> (\<forall>Y' s'. G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y',s')\<longrightarrow>
  (\<forall>L. s\<Colon>\<preceq>(G,L) \<longrightarrow> (\<forall>T C A. (normal s \<longrightarrow> (\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T \<and>
                            \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>dom (locals (store s))\<guillemotright>t\<guillemotright>A)) \<longrightarrow>
  Q Y' s' Z \<and> s'\<Colon>\<preceq>(G,L)))))"
apply (unfold triple_valid2_def)
apply (simp (no_asm) add: split_paired_All)
apply blast
done

lemma triple_valid2_eq [rule_format (no_asm)]:
  "wf_prog G ==> triple_valid2 G = triple_valid G"
apply (rule ext)
apply (rule ext)
apply (rule triple.induct)
apply (simp (no_asm) add: triple_valid_def2 triple_valid2_def2)
apply (rule iffI)
apply  fast
apply clarify
apply (tactic "smp_tac \<^context> 3 1")
apply (case_tac "normal s")
apply  clarsimp
apply  (elim conjE impE)
apply    blast

apply    (tactic "smp_tac \<^context> 2 1")
apply    (drule evaln_eval)
apply    (drule (1) eval_type_sound [THEN conjunct1],simp, assumption+)
apply    simp

apply    clarsimp
done

lemma ax_valids2_eq: "wf_prog G \ G,A|\\ts = G,A|\ts"
apply (unfold ax_valids_def ax_valids2_def)
apply (force simp add: triple_valid2_eq)
done

lemma triple_valid2_Suc [rule_format (no_asm)]: "G\Suc n\t \ G\n\t"
apply (induct_tac "t")
apply (subst triple_valid2_def2)
apply (subst triple_valid2_def2)
apply (fast intro: evaln_nonstrict_Suc)
done

lemma Methd_triple_valid2_0: "G\0\{Normal P} Methd C sig-\ {Q}"
by (auto elim!: evaln_elim_cases simp add: triple_valid2_def2)

lemma Methd_triple_valid2_SucI:
"\G\n\{Normal P} body G C sig-\{Q}\
  \<Longrightarrow> G\<Turnstile>Suc n\<Colon>{Normal P} Methd C sig-\<succ> {Q}"
apply (simp (no_asm_use) add: triple_valid2_def2)
apply (intro strip, tactic "smp_tac \<^context> 3 1", clarify)
apply (erule wt_elim_cases, erule da_elim_cases, erule evaln_elim_cases)
apply (unfold body_def Let_def)
apply (clarsimp simp add: inj_term_simps)
apply blast
done

lemma triples_valid2_Suc:
"Ball ts (triple_valid2 G (Suc n)) \ Ball ts (triple_valid2 G n)"
apply (fast intro: triple_valid2_Suc)
done

lemma "G|\n:insert t A = (G\n:t \ G|\n:A)"
oops

subsubsection "soundness"

lemma Methd_sound:
  assumes recursive: "G,A\ {{P} Methd-\ {Q} | ms}|\\{{P} body G-\ {Q} | ms}"
  shows "G,A|\\{{P} Methd-\ {Q} | ms}"
proof -
  have "\t\A. G\n\t \ \t\{{P} Methd-\ {Q} | ms}. G\n\t"
    if rec: "\n. \t\(A \ {{P} Methd-\ {Q} | ms}). G\n\t
                  \<Longrightarrow>  \<forall>t\<in>{{P} body G-\<succ> {Q} | ms}. G\<Turnstile>n\<Colon>t"
    for n
  proof (induct n)
    case 0
    show "\t\{{P} Methd-\ {Q} | ms}. G\0\t"
    proof -
      have "G\0\{Normal (P C sig)} Methd C sig-\ {Q C sig}"
        if "(C,sig) \ ms"
        for C sig
        by (rule Methd_triple_valid2_0)
      thus ?thesis
        by (simp add: mtriples_def split_def)
    qed
  next
    case (Suc m)
    note hyp = \<open>\<forall>t\<in>A. G\<Turnstile>m\<Colon>t \<Longrightarrow> \<forall>t\<in>{{P} Methd-\<succ> {Q} | ms}.  G\<Turnstile>m\<Colon>t\<close>
    note prem = \<open>\<forall>t\<in>A. G\<Turnstile>Suc m\<Colon>t\<close>
    show "\t\{{P} Methd-\ {Q} | ms}. G\Suc m\t"
    proof -
      have "G\Suc m\{Normal (P C sig)} Methd C sig-\ {Q C sig}"
        if m: "(C,sig) \ ms"
        for C sig
      proof -
        from prem have prem_m: "\t\A. G\m\t"
          by (rule triples_valid2_Suc)
        hence "\t\{{P} Methd-\ {Q} | ms}. G\m\t"
          by (rule hyp)
        with prem_m
        have "\t\(A \ {{P} Methd-\ {Q} | ms}). G\m\t"
          by (simp add: ball_Un)
        hence "\t\{{P} body G-\ {Q} | ms}. G\m\t"
          by (rule rec)
        with m have "G\m\{Normal (P C sig)} body G C sig-\ {Q C sig}"
          by (auto simp add: mtriples_def split_def)
        thus ?thesis
          by (rule Methd_triple_valid2_SucI)
      qed
      thus ?thesis
        by (simp add: mtriples_def split_def)
    qed
  qed
  with recursive show ?thesis
    by (unfold ax_valids2_def) blast
qed

lemma valids2_inductI: "\s t n Y' s'. G\s\t\\n\ (Y',s') \ t = c \
  Ball A (triple_valid2 G n) \<longrightarrow> (\<forall>Y Z. P Y s Z \<longrightarrow>
  (\<forall>L. s\<Colon>\<preceq>(G,L) \<longrightarrow>
    (\<forall>T C A. (normal s \<longrightarrow> (\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T) \<and>
                            \<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>dom (locals (store s))\<guillemotright>t\<guillemotright>A) \<longrightarrow>
    Q Y' s' Z \<and> s'\<Colon>\<preceq>(G, L)))) \<Longrightarrow>
  G,A|\<Turnstile>\<Colon>{ {P} c\<succ> {Q}}"
apply (simp (no_asm) add: ax_valids2_def triple_valid2_def2)
apply clarsimp
done

lemma da_good_approx_evalnE [consumes 4]:
  assumes evaln: "G\s0 \t\\n\ (v, s1)"
     and     wt: "\prg=G,cls=C,lcl=L\\t\T"
     and     da: "\prg=G,cls=C,lcl=L\\ dom (locals (store s0)) \t\ A"
     and     wf: "wf_prog G"
     and   elim: "\normal s1 \ nrm A \ dom (locals (store s1));
                  \<And> l. \<lbrakk>abrupt s1 = Some (Jump (Break l)); normal s0\<rbrakk>
                        \<Longrightarrow> brk A l \<subseteq> dom (locals (store s1));
                   \<lbrakk>abrupt s1 = Some (Jump Ret);normal s0\<rbrakk>
                   \<Longrightarrow>Result \<in> dom (locals (store s1))
                  \<rbrakk> \<Longrightarrow> P"
  shows "P"
proof -
  from evaln have "G\s0 \t\\ (v, s1)"
    by (rule evaln_eval)
  from this wt da wf elim show P
    by (rule da_good_approxE') iprover+
qed

