Theory State (Isabelle2005: October 2005)

# Theory State

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theory State
imports DeclConcepts
begin

    (*  Title:      HOL/Bali/State.thy
        ID:         $Id: State.thy,v 1.11 2005/06/17 14:13:06 haftmann Exp $
        Author:     David von Oheimb
    *)
    header {* State for evaluation of Java expressions and statements *}

    theory State imports DeclConcepts begin

    text {*
    design issues:
    \begin{itemize}
    \item all kinds of objects (class instances, arrays, and class objects)
      are handeled via a general object abstraction
    \item the heap and the map for class objects are combined into a single table
      @{text "(recall (loc, obj) table × (qtname, obj) table  ~=  (loc + qtname, obj) table)"}
    \end{itemize}
    *}

    section "objects"

    datatype  obj_tag =     --{* tag for generic object   *}
              CInst qtname  --{* class instance           *}
            | Arr  ty int   --{* array with component type and length *}
        --{* | CStat qtname   the tag is irrelevant for a class object,
                               i.e. the static fields of a class,
                               since its type is given already by the reference to
                               it (see below) *}

    types   vn   = "fspec + int"                    --{* variable name      *}
    record  obj  =
              tag :: "obj_tag"                      --{* generalized object *}
              values :: "(vn, val) table"

    translations
      "fspec" <= (type) "vname × qtname"
      "vn"    <= (type) "fspec + int"
      "obj"   <= (type) "(|tag::obj_tag, values::vn => val option|)),"
      "obj"   <= (type) "(|tag::obj_tag, values::vn => val option,…::'a|)),"

    constdefs

      the_Arr :: "obj option => ty × int × (vn, val) table"
     "the_Arr obj ≡ SOME (T,k,t). obj = Some (|tag=Arr T k,values=t|)),"

    lemma the_Arr_Arr [simp]: "the_Arr (Some (|tag=Arr T k,values=cs|)),) = (T,k,cs)"
    apply (auto simp: the_Arr_def)
    done

    lemma the_Arr_Arr1 [simp,intro,dest]:
     "[|tag obj = Arr T k|] ==> the_Arr (Some obj) = (T,k,values obj)"
    apply (auto simp add: the_Arr_def)
    done

    constdefs

      upd_obj       :: "vn => val => obj => obj"
     "upd_obj n v ≡ λ obj . obj (|values:=(values obj)(n\<mapsto>v)|)),"

    lemma upd_obj_def2 [simp]:
      "upd_obj n v obj = obj (|values:=(values obj)(n\<mapsto>v)|)),"
    apply (auto simp: upd_obj_def)
    done

    constdefs
      obj_ty        :: "obj => ty"
     "obj_ty obj    ≡ case tag obj of
                        CInst C => Class C
                      | Arr T k => T.[]"

    lemma obj_ty_eq [intro!]: "obj_ty (|tag=oi,values=x|)), = obj_ty (|tag=oi,values=y|)),"
    by (simp add: obj_ty_def)

    lemma obj_ty_eq1 [intro!,dest]:
      "tag obj = tag obj' ==> obj_ty obj = obj_ty obj'"
    by (simp add: obj_ty_def)

    lemma obj_ty_cong [simp]:
      "obj_ty (obj (|values:=vs|)),) = obj_ty obj"
    by auto

    lemma obj_ty_CInst [simp]:
     "obj_ty (|tag=CInst C,values=vs|)), = Class C"
    by (simp add: obj_ty_def)

    lemma obj_ty_CInst1 [simp,intro!,dest]:
     "[|tag obj = CInst C|] ==> obj_ty obj = Class C"
    by (simp add: obj_ty_def)

    lemma obj_ty_Arr [simp]:
     "obj_ty (|tag=Arr T i,values=vs|)), = T.[]"
    by (simp add: obj_ty_def)

    lemma obj_ty_Arr1 [simp,intro!,dest]:
     "[|tag obj = Arr T i|] ==> obj_ty obj = T.[]"
    by (simp add: obj_ty_def)

    lemma obj_ty_widenD:
     "G\<turnstile>obj_ty obj\<preceq>RefT t ==> (∃C. tag obj = CInst C) ∨ (∃T k. tag obj = Arr T k)"
    apply (unfold obj_ty_def)
    apply (auto split add: obj_tag.split_asm)
    done

    constdefs

      obj_class :: "obj => qtname"
     "obj_class obj ≡ case tag obj of
                        CInst C => C
                      | Arr T k => Object"

    lemma obj_class_CInst [simp]: "obj_class (|tag=CInst C,values=vs|)), = C"
    by (auto simp: obj_class_def)

    lemma obj_class_CInst1 [simp,intro!,dest]:
      "tag obj = CInst C ==> obj_class obj = C"
    by (auto simp: obj_class_def)

    lemma obj_class_Arr [simp]: "obj_class (|tag=Arr T k,values=vs|)), = Object"
    by (auto simp: obj_class_def)

    lemma obj_class_Arr1 [simp,intro!,dest]:
     "tag obj = Arr T k ==> obj_class obj = Object"
    by (auto simp: obj_class_def)

    lemma obj_ty_obj_class: "G\<turnstile>obj_ty obj\<preceq> Class statC = G\<turnstile>obj_class obj \<preceq>C statC"
    apply (case_tac "tag obj")
    apply (auto simp add: obj_ty_def obj_class_def)
    apply (case_tac "statC = Object")
    apply (auto dest: widen_Array_Class)
    done

