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

theory AxSound imports AxSem begin

subsubsection "validity"

definition
  triple_valid2 :: "prog \<Rightarrow> nat \<Rightarrow> 'a triple \<Rightarrow> bool"  ("_\<Turnstile>_\<Colon>_"[61,0, 58] 57)
  where
    "G\<Turnstile>n\<Colon>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 {* This definition differs from the ordinary  @{text triple_valid_def}
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.
*}

definition
  ax_valids2 :: "prog \<Rightarrow> 'a triples \<Rightarrow> 'a triples \<Rightarrow> bool"  ("_,_|\<Turnstile>\<Colon>_" [61,58,58] 57)
  where "G,A|\<Turnstile>\<Colon>ts = (\<forall>n. (\<forall>t\<in>A. G\<Turnstile>n\<Colon>t) \<longrightarrow> (\<forall>t\<in>ts. G\<Turnstile>n\<Colon>t))"

lemma triple_valid2_def2: "G\<Turnstile>n\<Colon>{P} t\<succ> {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 \<Longrightarrow> G,A|\<Turnstile>\<Colon>ts = G,A|\<Turnstile>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\<Turnstile>Suc n\<Colon>t \<longrightarrow> G\<Turnstile>n\<Colon>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\<Turnstile>0\<Colon>{Normal P} Methd C sig-\<succ> {Q}"
by (auto elim!: evaln_elim_cases simp add: triple_valid2_def2)

lemma Methd_triple_valid2_SucI:
"\<lbrakk>G\<Turnstile>n\<Colon>{Normal P} body G C sig-\<succ>{Q}\<rbrakk>
  \<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)) \<Longrightarrow> Ball ts (triple_valid2 G n)"
apply (fast intro: triple_valid2_Suc)
done

lemma "G|\<Turnstile>n:insert t A = (G\<Turnstile>n:t \<and> G|\<Turnstile>n:A)"
oops

subsubsection "soundness"

lemma Methd_sound:
  assumes recursive: "G,A\<union>  {{P} Methd-\<succ> {Q} | ms}|\<Turnstile>\<Colon>{{P} body G-\<succ> {Q} | ms}"
  shows "G,A|\<Turnstile>\<Colon>{{P} Methd-\<succ> {Q} | ms}"
proof -
  {
    fix n
    assume recursive: "\<And> n. \<forall>t\<in>(A \<union> {{P} Methd-\<succ> {Q} | ms}). G\<Turnstile>n\<Colon>t
                              \<Longrightarrow>  \<forall>t\<in>{{P} body G-\<succ> {Q} | ms}.  G\<Turnstile>n\<Colon>t"
    have "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t \<Longrightarrow> \<forall>t\<in>{{P} Methd-\<succ> {Q} | ms}.  G\<Turnstile>n\<Colon>t"
    proof (induct n)
      case 0
      show "\<forall>t\<in>{{P} Methd-\<succ> {Q} | ms}.  G\<Turnstile>0\<Colon>t"
      proof -
        {
          fix C sig
          assume "(C,sig) \<in> ms"
          have "G\<Turnstile>0\<Colon>{Normal (P C sig)} Methd C sig-\<succ> {Q C sig}"
            by (rule Methd_triple_valid2_0)
        }
        thus ?thesis
          by (simp add: mtriples_def split_def)
      qed
    next
      case (Suc m)
      note hyp = `\<forall>t\<in>A. G\<Turnstile>m\<Colon>t \<Longrightarrow> \<forall>t\<in>{{P} Methd-\<succ> {Q} | ms}.  G\<Turnstile>m\<Colon>t`
      note prem = `\<forall>t\<in>A. G\<Turnstile>Suc m\<Colon>t`
      show "\<forall>t\<in>{{P} Methd-\<succ> {Q} | ms}.  G\<Turnstile>Suc m\<Colon>t"
      proof -
        {
          fix C sig
          assume m: "(C,sig) \<in> ms"
          have "G\<Turnstile>Suc m\<Colon>{Normal (P C sig)} Methd C sig-\<succ> {Q C sig}"
          proof -
            from prem have prem_m: "\<forall>t\<in>A. G\<Turnstile>m\<Colon>t"
              by (rule triples_valid2_Suc)
            hence "\<forall>t\<in>{{P} Methd-\<succ> {Q} | ms}.  G\<Turnstile>m\<Colon>t"
              by (rule hyp)
            with prem_m
            have "\<forall>t\<in>(A \<union> {{P} Methd-\<succ> {Q} | ms}). G\<Turnstile>m\<Colon>t"
              by (simp add: ball_Un)
            hence "\<forall>t\<in>{{P} body G-\<succ> {Q} | ms}.  G\<Turnstile>m\<Colon>t"
              by (rule recursive)
            with m have "G\<Turnstile>m\<Colon>{Normal (P C sig)} body G C sig-\<succ> {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: "\<forall>s t n Y' s'. G\<turnstile>s\<midarrow>t\<succ>\<midarrow>n\<rightarrow> (Y',s') \<longrightarrow> t = c \<longrightarrow>
  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\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s1)"
     and     wt: "\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile>t\<Colon>T"
     and     da: "\<lparr>prg=G,cls=C,lcl=L\<rparr>\<turnstile> dom (locals (store s0)) \<guillemotright>t\<guillemotright> A"
     and     wf: "wf_prog G"
     and   elim: "\<lbrakk>normal s1 \<Longrightarrow> nrm A \<subseteq> 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\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (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: "\<And> 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|\<Turnstile>\<Colon>{ {P} t\<succ> {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: "\<And> 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|\<Turnstile>\<Colon>{ {P} \<langle>c\<rangle>\<^sub>s\<succ> {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: "\<And> 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|\<Turnstile>\<Colon>{ {Normal P} \<langle>c\<rangle>\<^sub>s\<succ> {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: "\<And> 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|\<Turnstile>\<Colon>{ {Normal P} \<langle>t\<rangle>\<^sub>v\<succ> {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: "\<And> 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|\<Turnstile>\<Colon>{ {Normal P} \<langle>t\<rangle>\<^sub>e\<succ> {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: "\<And> 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|\<Turnstile>\<Colon>{ {Normal P} \<langle>t\<rangle>\<^sub>l\<succ> {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|\<Turnstile>\<Colon>{ {P} t\<succ> {Q} }"
   and    P: "P Y s0 Z"
   and    valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
   and    conf: "s0\<Colon>\<preceq>(G,L)"
   and    eval: "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s1)"
   and    wt: "normal s0 \<Longrightarrow> \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>T"
   and    da: "normal s0 \<Longrightarrow> \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>t\<guillemotright>C"
   and    elim: "\<lbrakk>Q v s1 Z; s1\<Colon>\<preceq>(G,L)\<rbrakk> \<Longrightarrow> 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\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s1)" and
             wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>T" and
             da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0)) \<guillemotright>t\<guillemotright> A" and
        conf_s0: "s0\<Colon>\<preceq>(G,L)" and
             wf: "wf_prog G"
  shows "s1\<Colon>\<preceq>(G,L) \<and>  (normal s1 \<longrightarrow> G,L,store s1\<turnstile>t\<succ>v\<Colon>\<preceq>T) \<and>
         (error_free s0 = error_free s1)"
proof -
  from evaln have "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (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\<turnstile> s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v,s1)" and
    hyps: "\<lbrakk>dom (locals (store s0)) \<subseteq> 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\<turnstile> s0 \<midarrow>t\<succ>\<rightarrow> (v,s1)" by (rule evaln_eval)
  from this hyps show ?thesis
    by (rule dom_locals_eval_mono_elim) iprover+
qed

lemma evaln_no_abrupt:
   "\<And>s s'. \<lbrakk>G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (w,s'); normal s'\<rbrakk> \<Longrightarrow> normal s"
by (erule evaln_cases,auto)

