
diff -r 45be08fbdcff -r 6f0928a942d1 src/HOL/MicroJava/BV/JVM.thy
--- a/src/HOL/MicroJava/BV/JVM.thy	Wed Jun 19 11:48:01 2002 +0200
+++ b/src/HOL/MicroJava/BV/JVM.thy	Wed Jun 19 12:39:41 2002 +0200
@@ -6,14 +6,10 @@
 
 header {* \isaheader{Kildall for the JVM}\label{sec:JVM} *}
 
-theory JVM = Kildall_Lift + JVMType + EffectMono + BVSpec:
+theory JVM = Kildall + Typing_Framework_JVM:
 
 
 constdefs
-  exec :: "jvm_prog \<Rightarrow> nat \<Rightarrow> ty \<Rightarrow> exception_table \<Rightarrow> instr list \<Rightarrow> state step_type"
-  "exec G maxs rT et bs == 
-  err_step (size bs) (\<lambda>pc. app (bs!pc) G maxs rT pc et) (\<lambda>pc. eff (bs!pc) G pc et)"
-
   kiljvm :: "jvm_prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> ty \<Rightarrow> exception_table \<Rightarrow> 
              instr list \<Rightarrow> state list \<Rightarrow> state list"
   "kiljvm G maxs maxr rT et bs ==
@@ -34,251 +30,6 @@
 
 
 