    section "object references"

    types oref = "loc + qtname"         --{* generalized object reference *}
    syntax
      Heap  :: "loc   => oref"
      Stat  :: "qtname => oref"

    translations
      "Heap" => "Inl"
      "Stat" => "Inr"
      "oref" <= (type) "loc + qtname"

    constdefs
      fields_table::
        "prog => qtname => (fspec => field => bool)  => (fspec, ty) table"
     "fields_table G C P
        ≡ option_map type o table_of (filter (split P) (DeclConcepts.fields G C))"

    lemma fields_table_SomeI:
    "[|table_of (DeclConcepts.fields G C) n = Some f; P n f|]
     ==> fields_table G C P n = Some (type f)"
    apply (unfold fields_table_def)
    apply clarsimp
    apply (rule exI)
    apply (rule conjI)
    apply (erule map_of_filter_in)
    apply assumption
    apply simp
    done

    (* unused *)
    lemma fields_table_SomeD': "fields_table G C P fn = Some T ==>
      ∃f. (fn,f)∈set(DeclConcepts.fields G C) ∧ type f = T"
    apply (unfold fields_table_def)
    apply clarsimp
    apply (drule map_of_SomeD)
    apply auto
    done

    lemma fields_table_SomeD:
    "[|fields_table G C P fn = Some T; unique (DeclConcepts.fields G C)|] ==>
      ∃f. table_of (DeclConcepts.fields G C) fn = Some f ∧ type f = T"
    apply (unfold fields_table_def)
    apply clarsimp
    apply (rule exI)
    apply (rule conjI)
    apply (erule table_of_filter_unique_SomeD)
    apply assumption
    apply simp
    done

    constdefs
      in_bounds :: "int => int => bool"            ("(_/ in'_bounds _)" [50, 51] 50)
     "i in_bounds k ≡ 0 ≤ i ∧ i < k"

      arr_comps :: "'a => int => int => 'a option"
     "arr_comps T k ≡ λi. if i in_bounds k then Some T else None"

      var_tys       :: "prog => obj_tag => oref => (vn, ty) table"
    "var_tys G oi r
      ≡ case r of
          Heap a => (case oi of
                       CInst C => fields_table G C (λn f. ¬static f) (+) empty
                     | Arr T k => empty (+) arr_comps T k)
        | Stat C => fields_table G C (λfn f. declclassf fn = C ∧ static f)
                    (+) empty"

    lemma var_tys_Some_eq:
     "var_tys G oi r n = Some T
      = (case r of
           Inl a => (case oi of
                       CInst C => (∃nt. n = Inl nt ∧ fields_table G C (λn f.
                                   ¬static f) nt = Some T)
                     | Arr t k => (∃ i. n = Inr i  ∧ i in_bounds k ∧ t = T))
         | Inr C => (∃nt. n = Inl nt ∧
                     fields_table G C (λfn f. declclassf fn = C ∧ static f) nt
                      = Some T))"
    apply (unfold var_tys_def arr_comps_def)
    apply (force split add: sum.split_asm sum.split obj_tag.split)
    done

    section "stores"

    types   globs               --{* global variables: heap and static variables *}
            = "(oref , obj) table"
            heap
            = "(loc  , obj) table"
    (*      locals
            = "(lname, val) table" *) (* defined in Value.thy local variables *)

    translations
     "globs"  <= (type) "(oref , obj) table"
     "heap"   <= (type) "(loc  , obj) table"
    (*  "locals" <= (type) "(lname, val) table" *)

    datatype st = (* pure state, i.e. contents of all variables *)
             st globs locals

    subsection "access"

    constdefs

      globs  :: "st => globs"
     "globs  ≡ st_case (λg l. g)"

      locals :: "st => locals"
     "locals ≡ st_case (λg l. l)"

      heap   :: "st => heap"
     "heap s ≡ globs s o Heap"

    lemma globs_def2 [simp]: " globs (st g l) = g"
    by (simp add: globs_def)

    lemma locals_def2 [simp]: "locals (st g l) = l"
    by (simp add: locals_def)

    lemma heap_def2 [simp]:  "heap s a=globs s (Heap a)"
    by (simp add: heap_def)

    syntax
      val_this     :: "st => val"
      lookup_obj   :: "st => val => obj"

    translations
     "val_this s"       == "the (locals s This)"
     "lookup_obj s a'"  == "the (heap s (the_Addr a'))"

    subsection "memory allocation"

    constdefs
      new_Addr     :: "heap => loc option"
     "new_Addr h   ≡ if (∀a. h a ≠ None) then None else Some (SOME a. h a = None)"

    lemma new_AddrD: "new_Addr h = Some a ==> h a = None"
    apply (unfold new_Addr_def)
    apply auto
    apply (case_tac "h (SOME a::loc. h a = None)")
    apply simp
    apply (fast intro: someI2)
    done

    lemma new_AddrD2: "new_Addr h = Some a ==> ∀b. h b ≠ None --> b ≠ a"
    apply (drule new_AddrD)
    apply auto
    done

    lemma new_Addr_SomeI: "h a = None ==> ∃b. new_Addr h = Some b ∧ h b = None"
    apply (unfold new_Addr_def)
    apply (frule not_Some_eq [THEN iffD2])
    apply auto
    apply  (drule not_Some_eq [THEN iffD2])
    apply  auto
    apply (fast intro!: someI2)
    done