lemma validI:
   assumes I: "\ n s0 L accC T C v s1 Y Z.
               \<lbrakk>\<forall>t\<in>A. G\<Turnstile>n\<Colon>t; s0\<Colon>\<preceq>(G,L);
               normal s0 \<Longrightarrow> \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>T;
               normal s0 \<Longrightarrow> \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>t\<guillemotright>C;
               G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s1); P Y s0 Z\<rbrakk> \<Longrightarrow> Q v s1 Z \<and> s1\<Colon>\<preceq>(G,L)"
  shows "G,A|\\{ {P} t\ {Q} }"
apply (simp add: ax_valids2_def triple_valid2_def2)
apply (intro allI impI)
apply (case_tac "normal s")
apply   clarsimp
apply   (rule I,(assumption|simp)+)

apply   (rule I,auto)
done


declare [[simproc add: wt_expr wt_var wt_exprs wt_stmt]]

lemma valid_stmtI:
   assumes I: "\ n s0 L accC C s1 Y Z.
             \<lbrakk>\<forall>t\<in>A. G\<Turnstile>n\<Colon>t; s0\<Colon>\<preceq>(G,L);
              normal s0\<Longrightarrow> \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>c\<Colon>\<surd>;
              normal s0\<Longrightarrow>\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>c\<rangle>\<^sub>s\<guillemotright>C;
              G\<turnstile>s0 \<midarrow>c\<midarrow>n\<rightarrow> s1; P Y s0 Z\<rbrakk> \<Longrightarrow> Q \<diamondsuit> s1 Z \<and> s1\<Colon>\<preceq>(G,L)"
  shows "G,A|\\{ {P} \c\\<^sub>s\ {Q} }"
apply (simp add: ax_valids2_def triple_valid2_def2)
apply (intro allI impI)
apply (case_tac "normal s")
apply   clarsimp
apply   (rule I,(assumption|simp)+)

apply   (rule I,auto)
done

lemma valid_stmt_NormalI:
   assumes I: "\ n s0 L accC C s1 Y Z.
               \<lbrakk>\<forall>t\<in>A. G\<Turnstile>n\<Colon>t; s0\<Colon>\<preceq>(G,L); normal s0; \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>c\<Colon>\<surd>;
               \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>c\<rangle>\<^sub>s\<guillemotright>C;
               G\<turnstile>s0 \<midarrow>c\<midarrow>n\<rightarrow> s1; (Normal P) Y s0 Z\<rbrakk> \<Longrightarrow> Q \<diamondsuit> s1 Z \<and> s1\<Colon>\<preceq>(G,L)"
  shows "G,A|\\{ {Normal P} \c\\<^sub>s\ {Q} }"
apply (simp add: ax_valids2_def triple_valid2_def2)
apply (intro allI impI)
apply (elim exE conjE)
apply (rule I)
by auto

lemma valid_var_NormalI:
   assumes I: "\ n s0 L accC T C vf s1 Y Z.
               \<lbrakk>\<forall>t\<in>A. G\<Turnstile>n\<Colon>t; s0\<Colon>\<preceq>(G,L); normal s0;
                \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>=T;
                \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>t\<rangle>\<^sub>v\<guillemotright>C;
                G\<turnstile>s0 \<midarrow>t=\<succ>vf\<midarrow>n\<rightarrow> s1; (Normal P) Y s0 Z\<rbrakk>
               \<Longrightarrow> Q (In2 vf) s1 Z \<and> s1\<Colon>\<preceq>(G,L)"
   shows "G,A|\\{ {Normal P} \t\\<^sub>v\ {Q} }"
apply (simp add: ax_valids2_def triple_valid2_def2)
apply (intro allI impI)
apply (elim exE conjE)
apply simp
apply (rule I)
by auto

lemma valid_expr_NormalI:
   assumes I: "\ n s0 L accC T C v s1 Y Z.
               \<lbrakk>\<forall>t\<in>A. G\<Turnstile>n\<Colon>t; s0\<Colon>\<preceq>(G,L); normal s0;
                \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>-T;
                \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>t\<rangle>\<^sub>e\<guillemotright>C;
                G\<turnstile>s0 \<midarrow>t-\<succ>v\<midarrow>n\<rightarrow> s1; (Normal P) Y s0 Z\<rbrakk>
               \<Longrightarrow> Q (In1 v) s1 Z \<and> s1\<Colon>\<preceq>(G,L)"
   shows "G,A|\\{ {Normal P} \t\\<^sub>e\ {Q} }"
apply (simp add: ax_valids2_def triple_valid2_def2)
apply (intro allI impI)
apply (elim exE conjE)
apply simp
apply (rule I)
by auto

lemma valid_expr_list_NormalI:
   assumes I: "\ n s0 L accC T C vs s1 Y Z.
               \<lbrakk>\<forall>t\<in>A. G\<Turnstile>n\<Colon>t; s0\<Colon>\<preceq>(G,L); normal s0;
                \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>\<doteq>T;
                \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>t\<rangle>\<^sub>l\<guillemotright>C;
                G\<turnstile>s0 \<midarrow>t\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s1; (Normal P) Y s0 Z\<rbrakk>
                \<Longrightarrow> Q (In3 vs) s1 Z \<and> s1\<Colon>\<preceq>(G,L)"
   shows "G,A|\\{ {Normal P} \t\\<^sub>l\ {Q} }"
apply (simp add: ax_valids2_def triple_valid2_def2)
apply (intro allI impI)
apply (elim exE conjE)
apply simp
apply (rule I)
by auto

lemma validE [consumes 5]:
  assumes valid: "G,A|\\{ {P} t\ {Q} }"
   and    P: "P Y s0 Z"
   and    valid_A: "\t\A. G\n\t"
   and    conf: "s0\\(G,L)"
   and    eval: "G\s0 \t\\n\ (v,s1)"
   and    wt: "normal s0 \ \prg=G,cls=accC,lcl=L\\t\T"
   and    da: "normal s0 \ \prg=G,cls=accC,lcl=L\\dom (locals (store s0))\t\C"
   and    elim: "\Q v s1 Z; s1\\(G,L)\ \ concl"
  shows concl
using assms
by (simp add: ax_valids2_def triple_valid2_def2) fast
(* why consumes 5?. If I want to apply this lemma in a context wgere
   \<not> normal s0 holds,
   I can chain "\<not> normal s0" as fact number 6 and apply the rule with
   cases. Auto will then solve premise 6 and 7.
*)

lemma all_empty: "(\x. P) = P"
by simp

corollary evaln_type_sound:
  assumes evaln: "G\s0 \t\\n\ (v,s1)" and
             wt: "\prg=G,cls=accC,lcl=L\\t\T" and
             da: "\prg=G,cls=accC,lcl=L\\dom (locals (store s0)) \t\ A" and
        conf_s0: "s0\\(G,L)" and
             wf: "wf_prog G"
  shows "s1\\(G,L) \ (normal s1 \ G,L,store s1\t\v\\T) \
         (error_free s0 = error_free s1)"
proof -
  from evaln have "G\s0 \t\\ (v,s1)"
    by (rule evaln_eval)
  from this wt da wf conf_s0 show ?thesis
    by (rule eval_type_sound)
qed

corollary dom_locals_evaln_mono_elim [consumes 1]:
  assumes
  evaln: "G\ s0 \t\\n\ (v,s1)" and
    hyps: "\dom (locals (store s0)) \ dom (locals (store s1));
           \<And> vv s val. \<lbrakk>v=In2 vv; normal s1\<rbrakk>
                        \<Longrightarrow> dom (locals (store s))
                             \<subseteq> dom (locals (store ((snd vv) val s)))\<rbrakk> \<Longrightarrow> P"
shows "P"
proof -
  from evaln have "G\ s0 \t\\ (v,s1)" by (rule evaln_eval)
  from this hyps show ?thesis
    by (rule dom_locals_eval_mono_elim) iprover+
qed

lemma evaln_no_abrupt:
   "\s s'. \G\s \t\\n\ (w,s'); normal s'\ \ normal s"
by (erule evaln_cases,auto)