declare inj_term_simps [simp]
lemma ax_sound2:
  assumes    wf: "wf_prog G"
    and   deriv: "G,A|\<turnstile>ts"
  shows "G,A|\<Turnstile>\<Colon>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 = `G,A|\<Turnstile>\<Colon>{t}`
  moreover
  note valid_ts = `G,A|\<Turnstile>\<Colon>ts`
  {
    fix n assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    have "G\<Turnstile>n\<Colon>t" and "\<forall>t\<in>ts. G\<Turnstile>n\<Colon>t"
    proof -
      from valid_A valid_t show "G\<Turnstile>n\<Colon>t"
        by (simp add: ax_valids2_def)
    next
      from valid_A valid_ts show "\<forall>t\<in>ts. G\<Turnstile>n\<Colon>t"
        by (unfold ax_valids2_def) blast
    qed
    hence "\<forall>t'\<in>insert t ts. G\<Turnstile>n\<Colon>t'"
      by simp
  }
  thus ?case
    by (unfold ax_valids2_def) blast
next
  case (asm ts A)
  from `ts \<subseteq> A`
  show "G,A|\<Turnstile>\<Colon>ts"
    by (auto simp add: ax_valids2_def triple_valid2_def)
next
  case (weaken A ts' ts)
  note `G,A|\<Turnstile>\<Colon>ts'`
  moreover note `ts \<subseteq> ts'`
  ultimately show "G,A|\<Turnstile>\<Colon>ts"
    by (unfold ax_valids2_def triple_valid2_def) blast
next
  case (conseq P A t Q)
  note con = `\<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))`
  show "G,A|\<Turnstile>\<Colon>{ {P} t\<succ> {Q} }"
  proof (rule validI)
    fix n s0 L accC T C v s1 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf: "s0\<Colon>\<preceq>(G,L)"
    assume wt: "normal s0 \<Longrightarrow> \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>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\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s1)"
    assume P: "P Y s0 Z"
    show "Q v s1 Z \<and> s1\<Colon>\<preceq>(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\<Colon>\<preceq>(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|\<Turnstile>\<Colon>{ {P \<and>. Not \<circ> type_ok G t} t\<succ> {Q} }"
    by (simp add: ax_valids2_def triple_valid2_def2 type_ok_def) fast
next
  case (Abrupt A P t)
  show "G,A|\<Turnstile>\<Colon>{ {P\<leftarrow>undefined3 t \<and>. Not \<circ> normal} t\<succ> {P} }"
  proof (rule validI)
    fix n s0 L accC T C v s1 Y Z
    assume conf_s0: "s0\<Colon>\<preceq>(G, L)"
    assume eval: "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s1)"
    assume "(P\<leftarrow>undefined3 t \<and>. Not \<circ> normal) Y s0 Z"
    then obtain P: "P (undefined3 t) s0 Z" and abrupt_s0: "\<not> 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 \<and> s1\<Colon>\<preceq>(G, L)"
      by simp
  qed
next
  case (LVar A P vn)
  show "G,A|\<Turnstile>\<Colon>{ {Normal (\<lambda>s.. P\<leftarrow>In2 (lvar vn s))} LVar vn=\<succ> {P} }"
  proof (rule valid_var_NormalI)
    fix n s0 L accC T C vf s1 Y Z
    assume conf_s0: "s0\<Colon>\<preceq>(G, L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>LVar vn\<Colon>=T"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>LVar vn\<rangle>\<^sub>v\<guillemotright> C"
    assume eval: "G\<turnstile>s0 \<midarrow>LVar vn=\<succ>vf\<midarrow>n\<rightarrow> s1"
    assume P: "(Normal (\<lambda>s.. P\<leftarrow>In2 (lvar vn s))) Y s0 Z"
    show "P (In2 vf) s1 Z \<and> s1\<Colon>\<preceq>(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\<Colon>\<preceq>(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 = `G,A|\<Turnstile>\<Colon>{ {Normal P} .Init statDeclC. {Q} }`
  note valid_e = `G,A|\<Turnstile>\<Colon>{ {Q} e-\<succ> {\<lambda>Val:a:. fvar statDeclC stat fn a ..; R} }`
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} {accC,statDeclC,stat}e..fn=\<succ> {R} }"
  proof (rule valid_var_NormalI)
    fix n s0 L accC' T V vf s3 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0:  "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC',lcl=L\<rparr>\<turnstile>{accC,statDeclC,stat}e..fn\<Colon>=T"
    assume da: "\<lparr>prg=G,cls=accC',lcl=L\<rparr>
                  \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>{accC,statDeclC,stat}e..fn\<rangle>\<^sub>v\<guillemotright> V"
    assume eval: "G\<turnstile>s0 \<midarrow>{accC,statDeclC,stat}e..fn=\<succ>vf\<midarrow>n\<rightarrow> s3"
    assume P: "(Normal P) Y s0 Z"
    show "R \<lfloor>vf\<rfloor>\<^sub>v s3 Z \<and> s3\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain statC f where
        wt_e: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>e\<Colon>-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: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> V"
        by cases simp
      from eval obtain a s1 s2 s2' where
        eval_init: "G\<turnstile>s0 \<midarrow>Init statDeclC\<midarrow>n\<rightarrow> s1" and
        eval_e: "G\<turnstile>s1 \<midarrow>e-\<succ>a\<midarrow>n\<rightarrow> 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: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>(Init statDeclC)\<Colon>\<surd>"
      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: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                    \<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 \<diamondsuit> s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G, L)"
        by (rule validE)
      obtain
        R: "R \<lfloor>vf\<rfloor>\<^sub>v s2' Z" and
        conf_s2: "s2\<Colon>\<preceq>(G, L)" and
        conf_a: "normal s2 \<longrightarrow> G,store s2\<turnstile>a\<Colon>\<preceq>Class statC"
      proof (cases "normal s1")
        case True
        obtain V' where
          da_e':
          "\<lparr>prg=G,cls=accC,lcl=L\<rparr> \<turnstile>dom (locals (store s1))\<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> V'"
        proof -
          from eval_init
          have "(dom (locals (store s0))) \<subseteq> (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 \<lfloor>vf\<rfloor>\<^sub>v s2' Z" and "s2\<Colon>\<preceq>(G, L)"
          by (rule validE) (simp add: fvar [symmetric])
        moreover
        from eval_e wt_e da_e' conf_s1 wf
        have "normal s2 \<longrightarrow> G,store s2\<turnstile>a\<Colon>\<preceq>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 \<lfloor>vf\<rfloor>\<^sub>v s2' Z" and "s2\<Colon>\<preceq>(G, L)"
          by (cases rule: validE) (simp add: fvar [symmetric])+
        moreover from False eval_e have "\<not> normal s2"
          by auto
        hence "normal s2 \<longrightarrow> G,store s2\<turnstile>a\<Colon>\<preceq>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\<Colon>\<preceq>(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 = `G,A|\<Turnstile>\<Colon>{ {Normal P} e1-\<succ> {Q} }`
  have valid_e2: "\<And> a. G,A|\<Turnstile>\<Colon>{ {Q\<leftarrow>In1 a} e2-\<succ> {\<lambda>Val:i:. avar G i a ..; R} }"
    using AVar.hyps by simp
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} e1.[e2]=\<succ> {R} }"
  proof (rule valid_var_NormalI)
    fix n s0 L accC T V vf s2' Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e1.[e2]\<Colon>=T"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                  \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>e1.[e2]\<rangle>\<^sub>v\<guillemotright> V"
    assume eval: "G\<turnstile>s0 \<midarrow>e1.[e2]=\<succ>vf\<midarrow>n\<rightarrow> s2'"
    assume P: "(Normal P) Y s0 Z"
    show "R \<lfloor>vf\<rfloor>\<^sub>v s2' Z \<and> s2'\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain
        wt_e1: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e1\<Colon>-T.[]" and
        wt_e2: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e2\<Colon>-PrimT Integer"
        by (rule wt_elim_cases) simp
      from da obtain E1 where
        da_e1: "\<lparr>prg=G,cls=accC,lcl=L\<rparr> \<turnstile>dom (locals (store s0))\<guillemotright>\<langle>e1\<rangle>\<^sub>e\<guillemotright> E1" and
        da_e2: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> nrm E1 \<guillemotright>\<langle>e2\<rangle>\<^sub>e\<guillemotright> V"
        by (rule da_elim_cases) simp
      from eval obtain s1 a i s2 where
        eval_e1: "G\<turnstile>s0 \<midarrow>e1-\<succ>a\<midarrow>n\<rightarrow> s1" and
        eval_e2: "G\<turnstile>s1 \<midarrow>e2-\<succ>i\<midarrow>n\<rightarrow> 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 \<lfloor>a\<rfloor>\<^sub>e s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G, L)"
        by (rule validE)
      from Q have Q': "\<And> v. (Q\<leftarrow>In1 a) v s1 Z"
        by simp
      have "R \<lfloor>vf\<rfloor>\<^sub>v s2' Z"
      proof (cases "normal s1")
        case True
        obtain V' where
          "\<lparr>prg=G,cls=accC,lcl=L\<rparr> \<turnstile>dom (locals (store s1))\<guillemotright>\<langle>e2\<rangle>\<^sub>e\<guillemotright> V'"
        proof -
          from eval_e1  wt_e1 da_e1 wf True
          have "nrm E1 \<subseteq> 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'\<Colon>\<preceq>(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 = `G,A|\<Turnstile>\<Colon>{ {Normal P} .Init C. {Alloc G (CInst C) Q} }`
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} NewC C-\<succ> {Q} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s2 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>NewC C\<Colon>-T"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                  \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>NewC C\<rangle>\<^sub>e\<guillemotright> E"
    assume eval: "G\<turnstile>s0 \<midarrow>NewC C-\<succ>v\<midarrow>n\<rightarrow> s2"
    assume P: "(Normal P) Y s0 Z"
    show "Q \<lfloor>v\<rfloor>\<^sub>e s2 Z \<and> s2\<Colon>\<preceq>(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: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>Init C\<Colon>\<surd>"
        by auto
      obtain I where
        da_init: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>Init C\<rangle>\<^sub>s\<guillemotright> I"
        by (auto intro: da_Init [simplified] assigned.select_convs)
      from eval obtain s1 a where
        eval_init: "G\<turnstile>s0 \<midarrow>Init C\<midarrow>n\<rightarrow> s1" and
        alloc: "G\<turnstile>s1 \<midarrow>halloc CInst C\<succ>a\<rightarrow> 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) \<diamondsuit> s1 Z"
        by (rule validE)
      with alloc v have "Q \<lfloor>v\<rfloor>\<^sub>e s2 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s2\<Colon>\<preceq>(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 = `G,A|\<Turnstile>\<Colon>{ {Normal P} .init_comp_ty T. {Q} }`
  note valid_e = `G,A|\<Turnstile>\<Colon>{ {Q} e-\<succ> {\<lambda>Val:i:. abupd (check_neg i) .;
                                            Alloc G (Arr T (the_Intg i)) R}}`
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} New T[e]-\<succ> {R} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC arrT E v s3 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>New T[e]\<Colon>-arrT"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0)) \<guillemotright>\<langle>New T[e]\<rangle>\<^sub>e\<guillemotright> E"
    assume eval: "G\<turnstile>s0 \<midarrow>New T[e]-\<succ>v\<midarrow>n\<rightarrow> s3"
    assume P: "(Normal P) Y s0 Z"
    show "R \<lfloor>v\<rfloor>\<^sub>e s3 Z \<and> s3\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain
        wt_init: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>init_comp_ty T\<Colon>\<surd>" and
        wt_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e\<Colon>-PrimT Integer"
        by (rule wt_elim_cases) (auto intro: wt_init_comp_ty )
      from da obtain
        da_e:"\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E"
        by cases simp
      from eval obtain s1 i s2 a where
        eval_init: "G\<turnstile>s0 \<midarrow>init_comp_ty T\<midarrow>n\<rightarrow> s1" and
        eval_e: "G\<turnstile>s1 \<midarrow>e-\<succ>i\<midarrow>n\<rightarrow> s2" and
        alloc: "G\<turnstile>abupd (check_neg i) s2 \<midarrow>halloc Arr T (the_Intg i)\<succ>a\<rightarrow> s3" and
        v: "v=Addr a"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      obtain I where
        da_init:
        "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0)) \<guillemotright>\<langle>init_comp_ty T\<rangle>\<^sub>s\<guillemotright> I"
      proof (cases "\<exists>C. T = Class C")
        case True
        thus ?thesis
          by - (rule that, (auto intro: da_Init [simplified]
                                        assigned.select_convs
                              simp add: init_comp_ty_def))
         (* simplified: to rewrite \<langle>Init C\<rangle> to In1r (Init C) *)
      next
        case False
        thus ?thesis
          by - (rule that, (auto intro: da_Skip [simplified]
                                      assigned.select_convs
                           simp add: init_comp_ty_def))
         (* 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 \<diamondsuit> s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G, L)"
        by (rule validE)
      obtain E' where
       "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s1)) \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E'"
      proof -
        from eval_init
        have "dom (locals (store s0)) \<subseteq> 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 "(\<lambda>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 \<lfloor>v\<rfloor>\<^sub>e s3 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s3\<Colon>\<preceq>(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 = `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} }`
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} Cast T e-\<succ> {Q} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC castT E v s2 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>Cast T e\<Colon>-castT"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0)) \<guillemotright>\<langle>Cast T e\<rangle>\<^sub>e\<guillemotright> E"
    assume eval: "G\<turnstile>s0 \<midarrow>Cast T e-\<succ>v\<midarrow>n\<rightarrow> s2"
    assume P: "(Normal P) Y s0 Z"
    show "Q \<lfloor>v\<rfloor>\<^sub>e s2 Z \<and> s2\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain eT where
        wt_e: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-eT"
        by cases simp
      from da obtain
        da_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E"
        by cases simp
      from eval obtain s1 where
        eval_e: "G\<turnstile>s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1" and
        s2: "s2 = abupd (raise_if (\<not> G,snd s1\<turnstile>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 "(\<lambda>Val:v:. \<lambda>s.. abupd (raise_if (\<not> G,s\<turnstile>v fits T) ClassCast) .;
                  Q\<leftarrow>In1 v) \<lfloor>v\<rfloor>\<^sub>e s1 Z"
        by (rule validE)
      with s2 have "Q \<lfloor>v\<rfloor>\<^sub>e s2 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s2\<Colon>\<preceq>(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|\<Turnstile>\<Colon>{ {Normal P} e-\<succ>
               {\<lambda>Val:v:. \<lambda>s.. Q\<leftarrow>In1 (Bool (v \<noteq> Null \<and> G,s\<turnstile>v fits RefT T))} }"
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} e InstOf T-\<succ> {Q} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC instT E v s1 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e InstOf T\<Colon>-instT"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>e InstOf T\<rangle>\<^sub>e\<guillemotright> E"
    assume eval: "G\<turnstile>s0 \<midarrow>e InstOf T-\<succ>v\<midarrow>n\<rightarrow> s1"
    assume P: "(Normal P) Y s0 Z"
    show "Q \<lfloor>v\<rfloor>\<^sub>e s1 Z \<and> s1\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain eT where
        wt_e: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-eT"
        by cases simp
      from da obtain
        da_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E"
        by cases simp
      from eval obtain a where
        eval_e: "G\<turnstile>s0 \<midarrow>e-\<succ>a\<midarrow>n\<rightarrow> s1" and
        v: "v = Bool (a \<noteq> Null \<and> G,store s1\<turnstile>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 "(\<lambda>Val:v:. \<lambda>s.. Q\<leftarrow>In1 (Bool (v \<noteq> Null \<and> G,s\<turnstile>v fits RefT T)))
              \<lfloor>a\<rfloor>\<^sub>e s1 Z"
        by (rule validE)
      with v have "Q \<lfloor>v\<rfloor>\<^sub>e s1 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s1\<Colon>\<preceq>(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Lit A P v)
  show "G,A|\<Turnstile>\<Colon>{ {Normal (P\<leftarrow>In1 v)} Lit v-\<succ> {P} }"
  proof (rule valid_expr_NormalI)
    fix n L s0 s1 v'  Y Z
    assume conf_s0: "s0\<Colon>\<preceq>(G, L)"
    assume normal_s0: " normal s0"
    assume eval: "G\<turnstile>s0 \<midarrow>Lit v-\<succ>v'\<midarrow>n\<rightarrow> s1"
    assume P: "(Normal (P\<leftarrow>In1 v)) Y s0 Z"
    show "P \<lfloor>v'\<rfloor>\<^sub>e s1 Z \<and> s1\<Colon>\<preceq>(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|\<Turnstile>\<Colon>{ {Normal P}e-\<succ>{\<lambda>Val:v:. Q\<leftarrow>In1 (eval_unop unop v)} }"
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} UnOp unop e-\<succ> {Q} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s1 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>UnOp unop e\<Colon>-T"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>UnOp unop e\<rangle>\<^sub>e\<guillemotright>E"
    assume eval: "G\<turnstile>s0 \<midarrow>UnOp unop e-\<succ>v\<midarrow>n\<rightarrow> s1"
    assume P: "(Normal P) Y s0 Z"
    show "Q \<lfloor>v\<rfloor>\<^sub>e s1 Z \<and> s1\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain eT where
        wt_e: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-eT"
        by cases simp
      from da obtain
        da_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E"
        by cases simp
      from eval obtain ve where
        eval_e: "G\<turnstile>s0 \<midarrow>e-\<succ>ve\<midarrow>n\<rightarrow> 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 "(\<lambda>Val:v:. Q\<leftarrow>In1 (eval_unop unop v)) \<lfloor>ve\<rfloor>\<^sub>e s1 Z"
        by (rule validE)
      with v have "Q \<lfloor>v\<rfloor>\<^sub>e s1 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s1\<Colon>\<preceq>(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|\<Turnstile>\<Colon>{ {Normal P} e1-\<succ> {Q} }"
  have valid_e2: "\<And> v1.  G,A|\<Turnstile>\<Colon>{ {Q\<leftarrow>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|\<Turnstile>\<Colon>{ {Normal P} BinOp binop e1 e2-\<succ> {R} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s2 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>BinOp binop e1 e2\<Colon>-T"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                  \<turnstile>dom (locals (store s0)) \<guillemotright>\<langle>BinOp binop e1 e2\<rangle>\<^sub>e\<guillemotright> E"
    assume eval: "G\<turnstile>s0 \<midarrow>BinOp binop e1 e2-\<succ>v\<midarrow>n\<rightarrow> s2"
    assume P: "(Normal P) Y s0 Z"
    show "R \<lfloor>v\<rfloor>\<^sub>e s2 Z \<and> s2\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain e1T e2T where
        wt_e1: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e1\<Colon>-e1T" and
        wt_e2: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e2\<Colon>-e2T" and
        wt_binop: "wt_binop G binop e1T e2T"
        by cases simp
      have wt_Skip: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>Skip\<Colon>\<surd>"
        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: "\<lparr>prg=G,cls=accC,lcl=L\<rparr> \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>e1\<rangle>\<^sub>e\<guillemotright> E1"
        by cases simp+
      from eval obtain v1 s1 v2 where
        eval_e1: "G\<turnstile>s0 \<midarrow>e1-\<succ>v1\<midarrow>n\<rightarrow> s1" and
        eval_e2: "G\<turnstile>s1 \<midarrow>(if need_second_arg binop v1 then \<langle>e2\<rangle>\<^sub>e else \<langle>Skip\<rangle>\<^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 \<lfloor>v1\<rfloor>\<^sub>e s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G,L)"
        by (rule validE)
      from Q have Q': "\<And> v. (Q\<leftarrow>In1 v1) v s1 Z"
        by simp
      have "(\<lambda>Val:v2:. R\<leftarrow>In1 (eval_binop binop v1 v2)) \<lfloor>v2\<rfloor>\<^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\<turnstile>v1\<Colon>\<preceq>e1T"
          by (rule evaln_type_sound [elim_format]) (insert True,simp)
        from eval_e1
        have "G\<turnstile>s0 \<midarrow>e1-\<succ>v1\<rightarrow> 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: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> 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 "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                  \<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 \<lfloor>v\<rfloor>\<^sub>e s2 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s2\<Colon>\<preceq>(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Super A P)
  show "G,A|\<Turnstile>\<Colon>{ {Normal (\<lambda>s.. P\<leftarrow>In1 (val_this s))} Super-\<succ> {P} }"
  proof (rule valid_expr_NormalI)
    fix n L s0 s1 v  Y Z
    assume conf_s0: "s0\<Colon>\<preceq>(G, L)"
    assume normal_s0: " normal s0"
    assume eval: "G\<turnstile>s0 \<midarrow>Super-\<succ>v\<midarrow>n\<rightarrow> s1"
    assume P: "(Normal (\<lambda>s.. P\<leftarrow>In1 (val_this s))) Y s0 Z"
    show "P \<lfloor>v\<rfloor>\<^sub>e s1 Z \<and> s1\<Colon>\<preceq>(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 = `G,A|\<Turnstile>\<Colon>{ {Normal P} var=\<succ> {\<lambda>Var:(v, f):. Q\<leftarrow>In1 v} }`
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} Acc var-\<succ> {Q} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s1 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>Acc var\<Colon>-T"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>Acc var\<rangle>\<^sub>e\<guillemotright>E"
    assume eval: "G\<turnstile>s0 \<midarrow>Acc var-\<succ>v\<midarrow>n\<rightarrow> s1"
    assume P: "(Normal P) Y s0 Z"
    show "Q \<lfloor>v\<rfloor>\<^sub>e s1 Z \<and> s1\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain
        wt_var: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>var\<Colon>=T"
        by cases simp
      from da obtain V where
        da_var: "\<lparr>prg=G,cls=accC,lcl=L\<rparr> \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>var\<rangle>\<^sub>v\<guillemotright> V"
        by (cases "\<exists> n. var=LVar n") (insert da.LVar,auto elim!: da_elim_cases)
      from eval obtain upd where
        eval_var: "G\<turnstile>s0 \<midarrow>var=\<succ>(v, upd)\<midarrow>n\<rightarrow> 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 "(\<lambda>Var:(v, f):. Q\<leftarrow>In1 v) \<lfloor>(v, upd)\<rfloor>\<^sub>v s1 Z"
        by (rule validE)
      then have "Q \<lfloor>v\<rfloor>\<^sub>e s1 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s1\<Colon>\<preceq>(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 = `G,A|\<Turnstile>\<Colon>{ {Normal P} var=\<succ> {Q} }`
  have valid_e: "\<And> 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|\<Turnstile>\<Colon>{ {Normal P} var:=e-\<succ> {R} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s3 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>var:=e\<Colon>-T"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>var:=e\<rangle>\<^sub>e\<guillemotright>E"
    assume eval: "G\<turnstile>s0 \<midarrow>var:=e-\<succ>v\<midarrow>n\<rightarrow> s3"
    assume P: "(Normal P) Y s0 Z"
    show "R \<lfloor>v\<rfloor>\<^sub>e s3 Z \<and> s3\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain varT  where
        wt_var: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>var\<Colon>=varT" and
        wt_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e\<Colon>-T"
        by cases simp
      from eval obtain w upd s1 s2 where
        eval_var: "G\<turnstile>s0 \<midarrow>var=\<succ>(w, upd)\<midarrow>n\<rightarrow> s1" and
        eval_e: "G\<turnstile>s1 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s2" and
        s3: "s3=assign upd v s2"
        using normal_s0 by (auto elim: evaln_elim_cases)
      have "R \<lfloor>v\<rfloor>\<^sub>e s3 Z"
      proof (cases "\<exists> vn. var = LVar vn")
        case False
        with da obtain V where
          da_var: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                      \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>var\<rangle>\<^sub>v\<guillemotright> V" and
          da_e:   "\<lparr>prg=G,cls=accC,lcl=L\<rparr> \<turnstile> nrm V \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E"
          by cases simp+
        from valid_var P valid_A conf_s0 eval_var wt_var da_var
        obtain Q: "Q \<lfloor>(w,upd)\<rfloor>\<^sub>v s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G,L)"
          by (rule validE)
        hence Q': "\<And> v. (Q\<leftarrow>In2 (w,upd)) v s1 Z"
          by simp
        have "(\<lambda>Val:v:. assign (snd (w,upd)) v .; R) \<lfloor>v\<rfloor>\<^sub>e s2 Z"
        proof (cases "normal s1")
          case True
          obtain E' where
            da_e': "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s1)) \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E'"
          proof -
            from eval_var wt_var da_var wf True
            have "nrm V \<subseteq>  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 \<lfloor>v\<rfloor>\<^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:   "\<lparr>prg=G,cls=accC,lcl=L\<rparr> \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E"
          by cases simp+
        from da.LVar vn obtain  V where
          da_var: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                      \<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 \<lfloor>(w,upd)\<rfloor>\<^sub>v s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G,L)"
          by (rule validE)
        hence Q': "\<And> v. (Q\<leftarrow>In2 (w,upd)) v s1 Z"
          by simp
        have "(\<lambda>Val:v:. assign (snd (w,upd)) v .; R) \<lfloor>v\<rfloor>\<^sub>e s2 Z"
        proof (cases "normal s1")
          case True
          obtain E' where
            da_e': "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                       \<turnstile> dom (locals (store s1)) \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E'"
          proof -
            from eval_var
            have "dom (locals (store s0)) \<subseteq> 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 \<lfloor>v\<rfloor>\<^sub>e s3 Z"
          by simp
      qed
      moreover
      from eval wt da conf_s0 wf
      have "s3\<Colon>\<preceq>(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 = `G,A|\<Turnstile>\<Colon>{ {Normal P} e0-\<succ> {P'} }`
  have valid_then_else:"\<And> b.  G,A|\<Turnstile>\<Colon>{ {P'\<leftarrow>=b} (if b then e1 else e2)-\<succ> {Q} }"
    using Cond.hyps by simp
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} e0 ? e1 : e2-\<succ> {Q} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s2 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e0 ? e1 : e2\<Colon>-T"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>e0 ? e1:e2\<rangle>\<^sub>e\<guillemotright>E"
    assume eval: "G\<turnstile>s0 \<midarrow>e0 ? e1 : e2-\<succ>v\<midarrow>n\<rightarrow> s2"
    assume P: "(Normal P) Y s0 Z"
    show "Q \<lfloor>v\<rfloor>\<^sub>e s2 Z \<and> s2\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain T1 T2 where
        wt_e0: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e0\<Colon>-PrimT Boolean" and
        wt_e1: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e1\<Colon>-T1" and
        wt_e2: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e2\<Colon>-T2"
        by cases simp
      from da obtain E0 E1 E2 where
        da_e0: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>e0\<rangle>\<^sub>e\<guillemotright> E0" and
        da_e1: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                 \<turnstile>(dom (locals (store s0)) \<union> assigns_if True e0)\<guillemotright>\<langle>e1\<rangle>\<^sub>e\<guillemotright> E1" and
        da_e2: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                 \<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\<turnstile>s0 \<midarrow>e0-\<succ>b\<midarrow>n\<rightarrow> s1" and
        eval_then_else: "G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>v\<midarrow>n\<rightarrow> 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' \<lfloor>b\<rfloor>\<^sub>e s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G,L)"
        by (rule validE)
      hence P': "\<And> v. (P'\<leftarrow>=(the_Bool b)) v s1 Z"
        by (cases "normal s1") auto
      have "Q \<lfloor>v\<rfloor>\<^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:
          "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>(if the_Bool b then e1 else e2)\<Colon>-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\<turnstile>s0 \<midarrow>e0-\<succ>b\<rightarrow> s1"
            by (rule evaln_eval)
          hence
            "dom (locals (store s0)) \<subseteq> 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 \<subseteq> dom (locals (store s1))"
            by (rule assigns_if_good_approx')
          ultimately show ?thesis by (rule Un_least)
        qed
        obtain E' where
          da_then_else:
          "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
              \<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\<Colon>\<preceq>(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 = `G,A|\<Turnstile>\<Colon>{ {Normal P} e-\<succ> {Q} }`
  have valid_args: "\<And> a. G,A|\<Turnstile>\<Colon>{ {Q\<leftarrow>In1 a} args\<doteq>\<succ> {R a} }"
    using Call.hyps by simp
  have valid_methd: "\<And> 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|\<Turnstile>\<Colon>{ {Normal P} {accC',statT,mode}e\<cdot>mn( {pTs'}args)-\<succ> {S} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s5 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>{accC',statT,mode}e\<cdot>mn( {pTs'}args)\<Colon>-T"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))
                   \<guillemotright>\<langle>{accC',statT,mode}e\<cdot>mn( {pTs'}args)\<rangle>\<^sub>e\<guillemotright> E"
    assume eval: "G\<turnstile>s0 \<midarrow>{accC',statT,mode}e\<cdot>mn( {pTs'}args)-\<succ>v\<midarrow>n\<rightarrow> s5"
    assume P: "(Normal P) Y s0 Z"
    show "S \<lfloor>v\<rfloor>\<^sub>e s5 Z \<and> s5\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain pTs statDeclT statM where
                 wt_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e\<Colon>-RefT statT" and
              wt_args: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>args\<Colon>\<doteq>pTs" and
                statM: "max_spec G accC statT \<lparr>name=mn,parTs=pTs\<rparr>
                         = {((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: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> (dom (locals (store s0)))\<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> C" and
        da_args: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> nrm C \<guillemotright>\<langle>args\<rangle>\<^sub>l\<guillemotright> E"
        by cases simp
      from eval eq_accC_accC' obtain a s1 vs s2 s3 s3' s4 invDeclC where
        evaln_e: "G\<turnstile>s0 \<midarrow>e-\<succ>a\<midarrow>n\<rightarrow> s1" and
        evaln_args: "G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<midarrow>n\<rightarrow> 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 \<lparr>name=mn,parTs=pTs'\<rparr> mode a vs s2" and
        check: "s3' = check_method_access G
                           accC' statT mode \<lparr>name = mn, parTs = pTs'\<rparr> a s3" and
        evaln_methd:
           "G\<turnstile>s3' \<midarrow>Methd invDeclC  \<lparr>name=mn,parTs=pTs'\<rparr>-\<succ>v\<midarrow>n\<rightarrow> 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\<turnstile>s0 \<midarrow>e-\<succ>a\<rightarrow> s1"
        by (rule evaln_eval)