-text {*
-  Executability of @{term check_bounded}:
-*}
-consts
-  list_all'_rec :: "('a \<Rightarrow> nat \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> bool"
-primrec
-  "list_all'_rec P n []     = True"
-  "list_all'_rec P n (x#xs) = (P x n \<and> list_all'_rec P (Suc n) xs)"
-
-constdefs
-  list_all' :: "('a \<Rightarrow> nat \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
-  "list_all' P xs \<equiv> list_all'_rec P 0 xs"
-
-lemma list_all'_rec:
-  "\<And>n. list_all'_rec P n xs = (\<forall>p < size xs. P (xs!p) (p+n))"
-  apply (induct xs)
-  apply auto
-  apply (case_tac p)
-  apply auto
-  done
-
-lemma list_all' [iff]:
-  "list_all' P xs = (\<forall>n < size xs. P (xs!n) n)"
-  by (unfold list_all'_def) (simp add: list_all'_rec)
-
-lemma list_all_ball:
-  "list_all P xs = (\<forall>x \<in> set xs. P x)"
-  by (induct xs) auto
-
-lemma [code]:
-  "check_bounded ins et = 
-  (list_all' (\<lambda>i pc. list_all (\<lambda>pc'. pc' < length ins) (succs i pc)) ins \<and> 
-   list_all (\<lambda>e. fst (snd (snd e)) < length ins) et)"
-  by (simp add: list_all_ball check_bounded_def)
-  
-text {*
-  Lemmas for Kildall instantiation
-*}
-
-lemma check_bounded_is_bounded:
-  "check_bounded ins et \<Longrightarrow> bounded (\<lambda>pc. eff (ins!pc) G pc et) (length ins)"  
-  by (unfold bounded_def) (blast dest: check_boundedD)
-
-lemma special_ex_swap_lemma [iff]: 
-  "(? X. (? n. X = A n & P n) & Q X) = (? n. Q(A n) & P n)"
-  by blast
-
-lemmas [iff del] = not_None_eq
-
-theorem exec_pres_type:
-  "wf_prog wf_mb S \<Longrightarrow> 
-  pres_type (exec S maxs rT et bs) (size bs) (states S maxs maxr)"
-  apply (unfold exec_def JVM_states_unfold)
-  apply (rule pres_type_lift)
-  apply clarify
-  apply (case_tac s)
-   apply simp
-   apply (drule effNone)
-   apply simp  
-  apply (simp add: eff_def xcpt_eff_def norm_eff_def)
-  apply (case_tac "bs!p")
-
-  apply (clarsimp simp add: not_Err_eq)
-  apply (drule listE_nth_in, assumption)
-  apply fastsimp
-
-  apply (fastsimp simp add: not_None_eq)
-
-  apply (fastsimp simp add: not_None_eq typeof_empty_is_type)
-
-  apply clarsimp
-  apply (erule disjE)
-   apply fastsimp
-  apply clarsimp
-  apply (rule_tac x="1" in exI)
-  apply fastsimp
-
-  apply clarsimp
-  apply (erule disjE)
-   apply (fastsimp dest: field_fields fields_is_type)
-  apply (simp add: match_some_entry split: split_if_asm)
-  apply (rule_tac x=1 in exI)
-  apply fastsimp
-
-  apply clarsimp
-  apply (erule disjE)
-   apply fastsimp
-  apply (simp add: match_some_entry split: split_if_asm)
-  apply (rule_tac x=1 in exI)
-  apply fastsimp
-
-  apply clarsimp
-  apply (erule disjE)
-   apply fastsimp
-  apply clarsimp
-  apply (rule_tac x=1 in exI)
-  apply fastsimp
-
-  defer 
-
-  apply fastsimp
-  apply fastsimp
-
-  apply clarsimp
-  apply (rule_tac x="n'+2" in exI)  
-  apply simp
-  apply (drule listE_length)+
-  apply fastsimp
-
-  apply clarsimp
-  apply (rule_tac x="Suc (Suc (Suc (length ST)))" in exI)  
-  apply simp
-  apply (drule listE_length)+
-  apply fastsimp
-
-  apply clarsimp
-  apply (rule_tac x="Suc (Suc (Suc (Suc (length ST))))" in exI)  
-  apply simp
-  apply (drule listE_length)+
-  apply fastsimp
-
-  apply fastsimp
-  apply fastsimp
-  apply fastsimp
-  apply fastsimp
-
-  apply clarsimp
-  apply (erule disjE)
-   apply fastsimp
-  apply clarsimp
-  apply (rule_tac x=1 in exI)
-  apply fastsimp
-  
-  apply (erule disjE)
-   apply (clarsimp simp add: Un_subset_iff)  
-   apply (drule method_wf_mdecl, assumption+)
-   apply (clarsimp simp add: wf_mdecl_def wf_mhead_def)
-   apply fastsimp
-  apply clarsimp
-  apply (rule_tac x=1 in exI)
-  apply fastsimp
-  done
-
-lemmas [iff] = not_None_eq
-
-
-lemma sup_state_opt_unfold:
-  "sup_state_opt G \<equiv> Opt.le (Product.le (Listn.le (subtype G)) (Listn.le (Err.le (subtype G))))"
-  by (simp add: sup_state_opt_def sup_state_def sup_loc_def sup_ty_opt_def)
-
-constdefs
-  opt_states :: "'c prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> (ty list \<times> ty err list) option set"
-  "opt_states G maxs maxr \<equiv> opt (\<Union>{list n (types G) |n. n \<le> maxs} \<times> list maxr (err (types G)))"
-
-lemma app_mono:
-  "app_mono (sup_state_opt G) (\<lambda>pc. app (bs!pc) G maxs rT pc et) (length bs) (opt_states G maxs maxr)"
-  by (unfold app_mono_def lesub_def) (blast intro: EffectMono.app_mono)
-  
-
-lemma lesubstep_type_simple:
-  "a <=[Product.le (op =) r] b \<Longrightarrow> a <=|r| b"
-  apply (unfold lesubstep_type_def)
-  apply clarify
-  apply (simp add: set_conv_nth)
-  apply clarify
-  apply (drule le_listD, assumption)
-  apply (clarsimp simp add: lesub_def Product.le_def)
-  apply (rule exI)
-  apply (rule conjI)
-   apply (rule exI)
-   apply (rule conjI)
-    apply (rule sym)
-    apply assumption
-   apply assumption
-  apply assumption
-  done
-  
-
-lemma eff_mono:
-  "\<lbrakk>p < length bs; s <=_(sup_state_opt G) t; app (bs!p) G maxs rT pc et t\<rbrakk>
-  \<Longrightarrow> eff (bs!p) G p et s <=|sup_state_opt G| eff (bs!p) G p et t"
-  apply (unfold eff_def)
-  apply (rule lesubstep_type_simple)
-  apply (rule le_list_appendI)
-   apply (simp add: norm_eff_def)
-   apply (rule le_listI)
-    apply simp
-   apply simp
-   apply (simp add: lesub_def)
-   apply (case_tac s)
-    apply simp
-   apply (simp del: split_paired_All split_paired_Ex)
-   apply (elim exE conjE)
-   apply simp
-   apply (drule eff'_mono, assumption)
-   apply assumption
-  apply (simp add: xcpt_eff_def)
-  apply (rule le_listI)
-    apply simp
-  apply simp
-  apply (simp add: lesub_def)
-  apply (case_tac s)
-   apply simp
-  apply simp
-  apply (case_tac t)
-   apply simp
-  apply (clarsimp simp add: sup_state_conv)
-  done
-
-lemma order_sup_state_opt:
-  "wf_prog wf_mb G \<Longrightarrow> order (sup_state_opt G)"
-  by (unfold sup_state_opt_unfold) (blast dest: acyclic_subcls1 order_widen)
-
-theorem exec_mono:
-  "wf_prog wf_mb G \<Longrightarrow> bounded (exec G maxs rT et bs) (size bs) \<Longrightarrow>
-  mono (JVMType.le G maxs maxr) (exec G maxs rT et bs) (size bs) (states G maxs maxr)"  
-  apply (unfold exec_def JVM_le_unfold JVM_states_unfold)  
-  apply (rule mono_lift)
-     apply (fold sup_state_opt_unfold opt_states_def)
-     apply (erule order_sup_state_opt)
-    apply (rule app_mono)
-   apply assumption
-  apply clarify
-  apply (rule eff_mono)
-  apply assumption+
-  done
-
-theorem semilat_JVM_slI:
-  "wf_prog wf_mb G \<Longrightarrow> semilat (JVMType.sl G maxs maxr)"
-  apply (unfold JVMType.sl_def stk_esl_def reg_sl_def)
-  apply (rule semilat_opt)
-  apply (rule err_semilat_Product_esl)
-  apply (rule err_semilat_upto_esl)
-  apply (rule err_semilat_JType_esl, assumption+)
-  apply (rule err_semilat_eslI)
-  apply (rule Listn_sl)
-  apply (rule err_semilat_JType_esl, assumption+)
-  done
-
-lemma sl_triple_conv:
-  "JVMType.sl G maxs maxr == 
-  (states G maxs maxr, JVMType.le G maxs maxr, JVMType.sup G maxs maxr)"
-  by (simp (no_asm) add: states_def JVMType.le_def JVMType.sup_def)
-
-
 theorem is_bcv_kiljvm:
   "\<lbrakk> wf_prog wf_mb G; bounded (exec G maxs rT et bs) (size bs) \<rbrakk> \<Longrightarrow>
       is_bcv (JVMType.le G maxs maxr) Err (exec G maxs rT et bs)
@@ -296,23 +47,31 @@
   apply (erule exec_mono, assumption)  
   done
 