declare inj_term_simps [simp]
lemma ax_sound2:
  assumes    wf: "wf_prog G"
    and   deriv: "G,A|\ts"
  shows "G,A|\\ts"
using deriv
proof (induct)
  case (empty A)
  show ?case
    by (simp add: ax_valids2_def triple_valid2_def2)
next
  case (insert A t ts)
  note valid_t = \<open>G,A|\<Turnstile>\<Colon>{t}\<close>
  moreover
  note valid_ts = \<open>G,A|\<Turnstile>\<Colon>ts\<close>
  have "\t'\insert t ts. G\n\t'"
    if valid_A: "\t\A. G\n\t"
    for n
  proof -
    have "G\n\t"
      using valid_A valid_t by (simp add: ax_valids2_def)
    moreover
    have "\t\ts. G\n\t"
      using valid_A valid_ts by (unfold ax_valids2_def) blast
    ultimately show "\t'\insert t ts. G\n\t'"
      by simp
  qed
  thus ?case
    by (unfold ax_valids2_def) blast
next
  case (asm ts A)
  from \<open>ts \<subseteq> A\<close>
  show "G,A|\\ts"
    by (auto simp add: ax_valids2_def triple_valid2_def)
next
  case (weaken A ts' ts)
  note \<open>G,A|\<Turnstile>\<Colon>ts'\<close>
  moreover note \<open>ts \<subseteq> ts'\<close>
  ultimately show "G,A|\\ts"
    by (unfold ax_valids2_def triple_valid2_def) blast
next
  case (conseq P A t Q)
  note con = \<open>\<forall>Y s Z. P Y s Z \<longrightarrow>
              (\<exists>P' Q'.
                  (G,A\<turnstile>{P'} t\<succ> {Q'} \<and> G,A|\<Turnstile>\<Colon>{ {P'} t\<succ> {Q'} }) \<and>
                  (\<forall>Y' s'. (\<forall>Y Z'. P' Y s Z' \<longrightarrow> Q' Y' s' Z') \<longrightarrow> Q Y' s' Z))\<close>
  show "G,A|\\{ {P} t\ {Q} }"
  proof (rule validI)
    fix n s0 L accC T C v s1 Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf: "s0\\(G,L)"
    assume wt: "normal s0 \ \prg=G,cls=accC,lcl=L\\t\T"
    assume da: "normal s0
                 \<Longrightarrow> \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0)) \<guillemotright>t\<guillemotright> C"
    assume eval: "G\s0 \t\\n\ (v, s1)"
    assume P: "P Y s0 Z"
    show "Q v s1 Z \ s1\\(G, L)"
    proof -
      from valid_A conf wt da eval P con
      have "Q v s1 Z"
        apply (simp add: ax_valids2_def triple_valid2_def2)
        apply (tactic "smp_tac \<^context> 3 1")
        apply clarify
        apply (tactic "smp_tac \<^context> 1 1")
        apply (erule allE,erule allE, erule mp)
        apply (intro strip)
        apply (tactic "smp_tac \<^context> 3 1")
        apply (tactic "smp_tac \<^context> 2 1")
        apply (tactic "smp_tac \<^context> 1 1")
        by blast
      moreover have "s1\\(G, L)"
      proof (cases "normal s0")
        case True
        from eval wt [OF True] da [OF True] conf wf
        show ?thesis
          by (rule evaln_type_sound [elim_format]) simp
      next
        case False
        with eval have "s1=s0"
          by auto
        with conf show ?thesis by simp
      qed
      ultimately show ?thesis ..
    qed
  qed
next
  case (hazard A P t Q)
  show "G,A|\\{ {P \. Not \ type_ok G t} t\ {Q} }"
    by (simp add: ax_valids2_def triple_valid2_def2 type_ok_def) fast
next
  case (Abrupt A P t)
  show "G,A|\\{ {P\undefined3 t \. Not \ normal} t\ {P} }"
  proof (rule validI)
    fix n s0 L accC T C v s1 Y Z
    assume conf_s0: "s0\\(G, L)"
    assume eval: "G\s0 \t\\n\ (v, s1)"
    assume "(P\undefined3 t \. Not \ normal) Y s0 Z"
    then obtain P: "P (undefined3 t) s0 Z" and abrupt_s0: "\ normal s0"
      by simp
    from eval abrupt_s0 obtain "s1=s0" and "v=undefined3 t"
      by auto
    with P conf_s0
    show "P v s1 Z \ s1\\(G, L)"
      by simp
  qed
next
  case (LVar A P vn)
  show "G,A|\\{ {Normal (\s.. P\In2 (lvar vn s))} LVar vn=\ {P} }"
  proof (rule valid_var_NormalI)
    fix n s0 L accC T C vf s1 Y Z
    assume conf_s0: "s0\\(G, L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg = G, cls = accC, lcl = L\\LVar vn\=T"
    assume da: "\prg=G,cls=accC,lcl=L\\ dom (locals (store s0)) \\LVar vn\\<^sub>v\ C"
    assume eval: "G\s0 \LVar vn=\vf\n\ s1"
    assume P: "(Normal (\s.. P\In2 (lvar vn s))) Y s0 Z"
    show "P (In2 vf) s1 Z \ s1\\(G, L)"
    proof
      from eval normal_s0 obtain "s1=s0" "vf=lvar vn (store s0)"
        by (fastforce elim: evaln_elim_cases)
      with P show "P (In2 vf) s1 Z"
        by simp
    next
      from eval wt da conf_s0 wf
      show "s1\\(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
    qed
  qed
next
  case (FVar A P statDeclC Q e stat fn R accC)
  note valid_init = \<open>G,A|\<Turnstile>\<Colon>{ {Normal P} .Init statDeclC. {Q} }\<close>
  note valid_e = \<open>G,A|\<Turnstile>\<Colon>{ {Q} e-\<succ> {\<lambda>Val:a:. fvar statDeclC stat fn a ..; R} }\<close>
  show "G,A|\\{ {Normal P} {accC,statDeclC,stat}e..fn=\ {R} }"
  proof (rule valid_var_NormalI)
    fix n s0 L accC' T V vf s3 Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf_s0:  "s0\\(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg=G,cls=accC',lcl=L\\{accC,statDeclC,stat}e..fn\=T"
    assume da: "\prg=G,cls=accC',lcl=L\
                  \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>{accC,statDeclC,stat}e..fn\<rangle>\<^sub>v\<guillemotright> V"
    assume eval: "G\s0 \{accC,statDeclC,stat}e..fn=\vf\n\ s3"
    assume P: "(Normal P) Y s0 Z"
    show "R \vf\\<^sub>v s3 Z \ s3\\(G, L)"
    proof -
      from wt obtain statC f where
        wt_e: "\prg=G, cls=accC, lcl=L\\e\-Class statC" and
        accfield: "accfield G accC statC fn = Some (statDeclC,f)" and
        eq_accC: "accC=accC'" and
        stat: "stat=is_static f" and
        T: "T=(type f)"
        by (cases) (auto simp add: member_is_static_simp)
      from da eq_accC
      have da_e: "\prg=G, cls=accC, lcl=L\\dom (locals (store s0))\\e\\<^sub>e\ V"
        by cases simp
      from eval obtain a s1 s2 s2' where
        eval_init: "G\s0 \Init statDeclC\n\ s1" and
        eval_e: "G\s1 \e-\a\n\ s2" and
        fvar: "(vf,s2')=fvar statDeclC stat fn a s2" and
        s3: "s3 = check_field_access G accC statDeclC fn stat a s2'"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      have wt_init: "\prg=G, cls=accC, lcl=L\\(Init statDeclC)\\"
      proof -
        from wf wt_e
        have iscls_statC: "is_class G statC"
          by (auto dest: ty_expr_is_type type_is_class)
        with wf accfield
        have iscls_statDeclC: "is_class G statDeclC"
          by (auto dest!: accfield_fields dest: fields_declC)
        thus ?thesis by simp
      qed
      obtain I where
        da_init: "\prg=G,cls=accC,lcl=L\
                    \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>Init statDeclC\<rangle>\<^sub>s\<guillemotright> I"
        by (auto intro: da_Init [simplified] assigned.select_convs)
      from valid_init P valid_A conf_s0 eval_init wt_init da_init
      obtain Q: "Q \ s1 Z" and conf_s1: "s1\\(G, L)"
        by (rule validE)
      obtain
        R: "R \vf\\<^sub>v s2' Z" and
        conf_s2: "s2\\(G, L)" and
        conf_a: "normal s2 \ G,store s2\a\\Class statC"
      proof (cases "normal s1")
        case True
        obtain V' where
          da_e':
          "\prg=G,cls=accC,lcl=L\ \dom (locals (store s1))\\e\\<^sub>e\ V'"
        proof -
          from eval_init
          have "(dom (locals (store s0))) \ (dom (locals (store s1)))"
            by (rule dom_locals_evaln_mono_elim)
          with da_e show thesis
            by (rule da_weakenE) (rule that)
        qed