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

      from valid_e P valid_A conf_s0 evaln_e wt_e da_e
      obtain "Q \<lfloor>a\<rfloor>\<^sub>e s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G,L)"
        by (rule validE)
      hence Q: "\<And> v. (Q\<leftarrow>In1 a) v s1 Z"
        by simp
      obtain
        R: "(R a) \<lfloor>vs\<rfloor>\<^sub>l s2 Z" and
        conf_s2: "s2\<Colon>\<preceq>(G,L)" and
        s2_no_return: "abrupt s2 \<noteq> Some (Jump Ret)"
      proof (cases "normal s1")
        case True
        obtain E' where
          da_args':
          "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s1)) \<guillemotright>\<langle>args\<rangle>\<^sub>l\<guillemotright> E'"
        proof -
          from evaln_e wt_e da_e wf True
          have "nrm C \<subseteq>  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) \<lfloor>vs\<rfloor>\<^sub>l s2 Z" "s2\<Colon>\<preceq>(G,L)"
          by (rule validE)
        moreover
        from evaln_args
        have e: "G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<rightarrow> s2"
          by (rule evaln_eval)
        from this s1_no_return wt_args wf
        have "abrupt s2 \<noteq> Some (Jump Ret)"
          by (rule eval_expression_list_no_jump
                 [where ?Env="\<lparr>prg=G,cls=accC,lcl=L\<rparr>",simplified])
        ultimately show ?thesis ..
      next
        case False
        with valid_args Q valid_A conf_s1 evaln_args
        obtain "(R a) \<lfloor>vs\<rfloor>\<^sub>l s2 Z" "s2\<Colon>\<preceq>(G,L)"
          by (cases rule: validE) iprover+
        moreover
        from False evaln_args have "s2=s1"
          by auto
        with s1_no_return have "abrupt s2 \<noteq> 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)"
        by (cases s2,cases mode) (auto simp add: init_lvars_def2 )
      obtain l where
        l: "l = locals (store s2)"
        by simp