    subsection "initialization"

    syntax

      init_vals     :: "('a, ty) table => ('a, val) table"

    translations
     "init_vals vs"    == "option_map default_val o vs"

    lemma init_arr_comps_base [simp]: "init_vals (arr_comps T 0) = empty"
    apply (unfold arr_comps_def in_bounds_def)
    apply (rule ext)
    apply auto
    done

    lemma init_arr_comps_step [simp]:
    "0 < j ==> init_vals (arr_comps T  j    ) =
               init_vals (arr_comps T (j - 1))(j - 1\<mapsto>default_val T)"
    apply (unfold arr_comps_def in_bounds_def)
    apply (rule ext)
    apply auto
    done

    subsection "update"

    constdefs
      gupd       :: "oref  => obj => st => st"        ("gupd'(_\<mapsto>_')"[10,10]1000)
     "gupd r obj  ≡ st_case (λg l. st (g(r\<mapsto>obj)) l)"

      lupd       :: "lname => val => st => st"        ("lupd'(_\<mapsto>_')"[10,10]1000)
     "lupd vn v   ≡ st_case (λg l. st g (l(vn\<mapsto>v)))"

      upd_gobj   :: "oref => vn => val => st => st"
     "upd_gobj r n v ≡ st_case (λg l. st (chg_map (upd_obj n v) r g) l)"

      set_locals  :: "locals => st => st"
     "set_locals l ≡ st_case (λg l'. st g l)"

      init_obj    :: "prog => obj_tag => oref => st => st"
     "init_obj G oi r ≡ gupd(r\<mapsto>(|tag=oi, values=init_vals (var_tys G oi r)|)),)"

    syntax
      init_class_obj :: "prog => qtname => st => st"

    translations
     "init_class_obj G C" == "init_obj G arbitrary (Inr C)"

    lemma gupd_def2 [simp]: "gupd(r\<mapsto>obj) (st g l) = st (g(r\<mapsto>obj)) l"
    apply (unfold gupd_def)
    apply (simp (no_asm))
    done

    lemma lupd_def2 [simp]: "lupd(vn\<mapsto>v) (st g l) = st g (l(vn\<mapsto>v))"
    apply (unfold lupd_def)
    apply (simp (no_asm))
    done

    lemma globs_gupd [simp]: "globs  (gupd(r\<mapsto>obj) s) = globs s(r\<mapsto>obj)"
    apply (induct "s")
    by (simp add: gupd_def)

    lemma globs_lupd [simp]: "globs  (lupd(vn\<mapsto>v ) s) = globs  s"
    apply (induct "s")
    by (simp add: lupd_def)

    lemma locals_gupd [simp]: "locals (gupd(r\<mapsto>obj) s) = locals s"
    apply (induct "s")
    by (simp add: gupd_def)

    lemma locals_lupd [simp]: "locals (lupd(vn\<mapsto>v ) s) = locals s(vn\<mapsto>v )"
    apply (induct "s")
    by (simp add: lupd_def)

    lemma globs_upd_gobj_new [rule_format (no_asm), simp]:
      "globs s r = None --> globs (upd_gobj r n v s) = globs s"
    apply (unfold upd_gobj_def)
    apply (induct "s")
    apply auto
    done

    lemma globs_upd_gobj_upd [rule_format (no_asm), simp]:
    "globs s r=Some obj--> globs (upd_gobj r n v s) = globs s(r\<mapsto>upd_obj n v obj)"
    apply (unfold upd_gobj_def)
    apply (induct "s")
    apply auto
    done

    lemma locals_upd_gobj [simp]: "locals (upd_gobj r n v s) = locals s"
    apply (induct "s")
    by (simp add: upd_gobj_def)

    lemma globs_init_obj [simp]: "globs (init_obj G oi r s) t =
      (if t=r then Some (|tag=oi,values=init_vals (var_tys G oi r)|)), else globs s t)"
    apply (unfold init_obj_def)
    apply (simp (no_asm))
    done

    lemma locals_init_obj [simp]: "locals (init_obj G oi r s) = locals s"
    by (simp add: init_obj_def)

    lemma surjective_st [simp]: "st (globs s) (locals s) = s"
    apply (induct "s")
    by auto

    lemma surjective_st_init_obj:
     "st (globs (init_obj G oi r s)) (locals s) = init_obj G oi r s"
    apply (subst locals_init_obj [THEN sym])
    apply (rule surjective_st)
    done

    lemma heap_heap_upd [simp]:
      "heap (st (g(Inl a\<mapsto>obj)) l) = heap (st g l)(a\<mapsto>obj)"
    apply (rule ext)
    apply (simp (no_asm))
    done
    lemma heap_stat_upd [simp]: "heap (st (g(Inr C\<mapsto>obj)) l) = heap (st g l)"
    apply (rule ext)
    apply (simp (no_asm))
    done
    lemma heap_local_upd [simp]: "heap (st g (l(vn\<mapsto>v))) = heap (st g l)"
    apply (rule ext)
    apply (simp (no_asm))
    done

