Quelle SchorrWaite.thy Sprache: Isabelle

cr  ) iR    "
     xs
apply  :eq_sym_conv
*java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

sectionreachablerelS}root

theory SchorrWaite
  imports
begin[]: "\x. x \ set xs \

subsection  =root;p:c((:g iLRef = clriRx)xs

definition
                  <in (applyinduct
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  where "rel m x=)liL Refx)xs = stkOk (c(x := g)) l r iL iR (Ref x) xs = stkOk\ R) \ \ \\i5\\

definition
  relS                  \<in> reachable (relS{l,r}|m) ({t}\<union>set(map r stack))) \<and>
  where "relS M = (\m \ M. rel m)"

definition
  addrs :: "'a ref set \ 'a set"
  where          \<>x. m x <longrightarrow> x \<in> R) \<and>                                         \<comment> \<open>\<open>i5\<close>\<close>\<forall>x. x \<notin> set stack \<longrightarrow> r x = iR x \<and> l x = iL x) \<and>             \<comment> \<open>\<open>i6\<close>\<close>

definition
  reachable :: "('ajava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
whererPjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

lemmas (\<forall>x. (x \<in> R) = m x) \<and> (r = iR \<and> l = iL) }"

  is

lemma{           , ,qoot  lr}
apply java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
done

lemma oneStep_reachablevcg pjava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 3
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
done

lemma still_reachable: "\B\Ra\<^sup>*``A; \ (x,y) \ Rb-Ra. y\ (Ra\<^sup>*``A)\ \ Rb\<^sup>* `` B \ Ra\<^sup>* `` A "
apply (clarsimp            reachable     "cmlrroot " autoreachable_def
apply  hus     " ( reachable_def addrs_defjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
  blast
apply (subgoal_tac  mlrtpq
apply (erule UnE)
apply (auto intro:rtrancl_into_rtrancllet
applyapply \forall>x.m x longrightarrow> x \<in> R) \<and>                                         \<comment> \<open>\<open>i5\<close>\<close>   pq
done

lemma still_reachable_eq: "\ A\Rb\<^sup>*``B; B\Ra\<^sup>*``A; \ (x,y) \ Ra-Rb. y \(Rb\<^sup>*``B); \ (x,y) \ Rb-Ra. y\ (Ra\<^sup>*``A)\ \ Ra\<^sup>*``A = Rb\<^sup>*``B "
apply (rule equalityI)
apply (erule still_reachable ,assumption)+
done

lemma reachable_null: "reachable mS ELSEq:i1: ? and i4:"I4:I5i6"bysimp+
apply               m=;^.c :Falsejava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
done

lemma reachable_empty: "reachable mS {} = {}"
apply       i6 "(\x.(x \ R) = m x) \ r = iR \ l = iL" by(auto simp: stackEmpty fun_eq_iff intro:RisMarked)
    fixr   ot

lemmalet\<exists>stack. ?Inv stack"  =  "?inv c m l r t p"
apply (impm,lr      "
apply blast
done

          _"nv(
apply (simpstack " java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
     inv
done

lemma rel_upd1fix  let"=".java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
  classical
apply (simp add:    let <and> ?I2 \<and> ?I3 \<and> ?I4 \<and> ?I5 \<and> ?I6 \<and> ?I7" = "?Inv stack"a "(
done

lemma rel_upd2 i5inv:""andI2:?" and i4 ?"
apply (rule classical)
     "\stack. ?Inv stack" = "?inv c m l r t p"
done

definition
  \<comment> \<open>Restriction of a relation\<close>
java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
    (
  where " java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

text\<open>Rewrite rules for the restriction of a relation\<close>

lemma restr_identity[simp]:
"(\x. \ m x) \ (R |m) = R"
by (    "?(\?ifB1 \ (\stack.?puInv stack))"

         :"fromi1havestackEmpty:stack ] simp
(auto add elim)

lemma [    fromifB1=raddr_p_eqaddr_p
by(uto i5 show\<

lemma         ifB1 have pNotNull
apply( simp rel_defcmrpqroot
apply (    let ">stack           java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
apply (case_tac"java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
applyauto
done

lemma restr_un: "((r \ s)|m) = (r|m) \ (s|m)"
  by (auto "exists>.? stack" =

lemma      ?have distinct)"usingi1 (List_distinctjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
apply (rule classical)
     "?"  =  "p^."
done            let?\and> ?<java.lang.StringIndexOutOfBoundsException: Index 112 out of bounds for length 112

definition
  \<comment> \<open>A short form for the stack mapping function for List\<close>
  S :          <> \<open>List property is maintained:\<close>
  where  r=(lambda>x. if c x then r x else l x)"

text \<open>Rewrite rules for Lists using S as their mapping\<close>andi5              poI1 (java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74

lemmasimp
".a notin> set stack \ List (S c l r) p stack = List (S (c(a:=x)) (l(a:=y)) (r(a:=z))) p stack"
(stack
apply(simpjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
done

lemma [,simp]            \<not>?ifB1 \<longrightarrow> (\<exists>stack.?puInv stack))"
" let "(R = reachable ?Ra ?A)" = "?I3"
apply(induct_tac)
apply           "java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 7
done

lemma         fromifB1 have pNotNullp \<noteq> Null" by auto
"\p. a \ set stack \ List (S c (l(a:=z)) r) p stack = List (S c l r) p stack"
apply(induct_tac         i1  stack_tl ?\<^sup>* `` addrs ?A = ?Rb\<^sup>* `` addrs ?B" (is "?L = ?R")
apply(simp:fun_upd_apply)+            still_reachable
done