      from eval wt da conf_s0 wf
      have conf_s5: "s5\<Colon>\<preceq>(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      let "PROP ?R" = "\<And> v.
             (R a\<leftarrow>In3 vs \<and>.
                 (\<lambda>s. invDeclC = 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 invDeclC \<lparr>name = mn, parTs = pTs'\<rparr> mode a vs \<and>.
                  (\<lambda>s. normal s \<longrightarrow> G\<turnstile>mode\<rightarrow>invC\<preceq>statT)
               ) v s3' Z"
      {
        assume abrupt_s3: "\<not> normal s3"
        have "S \<lfloor>v\<rfloor>\<^sub>e s5 Z"
        proof -
          from abrupt_s3 check have eq_s3'_s3: "s3'=s3"
            by (auto simp add: check_method_access_def Let_def)
          with R s3 invDeclC invC l abrupt_s3
          have R': "PROP ?R"
            by auto
          have conf_s3': "s3'\<Colon>\<preceq>(G, empty)"
           (* we need an arbirary environment (here empty) that s2' conforms to
              to apply validE *)
          proof -
            from s2_no_return s3
            have "abrupt s3 \<noteq> Some (Jump Ret)"
              by (cases s2) (auto simp add: init_lvars_def2 split: split_if_asm)
            moreover
            obtain abr2 str2 where s2: "s2=(abr2,str2)"
              by (cases s2)
            from s3 s2 conf_s2 have "(abrupt s3,str2)\<Colon>\<preceq>(G, L)"
              by (auto simp add: init_lvars_def2 split: split_if_asm)
            ultimately show ?thesis
              using s3 s2 eq_s3'_s3
              apply (simp add: init_lvars_def2)
              apply (rule conforms_set_locals [OF _ wlconf_empty])
              by auto
          qed
          from valid_methd R' valid_A conf_s3' evaln_methd abrupt_s3 eq_s3'_s3
          have "(set_lvars l .; S) \<lfloor>v\<rfloor>\<^sub>e s4 Z"
            by (cases rule: validE) simp+
          with s5 l show ?thesis
            by simp
        qed
      } note abrupt_s3_lemma = this

      have "S \<lfloor>v\<rfloor>\<^sub>e s5 Z"
      proof (cases "normal s2")
        case False
        with s3 have abrupt_s3: "\<not> normal s3"
          by (cases s2) (simp add: init_lvars_def2)
        thus ?thesis
          by (rule abrupt_s3_lemma)
      next
        case True
        note normal_s2 = this
        with evaln_args
        have normal_s1: "normal s1"
          by (rule evaln_no_abrupt)
        obtain E' where
          da_args':
          "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s1)) \<guillemotright>\<langle>args\<rangle>\<^sub>l\<guillemotright> E'"
        proof -
          from evaln_e wt_e da_e wf normal_s1
          have "nrm C \<subseteq>  dom (locals (store s1))"
            by (cases rule: da_good_approx_evalnE) iprover
          with da_args show thesis
            by (rule da_weakenE) (rule that)
        qed
        from evaln_args
        have eval_args: "G\<turnstile>s1 \<midarrow>args\<doteq>\<succ>vs\<rightarrow> s2"
          by (rule evaln_eval)
        from evaln_e wt_e da_e conf_s0 wf
        have conf_a: "G, store s1\<turnstile>a\<Colon>\<preceq>RefT statT"
          by (rule evaln_type_sound [elim_format]) (insert normal_s1,simp)
        with normal_s1 normal_s2 eval_args
        have conf_a_s2: "G, store s2\<turnstile>a\<Colon>\<preceq>RefT statT"
          by (auto dest: eval_gext)
        from evaln_args wt_args da_args' conf_s1 wf
        have conf_args: "list_all2 (conf G (store s2)) vs pTs"
          by (rule evaln_type_sound [elim_format]) (insert normal_s2,simp)
        from statM
        obtain
          statM': "(statDeclT,statM)\<in>mheads G accC statT \<lparr>name=mn,parTs=pTs'\<rparr>"
          and
          pTs_widen: "G\<turnstile>pTs[\<preceq>]pTs'"
          by (blast dest: max_spec2mheads)
        show ?thesis
        proof (cases "normal s3")
          case False
          thus ?thesis
            by (rule abrupt_s3_lemma)
        next
          case True
          note normal_s3 = this
          with s3 have notNull: "mode = IntVir \<longrightarrow> a \<noteq> Null"
            by (cases s2) (auto simp add: init_lvars_def2)
          from conf_s2 conf_a_s2 wf notNull invC
          have dynT_prop: "G\<turnstile>mode\<rightarrow>invC\<preceq>statT"
            by (cases s2) (auto intro: DynT_propI)

          with wt_e statM' invC mode wf
          obtain dynM where
            dynM: "dynlookup G statT invC  \<lparr>name=mn,parTs=pTs'\<rparr> = Some dynM" and
            acc_dynM: "G \<turnstile>Methd  \<lparr>name=mn,parTs=pTs'\<rparr> dynM
                            in invC dyn_accessible_from accC"
            by (force dest!: call_access_ok)
          with invC' check eq_accC_accC'
          have eq_s3'_s3: "s3'=s3"
            by (auto simp add: check_method_access_def Let_def)

          with dynT_prop R s3 invDeclC invC l
          have R': "PROP ?R"
            by auto