        with valid_e Q valid_A conf_s1 eval_e wt_e
        obtain "R \vf\\<^sub>v s2' Z" and "s2\\(G, L)"
          by (rule validE) (simp add: fvar [symmetric])
        moreover
        from eval_e wt_e da_e' conf_s1 wf
        have "normal s2 \ G,store s2\a\\Class statC"
          by (rule evaln_type_sound [elim_format]) simp
        ultimately show ?thesis ..
      next
        case False
        with valid_e Q valid_A conf_s1 eval_e
        obtain  "R \vf\\<^sub>v s2' Z" and "s2\\(G, L)"
          by (cases rule: validE) (simp add: fvar [symmetric])+
        moreover from False eval_e have "\ normal s2"
          by auto
        hence "normal s2 \ G,store s2\a\\Class statC"
          by auto
        ultimately show ?thesis ..
      qed
      from accfield wt_e eval_init eval_e conf_s2 conf_a fvar stat s3 wf
      have eq_s3_s2': "s3=s2'"
        using normal_s0 by (auto dest!: error_free_field_access evaln_eval)
      moreover
      from eval wt da conf_s0 wf
      have "s3\\(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis using Q R by simp
    qed
  qed
next
  case (AVar A P e1 Q e2 R)
  note valid_e1 = \<open>G,A|\<Turnstile>\<Colon>{ {Normal P} e1-\<succ> {Q} }\<close>
  have valid_e2: "\ a. G,A|\\{ {Q\In1 a} e2-\ {\Val:i:. avar G i a ..; R} }"
    using AVar.hyps by simp
  show "G,A|\\{ {Normal P} e1.[e2]=\ {R} }"
  proof (rule valid_var_NormalI)
    fix n s0 L accC T V vf s2' Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf_s0: "s0\\(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg=G,cls=accC,lcl=L\\e1.[e2]\=T"
    assume da: "\prg=G,cls=accC,lcl=L\
                  \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>e1.[e2]\<rangle>\<^sub>v\<guillemotright> V"
    assume eval: "G\s0 \e1.[e2]=\vf\n\ s2'"
    assume P: "(Normal P) Y s0 Z"
    show "R \vf\\<^sub>v s2' Z \ s2'\\(G, L)"
    proof -
      from wt obtain
        wt_e1: "\prg=G,cls=accC,lcl=L\\e1\-T.[]" and
        wt_e2: "\prg=G,cls=accC,lcl=L\\e2\-PrimT Integer"
        by (rule wt_elim_cases) simp
      from da obtain E1 where
        da_e1: "\prg=G,cls=accC,lcl=L\ \dom (locals (store s0))\\e1\\<^sub>e\ E1" and
        da_e2: "\prg=G,cls=accC,lcl=L\\ nrm E1 \\e2\\<^sub>e\ V"
        by (rule da_elim_cases) simp
      from eval obtain s1 a i s2 where
        eval_e1: "G\s0 \e1-\a\n\ s1" and
        eval_e2: "G\s1 \e2-\i\n\ s2" and
        avar: "avar G i a s2 =(vf, s2')"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from valid_e1 P valid_A conf_s0 eval_e1 wt_e1 da_e1
      obtain Q: "Q \a\\<^sub>e s1 Z" and conf_s1: "s1\\(G, L)"
        by (rule validE)
      from Q have Q': "\ v. (Q\In1 a) v s1 Z"
        by simp
      have "R \vf\\<^sub>v s2' Z"
      proof (cases "normal s1")
        case True
        obtain V' where
          "\prg=G,cls=accC,lcl=L\ \dom (locals (store s1))\\e2\\<^sub>e\ V'"
        proof -
          from eval_e1  wt_e1 da_e1 wf True
          have "nrm E1 \ dom (locals (store s1))"
            by (cases rule: da_good_approx_evalnE) iprover
          with da_e2 show thesis
            by (rule da_weakenE) (rule that)
        qed
        with valid_e2 Q' valid_A conf_s1 eval_e2 wt_e2
        show ?thesis
          by (rule validE) (simp add: avar)
      next
        case False
        with valid_e2 Q' valid_A conf_s1 eval_e2
        show ?thesis
          by (cases rule: validE) (simp add: avar)+
      qed
      moreover
      from eval wt da conf_s0 wf
      have "s2'\\(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (NewC A P C Q)
  note valid_init = \<open>G,A|\<Turnstile>\<Colon>{ {Normal P} .Init C. {Alloc G (CInst C) Q} }\<close>
  show "G,A|\\{ {Normal P} NewC C-\ {Q} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s2 Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf_s0: "s0\\(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg=G,cls=accC,lcl=L\\NewC C\-T"
    assume da: "\prg=G,cls=accC,lcl=L\
                  \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>NewC C\<rangle>\<^sub>e\<guillemotright> E"
    assume eval: "G\s0 \NewC C-\v\n\ s2"
    assume P: "(Normal P) Y s0 Z"
    show "Q \v\\<^sub>e s2 Z \ s2\\(G, L)"
    proof -
      from wt obtain is_cls_C: "is_class G C"
        by (rule wt_elim_cases) (auto dest: is_acc_classD)
      hence wt_init: "\prg=G, cls=accC, lcl=L\\Init C\\"
        by auto
      obtain I where
        da_init: "\prg=G,cls=accC,lcl=L\\ dom (locals (store s0)) \\Init C\\<^sub>s\ I"
        by (auto intro: da_Init [simplified] assigned.select_convs)
      from eval obtain s1 a where
        eval_init: "G\s0 \Init C\n\ s1" and
        alloc: "G\s1 \halloc CInst C\a\ s2" and
        v: "v=Addr a"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from valid_init P valid_A conf_s0 eval_init wt_init da_init
      obtain "(Alloc G (CInst C) Q) \ s1 Z"
        by (rule validE)
      with alloc v have "Q \v\\<^sub>e s2 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s2\\(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (NewA A P T Q e R)
  note valid_init = \<open>G,A|\<Turnstile>\<Colon>{ {Normal P} .init_comp_ty T. {Q} }\<close>
  note valid_e = \<open>G,A|\<Turnstile>\<Colon>{ {Q} e-\<succ> {\<lambda>Val:i:. abupd (check_neg i) .;
                                            Alloc G (Arr T (the_Intg i)) R}}\<close>
  show "G,A|\\{ {Normal P} New T[e]-\ {R} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC arrT E v s3 Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf_s0: "s0\\(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg=G,cls=accC,lcl=L\\New T[e]\-arrT"
    assume da: "\prg=G,cls=accC,lcl=L\\dom (locals (store s0)) \\New T[e]\\<^sub>e\ E"
    assume eval: "G\s0 \New T[e]-\v\n\ s3"
    assume P: "(Normal P) Y s0 Z"
    show "R \v\\<^sub>e s3 Z \ s3\\(G, L)"
    proof -
      from wt obtain
        wt_init: "\prg=G,cls=accC,lcl=L\\init_comp_ty T\\" and
        wt_e: "\prg=G,cls=accC,lcl=L\\e\-PrimT Integer"
        by (rule wt_elim_cases) (auto intro: wt_init_comp_ty )
      from da obtain
        da_e:"\prg=G,cls=accC,lcl=L\\ dom (locals (store s0)) \\e\\<^sub>e\ E"
        by cases simp
      from eval obtain s1 i s2 a where
        eval_init: "G\s0 \init_comp_ty T\n\ s1" and
        eval_e: "G\s1 \e-\i\n\ s2" and
        alloc: "G\abupd (check_neg i) s2 \halloc Arr T (the_Intg i)\a\ s3" and
        v: "v=Addr a"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      obtain I where
        da_init:
        "\prg=G,cls=accC,lcl=L\\dom (locals (store s0)) \\init_comp_ty T\\<^sub>s\ I"
      proof (cases "\C. T = Class C")
        case True
        show ?thesis
          by (rule that)
            (use True in
              \<open>auto intro: da_Init [simplified] assigned.select_convs
                simp add: init_comp_ty_def\<close>)
         (* simplified: to rewrite \<langle>Init C\<rangle> to In1r (Init C) *)
      next
        case False
        show ?thesis
          by (rule that)
            (use False in
              \<open>auto intro: da_Skip [simplified] assigned.select_convs
                simp add: init_comp_ty_def\<close>)
         (* simplified: to rewrite \<langle>Skip\<rangle> to In1r (Skip) *)
      qed
      with valid_init P valid_A conf_s0 eval_init wt_init
      obtain Q: "Q \ s1 Z" and conf_s1: "s1\\(G, L)"
        by (rule validE)