-lemma map_id: "\<forall>x \<in> set xs. f (g x) = x \<Longrightarrow> map f (map g xs) = xs"
-  by (induct xs) auto
+lemma subset_replicate: "set (replicate n x) \<subseteq> {x}"
+  by (induct n) auto
+
+lemma in_set_replicate:
+  "x \<in> set (replicate n y) \<Longrightarrow> x = y"
+proof -
+  assume "x \<in> set (replicate n y)"
+  also have "set (replicate n y) \<subseteq> {y}" by (rule subset_replicate)
+  finally have "x \<in> {y}" .
+  thus ?thesis by simp
+qed
 
 theorem wt_kil_correct:
-  "\<lbrakk> wt_kil G C pTs rT maxs mxl et bs; wf_prog wf_mb G; 
-      is_class G C; \<forall>x \<in> set pTs. is_type G x \<rbrakk>
-  \<Longrightarrow> \<exists>phi. wt_method G C pTs rT maxs mxl bs et phi"
+  assumes wf:  "wf_prog wf_mb G"
+  assumes C:   "is_class G C"
+  assumes pTs: "set pTs \<subseteq> types G"
+  
+  assumes wtk: "wt_kil G C pTs rT maxs mxl et bs"
+  
+  shows "\<exists>phi. wt_method G C pTs rT maxs mxl bs et phi"
 proof -
   let ?start = "OK (Some ([],(OK (Class C))#((map OK pTs))@(replicate mxl Err)))
                 #(replicate (size bs - 1) (OK None))"
 