          from dynT_prop wf wt_e statM' mode invC invDeclC dynM
          obtain
            dynM: "dynlookup G statT invC  \<lparr>name=mn,parTs=pTs'\<rparr> = Some dynM" and
            wf_dynM: "wf_mdecl G invDeclC (\<lparr>name=mn,parTs=pTs'\<rparr>,mthd dynM)" and
              dynM': "methd G invDeclC \<lparr>name=mn,parTs=pTs'\<rparr> = Some dynM" and
            iscls_invDeclC: "is_class G invDeclC" and
                 invDeclC': "invDeclC = declclass dynM" and
              invC_widen: "G\<turnstile>invC\<preceq>\<^sub>C invDeclC" and
             resTy_widen: "G\<turnstile>resTy dynM\<preceq>resTy statM" and
            is_static_eq: "is_static dynM = is_static statM" and
            involved_classes_prop:
             "(if invmode statM e = IntVir
               then \<forall>statC. statT = ClassT statC \<longrightarrow> G\<turnstile>invC\<preceq>\<^sub>C statC
               else ((\<exists>statC. statT = ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C invDeclC) \<or>
                     (\<forall>statC. statT \<noteq> ClassT statC \<and> invDeclC = Object)) \<and>
                      statDeclT = ClassT invDeclC)"
            by (cases rule: DynT_mheadsE) simp
          obtain L' where
            L':"L'=(\<lambda> k.
                    (case k of
                       EName e
                       \<Rightarrow> (case e of
                             VNam v
                             \<Rightarrow>(table_of (lcls (mbody (mthd dynM)))
                                (pars (mthd dynM)[\<mapsto>]pTs')) v
                           | Res \<Rightarrow> Some (resTy dynM))
                     | This \<Rightarrow> if is_static statM
                               then None else Some (Class invDeclC)))"
            by simp
          from wf_dynM [THEN wf_mdeclD1, THEN conjunct1] normal_s2 conf_s2 wt_e
            wf eval_args conf_a mode notNull wf_dynM involved_classes_prop
          have conf_s3: "s3\<Colon>\<preceq>(G,L')"
            apply -
               (* FIXME confomrs_init_lvars should be
                  adjusted to be more directy applicable *)
            apply (drule conforms_init_lvars [of G invDeclC
                    "\<lparr>name=mn,parTs=pTs'\<rparr>" dynM "store s2" vs pTs "abrupt s2"
                    L statT invC a "(statDeclT,statM)" e])
            apply (rule wf)
            apply (rule conf_args)
            apply (simp add: pTs_widen)
            apply (cases s2,simp)
            apply (rule dynM')
            apply (force dest: ty_expr_is_type)
            apply (rule invC_widen)
            apply (force dest: eval_gext)
            apply simp
            apply simp
            apply (simp add: invC)
            apply (simp add: invDeclC)
            apply (simp add: normal_s2)
            apply (cases s2, simp add: L' init_lvars_def2 s3
                             cong add: lname.case_cong ename.case_cong)
            done
          with eq_s3'_s3 have conf_s3': "s3'\<Colon>\<preceq>(G,L')" by simp
          from is_static_eq wf_dynM L'
          obtain mthdT where
            "\<lparr>prg=G,cls=invDeclC,lcl=L'\<rparr>
               \<turnstile>Body invDeclC (stmt (mbody (mthd dynM)))\<Colon>-mthdT" and
            mthdT_widen: "G\<turnstile>mthdT\<preceq>resTy dynM"
            by - (drule wf_mdecl_bodyD,
                  auto simp add: callee_lcl_def
                       cong add: lname.case_cong ename.case_cong)
          with dynM' iscls_invDeclC invDeclC'
          have
            wt_methd:
            "\<lparr>prg=G,cls=invDeclC,lcl=L'\<rparr>
               \<turnstile>(Methd invDeclC \<lparr>name = mn, parTs = pTs'\<rparr>)\<Colon>-mthdT"
            by (auto intro: wt.Methd)
          obtain M where
            da_methd:
            "\<lparr>prg=G,cls=invDeclC,lcl=L'\<rparr>
               \<turnstile> dom (locals (store s3'))
                   \<guillemotright>\<langle>Methd invDeclC \<lparr>name=mn,parTs=pTs'\<rparr>\<rangle>\<^sub>e\<guillemotright> M"
          proof -
            from wf_dynM
            obtain M' where
              da_body:
              "\<lparr>prg=G, cls=invDeclC
               ,lcl=callee_lcl invDeclC \<lparr>name = mn, parTs = pTs'\<rparr> (mthd dynM)
               \<rparr> \<turnstile> parameters (mthd dynM) \<guillemotright>\<langle>stmt (mbody (mthd dynM))\<rangle>\<guillemotright> M'" and
              res: "Result \<in> nrm M'"
              by (rule wf_mdeclE) iprover
            from da_body is_static_eq L' have
              "\<lparr>prg=G, cls=invDeclC,lcl=L'\<rparr>
                 \<turnstile> parameters (mthd dynM) \<guillemotright>\<langle>stmt (mbody (mthd dynM))\<rangle>\<guillemotright> M'"
              by (simp add: callee_lcl_def
                  cong add: lname.case_cong ename.case_cong)
            moreover have "parameters (mthd dynM) \<subseteq>  dom (locals (store s3'))"
            proof -
              from is_static_eq
              have "(invmode (mthd dynM) e) = (invmode statM e)"
                by (simp add: invmode_def)
              moreover
              have "length (pars (mthd dynM)) = length vs"
              proof -
                from normal_s2 conf_args
                have "length vs = length pTs"
                  by (simp add: list_all2_iff)
                also from pTs_widen
                have "\<dots> = length pTs'"
                  by (simp add: widens_def list_all2_iff)
                also from wf_dynM
                have "\<dots> = length (pars (mthd dynM))"
                  by (simp add: wf_mdecl_def wf_mhead_def)
                finally show ?thesis ..
              qed
              moreover note s3 dynM' is_static_eq normal_s2 mode
              ultimately
              have "parameters (mthd dynM) = dom (locals (store s3))"
                using dom_locals_init_lvars
                  [of "mthd dynM" G invDeclC "\<lparr>name=mn,parTs=pTs'\<rparr>" vs e a s2]
                by simp
              thus ?thesis using eq_s3'_s3 by simp
            qed
            ultimately obtain M2 where
              da:
              "\<lparr>prg=G, cls=invDeclC,lcl=L'\<rparr>
                \<turnstile> dom (locals (store s3')) \<guillemotright>\<langle>stmt (mbody (mthd dynM))\<rangle>\<guillemotright> M2" and
              M2: "nrm M' \<subseteq> nrm M2"
              by (rule da_weakenE)
            from res M2 have "Result \<in> nrm M2"
              by blast
            moreover from wf_dynM
            have "jumpNestingOkS {Ret} (stmt (mbody (mthd dynM)))"
              by (rule wf_mdeclE)
            ultimately
            obtain M3 where
              "\<lparr>prg=G, cls=invDeclC,lcl=L'\<rparr> \<turnstile> dom (locals (store s3'))
                     \<guillemotright>\<langle>Body (declclass dynM) (stmt (mbody (mthd dynM)))\<rangle>\<guillemotright> M3"
              using da
              by (iprover intro: da.Body assigned.select_convs)
            from _ this [simplified]
            show thesis
              by (rule da.Methd [simplified,elim_format])
                 (auto intro: dynM' that)
          qed
          from valid_methd R' valid_A conf_s3' evaln_methd wt_methd da_methd
          have "(set_lvars l .; S) \<lfloor>v\<rfloor>\<^sub>e s4 Z"
            by (cases rule: validE) iprover+
          with s5 l show ?thesis
            by simp
        qed
      qed
      with conf_s5 show ?thesis by iprover
    qed
  qed
next
  case (Methd A P Q ms)
  note valid_body = `G,A \<union> {{P} Methd-\<succ> {Q} | ms}|\<Turnstile>\<Colon>{{P} body G-\<succ> {Q} | ms}`
  show "G,A|\<Turnstile>\<Colon>{{P} Methd-\<succ> {Q} | ms}"
    by (rule Methd_sound) (rule Methd.hyps)
next
  case (Body A P D Q c R)
  note valid_init = `G,A|\<Turnstile>\<Colon>{ {Normal P} .Init D. {Q} }`
  note valid_c = `G,A|\<Turnstile>\<Colon>{ {Q} .c.
              {\<lambda>s.. abupd (absorb Ret) .; R\<leftarrow>In1 (the (locals s Result))} }`
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} Body D c-\<succ> {R} }"
  proof (rule valid_expr_NormalI)
    fix n s0 L accC T E v s4 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>Body D c\<Colon>-T"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>Body D c\<rangle>\<^sub>e\<guillemotright>E"
    assume eval: "G\<turnstile>s0 \<midarrow>Body D c-\<succ>v\<midarrow>n\<rightarrow> s4"
    assume P: "(Normal P) Y s0 Z"
    show "R \<lfloor>v\<rfloor>\<^sub>e s4 Z \<and> s4\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain
        iscls_D: "is_class G D" and
        wt_init: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>Init D\<Colon>\<surd>" and
        wt_c: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>c\<Colon>\<surd>"
        by cases auto
      obtain I where
        da_init:"\<lparr>prg=G,cls=accC,lcl=L\<rparr> \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>Init D\<rangle>\<^sub>s\<guillemotright> I"
        by (auto intro: da_Init [simplified] assigned.select_convs)
      from da obtain C where
        da_c: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> (dom (locals (store s0)))\<guillemotright>\<langle>c\<rangle>\<^sub>s\<guillemotright> C" and
        jmpOk: "jumpNestingOkS {Ret} c"
        by cases simp
      from eval obtain s1 s2 s3 where
        eval_init: "G\<turnstile>s0 \<midarrow>Init D\<midarrow>n\<rightarrow> s1" and
        eval_c: "G\<turnstile>s1 \<midarrow>c\<midarrow>n\<rightarrow> s2" and
        v: "v = the (locals (store s2) Result)" and
        s3: "s3 =(if \<exists>l. abrupt s2 = Some (Jump (Break l)) \<or>
                         abrupt s2 = Some (Jump (Cont l))
                  then abupd (\<lambda>x. Some (Error CrossMethodJump)) s2 else s2)"and
        s4: "s4 = abupd (absorb Ret) s3"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      obtain C' where
        da_c': "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> (dom (locals (store s1)))\<guillemotright>\<langle>c\<rangle>\<^sub>s\<guillemotright> C'"
      proof -
        from eval_init
        have "(dom (locals (store s0))) \<subseteq> (dom (locals (store s1)))"
          by (rule dom_locals_evaln_mono_elim)
        with da_c show thesis by (rule da_weakenE) (rule that)
      qed
      from valid_init P valid_A conf_s0 eval_init wt_init da_init
      obtain Q: "Q \<diamondsuit> s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G,L)"
        by (rule validE)
      from valid_c Q valid_A conf_s1 eval_c wt_c da_c'
      have R: "(\<lambda>s.. abupd (absorb Ret) .; R\<leftarrow>In1 (the (locals s Result)))
                \<diamondsuit> s2 Z"
        by (rule validE)
      have "s3=s2"
      proof -
        from eval_init [THEN evaln_eval] wf
        have s1_no_jmp: "\<And> j. abrupt s1 \<noteq> Some (Jump j)"
          by - (rule eval_statement_no_jump [OF _ _ _ wt_init],
                insert normal_s0,auto)
        from eval_c [THEN evaln_eval] _ wt_c wf
        have "\<And> j. abrupt s2 = Some (Jump j) \<Longrightarrow> j=Ret"
          by (rule jumpNestingOk_evalE) (auto intro: jmpOk simp add: s1_no_jmp)
        moreover note s3
        ultimately show ?thesis
          by (force split: split_if)
      qed
      with R v s4
      have "R \<lfloor>v\<rfloor>\<^sub>e s4 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s4\<Colon>\<preceq>(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Nil A P)
  show "G,A|\<Turnstile>\<Colon>{ {Normal (P\<leftarrow>\<lfloor>[]\<rfloor>\<^sub>l)} []\<doteq>\<succ> {P} }"
  proof (rule valid_expr_list_NormalI)
    fix s0 s1 vs n L Y Z
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume eval: "G\<turnstile>s0 \<midarrow>[]\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s1"
    assume P: "(Normal (P\<leftarrow>\<lfloor>[]\<rfloor>\<^sub>l)) Y s0 Z"
    show "P \<lfloor>vs\<rfloor>\<^sub>l s1 Z \<and> s1\<Colon>\<preceq>(G, L)"
    proof -
      from eval obtain "vs=[]" "s1=s0"
        using normal_s0 by (auto elim: evaln_elim_cases)
      with P conf_s0 show ?thesis
        by simp
    qed
  qed
next
  case (Cons A P e Q es R)
  note valid_e = `G,A|\<Turnstile>\<Colon>{ {Normal P} e-\<succ> {Q} }`
  have valid_es: "\<And> v. G,A|\<Turnstile>\<Colon>{ {Q\<leftarrow>\<lfloor>v\<rfloor>\<^sub>e} es\<doteq>\<succ> {\<lambda>Vals:vs:. R\<leftarrow>\<lfloor>(v # vs)\<rfloor>\<^sub>l} }"
    using Cons.hyps by simp
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} e # es\<doteq>\<succ> {R} }"
  proof (rule valid_expr_list_NormalI)
    fix n s0 L accC T E v s2 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e # es\<Colon>\<doteq>T"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0)) \<guillemotright>\<langle>e # es\<rangle>\<^sub>l\<guillemotright> E"
    assume eval: "G\<turnstile>s0 \<midarrow>e # es\<doteq>\<succ>v\<midarrow>n\<rightarrow> s2"
    assume P: "(Normal P) Y s0 Z"
    show "R \<lfloor>v\<rfloor>\<^sub>l s2 Z \<and> s2\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain eT esT where
        wt_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e\<Colon>-eT" and
        wt_es: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>es\<Colon>\<doteq>esT"
        by cases simp
      from da obtain E1 where
        da_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> (dom (locals (store s0)))\<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E1" and
        da_es: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> nrm E1 \<guillemotright>\<langle>es\<rangle>\<^sub>l\<guillemotright> E"
        by cases simp
      from eval obtain s1 ve vs where
        eval_e: "G\<turnstile>s0 \<midarrow>e-\<succ>ve\<midarrow>n\<rightarrow> s1" and
        eval_es: "G\<turnstile>s1 \<midarrow>es\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s2" and
        v: "v=ve#vs"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from valid_e P valid_A conf_s0 eval_e wt_e da_e
      obtain Q: "Q \<lfloor>ve\<rfloor>\<^sub>e s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G,L)"
        by (rule validE)
      from Q have Q': "\<And> v. (Q\<leftarrow>\<lfloor>ve\<rfloor>\<^sub>e) v s1 Z"
        by simp
      have "(\<lambda>Vals:vs:. R\<leftarrow>\<lfloor>(ve # vs)\<rfloor>\<^sub>l) \<lfloor>vs\<rfloor>\<^sub>l s2 Z"
      proof (cases "normal s1")
        case True
        obtain E' where
          da_es': "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s1)) \<guillemotright>\<langle>es\<rangle>\<^sub>l\<guillemotright> E'"
        proof -
          from eval_e wt_e da_e wf True
          have "nrm E1 \<subseteq> dom (locals (store s1))"
            by (cases rule: da_good_approx_evalnE) iprover
          with da_es show thesis
            by (rule da_weakenE) (rule that)
        qed
        from valid_es Q' valid_A conf_s1 eval_es wt_es da_es'
        show ?thesis
          by (rule validE)
      next
        case False
        with valid_es Q' valid_A conf_s1 eval_es
        show ?thesis
          by (cases rule: validE) iprover+
      qed
      with v have "R \<lfloor>v\<rfloor>\<^sub>l s2 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s2\<Colon>\<preceq>(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Skip A P)
  show "G,A|\<Turnstile>\<Colon>{ {Normal (P\<leftarrow>\<diamondsuit>)} .Skip. {P} }"
  proof (rule valid_stmt_NormalI)
    fix s0 s1 n L Y Z
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume eval: "G\<turnstile>s0 \<midarrow>Skip\<midarrow>n\<rightarrow> s1"
    assume P: "(Normal (P\<leftarrow>\<diamondsuit>)) Y s0 Z"
    show "P \<diamondsuit> s1 Z \<and> s1\<Colon>\<preceq>(G, L)"
    proof -
      from eval obtain "s1=s0"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      with P conf_s0 show ?thesis
        by simp
    qed
  qed
next
  case (Expr A P e Q)
  note valid_e = `G,A|\<Turnstile>\<Colon>{ {Normal P} e-\<succ> {Q\<leftarrow>\<diamondsuit>} }`
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} .Expr e. {Q} }"
  proof (rule valid_stmt_NormalI)
    fix n s0 L accC C s1 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>Expr e\<Colon>\<surd>"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0)) \<guillemotright>\<langle>Expr e\<rangle>\<^sub>s\<guillemotright> C"
    assume eval: "G\<turnstile>s0 \<midarrow>Expr e\<midarrow>n\<rightarrow> s1"
    assume P: "(Normal P) Y s0 Z"
    show "Q \<diamondsuit> s1 Z \<and> s1\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain eT where
        wt_e: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-eT"
        by cases simp
      from da obtain E where
        da_e: "\<lparr>prg=G,cls=accC, lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright>E"
        by cases simp
      from eval obtain v where
        eval_e: "G\<turnstile>s0 \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s1"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from valid_e P valid_A conf_s0 eval_e wt_e da_e
      obtain Q: "(Q\<leftarrow>\<diamondsuit>) \<lfloor>v\<rfloor>\<^sub>e s1 Z" and "s1\<Colon>\<preceq>(G,L)"
        by (rule validE)
      thus ?thesis by simp
    qed
  qed
next
  case (Lab A P c l Q)
  note valid_c = `G,A|\<Turnstile>\<Colon>{ {Normal P} .c. {abupd (absorb l) .; Q} }`
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} .l\<bullet> c. {Q} }"
  proof (rule valid_stmt_NormalI)
    fix n s0 L accC C s2 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0: "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>l\<bullet> c\<Colon>\<surd>"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0)) \<guillemotright>\<langle>l\<bullet> c\<rangle>\<^sub>s\<guillemotright> C"
    assume eval: "G\<turnstile>s0 \<midarrow>l\<bullet> c\<midarrow>n\<rightarrow> s2"
    assume P: "(Normal P) Y s0 Z"
    show "Q \<diamondsuit> s2 Z \<and> s2\<Colon>\<preceq>(G, L)"
    proof -
      from wt obtain
        wt_c: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c\<Colon>\<surd>"
        by cases simp
      from da obtain E where
        da_c: "\<lparr>prg=G,cls=accC, lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>c\<rangle>\<^sub>s\<guillemotright>E"
        by cases simp
      from eval obtain s1 where
        eval_c: "G\<turnstile>s0 \<midarrow>c\<midarrow>n\<rightarrow> s1" and
        s2: "s2 = abupd (absorb l) s1"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from valid_c P valid_A conf_s0 eval_c wt_c da_c
      obtain Q: "(abupd (absorb l) .; Q) \<diamondsuit> s1 Z"
        by (rule validE)
      with s2 have "Q \<diamondsuit> s2 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s2\<Colon>\<preceq>(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Comp A P c1 Q c2 R)
  note valid_c1 = `G,A|\<Turnstile>\<Colon>{ {Normal P} .c1. {Q} }`
  note valid_c2 = `G,A|\<Turnstile>\<Colon>{ {Q} .c2. {R} }`
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} .c1;; c2. {R} }"
  proof (rule valid_stmt_NormalI)
    fix n s0 L accC C s2 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0:  "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>(c1;; c2)\<Colon>\<surd>"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s0))\<guillemotright>\<langle>c1;;c2\<rangle>\<^sub>s\<guillemotright>C"
    assume eval: "G\<turnstile>s0 \<midarrow>c1;; c2\<midarrow>n\<rightarrow> s2"
    assume P: "(Normal P) Y s0 Z"
    show "R \<diamondsuit> s2 Z \<and> s2\<Colon>\<preceq>(G,L)"
    proof -
      from eval  obtain s1 where
        eval_c1: "G\<turnstile>s0 \<midarrow>c1 \<midarrow>n\<rightarrow> s1" and
        eval_c2: "G\<turnstile>s1 \<midarrow>c2 \<midarrow>n\<rightarrow> s2"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from wt obtain
        wt_c1: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c1\<Colon>\<surd>" and
        wt_c2: "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c2\<Colon>\<surd>"
        by cases simp
      from da obtain C1 C2 where
        da_c1: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>c1\<rangle>\<^sub>s\<guillemotright> C1" and
        da_c2: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>nrm C1 \<guillemotright>\<langle>c2\<rangle>\<^sub>s\<guillemotright> C2"
        by cases simp
      from valid_c1 P valid_A conf_s0 eval_c1 wt_c1 da_c1
      obtain Q: "Q \<diamondsuit> s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G,L)"
        by (rule validE)
      have "R \<diamondsuit> s2 Z"
      proof (cases "normal s1")
        case True
        obtain C2' where
          "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s1)) \<guillemotright>\<langle>c2\<rangle>\<^sub>s\<guillemotright> C2'"
        proof -
          from eval_c1 wt_c1 da_c1 wf True
          have "nrm C1 \<subseteq> dom (locals (store s1))"
            by (cases rule: da_good_approx_evalnE) iprover
          with da_c2 show thesis
            by (rule da_weakenE) (rule that)
        qed
        with valid_c2 Q valid_A conf_s1 eval_c2 wt_c2
        show ?thesis
          by (rule validE)
      next
        case False
        from valid_c2 Q valid_A conf_s1 eval_c2 False
        show ?thesis
          by (cases rule: validE) iprover+
      qed
      moreover
      from eval wt da conf_s0 wf
      have "s2\<Colon>\<preceq>(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (If A P e P' c1 c2 Q)
  note valid_e = `G,A|\<Turnstile>\<Colon>{ {Normal P} e-\<succ> {P'} }`
  have valid_then_else: "\<And> b. G,A|\<Turnstile>\<Colon>{ {P'\<leftarrow>=b} .(if b then c1 else c2). {Q} }"
    using If.hyps by simp
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} .If(e) c1 Else c2. {Q} }"
  proof (rule valid_stmt_NormalI)
    fix n s0 L accC C s2 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0:  "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>If(e) c1 Else c2\<Colon>\<surd>"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                    \<turnstile>dom (locals (store s0))\<guillemotright>\<langle>If(e) c1 Else c2\<rangle>\<^sub>s\<guillemotright>C"
    assume eval: "G\<turnstile>s0 \<midarrow>If(e) c1 Else c2\<midarrow>n\<rightarrow> s2"
    assume P: "(Normal P) Y s0 Z"
    show "Q \<diamondsuit> s2 Z \<and> s2\<Colon>\<preceq>(G,L)"
    proof -
      from eval obtain b s1 where
        eval_e: "G\<turnstile>s0 \<midarrow>e-\<succ>b\<midarrow>n\<rightarrow> s1" and
        eval_then_else: "G\<turnstile>s1 \<midarrow>(if the_Bool b then c1 else c2)\<midarrow>n\<rightarrow> s2"
        using normal_s0 by (auto elim: evaln_elim_cases)
      from wt obtain
        wt_e: "\<lparr>prg=G, cls=accC, lcl=L\<rparr>\<turnstile>e\<Colon>-PrimT Boolean" and
        wt_then_else: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>(if the_Bool b then c1 else c2)\<Colon>\<surd>"
        by cases (simp split: split_if)
      from da obtain E S where
        da_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E" and
        da_then_else:
        "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>
             (dom (locals (store s0)) \<union> assigns_if (the_Bool b) e)
               \<guillemotright>\<langle>if the_Bool b then c1 else c2\<rangle>\<^sub>s\<guillemotright> S"
        by cases (cases "the_Bool b",auto)
      from valid_e P valid_A conf_s0 eval_e wt_e da_e
      obtain "P' \<lfloor>b\<rfloor>\<^sub>e s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G,L)"
        by (rule validE)
      hence P': "\<And>v. (P'\<leftarrow>=the_Bool b) v s1 Z"
        by (cases "normal s1") auto
      have "Q \<diamondsuit> s2 Z"
      proof (cases "normal s1")
        case True
        have s0_s1: "dom (locals (store s0))
                      \<union> assigns_if (the_Bool b) e \<subseteq> dom (locals (store s1))"
        proof -
          from eval_e
          have eval_e': "G\<turnstile>s0 \<midarrow>e-\<succ>b\<rightarrow> s1"
            by (rule evaln_eval)
          hence
            "dom (locals (store s0)) \<subseteq> dom (locals (store s1))"
            by (rule dom_locals_eval_mono_elim)
          moreover
          from eval_e' True wt_e
          have "assigns_if (the_Bool b) e \<subseteq> dom (locals (store s1))"
            by (rule assigns_if_good_approx')
          ultimately show ?thesis by (rule Un_least)
        qed
        with da_then_else
        obtain S' where
          "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
              \<turnstile>dom (locals (store s1))\<guillemotright>\<langle>if the_Bool b then c1 else c2\<rangle>\<^sub>s\<guillemotright> S'"
          by (rule da_weakenE)
        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\<Colon>\<preceq>(G, L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Loop A P e P' c l)
  note valid_e = `G,A|\<Turnstile>\<Colon>{ {P} e-\<succ> {P'} }`
  note valid_c = `G,A|\<Turnstile>\<Colon>{ {Normal (P'\<leftarrow>=True)}
                         .c.
                         {abupd (absorb (Cont l)) .; P} }`
  show "G,A|\<Turnstile>\<Colon>{ {P} .l\<bullet> While(e) c. {P'\<leftarrow>=False\<down>=\<diamondsuit>} }"
  proof (rule valid_stmtI)
    fix n s0 L accC C s3 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0:  "s0\<Colon>\<preceq>(G,L)"
    assume wt: "normal s0 \<Longrightarrow> \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>l\<bullet> While(e) c\<Colon>\<surd>"
    assume da: "normal s0 \<Longrightarrow> \<lparr>prg=G,cls=accC,lcl=L\<rparr>
                    \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>l\<bullet> While(e) c\<rangle>\<^sub>s\<guillemotright> C"
    assume eval: "G\<turnstile>s0 \<midarrow>l\<bullet> While(e) c\<midarrow>n\<rightarrow> s3"
    assume P: "P Y s0 Z"
    show "(P'\<leftarrow>=False\<down>=\<diamondsuit>) \<diamondsuit> s3 Z \<and> s3\<Colon>\<preceq>(G,L)"
    proof -
        --{* From the given hypothesises @{text valid_e} and @{text valid_c}
           we can only reach the state after unfolding the loop once, i.e.
           @{term "P \<diamondsuit> s2 Z"}, where @{term s2} is the state after executing
           @{term c}. To gain validity of the further execution of while, to
           finally get @{term "(P'\<leftarrow>=False\<down>=\<diamondsuit>) \<diamondsuit> s3 Z"} we have to get
           a hypothesis about the subsequent unfoldings (the whole loop again),
           too. We can achieve this, by performing induction on the
           evaluation relation, with all
           the necessary preconditions to apply @{text valid_e} and
           @{text valid_c} in the goal.
        *}
      {
        fix t s s' v
        assume "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s')"
        hence "\<And> Y' T E.