      obtain E' where
       "\prg=G,cls=accC,lcl=L\\ dom (locals (store s1)) \\e\\<^sub>e\ E'"
      proof -
        from eval_init
        have "dom (locals (store s0)) \ dom (locals (store s1))"
          by (rule dom_locals_evaln_mono_elim)
        with da_e show thesis
          by (rule da_weakenE) (rule that)
      qed
      with valid_e Q valid_A conf_s1 eval_e wt_e
      have "(\Val:i:. abupd (check_neg i) .;
                      Alloc G (Arr T (the_Intg i)) R) \<lfloor>i\<rfloor>\<^sub>e s2 Z"
        by (rule validE)
      with alloc v have "R \v\\<^sub>e s3 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s3\\(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Cast A P e T Q)
  note valid_e = \<open>G,A|\<Turnstile>\<Colon>{ {Normal P} e-\<succ>
                 {\<lambda>Val:v:. \<lambda>s.. abupd (raise_if (\<not> G,s\<turnstile>v fits T) ClassCast) .;
                  Q\<leftarrow>In1 v} }\<close>
  show "G,A|\\{ {Normal P} Cast T e-\ {Q} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC castT E v s2 Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf_s0: "s0\\(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg=G,cls=accC,lcl=L\\Cast T e\-castT"
    assume da: "\prg=G,cls=accC,lcl=L\\dom (locals (store s0)) \\Cast T e\\<^sub>e\ E"
    assume eval: "G\s0 \Cast T e-\v\n\ s2"
    assume P: "(Normal P) Y s0 Z"
    show "Q \v\\<^sub>e s2 Z \ s2\\(G, L)"
    proof -
      from wt obtain eT where
        wt_e: "\prg = G, cls = accC, lcl = L\\e\-eT"
        by cases simp
      from da obtain
        da_e: "\prg=G,cls=accC,lcl=L\\ dom (locals (store s0)) \\e\\<^sub>e\ E"
        by cases simp
      from eval obtain s1 where
        eval_e: "G\s0 \e-\v\n\ s1" and
        s2: "s2 = abupd (raise_if (\ G,snd s1\v fits T) ClassCast) s1"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from valid_e P valid_A conf_s0 eval_e wt_e da_e
      have "(\Val:v:. \s.. abupd (raise_if (\ G,s\v fits T) ClassCast) .;
                  Q\<leftarrow>In1 v) \<lfloor>v\<rfloor>\<^sub>e s1 Z"
        by (rule validE)
      with s2 have "Q \v\\<^sub>e s2 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s2\\(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Inst A P e Q T)
  assume valid_e: "G,A|\\{ {Normal P} e-\
               {\<lambda>Val:v:. \<lambda>s.. Q\<leftarrow>In1 (Bool (v \<noteq> Null \<and> G,s\<turnstile>v fits RefT T))} }"
  show "G,A|\\{ {Normal P} e InstOf T-\ {Q} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC instT E v s1 Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf_s0: "s0\\(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg=G,cls=accC,lcl=L\\e InstOf T\-instT"
    assume da: "\prg=G,cls=accC,lcl=L\\dom (locals (store s0))\\e InstOf T\\<^sub>e\ E"
    assume eval: "G\s0 \e InstOf T-\v\n\ s1"
    assume P: "(Normal P) Y s0 Z"
    show "Q \v\\<^sub>e s1 Z \ s1\\(G, L)"
    proof -
      from wt obtain eT where
        wt_e: "\prg = G, cls = accC, lcl = L\\e\-eT"
        by cases simp
      from da obtain
        da_e: "\prg=G,cls=accC,lcl=L\\ dom (locals (store s0)) \\e\\<^sub>e\ E"
        by cases simp
      from eval obtain a where
        eval_e: "G\s0 \e-\a\n\ s1" and
        v: "v = Bool (a \ Null \ G,store s1\a fits RefT T)"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from valid_e P valid_A conf_s0 eval_e wt_e da_e
      have "(\Val:v:. \s.. Q\In1 (Bool (v \ Null \ G,s\v fits RefT T)))
              \<lfloor>a\<rfloor>\<^sub>e s1 Z"
        by (rule validE)
      with v have "Q \v\\<^sub>e s1 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s1\\(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Lit A P v)
  show "G,A|\\{ {Normal (P\In1 v)} Lit v-\ {P} }"
  proof (rule valid_expr_NormalI)
    fix n L s0 s1 v' Y Z
    assume conf_s0: "s0\\(G, L)"
    assume normal_s0: " normal s0"
    assume eval: "G\s0 \Lit v-\v'\n\ s1"
    assume P: "(Normal (P\In1 v)) Y s0 Z"
    show "P \v'\\<^sub>e s1 Z \ s1\\(G, L)"
    proof -
      from eval have "s1=s0" and  "v'=v"
        using normal_s0 by (auto elim: evaln_elim_cases)
      with P conf_s0 show ?thesis by simp
    qed
  qed
next
  case (UnOp A P e Q unop)
  assume valid_e: "G,A|\\{ {Normal P}e-\{\Val:v:. Q\In1 (eval_unop unop v)} }"
  show "G,A|\\{ {Normal P} UnOp unop e-\ {Q} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s1 Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf_s0: "s0\\(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg=G,cls=accC,lcl=L\\UnOp unop e\-T"
    assume da: "\prg=G,cls=accC,lcl=L\\dom (locals (store s0))\\UnOp unop e\\<^sub>e\E"
    assume eval: "G\s0 \UnOp unop e-\v\n\ s1"
    assume P: "(Normal P) Y s0 Z"
    show "Q \v\\<^sub>e s1 Z \ s1\\(G, L)"
    proof -
      from wt obtain eT where
        wt_e: "\prg = G, cls = accC, lcl = L\\e\-eT"
        by cases simp
      from da obtain
        da_e: "\prg=G,cls=accC,lcl=L\\ dom (locals (store s0)) \\e\\<^sub>e\ E"
        by cases simp
      from eval obtain ve where
        eval_e: "G\s0 \e-\ve\n\ s1" and
        v: "v = eval_unop unop ve"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from valid_e P valid_A conf_s0 eval_e wt_e da_e
      have "(\Val:v:. Q\In1 (eval_unop unop v)) \ve\\<^sub>e s1 Z"
        by (rule validE)
      with v have "Q \v\\<^sub>e s1 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s1\\(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (BinOp A P e1 Q binop e2 R)
  assume valid_e1: "G,A|\\{ {Normal P} e1-\ {Q} }"
  have valid_e2: "\ v1. G,A|\\{ {Q\In1 v1}
              (if need_second_arg binop v1 then In1l e2 else In1r Skip)\<succ>
              {\<lambda>Val:v2:. R\<leftarrow>In1 (eval_binop binop v1 v2)} }"
    using BinOp.hyps by simp
  show "G,A|\\{ {Normal P} BinOp binop e1 e2-\ {R} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s2 Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf_s0: "s0\\(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg=G,cls=accC,lcl=L\\BinOp binop e1 e2\-T"
    assume da: "\prg=G,cls=accC,lcl=L\
                  \<turnstile>dom (locals (store s0)) \<guillemotright>\<langle>BinOp binop e1 e2\<rangle>\<^sub>e\<guillemotright> E"
    assume eval: "G\s0 \BinOp binop e1 e2-\v\n\ s2"
    assume P: "(Normal P) Y s0 Z"
    show "R \v\\<^sub>e s2 Z \ s2\\(G, L)"
    proof -
      from wt obtain e1T e2T where
        wt_e1: "\prg=G,cls=accC,lcl=L\\e1\-e1T" and
        wt_e2: "\prg=G,cls=accC,lcl=L\\e2\-e2T" and
        wt_binop: "wt_binop G binop e1T e2T"
        by cases simp
      have wt_Skip: "\prg = G, cls = accC, lcl = L\\Skip\\"
        by simp
      (*
      obtain S where
        daSkip: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                   \<turnstile> dom (locals (store s1)) \<guillemotright>In1r Skip\<guillemotright> S"
        by (auto intro: da_Skip [simplified] assigned.select_convs) *)
      from da obtain E1 where
        da_e1: "\prg=G,cls=accC,lcl=L\ \ dom (locals (store s0)) \\e1\\<^sub>e\ E1"
        by cases simp+
      from eval obtain v1 s1 v2 where
        eval_e1: "G\s0 \e1-\v1\n\ s1" and
        eval_e2: "G\s1 \(if need_second_arg binop v1 then \e2\\<^sub>e else \Skip\\<^sub>s)
                        \<succ>\<midarrow>n\<rightarrow> (\<lfloor>v2\<rfloor>\<^sub>e, s2)" and