    lemma heap_gupd_Heap [simp]: "heap (gupd(Heap a\<mapsto>obj) s) = heap s(a\<mapsto>obj)"
    apply (rule ext)
    apply (simp (no_asm))
    done
    lemma heap_gupd_Stat [simp]: "heap (gupd(Stat C\<mapsto>obj) s) = heap s"
    apply (rule ext)
    apply (simp (no_asm))
    done
    lemma heap_lupd [simp]: "heap (lupd(vn\<mapsto>v) s) = heap s"
    apply (rule ext)
    apply (simp (no_asm))
    done

    lemma heap_upd_gobj_Stat [simp]: "heap (upd_gobj (Stat C) n v s) = heap s"
    apply (rule ext)
    apply (simp (no_asm))
    apply (case_tac "globs s (Stat C)")
    apply  auto
    done

    lemma set_locals_def2 [simp]: "set_locals l (st g l') = st g l"
    apply (unfold set_locals_def)
    apply (simp (no_asm))
    done

    lemma set_locals_id [simp]: "set_locals (locals s) s = s"
    apply (unfold set_locals_def)
    apply (induct_tac "s")
    apply (simp (no_asm))
    done

    lemma set_set_locals [simp]: "set_locals l (set_locals l' s) = set_locals l s"
    apply (unfold set_locals_def)
    apply (induct_tac "s")
    apply (simp (no_asm))
    done

    lemma locals_set_locals [simp]: "locals (set_locals l s) = l"
    apply (unfold set_locals_def)
    apply (induct_tac "s")
    apply (simp (no_asm))
    done

    lemma globs_set_locals [simp]: "globs (set_locals l s) = globs s"
    apply (unfold set_locals_def)
    apply (induct_tac "s")
    apply (simp (no_asm))
    done

    lemma heap_set_locals [simp]: "heap (set_locals l s) = heap s"
    apply (unfold heap_def)
    apply (induct_tac "s")
    apply (simp (no_asm))
    done

    section "abrupt completion"

    consts

      the_Xcpt :: "abrupt => xcpt"
      the_Jump :: "abrupt => jump"
      the_Loc  :: "xcpt => loc"
      the_Std  :: "xcpt => xname"

    primrec "the_Xcpt (Xcpt x) = x"
    primrec "the_Jump (Jump j) = j"
    primrec "the_Loc (Loc a) = a"
    primrec "the_Std (Std x) = x"

    constdefs
      abrupt_if    :: "bool => abopt => abopt => abopt"
     "abrupt_if c x' x ≡ if c ∧ (x = None) then x' else x"

    lemma abrupt_if_True_None [simp]: "abrupt_if True x None = x"
    by (simp add: abrupt_if_def)

    lemma abrupt_if_True_not_None [simp]: "x ≠ None ==> abrupt_if True x y ≠ None"
    by (simp add: abrupt_if_def)

    lemma abrupt_if_False [simp]: "abrupt_if False x y = y"
    by (simp add: abrupt_if_def)

    lemma abrupt_if_Some [simp]: "abrupt_if c x (Some y) = Some y"
    by (simp add: abrupt_if_def)

    lemma abrupt_if_not_None [simp]: "y ≠ None ==> abrupt_if c x y = y"
    apply (simp add: abrupt_if_def)
    by auto

    lemma split_abrupt_if:
    "P (abrupt_if c x' x) =
          ((c ∧ x = None --> P x') ∧ (¬ (c ∧ x = None) --> P x))"
    apply (unfold abrupt_if_def)
    apply (split split_if)
    apply auto
    done

    syntax

      raise_if :: "bool => xname => abopt => abopt"
      np       :: "val            => abopt => abopt"
      check_neg:: "val            => abopt => abopt"
      error_if :: "bool => error => abopt => abopt"

    translations

     "raise_if c xn" == "abrupt_if c (Some (Xcpt (Std xn)))"
     "np v"          == "raise_if (v = Null)      NullPointer"
     "check_neg i'"  == "raise_if (the_Intg i'<0) NegArrSize"
     "error_if c e"  == "abrupt_if c (Some (Error e))"

    lemma raise_if_None [simp]: "(raise_if c x y = None) = (¬c ∧ y = None)"
    apply (simp add: abrupt_if_def)
    by auto
    declare raise_if_None [THEN iffD1, dest!]

    lemma if_raise_if_None [simp]:
      "((if b then y else raise_if c x y) = None) = ((c --> b) ∧ y = None)"
    apply (simp add: abrupt_if_def)
    apply auto
    done

    lemma raise_if_SomeD [dest!]:
      "raise_if c x y = Some z ==> c ∧ z=(Xcpt (Std x)) ∧ y=None ∨ (y=Some z)"
    apply (case_tac y)
    apply (case_tac c)
    apply (simp add: abrupt_if_def)
    apply (simp add: abrupt_if_def)
    apply auto
    done

    lemma error_if_None [simp]: "(error_if c e y = None) = (¬c ∧ y = None)"
    apply (simp add: abrupt_if_def)
    by auto
    declare error_if_None [THEN iffD1, dest!]