                \<lbrakk>t =  \<langle>l\<bullet> While(e) c\<rangle>\<^sub>s; \<forall>t\<in>A. G\<Turnstile>n\<Colon>t; P Y' s Z; s\<Colon>\<preceq>(G, L);
                 normal s \<Longrightarrow> \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>t\<Colon>T;
                 normal s \<Longrightarrow> \<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>dom (locals (store s))\<guillemotright>t\<guillemotright>E
                \<rbrakk>\<Longrightarrow> (P'\<leftarrow>=False\<down>=\<diamondsuit>) v s' Z"
          (is "PROP ?Hyp n t s v s'")
        proof (induct)
          case (Loop s0' e' b n' s1' c' s2' l' s3' Y' T E)
          note while = `(\<langle>l'\<bullet> While(e') c'\<rangle>\<^sub>s::term) = \<langle>l\<bullet> While(e) c\<rangle>\<^sub>s`
          hence eqs: "l'=l" "e'=e" "c'=c" by simp_all
          note valid_A = `\<forall>t\<in>A. G\<Turnstile>n'\<Colon>t`
          note P = `P Y' (Norm s0') Z`
          note conf_s0' = `Norm s0'\<Colon>\<preceq>(G, L)`
          have wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>\<langle>l\<bullet> While(e) c\<rangle>\<^sub>s\<Colon>T"
            using Loop.prems eqs by simp
          have da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>
                    dom (locals (store ((Norm s0')::state)))\<guillemotright>\<langle>l\<bullet> While(e) c\<rangle>\<^sub>s\<guillemotright>E"
            using Loop.prems eqs by simp
          have evaln_e: "G\<turnstile>Norm s0' \<midarrow>e-\<succ>b\<midarrow>n'\<rightarrow> s1'"
            using Loop.hyps eqs by simp
          show "(P'\<leftarrow>=False\<down>=\<diamondsuit>) \<diamondsuit> s3' Z"
          proof -
            from wt  obtain
              wt_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e\<Colon>-PrimT Boolean" and
              wt_c: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>c\<Colon>\<surd>"
              by cases (simp add: eqs)
            from da obtain E S where
              da_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                     \<turnstile> dom (locals (store ((Norm s0')::state))) \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E" and
              da_c: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                     \<turnstile> (dom (locals (store ((Norm s0')::state)))
                            \<union> assigns_if True e) \<guillemotright>\<langle>c\<rangle>\<^sub>s\<guillemotright> S"
              by cases (simp add: eqs)
            from evaln_e
            have eval_e: "G\<turnstile>Norm s0' \<midarrow>e-\<succ>b\<rightarrow> s1'"
              by (rule evaln_eval)
            from valid_e P valid_A conf_s0' evaln_e wt_e da_e
            obtain P': "P' \<lfloor>b\<rfloor>\<^sub>e s1' Z" and conf_s1': "s1'\<Colon>\<preceq>(G,L)"
              by (rule validE)
            show "(P'\<leftarrow>=False\<down>=\<diamondsuit>) \<diamondsuit> s3' Z"
            proof (cases "normal s1'")
              case True
              note normal_s1'=this
              show ?thesis
              proof (cases "the_Bool b")
                case True
                with P' normal_s1' have P'': "(Normal (P'\<leftarrow>=True)) \<lfloor>b\<rfloor>\<^sub>e s1' Z"
                  by auto
                from True Loop.hyps obtain
                  eval_c: "G\<turnstile>s1' \<midarrow>c\<midarrow>n'\<rightarrow> s2'" and
                  eval_while:
                     "G\<turnstile>abupd (absorb (Cont l)) s2' \<midarrow>l\<bullet> While(e) c\<midarrow>n'\<rightarrow> s3'"
                  by (simp add: eqs)
                from True Loop.hyps have
                  hyp: "PROP ?Hyp n' \<langle>l\<bullet> While(e) c\<rangle>\<^sub>s
                          (abupd (absorb (Cont l')) s2') \<diamondsuit> s3'"
                  apply (simp only: True if_True eqs)
                  apply (elim conjE)
                  apply (tactic "smp_tac @{context} 3 1")
                  apply fast
                  done
                from eval_e
                have s0'_s1': "dom (locals (store ((Norm s0')::state)))
                                  \<subseteq> dom (locals (store s1'))"
                  by (rule dom_locals_eval_mono_elim)
                obtain S' where
                  da_c':
                   "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>(dom (locals (store s1')))\<guillemotright>\<langle>c\<rangle>\<^sub>s\<guillemotright> S'"
                proof -
                  note s0'_s1'
                  moreover
                  from eval_e normal_s1' wt_e
                  have "assigns_if True e \<subseteq> dom (locals (store s1'))"
                    by (rule assigns_if_good_approx' [elim_format])
                       (simp add: True)
                  ultimately
                  have "dom (locals (store ((Norm s0')::state)))
                           \<union> assigns_if True e \<subseteq> dom (locals (store s1'))"
                    by (rule Un_least)
                  with da_c show thesis
                    by (rule da_weakenE) (rule that)
                qed
                with valid_c P'' valid_A conf_s1' eval_c wt_c
                obtain "(abupd (absorb (Cont l)) .; P) \<diamondsuit> s2' Z" and
                  conf_s2': "s2'\<Colon>\<preceq>(G,L)"
                  by (rule validE)
                hence P_s2': "P \<diamondsuit> (abupd (absorb (Cont l)) s2') Z"
                  by simp
                from conf_s2'
                have conf_absorb: "abupd (absorb (Cont l)) s2' \<Colon>\<preceq>(G, L)"
                  by (cases s2') (auto intro: conforms_absorb)
                moreover
                obtain E' where
                  da_while':
                   "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>
                     dom (locals(store (abupd (absorb (Cont l)) s2')))
                      \<guillemotright>\<langle>l\<bullet> While(e) c\<rangle>\<^sub>s\<guillemotright> E'"
                proof -
                  note s0'_s1'
                  also
                  from eval_c
                  have "G\<turnstile>s1' \<midarrow>c\<rightarrow> s2'"
                    by (rule evaln_eval)
                  hence "dom (locals (store s1')) \<subseteq> dom (locals (store s2'))"
                    by (rule dom_locals_eval_mono_elim)
                  also
                  have "\<dots>\<subseteq>dom (locals (store (abupd (absorb (Cont l)) s2')))"
                    by simp
                  finally
                  have "dom (locals (store ((Norm s0')::state))) \<subseteq> \<dots>" .
                  with da show thesis
                    by (rule da_weakenE) (rule that)
                qed
                from valid_A P_s2' conf_absorb wt da_while'
                show "(P'\<leftarrow>=False\<down>=\<diamondsuit>) \<diamondsuit> s3' Z"
                  using hyp by (simp add: eqs)
              next
                case False
                with Loop.hyps obtain "s3'=s1'"
                  by simp
                with P' False show ?thesis
                  by auto
              qed
            next
              case False
              note abnormal_s1'=this
              have "s3'=s1'"
              proof -
                from False obtain abr where abr: "abrupt s1' = Some abr"
                  by (cases s1') auto
                from eval_e _ wt_e wf
                have no_jmp: "\<And> j. abrupt s1' \<noteq> Some (Jump j)"
                  by (rule eval_expression_no_jump
                       [where ?Env="\<lparr>prg=G,cls=accC,lcl=L\<rparr>",simplified])
                     simp
                show ?thesis
                proof (cases "the_Bool b")
                  case True
                  with Loop.hyps obtain
                    eval_c: "G\<turnstile>s1' \<midarrow>c\<midarrow>n'\<rightarrow> s2'" and
                    eval_while:
                     "G\<turnstile>abupd (absorb (Cont l)) s2' \<midarrow>l\<bullet> While(e) c\<midarrow>n'\<rightarrow> s3'"
                    by (simp add: eqs)
                  from eval_c abr have "s2'=s1'" by auto
                  moreover from calculation no_jmp
                  have "abupd (absorb (Cont l)) s2'=s2'"
                    by (cases s1') (simp add: absorb_def)
                  ultimately show ?thesis
                    using eval_while abr
                    by auto
                next
                  case False
                  with Loop.hyps show ?thesis by simp
                qed
              qed
              with P' False show ?thesis
                by auto
            qed
          qed
        next
          case (Abrupt abr s t' n' Y' T E)
          note t' = `t' = \<langle>l\<bullet> While(e) c\<rangle>\<^sub>s`
          note conf = `(Some abr, s)\<Colon>\<preceq>(G, L)`
          note P = `P Y' (Some abr, s) Z`
          note valid_A = `\<forall>t\<in>A. G\<Turnstile>n'\<Colon>t`
          show "(P'\<leftarrow>=False\<down>=\<diamondsuit>) (undefined3 t') (Some abr, s) Z"
          proof -
            have eval_e:
              "G\<turnstile>(Some abr,s) \<midarrow>\<langle>e\<rangle>\<^sub>e\<succ>\<midarrow>n'\<rightarrow> (undefined3 \<langle>e\<rangle>\<^sub>e,(Some abr,s))"
              by auto
            from valid_e P valid_A conf eval_e
            have "P' (undefined3 \<langle>e\<rangle>\<^sub>e) (Some abr,s) Z"
              by (cases rule: validE [where ?P="P"]) simp+
            with t' show ?thesis
              by auto
          qed
        qed simp_all
      } note generalized=this
      from eval _ valid_A P conf_s0 wt da
      have "(P'\<leftarrow>=False\<down>=\<diamondsuit>) \<diamondsuit> s3 Z"
        by (rule generalized)  simp_all
      moreover
      have "s3\<Colon>\<preceq>(G, L)"
      proof (cases "normal s0")
        case True
        from eval wt [OF True] da [OF True] conf_s0 wf
        show ?thesis
          by (rule evaln_type_sound [elim_format]) simp
      next
        case False
        with eval have "s3=s0"
          by auto
        with conf_s0 show ?thesis
          by simp
      qed
      ultimately show ?thesis ..
    qed
  qed
next
  case (Jmp A j P)
  show "G,A|\<Turnstile>\<Colon>{ {Normal (abupd (\<lambda>a. Some (Jump j)) .; P\<leftarrow>\<diamondsuit>)} .Jmp j. {P} }"
  proof (rule valid_stmt_NormalI)
    fix n s0 L accC C s1 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0:  "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>Jmp j\<Colon>\<surd>"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                    \<turnstile>dom (locals (store s0))\<guillemotright>\<langle>Jmp j\<rangle>\<^sub>s\<guillemotright>C"
    assume eval: "G\<turnstile>s0 \<midarrow>Jmp j\<midarrow>n\<rightarrow> s1"
    assume P: "(Normal (abupd (\<lambda>a. Some (Jump j)) .; P\<leftarrow>\<diamondsuit>)) Y s0 Z"
    show "P \<diamondsuit> s1 Z \<and> s1\<Colon>\<preceq>(G,L)"
    proof -
      from eval obtain s where
        s: "s0=Norm s" "s1=(Some (Jump j), s)"
        using normal_s0 by (auto elim: evaln_elim_cases)
      with P have "P \<diamondsuit> s1 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s1\<Colon>\<preceq>(G,L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Throw A P e Q)
  note valid_e = `G,A|\<Turnstile>\<Colon>{ {Normal P} e-\<succ> {\<lambda>Val:a:. abupd (throw a) .; Q\<leftarrow>\<diamondsuit>} }`
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} .Throw e. {Q} }"
  proof (rule valid_stmt_NormalI)
    fix n s0 L accC C s2 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0:  "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>Throw e\<Colon>\<surd>"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                    \<turnstile>dom (locals (store s0))\<guillemotright>\<langle>Throw e\<rangle>\<^sub>s\<guillemotright>C"
    assume eval: "G\<turnstile>s0 \<midarrow>Throw e\<midarrow>n\<rightarrow> s2"
    assume P: "(Normal P) Y s0 Z"
    show "Q \<diamondsuit> s2 Z \<and> s2\<Colon>\<preceq>(G,L)"
    proof -
      from eval obtain s1 a where
        eval_e: "G\<turnstile>s0 \<midarrow>e-\<succ>a\<midarrow>n\<rightarrow> s1" and
        s2: "s2 = abupd (throw a) s1"
        using normal_s0 by (auto elim: evaln_elim_cases)
      from wt obtain T where
        wt_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>e\<Colon>-T"
        by cases simp
      from da obtain E where
        da_e: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>e\<rangle>\<^sub>e\<guillemotright> E"
        by cases simp
      from valid_e P valid_A conf_s0 eval_e wt_e da_e
      obtain "(\<lambda>Val:a:. abupd (throw a) .; Q\<leftarrow>\<diamondsuit>) \<lfloor>a\<rfloor>\<^sub>e s1 Z"
        by (rule validE)
      with s2 have "Q \<diamondsuit> s2 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s2\<Colon>\<preceq>(G,L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Try A P c1 Q C vn c2 R)
  note valid_c1 = `G,A|\<Turnstile>\<Colon>{ {Normal P} .c1. {SXAlloc G Q} }`
  note valid_c2 = `G,A|\<Turnstile>\<Colon>{ {Q \<and>. (\<lambda>s. G,s\<turnstile>catch C) ;. new_xcpt_var vn}
                           .c2.
                          {R} }`
  note Q_R = `(Q \<and>. (\<lambda>s. \<not> G,s\<turnstile>catch C)) \<Rightarrow> R`
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} .Try c1 Catch(C vn) c2. {R} }"
  proof (rule valid_stmt_NormalI)
    fix n s0 L accC E s3 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0:  "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>Try c1 Catch(C vn) c2\<Colon>\<surd>"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                    \<turnstile>dom (locals (store s0)) \<guillemotright>\<langle>Try c1 Catch(C vn) c2\<rangle>\<^sub>s\<guillemotright> E"
    assume eval: "G\<turnstile>s0 \<midarrow>Try c1 Catch(C vn) c2\<midarrow>n\<rightarrow> s3"
    assume P: "(Normal P) Y s0 Z"
    show "R \<diamondsuit> s3 Z \<and> s3\<Colon>\<preceq>(G,L)"
    proof -
      from eval obtain s1 s2 where
        eval_c1: "G\<turnstile>s0 \<midarrow>c1\<midarrow>n\<rightarrow> s1" and
        sxalloc: "G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2" and
        s3: "if G,s2\<turnstile>catch C
                then G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<midarrow>n\<rightarrow> s3
                else s3 = s2"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from wt obtain
        wt_c1: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>c1\<Colon>\<surd>" and
        wt_c2: "\<lparr>prg=G,cls=accC,lcl=L(VName vn\<mapsto>Class C)\<rparr>\<turnstile>c2\<Colon>\<surd>"
        by cases simp
      from da obtain C1 C2 where
        da_c1: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>c1\<rangle>\<^sub>s\<guillemotright> C1" and
        da_c2: "\<lparr>prg=G,cls=accC,lcl=L(VName vn\<mapsto>Class C)\<rparr>
                   \<turnstile> (dom (locals (store s0)) \<union> {VName vn}) \<guillemotright>\<langle>c2\<rangle>\<^sub>s\<guillemotright> C2"
        by cases simp
      from valid_c1 P valid_A conf_s0 eval_c1 wt_c1 da_c1
      obtain sxQ: "(SXAlloc G Q) \<diamondsuit> s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G,L)"
        by (rule validE)
      from sxalloc sxQ
      have Q: "Q \<diamondsuit> s2 Z"
        by auto
      have "R \<diamondsuit> s3 Z"
      proof (cases "\<exists> x. abrupt s1 = Some (Xcpt x)")
        case False
        from sxalloc wf
        have "s2=s1"
          by (rule sxalloc_type_sound [elim_format])
             (insert False, auto split: option.splits abrupt.splits )
        with False
        have no_catch: "\<not>  G,s2\<turnstile>catch C"
          by (simp add: catch_def)
        moreover
        from no_catch s3
        have "s3=s2"
          by simp
        ultimately show ?thesis
          using Q Q_R by simp
      next
        case True
        note exception_s1 = this
        show ?thesis
        proof (cases "G,s2\<turnstile>catch C")
          case False
          with s3
          have "s3=s2"
            by simp
          with False Q Q_R show ?thesis
            by simp
        next
          case True
          with s3 have eval_c2: "G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<midarrow>n\<rightarrow> s3"
            by simp
          from conf_s1 sxalloc wf
          have conf_s2: "s2\<Colon>\<preceq>(G, L)"
            by (auto dest: sxalloc_type_sound
                    split: option.splits abrupt.splits)
          from exception_s1 sxalloc wf
          obtain a
            where xcpt_s2: "abrupt s2 = Some (Xcpt (Loc a))"
            by (auto dest!: sxalloc_type_sound
                            split: option.splits abrupt.splits)
          with True
          have "G\<turnstile>obj_ty (the (globs (store s2) (Heap a)))\<preceq>Class C"
            by (cases s2) simp
          with xcpt_s2 conf_s2 wf
          have conf_new_xcpt: "new_xcpt_var vn s2 \<Colon>\<preceq>(G, L(VName vn\<mapsto>Class C))"
            by (auto dest: Try_lemma)
          obtain C2' where
            da_c2':
            "\<lparr>prg=G,cls=accC,lcl=L(VName vn\<mapsto>Class C)\<rparr>
              \<turnstile> (dom (locals (store (new_xcpt_var vn s2)))) \<guillemotright>\<langle>c2\<rangle>\<^sub>s\<guillemotright> C2'"
          proof -
            have "(dom (locals (store s0)) \<union> {VName vn})
                    \<subseteq> dom (locals (store (new_xcpt_var vn s2)))"
            proof -
              from eval_c1
              have "dom (locals (store s0))
                      \<subseteq> dom (locals (store s1))"
                by (rule dom_locals_evaln_mono_elim)
              also
              from sxalloc
              have "\<dots> \<subseteq> dom (locals (store s2))"
                by (rule dom_locals_sxalloc_mono)
              also
              have "\<dots> \<subseteq> dom (locals (store (new_xcpt_var vn s2)))"
                by (cases s2) (simp add: new_xcpt_var_def, blast)
              also
              have "{VName vn} \<subseteq> \<dots>"
                by (cases s2) simp
              ultimately show ?thesis
                by (rule Un_least)
            qed
            with da_c2 show thesis
              by (rule da_weakenE) (rule that)
          qed
          from Q eval_c2 True
          have "(Q \<and>. (\<lambda>s. G,s\<turnstile>catch C) ;. new_xcpt_var vn)
                   \<diamondsuit> (new_xcpt_var vn s2) Z"
            by auto
          from valid_c2 this valid_A conf_new_xcpt eval_c2 wt_c2 da_c2'
          show "R \<diamondsuit> s3 Z"
            by (rule validE)
        qed
      qed
      moreover
      from eval wt da conf_s0 wf
      have "s3\<Colon>\<preceq>(G,L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Fin A P c1 Q c2 R)
  note valid_c1 = `G,A|\<Turnstile>\<Colon>{ {Normal P} .c1. {Q} }`
  have valid_c2: "\<And> abr. G,A|\<Turnstile>\<Colon>{ {Q \<and>. (\<lambda>s. abr = fst s) ;. abupd (\<lambda>x. None)}
                                  .c2.
                                  {abupd (abrupt_if (abr \<noteq> None) abr) .; R} }"
    using Fin.hyps by simp
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} .c1 Finally c2. {R} }"
  proof (rule valid_stmt_NormalI)
    fix n s0 L accC E s3 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0:  "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>c1 Finally c2\<Colon>\<surd>"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                    \<turnstile>dom (locals (store s0)) \<guillemotright>\<langle>c1 Finally c2\<rangle>\<^sub>s\<guillemotright> E"
    assume eval: "G\<turnstile>s0 \<midarrow>c1 Finally c2\<midarrow>n\<rightarrow> s3"
    assume P: "(Normal P) Y s0 Z"
    show "R \<diamondsuit> s3 Z \<and> s3\<Colon>\<preceq>(G,L)"
    proof -
      from eval obtain s1 abr1 s2 where
        eval_c1: "G\<turnstile>s0 \<midarrow>c1\<midarrow>n\<rightarrow> (abr1, s1)" and
        eval_c2: "G\<turnstile>Norm s1 \<midarrow>c2\<midarrow>n\<rightarrow> s2" and
        s3: "s3 = (if \<exists>err. abr1 = Some (Error err)
                      then (abr1, s1)
                      else abupd (abrupt_if (abr1 \<noteq> None) abr1) s2)"
        using normal_s0 by (fastforce elim: evaln_elim_cases)
      from wt obtain
        wt_c1: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>c1\<Colon>\<surd>" and
        wt_c2: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>c2\<Colon>\<surd>"
        by cases simp
      from da obtain C1 C2 where
        da_c1: "\<lparr>prg=G,cls=accC,lcl=L\<rparr> \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>c1\<rangle>\<^sub>s\<guillemotright> C1" and
        da_c2: "\<lparr>prg=G,cls=accC,lcl=L\<rparr> \<turnstile> dom (locals (store s0)) \<guillemotright>\<langle>c2\<rangle>\<^sub>s\<guillemotright> C2"
        by cases simp
      from valid_c1 P valid_A conf_s0 eval_c1 wt_c1 da_c1
      obtain Q: "Q \<diamondsuit> (abr1,s1) Z" and conf_s1: "(abr1,s1)\<Colon>\<preceq>(G,L)"
        by (rule validE)
      from Q
      have Q': "(Q \<and>. (\<lambda>s. abr1 = fst s) ;. abupd (\<lambda>x. None)) \<diamondsuit> (Norm s1) Z"
        by auto
      from eval_c1 wt_c1 da_c1 conf_s0 wf
      have  "error_free (abr1,s1)"
        by (rule evaln_type_sound  [elim_format]) (insert normal_s0,simp)
      with s3 have s3': "s3 = abupd (abrupt_if (abr1 \<noteq> None) abr1) s2"
        by (simp add: error_free_def)
      from conf_s1
      have conf_Norm_s1: "Norm s1\<Colon>\<preceq>(G,L)"
        by (rule conforms_NormI)
      obtain C2' where
        da_c2': "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                   \<turnstile> dom (locals (store ((Norm s1)::state))) \<guillemotright>\<langle>c2\<rangle>\<^sub>s\<guillemotright> C2'"
      proof -
        from eval_c1
        have "dom (locals (store s0)) \<subseteq> dom (locals (store (abr1,s1)))"
          by (rule dom_locals_evaln_mono_elim)
        hence "dom (locals (store s0))
                 \<subseteq> dom (locals (store ((Norm s1)::state)))"
          by simp
        with da_c2 show thesis
          by (rule da_weakenE) (rule that)
      qed
      from valid_c2 Q' valid_A conf_Norm_s1 eval_c2 wt_c2 da_c2'
      have "(abupd (abrupt_if (abr1 \<noteq> None) abr1) .; R) \<diamondsuit> s2 Z"
        by (rule validE)
      with s3' have "R \<diamondsuit> s3 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s3\<Colon>\<preceq>(G,L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (Done A P C)
  show "G,A|\<Turnstile>\<Colon>{ {Normal (P\<leftarrow>\<diamondsuit> \<and>. initd C)} .Init C. {P} }"
  proof (rule valid_stmt_NormalI)
    fix n s0 L accC E s3 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0:  "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>Init C\<Colon>\<surd>"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                    \<turnstile>dom (locals (store s0)) \<guillemotright>\<langle>Init C\<rangle>\<^sub>s\<guillemotright> E"
    assume eval: "G\<turnstile>s0 \<midarrow>Init C\<midarrow>n\<rightarrow> s3"
    assume P: "(Normal (P\<leftarrow>\<diamondsuit> \<and>. initd C)) Y s0 Z"
    show "P \<diamondsuit> s3 Z \<and> s3\<Colon>\<preceq>(G,L)"
    proof -
      from P have inited: "inited C (globs (store s0))"
        by simp
      with eval have "s3=s0"
        using normal_s0 by (auto elim: evaln_elim_cases)
      with P conf_s0 show ?thesis
        by simp
    qed
  qed
next
  case (Init C c A P Q R)
  note c = `the (class G C) = c`
  note valid_super =
        `G,A|\<Turnstile>\<Colon>{ {Normal (P \<and>. Not \<circ> initd C ;. supd (init_class_obj G C))}
                 .(if C = Object then Skip else Init (super c)).
                 {Q} }`
  have valid_init:
        "\<And> l.  G,A|\<Turnstile>\<Colon>{ {Q \<and>. (\<lambda>s. l = locals (snd s)) ;. set_lvars empty}
                        .init c.
                        {set_lvars l .; R} }"
    using Init.hyps by simp
  show "G,A|\<Turnstile>\<Colon>{ {Normal (P \<and>. Not \<circ> initd C)} .Init C. {R} }"
  proof (rule valid_stmt_NormalI)
    fix n s0 L accC E s3 Y Z
    assume valid_A: "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"
    assume conf_s0:  "s0\<Colon>\<preceq>(G,L)"
    assume normal_s0: "normal s0"
    assume wt: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>\<turnstile>Init C\<Colon>\<surd>"
    assume da: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                    \<turnstile>dom (locals (store s0)) \<guillemotright>\<langle>Init C\<rangle>\<^sub>s\<guillemotright> E"
    assume eval: "G\<turnstile>s0 \<midarrow>Init C\<midarrow>n\<rightarrow> s3"
    assume P: "(Normal (P \<and>. Not \<circ> initd C)) Y s0 Z"
    show "R \<diamondsuit> s3 Z \<and> s3\<Colon>\<preceq>(G,L)"
    proof -
      from P have not_inited: "\<not> inited C (globs (store s0))" by simp
      with eval c obtain s1 s2 where
        eval_super:
        "G\<turnstile>Norm ((init_class_obj G C) (store s0))
           \<midarrow>(if C = Object then Skip else Init (super c))\<midarrow>n\<rightarrow> s1" and
        eval_init: "G\<turnstile>(set_lvars empty) s1 \<midarrow>init c\<midarrow>n\<rightarrow> s2" and
        s3: "s3 = (set_lvars (locals (store s1))) s2"
        using normal_s0 by (auto elim!: evaln_elim_cases)
      from wt c have
        cls_C: "class G C = Some c"
        by cases auto
      from wf cls_C have
        wt_super: "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
                         \<turnstile>(if C = Object then Skip else Init (super c))\<Colon>\<surd>"
        by (cases "C=Object")
           (auto dest: wf_prog_cdecl wf_cdecl_supD is_acc_classD)
      obtain S where
        da_super:
        "\<lparr>prg=G,cls=accC,lcl=L\<rparr>
          \<turnstile> dom (locals (store ((Norm
                            ((init_class_obj G C) (store s0)))::state)))
               \<guillemotright>\<langle>if C = Object then Skip else Init (super c)\<rangle>\<^sub>s\<guillemotright> S"
      proof (cases "C=Object")
        case True
        with da_Skip show ?thesis
          using that by (auto intro: assigned.select_convs)
      next
        case False
        with da_Init show ?thesis
          by - (rule that, auto intro: assigned.select_convs)
      qed
      from normal_s0 conf_s0 wf cls_C not_inited
      have conf_init_cls: "(Norm ((init_class_obj G C) (store s0)))\<Colon>\<preceq>(G, L)"
        by (auto intro: conforms_init_class_obj)
      from P
      have P': "(Normal (P \<and>. Not \<circ> initd C ;. supd (init_class_obj G C)))
                   Y (Norm ((init_class_obj G C) (store s0))) Z"
        by auto