        v: "v=eval_binop binop v1 v2"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from valid_e1 P valid_A conf_s0 eval_e1 wt_e1 da_e1
      obtain Q: "Q \v1\\<^sub>e s1 Z" and conf_s1: "s1\\(G,L)"
        by (rule validE)
      from Q have Q': "\ v. (Q\In1 v1) v s1 Z"
        by simp
      have "(\Val:v2:. R\In1 (eval_binop binop v1 v2)) \v2\\<^sub>e s2 Z"
      proof (cases "normal s1")
        case True
        from eval_e1 wt_e1 da_e1 conf_s0 wf
        have conf_v1: "G,store s1\v1\\e1T"
          by (rule evaln_type_sound [elim_format]) (use True in simp)
        from eval_e1
        have "G\s0 \e1-\v1\ s1"
          by (rule evaln_eval)
        from da wt_e1 wt_e2 wt_binop conf_s0 True this conf_v1 wf
        obtain E2 where
          da_e2: "\prg=G,cls=accC,lcl=L\\ dom (locals (store s1))
                   \<guillemotright>(if need_second_arg binop v1 then \<langle>e2\<rangle>\<^sub>e else \<langle>Skip\<rangle>\<^sub>s)\<guillemotright> E2"
          by (rule da_e2_BinOp [elim_format]) iprover
        from wt_e2 wt_Skip obtain T2
          where "\prg=G,cls=accC,lcl=L\
                  \<turnstile>(if need_second_arg binop v1 then \<langle>e2\<rangle>\<^sub>e else \<langle>Skip\<rangle>\<^sub>s)\<Colon>T2"
          by (cases "need_second_arg binop v1") auto
        note ve=validE [OF valid_e2,OF  Q' valid_A conf_s1 eval_e2 this da_e2]
        (* chaining Q', without extra OF causes unification error *)
        thus ?thesis
          by (rule ve)
      next
        case False
        note ve=validE [OF valid_e2,OF Q' valid_A conf_s1 eval_e2]
        with False show ?thesis
          by iprover
      qed
      with v have "R \v\\<^sub>e s2 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s2\\(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Super A P)
  show "G,A|\\{ {Normal (\s.. P\In1 (val_this s))} Super-\ {P} }"
  proof (rule valid_expr_NormalI)
    fix n L s0 s1 v  Y Z
    assume conf_s0: "s0\\(G, L)"
    assume normal_s0: " normal s0"
    assume eval: "G\s0 \Super-\v\n\ s1"
    assume P: "(Normal (\s.. P\In1 (val_this s))) Y s0 Z"
    show "P \v\\<^sub>e s1 Z \ s1\\(G, L)"
    proof -
      from eval have "s1=s0" and  "v=val_this (store s0)"
        using normal_s0 by (auto elim: evaln_elim_cases)
      with P conf_s0 show ?thesis by simp
    qed
  qed
next
  case (Acc A P var Q)
  note valid_var = \<open>G,A|\<Turnstile>\<Colon>{ {Normal P} var=\<succ> {\<lambda>Var:(v, f):. Q\<leftarrow>In1 v} }\<close>
  show "G,A|\\{ {Normal P} Acc var-\ {Q} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s1 Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf_s0: "s0\\(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg=G,cls=accC,lcl=L\\Acc var\-T"
    assume da: "\prg=G,cls=accC,lcl=L\\dom (locals (store s0))\\Acc var\\<^sub>e\E"
    assume eval: "G\s0 \Acc var-\v\n\ s1"
    assume P: "(Normal P) Y s0 Z"
    show "Q \v\\<^sub>e s1 Z \ s1\\(G, L)"
    proof -
      from wt obtain
        wt_var: "\prg=G,cls=accC,lcl=L\\var\=T"
        by cases simp
      from da obtain V where
        da_var: "\prg=G,cls=accC,lcl=L\ \ dom (locals (store s0)) \\var\\<^sub>v\ V"
        by (cases "\ n. var=LVar n") (use da.LVar in \auto elim!: da_elim_cases\)
      from eval obtain upd where
        eval_var: "G\s0 \var=\(v, upd)\n\ s1"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from valid_var P valid_A conf_s0 eval_var wt_var da_var
      have "(\Var:(v, f):. Q\In1 v) \(v, upd)\\<^sub>v s1 Z"
        by (rule validE)
      then have "Q \v\\<^sub>e s1 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s1\\(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Ass A P var Q e R)
  note valid_var = \<open>G,A|\<Turnstile>\<Colon>{ {Normal P} var=\<succ> {Q} }\<close>
  have valid_e: "\ vf.
                  G,A|\<Turnstile>\<Colon>{ {Q\<leftarrow>In2 vf} e-\<succ> {\<lambda>Val:v:. assign (snd vf) v .; R} }"
    using Ass.hyps by simp
  show "G,A|\\{ {Normal P} var:=e-\ {R} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s3 Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf_s0: "s0\\(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg=G,cls=accC,lcl=L\\var:=e\-T"
    assume da: "\prg=G,cls=accC,lcl=L\\dom (locals (store s0))\\var:=e\\<^sub>e\E"
    assume eval: "G\s0 \var:=e-\v\n\ s3"
    assume P: "(Normal P) Y s0 Z"
    show "R \v\\<^sub>e s3 Z \ s3\\(G, L)"
    proof -
      from wt obtain varT  where
        wt_var: "\prg=G,cls=accC,lcl=L\\var\=varT" and
        wt_e: "\prg=G,cls=accC,lcl=L\\e\-T"
        by cases simp
      from eval obtain w upd s1 s2 where
        eval_var: "G\s0 \var=\(w, upd)\n\ s1" and
        eval_e: "G\s1 \e-\v\n\ s2" and
        s3: "s3=assign upd v s2"
        using normal_s0 by (auto elim: evaln_elim_cases)
      have "R \v\\<^sub>e s3 Z"
      proof (cases "\ vn. var = LVar vn")
        case False
        with da obtain V where
          da_var: "\prg=G,cls=accC,lcl=L\
                      \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>var\<rangle>\<^sub>v\<guillemotright> V" and
          da_e:   "\prg=G,cls=accC,lcl=L\ \ nrm V \\e\\<^sub>e\ E"
          by cases simp+
        from valid_var P valid_A conf_s0 eval_var wt_var da_var
        obtain Q: "Q \(w,upd)\\<^sub>v s1 Z" and conf_s1: "s1\\(G,L)"
          by (rule validE)
        hence Q': "\ v. (Q\In2 (w,upd)) v s1 Z"
          by simp
        have "(\Val:v:. assign (snd (w,upd)) v .; R) \v\\<^sub>e s2 Z"
        proof (cases "normal s1")
          case True
          obtain E' where
            da_e': "\prg=G,cls=accC,lcl=L\\ dom (locals (store s1)) \\e\\<^sub>e\ E'"
          proof -
            from eval_var wt_var da_var wf True
            have "nrm V \ dom (locals (store s1))"
              by (cases rule: da_good_approx_evalnE) iprover
            with da_e show thesis
              by (rule da_weakenE) (rule that)
          qed
          note ve=validE [OF valid_e,OF Q' valid_A conf_s1 eval_e wt_e da_e']
          show ?thesis
            by (rule ve)
        next
          case False
          note ve=validE [OF valid_e,OF Q' valid_A conf_s1 eval_e]
          with False show ?thesis
            by iprover
        qed
        with s3 show "R \v\\<^sub>e s3 Z"
          by simp
      next
        case True
        then obtain vn where
          vn: "var = LVar vn"
          by auto
        with da obtain E where
            da_e:   "\prg=G,cls=accC,lcl=L\ \ dom (locals (store s0)) \\e\\<^sub>e\ E"
          by cases simp+
        from da.LVar vn obtain  V where
          da_var: "\prg=G,cls=accC,lcl=L\
                      \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>var\<rangle>\<^sub>v\<guillemotright> V"
          by auto
        from valid_var P valid_A conf_s0 eval_var wt_var da_var
        obtain Q: "Q \(w,upd)\\<^sub>v s1 Z" and conf_s1: "s1\\(G,L)"
          by (rule validE)
        hence Q': "\ v. (Q\In2 (w,upd)) v s1 Z"
          by simp
        have "(\Val:v:. assign (snd (w,upd)) v .; R) \v\\<^sub>e s2 Z"
        proof (cases "normal s1")
          case True
          obtain E' where
            da_e': "\prg=G,cls=accC,lcl=L\
                       \<turnstile> dom (locals (store s1)) \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E'"
          proof -
            from eval_var
            have "dom (locals (store s0)) \ dom (locals (store (s1)))"