-  assume wf:      "wf_prog wf_mb G"
-  assume isclass: "is_class G C"
-  assume istype:  "\<forall>x \<in> set pTs. is_type G x"
-
-  assume "wt_kil G C pTs rT maxs mxl et bs"
-  then obtain maxr r where    
+  from wtk obtain maxr r where    
     bounded: "check_bounded bs et" and
     result:  "r = kiljvm G maxs maxr rT et bs ?start" and
     success: "\<forall>n < size bs. r!n \<noteq> Err" and
@@ -327,23 +86,14 @@
     (size bs) (states G maxs maxr) (kiljvm G maxs maxr rT et bs)"
     by (rule is_bcv_kiljvm)
     
-  { fix l x have "set (replicate l x) \<subseteq> {x}" by (cases "0 < l") simp+
-  } note subset_replicate = this
-  from istype have "set pTs \<subseteq> types G" by auto
-  hence "OK ` set pTs \<subseteq> err (types G)" by auto
-  with instrs maxr isclass 
+  from C pTs instrs maxr
   have "?start \<in> list (length bs) (states G maxs maxr)"
-    apply (unfold list_def JVM_states_unfold)
-    apply simp
-    apply (rule conjI)
-     apply (simp add: Un_subset_iff)
-     apply (rule_tac B = "{Err}" in subset_trans)
-      apply (simp add: subset_replicate)
-     apply simp
-    apply (rule_tac B = "{OK None}" in subset_trans)
-     apply (simp add: subset_replicate)
-    apply simp
-    done
+    apply (unfold JVM_states_unfold)
+    apply (rule listI)
+    apply (auto intro: list_appendI dest!: in_set_replicate)
+    apply force
+    done    
+
   with bcv success result have 
     "\<exists>ts\<in>list (length bs) (states G maxs maxr).
          ?start <=[JVMType.le G maxs maxr] ts \<and>
@@ -368,11 +118,10 @@
 
   from phi' have "check_types G maxs maxr phi'" by(simp add: check_types_def)
   also from w have "phi' = map OK (map ok_val phi')" 
-    apply (clarsimp simp add: wt_step_def)
+    apply (clarsimp simp add: wt_step_def not_Err_eq) 
     apply (rule map_id [symmetric])
-    apply (clarsimp simp add: in_set_conv_decomp)
-    apply (erule_tac x = "length ys" in allE)
-    apply (clarsimp simp add: nth_append not_Err_eq)
+    apply (erule allE, erule impE, assumption)
+    apply clarsimp
     done    
   finally 
   have check_types:
@@ -409,19 +158,17 @@
 
 
 theorem wt_kil_complete:
-  "\<lbrakk> wt_method G C pTs rT maxs mxl bs et phi; wf_prog wf_mb G; 
-      is_class G C; 
-      \<forall>x \<in> set pTs. is_type G x \<rbrakk>
-  \<Longrightarrow> wt_kil G C pTs rT maxs mxl et bs"
+  assumes wf:  "wf_prog wf_mb G"  
+  assumes C:   "is_class G C"
+  assumes pTs: "set pTs \<subseteq> types G"
+
+  assumes wtm: "wt_method G C pTs rT maxs mxl bs et phi"
+
+  shows "wt_kil G C pTs rT maxs mxl et bs"
 proof -
-  assume wf: "wf_prog wf_mb G"  
-  assume isclass: "is_class G C"
-  assume istype: "\<forall>x \<in> set pTs. is_type G x"
-  
   let ?mxr = "1+size pTs+mxl"
   
-  assume "wt_method G C pTs rT maxs mxl bs et phi"
-  then obtain
+  from wtm obtain
     instrs:   "0 < length bs" and
     len:      "length phi = length bs" and
     bounded:  "check_bounded bs et" and
@@ -456,37 +203,25 @@
     "wt_err_step (sup_state_opt G) (exec G maxs rT et bs) (map OK phi)"
     by (unfold exec_def) (simp add: len)
  