      from valid_super P' valid_A conf_init_cls eval_super wt_super da_super
      obtain Q: "Q \<diamondsuit> s1 Z" and conf_s1: "s1\<Colon>\<preceq>(G,L)"
        by (rule validE)

      from cls_C wf have wt_init: "\<lparr>prg=G, cls=C,lcl=empty\<rparr>\<turnstile>(init c)\<Colon>\<surd>"
        by (rule wf_prog_cdecl [THEN wf_cdecl_wt_init])
      from cls_C wf obtain I where
        "\<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile> {} \<guillemotright>\<langle>init c\<rangle>\<^sub>s\<guillemotright> I"
        by (rule wf_prog_cdecl [THEN wf_cdeclE,simplified]) blast
       (*  simplified: to rewrite \<langle>init c\<rangle> to In1r (init c) *)
      then obtain I' where
        da_init:
        "\<lparr>prg=G,cls=C,lcl=empty\<rparr>\<turnstile>dom (locals (store ((set_lvars empty) s1)))
            \<guillemotright>\<langle>init c\<rangle>\<^sub>s\<guillemotright> I'"
        by (rule da_weakenE) simp
      have conf_s1_empty: "(set_lvars empty) s1\<Colon>\<preceq>(G, empty)"
      proof -
        from eval_super have
          "G\<turnstile>Norm ((init_class_obj G C) (store s0))
             \<midarrow>(if C = Object then Skip else Init (super c))\<rightarrow> s1"
          by (rule evaln_eval)
        from this wt_super wf
        have s1_no_ret: "\<And> j. abrupt s1 \<noteq> Some (Jump j)"
          by - (rule eval_statement_no_jump
                 [where ?Env="\<lparr>prg=G,cls=accC,lcl=L\<rparr>"], auto split: split_if)
        with conf_s1
        show ?thesis
          by (cases s1) (auto intro: conforms_set_locals)
      qed