              by (rule dom_locals_evaln_mono_elim)
            with da_e show thesis
              by (rule da_weakenE) (rule that)
          qed
          note ve=validE [OF valid_e,OF Q' valid_A conf_s1 eval_e wt_e da_e']
          show ?thesis
            by (rule ve)
        next
          case False
          note ve=validE [OF valid_e,OF Q' valid_A conf_s1 eval_e]
          with False show ?thesis
            by iprover
        qed
        with s3 show "R \v\\<^sub>e s3 Z"
          by simp
      qed
      moreover
      from eval wt da conf_s0 wf
      have "s3\\(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Cond A P e0 P' e1 e2 Q)
  note valid_e0 = \<open>G,A|\<Turnstile>\<Colon>{ {Normal P} e0-\<succ> {P'} }\<close>
  have valid_then_else:"\ b. G,A|\\{ {P'\=b} (if b then e1 else e2)-\ {Q} }"
    using Cond.hyps by simp
  show "G,A|\\{ {Normal P} e0 ? e1 : e2-\ {Q} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s2 Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf_s0: "s0\\(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg=G,cls=accC,lcl=L\\e0 ? e1 : e2\-T"
    assume da: "\prg=G,cls=accC,lcl=L\\dom (locals (store s0))\\e0 ? e1:e2\\<^sub>e\E"
    assume eval: "G\s0 \e0 ? e1 : e2-\v\n\ s2"
    assume P: "(Normal P) Y s0 Z"
    show "Q \v\\<^sub>e s2 Z \ s2\\(G, L)"
    proof -
      from wt obtain T1 T2 where
        wt_e0: "\prg=G,cls=accC,lcl=L\\e0\-PrimT Boolean" and
        wt_e1: "\prg=G,cls=accC,lcl=L\\e1\-T1" and
        wt_e2: "\prg=G,cls=accC,lcl=L\\e2\-T2"
        by cases simp
      from da obtain E0 E1 E2 where
        da_e0: "\prg=G,cls=accC,lcl=L\\ dom (locals (store s0)) \\e0\\<^sub>e\ E0" and
        da_e1: "\prg=G,cls=accC,lcl=L\
                 \<turnstile>(dom (locals (store s0)) \<union> assigns_if True e0)\<guillemotright>\<langle>e1\<rangle>\<^sub>e\<guillemotright> E1" and
        da_e2: "\prg=G,cls=accC,lcl=L\
                 \<turnstile>(dom (locals (store s0)) \<union> assigns_if False e0)\<guillemotright>\<langle>e2\<rangle>\<^sub>e\<guillemotright> E2"
        by cases simp+
      from eval obtain b s1 where
        eval_e0: "G\s0 \e0-\b\n\ s1" and
        eval_then_else: "G\s1 \(if the_Bool b then e1 else e2)-\v\n\ s2"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from valid_e0 P valid_A conf_s0 eval_e0 wt_e0 da_e0
      obtain "P' \b\\<^sub>e s1 Z" and conf_s1: "s1\\(G,L)"
        by (rule validE)
      hence P': "\ v. (P'\=(the_Bool b)) v s1 Z"
        by (cases "normal s1") auto
      have "Q \v\\<^sub>e s2 Z"
      proof (cases "normal s1")
        case True
        note normal_s1=this
        from wt_e1 wt_e2 obtain T' where
          wt_then_else:
          "\prg=G,cls=accC,lcl=L\\(if the_Bool b then e1 else e2)\-T'"
          by (cases "the_Bool b") simp+
        have s0_s1: "dom (locals (store s0))
                      \<union> assigns_if (the_Bool b) e0 \<subseteq> dom (locals (store s1))"
        proof -
          from eval_e0
          have eval_e0': "G\s0 \e0-\b\ s1"
            by (rule evaln_eval)
          hence
            "dom (locals (store s0)) \ dom (locals (store s1))"
            by (rule dom_locals_eval_mono_elim)
          moreover
          from eval_e0' True wt_e0
          have "assigns_if (the_Bool b) e0 \ dom (locals (store s1))"
            by (rule assigns_if_good_approx')
          ultimately show ?thesis by (rule Un_least)
        qed
        obtain E' where
          da_then_else:
          "\prg=G,cls=accC,lcl=L\
              \<turnstile>dom (locals (store s1))\<guillemotright>\<langle>if the_Bool b then e1 else e2\<rangle>\<^sub>e\<guillemotright> E'"
        proof (cases "the_Bool b")
          case True
          with that da_e1 s0_s1 show ?thesis
            by simp (erule da_weakenE,auto)
        next
          case False
          with that da_e2 s0_s1 show ?thesis
            by simp (erule da_weakenE,auto)
        qed
        with valid_then_else P' valid_A conf_s1 eval_then_else wt_then_else
        show ?thesis
          by (rule validE)
      next
        case False
        with valid_then_else P' valid_A conf_s1 eval_then_else
        show ?thesis
          by (cases rule: validE) iprover+
      qed
      moreover
      from eval wt da conf_s0 wf
      have "s2\\(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Call A P e Q args R mode statT mn pTs' S accC')
  note valid_e = \<open>G,A|\<Turnstile>\<Colon>{ {Normal P} e-\<succ> {Q} }\<close>
  have valid_args: "\ a. G,A|\\{ {Q\In1 a} args\\ {R a} }"
    using Call.hyps by simp
  have valid_methd: "\ a vs invC declC l.
        G,A|\<Turnstile>\<Colon>{ {R a\<leftarrow>In3 vs \<and>.
                 (\<lambda>s. declC =
                    invocation_declclass G mode (store s) a statT
                     \<lparr>name = mn, parTs = pTs'\<rparr> \<and>
                    invC = invocation_class mode (store s) a statT \<and>
                    l = locals (store s)) ;.
                 init_lvars G declC \<lparr>name = mn, parTs = pTs'\<rparr> mode a vs \<and>.
                 (\<lambda>s. normal s \<longrightarrow> G\<turnstile>mode\<rightarrow>invC\<preceq>statT)}
            Methd declC \<lparr>name=mn,parTs=pTs'\<rparr>-\<succ> {set_lvars l .; S} }"
    using Call.hyps by simp
  show "G,A|\\{ {Normal P} {accC',statT,mode}e\mn( {pTs'}args)-\ {S} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s5 Y Z
    assume valid_A: "\t\A. G\n\t"
    assume conf_s0: "s0\\(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\prg=G,cls=accC,lcl=L\\{accC',statT,mode}e\mn( {pTs'}args)\-T"
    assume da: "\prg=G,cls=accC,lcl=L\\dom (locals (store s0))
                   \<guillemotright>\<langle>{accC',statT,mode}e\<cdot>mn( {pTs'}args)\<rangle>\<^sub>e\<guillemotright> E"
    assume eval: "G\s0 \{accC',statT,mode}e\mn( {pTs'}args)-\v\n\ s5"
    assume P: "(Normal P) Y s0 Z"
    show "S \v\\<^sub>e s5 Z \ s5\\(G, L)"
    proof -
      from wt obtain pTs statDeclT statM where
                 wt_e: "\prg=G,cls=accC,lcl=L\\e\-RefT statT" and
              wt_args: "\prg=G,cls=accC,lcl=L\\args\\pTs" and
                statM: "max_spec G accC statT \name=mn,parTs=pTs\
                         = {((statDeclT,statM),pTs')}" and
                 mode: "mode = invmode statM e" and
                    T: "T =(resTy statM)" and
        eq_accC_accC': "accC=accC'"
        by cases fastforce+
      from da obtain C where
        da_e: "\prg=G,cls=accC,lcl=L\\ (dom (locals (store s0)))\\e\\<^sub>e\ C" and
        da_args: "\prg=G,cls=accC,lcl=L\\ nrm C \\args\\<^sub>l\ E"
        by cases simp
      from eval eq_accC_accC' obtain a s1 vs s2 s3 s3' s4 invDeclC where
        evaln_e: "G\s0 \e-\a\n\ s1" and
        evaln_args: "G\s1 \args\\vs\n\ s2" and
        invDeclC: "invDeclC = invocation_declclass
                G mode (store s2) a statT \<lparr>name=mn,parTs=pTs'\<rparr>" and
        s3: "s3 = init_lvars G invDeclC \name=mn,parTs=pTs'\ mode a vs s2" and
        check: "s3' = check_method_access G
                           accC' statT mode \name = mn, parTs = pTs'\ a s3" and
        evaln_methd:
           "G\s3' \Methd invDeclC \name=mn,parTs=pTs'\-\v\n\ s4" and
        s5: "s5=(set_lvars (locals (store s2))) s4"
        using normal_s0 by (auto elim: evaln_elim_cases)