-  let ?maxr = "1+size pTs+mxl"
   from wf bounded_exec
   have is_bcv: 
-    "is_bcv (JVMType.le G maxs ?maxr) Err (exec G maxs rT et bs) 
-            (size bs) (states G maxs ?maxr) (kiljvm G maxs ?maxr rT et bs)"
+    "is_bcv (JVMType.le G maxs ?mxr) Err (exec G maxs rT et bs) 
+            (size bs) (states G maxs ?mxr) (kiljvm G maxs ?mxr rT et bs)"
     by (rule is_bcv_kiljvm)
 
   let ?start = "OK (Some ([],(OK (Class C))#((map OK pTs))@(replicate mxl Err)))
                 #(replicate (size bs - 1) (OK None))"
 
-  { fix l x have "set (replicate l x) \<subseteq> {x}" by (cases "0 < l") simp+
-  } note subset_replicate = this
-
-  from istype have "set pTs \<subseteq> types G" by auto
-  hence "OK ` set pTs \<subseteq> err (types G)" by auto
-  with instrs isclass have start:
-    "?start \<in> list (length bs) (states G maxs ?maxr)"
-    apply (unfold list_def JVM_states_unfold)
-    apply simp
-    apply (rule conjI)
-     apply (simp add: Un_subset_iff)
-     apply (rule_tac B = "{Err}" in subset_trans)
-      apply (simp add: subset_replicate)
-     apply simp
-    apply (rule_tac B = "{OK None}" in subset_trans)
-     apply (simp add: subset_replicate)
-    apply simp
-    done
+  from C pTs instrs
+  have start: "?start \<in> list (length bs) (states G maxs ?mxr)"
+    apply (unfold JVM_states_unfold)
+    apply (rule listI)
+    apply (auto intro!: list_appendI dest!: in_set_replicate)
+    apply force
+    done    
 
   let ?phi = "map OK phi"  
-  have less_phi: "?start <=[JVMType.le G maxs ?maxr] ?phi"
+  have less_phi: "?start <=[JVMType.le G maxs ?mxr] ?phi"
   proof -
     from len instrs
     have "length ?start = length (map OK phi)" by simp
@@ -499,112 +234,55 @@
       from instrs len
       have "0 < length phi" by simp
       ultimately
-      have "JVMType.le G maxs ?maxr (?start!0) (?phi!0)"
+      have "JVMType.le G maxs ?mxr (?start!0) (?phi!0)"
         by (simp add: JVM_le_Err_conv Err.le_def lesub_def)
       moreover
       { fix n'
-        have "JVMType.le G maxs ?maxr (OK None) (?phi!n)"
+        have "JVMType.le G maxs ?mxr (OK None) (?phi!n)"
           by (auto simp add: JVM_le_Err_conv Err.le_def lesub_def 
             split: err.splits)        
         hence "\<lbrakk> n = Suc n'; n < length ?start \<rbrakk> 
-          \<Longrightarrow> JVMType.le G maxs ?maxr (?start!n) (?phi!n)"
+          \<Longrightarrow> JVMType.le G maxs ?mxr (?start!n) (?phi!n)"
           by simp
       }
       ultimately
-      have "n < length ?start \<Longrightarrow> (?start!n) <=_(JVMType.le G maxs ?maxr) (?phi!n)"
+      have "n < length ?start \<Longrightarrow> (?start!n) <=_(JVMType.le G maxs ?mxr) (?phi!n)"
         by (unfold lesub_def) (cases n, blast+)
     } 
     ultimately show ?thesis by (rule le_listI)
   qed         
 
   from wt_err
-  have "wt_step (JVMType.le G maxs ?maxr) Err (exec G maxs rT et bs) ?phi"
+  have "wt_step (JVMType.le G maxs ?mxr) Err (exec G maxs rT et bs) ?phi"
     by (simp add: wt_err_step_def JVM_le_Err_conv)  
   with start istype_phi less_phi is_bcv
-  have "\<forall>p. p < length bs \<longrightarrow> kiljvm G maxs ?maxr rT et bs ?start ! p \<noteq> Err"
+  have "\<forall>p. p < length bs \<longrightarrow> kiljvm G maxs ?mxr rT et bs ?start ! p \<noteq> Err"
     by (unfold is_bcv_def) auto
   with bounded instrs
   show "wt_kil G C pTs rT maxs mxl et bs" by (unfold wt_kil_def) simp
 qed
 