      obtain l where l: "l = locals (store s1)"
        by simp
      with Q
      have Q': "(Q \<and>. (\<lambda>s. l = locals (snd s)) ;. set_lvars empty)
                  \<diamondsuit> ((set_lvars empty) s1) Z"
        by auto
      from valid_init Q' valid_A conf_s1_empty eval_init wt_init da_init
      have "(set_lvars l .; R) \<diamondsuit> s2 Z"
        by (rule validE)
      with s3 l have "R \<diamondsuit> s3 Z"
        by simp
      moreover
      from eval wt da conf_s0 wf
      have "s3\<Colon>\<preceq>(G,L)"
        by (rule evaln_type_sound [elim_format]) simp
      ultimately show ?thesis ..
    qed
  qed
next
  case (InsInitV A P c v Q)
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} InsInitV c v=\<succ> {Q} }"
  proof (rule valid_var_NormalI)
    fix s0 vf n s1 L Z
    assume "normal s0"
    moreover
    assume "G\<turnstile>s0 \<midarrow>InsInitV c v=\<succ>vf\<midarrow>n\<rightarrow> s1"
    ultimately have "False"
      by (cases s0) (simp add: evaln_InsInitV)
    thus "Q \<lfloor>vf\<rfloor>\<^sub>v s1 Z \<and> s1\<Colon>\<preceq>(G, L)"..
  qed
next
  case (InsInitE A P c e Q)
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} InsInitE c e-\<succ> {Q} }"
  proof (rule valid_expr_NormalI)
    fix s0 v n s1 L Z
    assume "normal s0"
    moreover
    assume "G\<turnstile>s0 \<midarrow>InsInitE c e-\<succ>v\<midarrow>n\<rightarrow> s1"
    ultimately have "False"
      by (cases s0) (simp add: evaln_InsInitE)
    thus "Q \<lfloor>v\<rfloor>\<^sub>e s1 Z \<and> s1\<Colon>\<preceq>(G, L)"..
  qed
next
  case (Callee A P l e Q)
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} Callee l e-\<succ> {Q} }"
  proof (rule valid_expr_NormalI)
    fix s0 v n s1 L Z
    assume "normal s0"
    moreover
    assume "G\<turnstile>s0 \<midarrow>Callee l e-\<succ>v\<midarrow>n\<rightarrow> s1"
    ultimately have "False"
      by (cases s0) (simp add: evaln_Callee)
    thus "Q \<lfloor>v\<rfloor>\<^sub>e s1 Z \<and> s1\<Colon>\<preceq>(G, L)"..
  qed
next
  case (FinA A P a c Q)
  show "G,A|\<Turnstile>\<Colon>{ {Normal P} .FinA a c. {Q} }"
  proof (rule valid_stmt_NormalI)
    fix s0 v n s1 L Z
    assume "normal s0"
    moreover
    assume "G\<turnstile>s0 \<midarrow>FinA a c\<midarrow>n\<rightarrow> s1"
    ultimately have "False"
      by (cases s0) (simp add: evaln_FinA)
    thus "Q \<diamondsuit> s1 Z \<and> s1\<Colon>\<preceq>(G, L)"..
  qed
qed
declare inj_term_simps [simp del]

theorem ax_sound:
 "wf_prog G \<Longrightarrow> G,(A::'a triple set)|\<turnstile>(ts::'a triple set) \<Longrightarrow> G,A|\<Turnstile>ts"
apply (subst ax_valids2_eq [symmetric])
apply  assumption
apply (erule (1) ax_sound2)
done

lemma sound_valid2_lemma:
"\<lbrakk>\<forall>v n. Ball A (triple_valid2 G n) \<longrightarrow> P v n; Ball A (triple_valid2 G n)\<rbrakk>
 \<Longrightarrow>P v n"
by blast

end