      from evaln_e
      have eval_e: "G\s0 \e-\a\ s1"
        by (rule evaln_eval)

      from eval_e _ wt_e wf
      have s1_no_return: "abrupt s1 \ Some (Jump Ret)"
        by (rule eval_expression_no_jump
                 [where ?Env="\prg=G,cls=accC,lcl=L\",simplified])
           (use normal_s0 in auto)

      from valid_e P valid_A conf_s0 evaln_e wt_e da_e
      obtain "Q \a\\<^sub>e s1 Z" and conf_s1: "s1\\(G,L)"
        by (rule validE)
      hence Q: "\ v. (Q\In1 a) v s1 Z"
        by simp
      obtain
        R: "(R a) \vs\\<^sub>l s2 Z" and
        conf_s2: "s2\\(G,L)" and
        s2_no_return: "abrupt s2 \ Some (Jump Ret)"
      proof (cases "normal s1")
        case True
        obtain E' where
          da_args':
          "\prg=G,cls=accC,lcl=L\\ dom (locals (store s1)) \\args\\<^sub>l\ E'"
        proof -
          from evaln_e wt_e da_e wf True
          have "nrm C \ dom (locals (store s1))"
            by (cases rule: da_good_approx_evalnE) iprover
          with da_args show thesis
            by (rule da_weakenE) (rule that)
        qed
        with valid_args Q valid_A conf_s1 evaln_args wt_args
        obtain "(R a) \vs\\<^sub>l s2 Z" "s2\\(G,L)"
          by (rule validE)
        moreover
        from evaln_args
        have e: "G\s1 \args\\vs\ s2"
          by (rule evaln_eval)
        from this s1_no_return wt_args wf
        have "abrupt s2 \ Some (Jump Ret)"
          by (rule eval_expression_list_no_jump
                 [where ?Env="\prg=G,cls=accC,lcl=L\",simplified])
        ultimately show ?thesis ..
      next
        case False
        with valid_args Q valid_A conf_s1 evaln_args
        obtain "(R a) \vs\\<^sub>l s2 Z" "s2\\(G,L)"
          by (cases rule: validE) iprover+
        moreover
        from False evaln_args have "s2=s1"
          by auto
        with s1_no_return have "abrupt s2 \ Some (Jump Ret)"
          by simp
        ultimately show ?thesis ..
      qed

      obtain invC where
        invC: "invC = invocation_class mode (store s2) a statT"
        by simp
      with s3
      have invC': "invC = (invocation_class mode (store s3) a statT)"
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