    lemma if_error_if_None [simp]:
      "((if b then y else error_if c e y) = None) = ((c --> b) ∧ y = None)"
    apply (simp add: abrupt_if_def)
    apply auto
    done

    lemma error_if_SomeD [dest!]:
      "error_if c e y = Some z ==> c ∧ z=(Error e) ∧ y=None ∨ (y=Some z)"
    apply (case_tac y)
    apply (case_tac c)
    apply (simp add: abrupt_if_def)
    apply (simp add: abrupt_if_def)
    apply auto
    done

    constdefs
       absorb :: "jump => abopt => abopt"
      "absorb j a ≡ if a=Some (Jump j) then None else a"

    lemma absorb_SomeD [dest!]: "absorb j a = Some x ==> a = Some x"
    by (auto simp add: absorb_def)

    lemma absorb_same [simp]: "absorb j (Some (Jump j)) = None"
    by (auto simp add: absorb_def)

    lemma absorb_other [simp]: "a ≠ Some (Jump j) ==> absorb j a = a"
    by (auto simp add: absorb_def)

    lemma absorb_Some_NoneD: "absorb j (Some abr) = None ==> abr = Jump j"
      by (simp add: absorb_def)

    lemma absorb_Some_JumpD: "absorb j s = Some (Jump j') ==> j'≠j"
      by (simp add: absorb_def)

    section "full program state"

    types
      state = "abopt × st"          --{* state including abruption information *}

    syntax
      Norm   :: "st => state"
      abrupt :: "state => abopt"
      store  :: "state => st"

    translations

      "Norm s"     == "(None,s)"
      "abrupt"     => "fst"
      "store"      => "snd"
      "abopt"       <= (type) "State.abrupt option"
      "abopt"       <= (type) "abrupt option"
      "state"      <= (type) "abopt × State.st"
      "state"      <= (type) "abopt × st"

    lemma single_stateE: "∀Z. Z = (s::state) ==> False"
    apply (erule_tac x = "(Some k,y)" in all_dupE)
    apply (erule_tac x = "(None,y)" in allE)
    apply clarify
    done

    lemma state_not_single: "All (op = (x::state)) ==> R"
    apply (drule_tac x = "(if abrupt x = None then Some ?x else None,?y)" in spec)
    apply clarsimp
    done

    constdefs

      normal     :: "state => bool"
     "normal ≡ λs. abrupt s = None"

    lemma normal_def2 [simp]: "normal s = (abrupt s = None)"
    apply (unfold normal_def)
    apply (simp (no_asm))
    done

    constdefs
      heap_free :: "nat => state => bool"
     "heap_free n ≡ λs. atleast_free (heap (store s)) n"

    lemma heap_free_def2 [simp]: "heap_free n s = atleast_free (heap (store s)) n"
    apply (unfold heap_free_def)
    apply simp
    done

    subsection "update"

    constdefs

      abupd     :: "(abopt => abopt) => state => state"
     "abupd f ≡ prod_fun f id"

      supd     :: "(st => st) => state => state"
     "supd ≡ prod_fun id"

    lemma abupd_def2 [simp]: "abupd f (x,s) = (f x,s)"
    by (simp add: abupd_def)

    lemma abupd_abrupt_if_False [simp]: "!! s. abupd (abrupt_if False xo) s = s"
    by simp

    lemma supd_def2 [simp]: "supd f (x,s) = (x,f s)"
    by (simp add: supd_def)

    lemma supd_lupd [simp]:
     "!! s. supd (lupd vn v ) s = (abrupt s,lupd vn v (store s))"
    apply (simp (no_asm_simp) only: split_tupled_all)
    apply (simp (no_asm))
    done

    lemma supd_gupd [simp]:
     "!! s. supd (gupd r obj) s = (abrupt s,gupd r obj (store s))"
    apply (simp (no_asm_simp) only: split_tupled_all)
    apply (simp (no_asm))
    done

    lemma supd_init_obj [simp]:
     "supd (init_obj G oi r) s = (abrupt s,init_obj G oi r (store s))"
    apply (unfold init_obj_def)
    apply (simp (no_asm))
    done

    lemma abupd_store_invariant [simp]: "store (abupd f s) = store s"
      by (cases s) simp

    lemma supd_abrupt_invariant [simp]: "abrupt (supd f s) = abrupt s"
      by (cases s) simp

    syntax

      set_lvars     :: "locals => state => state"
      restore_lvars :: "state  => state => state"

    translations

     "set_lvars l" == "supd (set_locals l)"
     "restore_lvars s' s" == "set_lvars (locals (store s')) s"

    lemma set_set_lvars [simp]: "!! s. set_lvars l (set_lvars l' s) = set_lvars l s"
    apply (simp (no_asm_simp) only: split_tupled_all)
    apply (simp (no_asm))
    done

    lemma set_lvars_id [simp]: "!! s. set_lvars (locals (store s)) s = s"
    apply (simp (no_asm_simp) only: split_tupled_all)
    apply (simp (no_asm))
    done

    section "initialisation test"

    constdefs

      inited   :: "qtname => globs => bool"
     "inited C g ≡ g (Stat C) ≠ None"

      initd    :: "qtname => state => bool"
     "initd C ≡ inited C o globs o store"

    lemma not_inited_empty [simp]: "¬inited C empty"
    apply (unfold inited_def)
    apply (simp (no_asm))
    done

    lemma inited_gupdate [simp]: "inited C (g(r\<mapsto>obj)) = (inited C g ∨ r = Stat C)"
    apply (unfold inited_def)
    apply (auto split add: st.split)
    done

    lemma inited_init_class_obj [intro!]: "inited C (globs (init_class_obj G C s))"
    apply (unfold inited_def)
    apply (simp (no_asm))
    done

    lemma not_initedD: "¬ inited C g ==> g (Stat C) = None"
    apply (unfold inited_def)
    apply (erule notnotD)
    done

