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Datei: Index.thy Sprache: Isabelle

Original von: Isabelle^©

(*  Title:      HOL/MicroJava/Comp/Index.thy
    Author:     Martin Strecker
*)

(* Index of variable in list of parameter names and local variables *)

theory Index
imports AuxLemmas DefsComp
begin

(*indexing a variable name among all variable declarations in a method body*)
definition index :: "java_mb => vname => nat" where
"index == \ (pn,lv,blk,res) v.
  if v = This
  then 0
  else Suc (length (takeWhile (\<lambda> z. z~=v) (pn @ map fst lv)))"

lemma index_length_pns: "
  \<lbrakk> i = index (pns,lvars,blk,res) vn;
  wf_java_mdecl G C ((mn,pTs),rT, (pns,lvars,blk,res));
  vn \<in> set pns\<rbrakk>
  \<Longrightarrow> 0 < i \<and> i < Suc (length pns)"
  apply (simp add: wf_java_mdecl_def index_def)
  apply (subgoal_tac "vn \ This")
   apply (auto intro: length_takeWhile)
  done

lemma index_length_lvars: "
  \<lbrakk> i = index (pns,lvars,blk,res) vn;
  wf_java_mdecl G C ((mn,pTs),rT, (pns,lvars,blk,res));
  vn \<in> set (map fst lvars)\<rbrakk>
  \<Longrightarrow> (length pns) < i \<and> i < Suc((length pns) + (length lvars))"
  apply (simp add: wf_java_mdecl_def index_def)
  apply (subgoal_tac "vn \ This")
   apply simp
   apply (subgoal_tac "\ x \ set pns. (\z. z \ vn) x")
    apply simp
    apply (subgoal_tac "length (takeWhile (\z. z \ vn) (map fst lvars)) < length (map fst lvars)")
     apply simp
    apply (rule length_takeWhile)
    apply simp
   apply (simp add: map_of_in_set)
   apply (intro strip notI)
   apply simp
  apply blast
  done

(***  index  ***)

lemma select_at_index :
  "x \ set (gjmb_plns (gmb G C S)) \ x = This
  \<Longrightarrow> (the (loc This) # glvs (gmb G C S) loc) ! (index (gmb G C S) x) = the (loc x)"
  apply (simp only: index_def gjmb_plns_def)
  apply (case_tac "gmb G C S" rule: prod.exhaust)
  apply (simp add: galldefs del: set_append map_append)
  apply (rename_tac a b)
  apply (case_tac b, simp add: gmb_def gjmb_lvs_def del: set_append map_append)
  apply (intro strip)
  apply (simp del: set_append map_append)
  apply (frule length_takeWhile)
  apply (frule_tac f = "(the \ loc)" in nth_map)
  apply simp
  done

lemma lift_if: "(f (if b then t else e)) = (if b then (f t) else (f e))"
  by auto

lemma update_at_index: "
  \<lbrakk> distinct (gjmb_plns (gmb G C S));
  x \<in> set (gjmb_plns (gmb G C S)); x \<noteq> This \<rbrakk> \<Longrightarrow>
  (locvars_xstate G C S (Norm (h, l)))[index (gmb G C S) x := val] =
  locvars_xstate G C S (Norm (h, l(x\<mapsto>val)))"
  apply (simp only: locvars_xstate_def locvars_locals_def index_def)
  apply (case_tac "gmb G C S" rule: prod.exhaust, simp)
  apply (rename_tac a b)
  apply (case_tac b, simp)
  apply (rule conjI)
   apply (simp add: gl_def)
  apply (simp add: galldefs del: set_append map_append)
  done

(* !!!! incomprehensible: why can't List.takeWhile_append2 be applied the same
  way in the second case as in the first case ? *)
lemma index_of_var: "\ xvar \ set pns; xvar \ set (map fst zs); xvar \ This \
  \<Longrightarrow> index (pns, zs @ ((xvar, xval) # xys), blk, res) xvar = Suc (length pns + length zs)"
  apply (simp add: index_def)
  apply (subgoal_tac "(\x. ((x \ (set pns)) \ ((\z. (z \ xvar))x)))")
   apply simp
   apply (subgoal_tac "(takeWhile (\z. z \ xvar) (map fst zs @ xvar # map fst xys)) = map fst zs @ (takeWhile (\z. z \ xvar) (xvar # map fst xys))")
    apply simp
   apply (rule List.takeWhile_append2)
   apply auto
  done

(* The following definition should replace the conditions in WellType.thy / wf_java_mdecl
*)
definition disjoint_varnames :: "[vname list, (vname \ ty) list] \ bool" where
  "disjoint_varnames pns lvars \
  distinct pns \<and> unique lvars \<and> This \<notin> set pns \<and> This \<notin> set (map fst lvars) \<and>
  (\<forall>pn\<in>set pns. pn \<notin> set (map fst lvars))"

lemma index_of_var2: "
  disjoint_varnames pns (lvars_pre @ (vn, ty) # lvars_post)
  \<Longrightarrow> index (pns, lvars_pre @ (vn, ty) # lvars_post, blk, res) vn =
  Suc (length pns + length lvars_pre)"
  apply (simp add: disjoint_varnames_def index_def unique_def)
  apply (subgoal_tac "vn \ This", simp)
   apply (subgoal_tac
    "takeWhile (\z. z \ vn) (map fst lvars_pre @ vn # map fst lvars_post) =
    map fst lvars_pre @ takeWhile (\<lambda>z. z \<noteq> vn) (vn # map fst lvars_post)")
    apply simp
   apply (rule List.takeWhile_append2)
   apply auto
  done

lemma wf_java_mdecl_disjoint_varnames:
  "wf_java_mdecl G C (S,rT,(pns,lvars,blk,res))
  \<Longrightarrow> disjoint_varnames pns lvars"
  apply (cases S)
  apply (simp add: wf_java_mdecl_def disjoint_varnames_def  map_of_in_set)
  done

lemma wf_java_mdecl_length_pTs_pns:
  "wf_java_mdecl G C ((mn, pTs), rT, pns, lvars, blk, res)
  \<Longrightarrow> length pTs = length pns"
  by (simp add: wf_java_mdecl_def)

end

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