-lemma is_type_pTs:
-  "\<lbrakk> wf_prog wf_mb G; (C,S,fs,mdecls) \<in> set G; (sig,rT,code) \<in> set mdecls; 
-      t \<in> set (snd sig) \<rbrakk>
-  \<Longrightarrow> is_type G t"
-proof -
-  assume "wf_prog wf_mb G" 
-         "(C,S,fs,mdecls) \<in> set G"
-         "(sig,rT,code) \<in> set mdecls"
-  hence "wf_mdecl wf_mb G C (sig,rT,code)"
-    by (unfold wf_prog_def wf_cdecl_def) auto
-  hence "\<forall>t \<in> set (snd sig). is_type G t" 
-    by (unfold wf_mdecl_def wf_mhead_def) auto
-  moreover
-  assume "t \<in> set (snd sig)"
-  ultimately
-  show ?thesis by blast
-qed
-
 
 theorem jvm_kildall_sound_complete:
   "wt_jvm_prog_kildall G = (\<exists>Phi. wt_jvm_prog G Phi)"
 proof 
-  assume wtk: "wt_jvm_prog_kildall G"
-
-  then obtain wf_mb where
-    wf: "wf_prog wf_mb G"
-    by (auto simp add: wt_jvm_prog_kildall_def)
-
   let ?Phi = "\<lambda>C sig. let (C,rT,(maxs,maxl,ins,et)) = the (method (G,C) sig) in 
               SOME phi. wt_method G C (snd sig) rT maxs maxl ins et phi"
-   
-  { fix C S fs mdecls sig rT code
-    assume "(C,S,fs,mdecls) \<in> set G" "(sig,rT,code) \<in> set mdecls"
-    with wf
-    have "method (G,C) sig = Some (C,rT,code) \<and> is_class G C \<and> (\<forall>t \<in> set (snd sig). is_type G t)"
-      by (simp add: methd is_type_pTs)
-  } note this [simp]
- 
-  from wtk
-  have "wt_jvm_prog G ?Phi"
-    apply (unfold wt_jvm_prog_def wt_jvm_prog_kildall_def wf_prog_def wf_cdecl_def)
-    apply clarsimp
-    apply (drule bspec, assumption)
-    apply (unfold wf_mdecl_def)
-    apply clarsimp
-    apply (drule bspec, assumption)
-    apply clarsimp
-    apply (drule wt_kil_correct [OF _ wf])
-    apply (auto intro: someI)
+  
+  assume "wt_jvm_prog_kildall G"
+  hence "wt_jvm_prog G ?Phi"
+    apply (unfold wt_jvm_prog_def wt_jvm_prog_kildall_def)
+    apply (erule jvm_prog_lift)
+    apply (auto dest!: wt_kil_correct intro: someI)
     done
-
-  thus "\<exists>Phi. wt_jvm_prog G Phi" by blast
+  thus "\<exists>Phi. wt_jvm_prog G Phi" by fast
 next
   assume "\<exists>Phi. wt_jvm_prog G Phi"
-  then obtain Phi where wt: "wt_jvm_prog G Phi" ..
-
-  then obtain wf_mb where
-    wf: "wf_prog wf_mb G"
-    by (auto simp add: wt_jvm_prog_def)
-
-  { fix C S fs mdecls sig rT code
-    assume "(C,S,fs,mdecls) \<in> set G" "(sig,rT,code) \<in> set mdecls"
-    with wf
-    have "method (G,C) sig = Some (C,rT,code) \<and> is_class G C \<and> (\<forall>t \<in> set (snd sig). is_type G t)"
-      by (simp add: methd is_type_pTs)
-  } note this [simp]
- 
-  from wt
-  show "wt_jvm_prog_kildall G"
-    apply (unfold wt_jvm_prog_def wt_jvm_prog_kildall_def wf_prog_def wf_cdecl_def)
-    apply clarsimp
-    apply (drule bspec, assumption)
-    apply (unfold wf_mdecl_def)
-    apply clarsimp
-    apply (drule bspec, assumption)
-    apply clarsimp
-    apply (drule wt_kil_complete [OF _ wf])
-    apply (auto intro: someI)
+  thus "wt_jvm_prog_kildall G"
+    apply (clarify)
+    apply (unfold wt_jvm_prog_def wt_jvm_prog_kildall_def)
+    apply (erule jvm_prog_lift)
+    apply (auto intro: wt_kil_complete)
     done
 qed
 