    lemma initedD: "inited C g ==> ∃ obj. g (Stat C) = Some obj"
    apply (unfold inited_def)
    apply auto
    done

    lemma initd_def2 [simp]: "initd C s = inited C (globs (store s))"
    apply (unfold initd_def)
    apply (simp (no_asm))
    done

    section {* @{text error_free} *}
    constdefs error_free:: "state => bool"
    "error_free s ≡ ¬ (∃ err. abrupt s = Some (Error err))"

    lemma error_free_Norm [simp,intro]: "error_free (Norm s)"
    by (simp add: error_free_def)

    lemma error_free_normal [simp,intro]: "normal s ==> error_free s"
    by (simp add: error_free_def)

    lemma error_free_Xcpt [simp]: "error_free (Some (Xcpt x),s)"
    by (simp add: error_free_def)

    lemma error_free_Jump [simp,intro]: "error_free (Some (Jump j),s)"
    by (simp add: error_free_def)

    lemma error_free_Error [simp]: "error_free (Some (Error e),s) = False"
    by (simp add: error_free_def)

    lemma error_free_Some [simp,intro]:
     "¬ (∃ err. x=Error err) ==> error_free ((Some x),s)"
    by (auto simp add: error_free_def)

    lemma error_free_abupd_absorb [simp,intro]:
     "error_free s ==> error_free (abupd (absorb j) s)"
    by (cases s)
       (auto simp add: error_free_def absorb_def
             split: split_if_asm)

    lemma error_free_absorb [simp,intro]:
     "error_free (a,s) ==> error_free (absorb j a, s)"
    by (auto simp add: error_free_def absorb_def
                split: split_if_asm)

    lemma error_free_abrupt_if [simp,intro]:
    "[|error_free s; ¬ (∃ err. x=Error err)|]
     ==> error_free (abupd (abrupt_if p (Some x)) s)"
    by (cases s)
       (auto simp add: abrupt_if_def
                split: split_if)

    lemma error_free_abrupt_if1 [simp,intro]:
    "[|error_free (a,s); ¬ (∃ err. x=Error err)|]
     ==> error_free (abrupt_if p (Some x) a, s)"
    by  (auto simp add: abrupt_if_def
                split: split_if)

    lemma error_free_abrupt_if_Xcpt [simp,intro]:
     "error_free s
      ==> error_free (abupd (abrupt_if p (Some (Xcpt x))) s)"
    by simp

    lemma error_free_abrupt_if_Xcpt1 [simp,intro]:
     "error_free (a,s)
      ==> error_free (abrupt_if p (Some (Xcpt x)) a, s)"
    by simp

    lemma error_free_abrupt_if_Jump [simp,intro]:
     "error_free s
      ==> error_free (abupd (abrupt_if p (Some (Jump j))) s)"
    by simp

    lemma error_free_abrupt_if_Jump1 [simp,intro]:
     "error_free (a,s)
      ==> error_free (abrupt_if p (Some (Jump j)) a, s)"
    by simp

    lemma error_free_raise_if [simp,intro]:
     "error_free s ==> error_free (abupd (raise_if p x) s)"
    by simp

    lemma error_free_raise_if1 [simp,intro]:
     "error_free (a,s) ==> error_free ((raise_if p x a), s)"
    by simp

    lemma error_free_supd [simp,intro]:
     "error_free s ==> error_free (supd f s)"
    by (cases s) (simp add: error_free_def)

    lemma error_free_supd1 [simp,intro]:
     "error_free (a,s) ==> error_free (a,f s)"
    by (simp add: error_free_def)

    lemma error_free_set_lvars [simp,intro]:
    "error_free s ==> error_free ((set_lvars l) s)"
    by (cases s) simp

    lemma error_free_set_locals [simp,intro]:
    "error_free (x, s)
           ==> error_free (x, set_locals l s')"
    by (simp add: error_free_def)

    end

## objects

lemma the_Arr_Arr:

      the_Arr (Some (| obj.tag = Arr T k, values = cs |)) = (T, k, cs)

lemma the_Arr_Arr1:

      obj.tag obj = Arr T k ==> the_Arr (Some obj) = (T, k, values obj)

lemma upd_obj_def2:

      upd_obj n v obj = obj(| values := values obj(n |-> v) |)

lemma obj_ty_eq:

      obj_ty (| obj.tag = oi, values = x |) = obj_ty (| obj.tag = oi, values = y |)

lemma obj_ty_eq1:

      obj.tag obj = obj.tag obj' ==> obj_ty obj = obj_ty obj'

lemma obj_ty_cong:

      obj_ty (obj(| values := vs |)) = obj_ty obj

lemma obj_ty_CInst:

      obj_ty (| obj.tag = CInst C, values = vs |) = Class C

lemma obj_ty_CInst1:

      obj.tag obj = CInst C ==> obj_ty obj = Class C

lemma obj_ty_Arr:

      obj_ty (| obj.tag = Arr T i, values = vs |) = T.[]

lemma obj_ty_Arr1:

      obj.tag obj = Arr T i ==> obj_ty obj = T.[]

lemma obj_ty_widenD:

      G|-obj_ty obj<=:RefT t
      ==> (∃C. obj.tag obj = CInst C) ∨ (∃T k. obj.tag obj = Arr T k)

lemma obj_class_CInst:

      obj_class (| obj.tag = CInst C, values = vs |) = C

lemma obj_class_CInst1:

      obj.tag obj = CInst C ==> obj_class obj = C

lemma obj_class_Arr:

      obj_class (| obj.tag = Arr T k, values = vs |) = Object

lemma obj_class_Arr1:

      obj.tag obj = Arr T k ==> obj_class obj = Object

lemma obj_ty_obj_class:

      G|-obj_ty obj<=:Class statC = G|-obj_class obj<=:C statC

## object references

lemma fields_table_SomeI:

      [| table_of (DeclConcepts.fields G C) n = Some f; P n f |]
      ==> fields_table G C P n = Some (type f)

lemma fields_table_SomeD':

      fields_table G C P fn = Some T
      ==> ∃f. (fn, f) ∈ set (DeclConcepts.fields G C) ∧ type f = T

lemma fields_table_SomeD:

      [| fields_table G C P fn = Some T; unique (DeclConcepts.fields G C) |]
      ==> ∃f. table_of (DeclConcepts.fields G C) fn = Some f ∧ type f = T

lemma var_tys_Some_eq:

      (var_tys G oi r n = Some T) =
      (case r of
       Inl a =>
         case oi of
         CInst C => ∃nt. n = Inl nt ∧ fields_table G C (%n f. ¬ static f) nt = Some T
         | Arr t k => ∃i. n = Inr i ∧ i in_bounds k ∧ t = T
       | Inr C =>
           ∃nt. n = Inl nt ∧
                fields_table G C (%fn f. declclassf fn = C ∧ static f) nt = Some T)

## stores

### access

lemma globs_def2:

      globs (st g l) = g

lemma locals_def2:

      locals (st g l) = l

lemma heap_def2:

      heap s a = globs s (Inl a)

### memory allocation

lemma new_AddrD:

      new_Addr h = Some a ==> h a = None

lemma new_AddrD2:

      new_Addr h = Some a ==> ∀b. h b ≠ None --> b ≠ a

lemma new_Addr_SomeI:

      h a = None ==> ∃b. new_Addr h = Some b ∧ h b = None

### initialization

lemma init_arr_comps_base:

      init_vals (arr_comps T 0) = empty

lemma init_arr_comps_step:

      0 < j
      ==> init_vals (arr_comps T j) = init_vals (arr_comps T (j - 1))(j - 1 |->
          default_val T)

### update

lemma gupd_def2:

      gupd(r\<mapsto>obj) (st g l) = st (g(r |-> obj)) l

lemma lupd_def2:

      lupd(vn\<mapsto>v) (st g l) = st g (l(vn |-> v))

lemma globs_gupd:

      globs (gupd(r\<mapsto>obj) s) = globs s(r |-> obj)

lemma globs_lupd:

      globs (lupd(vn\<mapsto>v) s) = globs s

lemma locals_gupd:

      locals (gupd(r\<mapsto>obj) s) = locals s

lemma locals_lupd:

      locals (lupd(vn\<mapsto>v) s) = locals s(vn |-> v)

lemma globs_upd_gobj_new:

      globs s r = None ==> globs (upd_gobj r n v s) = globs s

lemma globs_upd_gobj_upd:

      globs s r = Some obj
      ==> globs (upd_gobj r n v s) = globs s(r |-> upd_obj n v obj)

lemma locals_upd_gobj:

      locals (upd_gobj r n v s) = locals s

lemma globs_init_obj:

      globs (init_obj G oi r s) t =
      (if t = r then Some (| obj.tag = oi, values = init_vals (var_tys G oi r) |)
       else globs s t)

lemma locals_init_obj:

      locals (init_obj G oi r s) = locals s

lemma surjective_st:

      st (globs s) (locals s) = s

lemma surjective_st_init_obj:

      st (globs (init_obj G oi r s)) (locals s) = init_obj G oi r s

lemma heap_heap_upd:

      heap (st (g(Inl a |-> obj)) l) = heap (st g l)(a |-> obj)

lemma heap_stat_upd:

      heap (st (g(Inr C |-> obj)) l) = heap (st g l)

lemma heap_local_upd:

      heap (st g (l(vn |-> v))) = heap (st g l)

lemma heap_gupd_Heap:

      heap (gupd(Inl a\<mapsto>obj) s) = heap s(a |-> obj)

lemma heap_gupd_Stat:

      heap (gupd(Inr C\<mapsto>obj) s) = heap s

lemma heap_lupd:

      heap (lupd(vn\<mapsto>v) s) = heap s

lemma heap_upd_gobj_Stat:

      heap (upd_gobj (Inr C) n v s) = heap s

lemma set_locals_def2:

      set_locals l (st g l') = st g l

lemma set_locals_id:

      set_locals (locals s) s = s

lemma set_set_locals:

      set_locals l (set_locals l' s) = set_locals l s

lemma locals_set_locals:

      locals (set_locals l s) = l

lemma globs_set_locals:

      globs (set_locals l s) = globs s

lemma heap_set_locals:

      heap (set_locals l s) = heap s

## abrupt completion

lemma abrupt_if_True_None:

      abrupt_if True x None = x

lemma abrupt_if_True_not_None:

      x ≠ None ==> abrupt_if True x y ≠ None

lemma abrupt_if_False:

      abrupt_if False x y = y

lemma abrupt_if_Some:

      abrupt_if c x (Some y) = Some y

lemma abrupt_if_not_None:

      y ≠ None ==> abrupt_if c x y = y

lemma split_abrupt_if:

      P (abrupt_if c x' x) = ((c ∧ x = None --> P x') ∧ (¬ (c ∧ x = None) --> P x))

lemma raise_if_None:

      ((raise_if c x) y = None) = (¬ c ∧ y = None)

lemma if_raise_if_None:

      ((if b then y else (raise_if c x) y) = None) = ((c --> b) ∧ y = None)

lemma raise_if_SomeD:

      (raise_if c x) y = Some z ==> c ∧ z = Xcpt (Std x) ∧ y = None ∨ y = Some z

lemma error_if_None:

      ((error_if c e) y = None) = (¬ c ∧ y = None)

lemma if_error_if_None:

      ((if b then y else (error_if c e) y) = None) = ((c --> b) ∧ y = None)

lemma error_if_SomeD:

      (error_if c e) y = Some z ==> c ∧ z = Error e ∧ y = None ∨ y = Some z

lemma absorb_SomeD:

      absorb j a = Some x ==> a = Some x

lemma absorb_same:

      absorb j (Some (Jump j)) = None

lemma absorb_other:

      a ≠ Some (Jump j) ==> absorb j a = a

lemma absorb_Some_NoneD:

      absorb j (Some abr) = None ==> abr = Jump j

lemma absorb_Some_JumpD:

      absorb j s = Some (Jump j') ==> j' ≠ j

## full program state

lemma single_stateE:

      ∀Z. Z = s ==> False

lemma state_not_single:

      All (op = x) ==> R

lemma normal_def2:

      normal s = (fst s = None)

lemma heap_free_def2:

      heap_free n s = atleast_free (heap (snd s)) n

### update

lemma abupd_def2:

      abupd f (x, s) = (f x, s)

lemma abupd_abrupt_if_False:

      abupd (abrupt_if False xo) s = s

lemma supd_def2:

      supd f (x, s) = (x, f s)

lemma supd_lupd:

      supd lupd(vn\<mapsto>v) s = (fst s, lupd(vn\<mapsto>v) (snd s))

lemma supd_gupd:

      supd gupd(r\<mapsto>obj) s = (fst s, gupd(r\<mapsto>obj) (snd s))

lemma supd_init_obj:

      supd (init_obj G oi r) s = (fst s, init_obj G oi r (snd s))

lemma abupd_store_invariant:

      snd (abupd f s) = snd s

lemma supd_abrupt_invariant:

      fst (supd f s) = fst s

lemma set_set_lvars:

      (set_lvars l) ((set_lvars l') s) = (set_lvars l) s

lemma set_lvars_id:

      (set_lvars (locals (snd s))) s = s

## initialisation test

lemma not_inited_empty:

      ¬ inited C empty

lemma inited_gupdate:

      inited C (g(r |-> obj)) = (inited C g ∨ r = Inr C)

lemma inited_init_class_obj:

      inited C (globs ((init_class_obj G C) s))

lemma not_initedD:

      ¬ inited C g ==> g (Inr C) = None

lemma initedD:

      inited C g ==> ∃obj. g (Inr C) = Some obj

lemma initd_def2:

      initd C s = inited C (globs (snd s))

## @{text error_free}

lemma error_free_Norm:

      error_free (Norm s)

lemma error_free_normal:

      normal s ==> error_free s

lemma error_free_Xcpt:

      error_free (Some (Xcpt x), s)

lemma error_free_Jump:

      error_free (Some (Jump j), s)

lemma error_free_Error:

      error_free (Some (Error e), s) = False

lemma error_free_Some:

      ¬ (∃err. x = Error err) ==> error_free (Some x, s)

lemma error_free_abupd_absorb:

      error_free s ==> error_free (abupd (absorb j) s)

lemma error_free_absorb:

      error_free (a, s) ==> error_free (absorb j a, s)

lemma error_free_abrupt_if:

      [| error_free s; ¬ (∃err. x = Error err) |]
      ==> error_free (abupd (abrupt_if p (Some x)) s)

lemma error_free_abrupt_if1:

      [| error_free (a, s); ¬ (∃err. x = Error err) |]
      ==> error_free (abrupt_if p (Some x) a, s)

lemma error_free_abrupt_if_Xcpt:

      error_free s ==> error_free (abupd (abrupt_if p (Some (Xcpt x))) s)

lemma error_free_abrupt_if_Xcpt1:

      error_free (a, s) ==> error_free (abrupt_if p (Some (Xcpt x)) a, s)

lemma error_free_abrupt_if_Jump:

      error_free s ==> error_free (abupd (abrupt_if p (Some (Jump j))) s)

lemma error_free_abrupt_if_Jump1:

      error_free (a, s) ==> error_free (abrupt_if p (Some (Jump j)) a, s)

lemma error_free_raise_if:

      error_free s ==> error_free (abupd (raise_if p x) s)

lemma error_free_raise_if1:

      error_free (a, s) ==> error_free ((raise_if p x) a, s)

lemma error_free_supd:

      error_free s ==> error_free (supd f s)

lemma error_free_supd1:

      error_free (a, s) ==> error_free (a, f s)

lemma error_free_set_lvars:

      error_free s ==> error_free ((set_lvars l) s)

lemma error_free_set_locals:

      error_free (x, s) ==> error_free (x, set_locals l s')