lemmarule_format]:
"\p. a \ set stack \ List (S (c(a:=z)) l r) p stack = List (S c l r) p stack"
apply(induct_tac:neStep_reachableTHEN)
apply(simp add:fun_upd_apply               "\(x,y) \ ?Ra-?Rb. y \ (?Rb\<^sup>* `` addrs ?B)" by (clarsimp simp:relS_def)
done

primrec
         "poI1 ?poI3\ ?poI4\ ?poI5\ ?poI6\ ?poI7" = "?popInv stack_tl"
stkOkhavepopInvjava.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 36
where
p (ulejava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
|
             ifB2
              byfastforce rel_defs
      iR"R ?L"

            proof still_reachable

lemmajava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
          <comment> \<open>Everything on the stack is marked:\<close>
apply (induct xs :  :
apply (auto
done

lemma                 reachable
stkOk  ) iRclarsimp:)
apply(fastforce addmoreover
apply (autoqed
done

i3 poI3R=           <forall>x. x \<in> R \<and> \<not> m x \<longrightarrow> x \<in> reachable ?Ra ?A"  =  ?I4
  " p< (p t)) stack_tl)"
apply (induct xs)
apply (auto> \<open>Our goal is \<open>\<forall>x. x \<in> R \<and> \<not> m x \<longrightarrow> x \<in> reachable ?Rb ?B\<close>.\<close>"tp^r"
done

lemma stkOk_r_rewrite [          :            rewriteforall>s\<in>set stack_tl. (r(p \<rightarrow> t)) s = r s"
  stkOk c l(( =fastforce : Image_iffdest)
apply (induct xs)
apply (let={, ^}java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 4

lemmashow
stkOk c (l(x :=               ( simprestr_def)
apply ( xs)
apply (uto:eq_sym_conv)
done

lemma [simp]: "\x. x \ set xs \
stkOkc(=g)lriL (Ref  = stkOk           the subset relation.\<close>show
(xs
apply (auto                    simp             java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
done

subsection \<open>The Schorr-Waite algorithm\<close>

theoremjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
" havepDisj".=Null
{R =          let?"=" l ( <
t := root;  : Null
WHILE p           ?T = "t,p^r}java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
INV {\<exists>stack.
           (rule)
          (\<forall>x \<in> set stack. m x) \<and>                                       \<comment> \<open>\<open>i2\<close>\<close>: "Ra
          R =byastforcesimp rewriteself_reachable
            a"
                 x \<in> reachable (relS{l,r}|m) ({t}\<union>set(map r stack))) \<and>
          \forallx             from  subset"
                    qed
          (stkOk c lr iLjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
DOt =Null
      THENhence: "?Ra\<^sup>* `` addrs ?A - ?Rb\<^sup>* `` addrs ?T \ ?Rb\<^sup>* `` addrs ?B"
                q :=t; t: ;p: ^.;t. =\<comment> \<open>\<open>pop\<close>\<close>byjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
               ELSE<> \<open>which corresponds to our goal.\<close>
                        p^.l           (java.lang.StringIndexOutOfBoundsException: Range [0, 21) out of bounds for length 13
      ELSE;:"
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
{(\<forall>x. (x \<in> R) = m x) \<and> (r = iR \<and> l = iL) }"
  (is
         {from i7<comment
         (Seq           i5poI5java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
         (Aseq _ (Aseq
proofushow "simp
{
    fix c m l r mment open therefore to right qed
    assume "?Pre c m l r root"
     "?inv c inclexcl subsetshowjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 7

    fix c m l
     " let "\stack. ?Inv stack" = "?inv c m l r t p"
    assume      <comment> \<open>Since they are in principle simmilar to the Pop arm proof,\<close>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    then obtain stack where inv: "?Inv stack" by blast
fromahavepNullmoreover
    let "?I1 \ _ \ _ \ ?I4 \ ?I5 \ ?I6 \ _" = "?Inv stack"
    fromfrom  i7
             ( simp addr_p_eq
             obtain where          show?  byhavepoI6<> x<notin> set stack_tl \<longrightarrow> (r(p \<rightarrow> t)) x = iR x \<and> l x = iL x"
            java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  next
      \<comment> \<open>Since they are in principle simmilar to the Pop arm proof,\<close>
    java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
let"swInvstackjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
    let "\stack. ?swInv stack" =
      "?inv (c(p nt>\<>List property qed
    let "\stack. ?puInv stack" =
" (( rightarrow> False)) (m(t \ True)) (l(t \ p)) r (t^.l) t"
    let "?ifB1"="t \ t^.m)"
    let "?ifB2"java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 14

    assume "(\stack.?Inv stack) \ ?whileB m t p"
p" by blast
     "I1<>?I2\ ?I3 \ ?I4 \ ?I5 \ ?I6 \ ?I7" = "?Inv stack"
    from inv have i1         i1obtain where      \<comment> \<open>we show fewer comments and use frequent pattern matching.\<close>
                and i5from whileB simp
    havestackDists)using  rule

    show "(?ifB1 \ (?ifB2 \ (\stack.?popInv stack)) \
                          (<not>?ifB2 \<longrightarrow> (\<exists>stack.?swInv stack)) ) \<and> i2 m_addr_p:"^mjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
(\<not>?ifB1 \<longrightarrow> (\<exists>stack.?puInv stack))"
    proof -
             showaddrs\<subseteq> ?Rb\<^sup>* `` addrs ?B"
assumeifB1 Null
        from ifB1 whileB have pNotNull: "p \ Null" by auto
        then obtain
( simp
          by next
        i2:"^m"
         stackDistdistinct"using i1 by( java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
        from stack_eqhave \<forall>(x, y)\<in>?Rb-?Ra. y\<in>(?Ra\<^sup>*``addrs ?A)"
let
        have "?popInv stack_tl"
        proof          java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13

          \<comment> \<open>List property is maintained:\<close>
          from i1 p_notin_stack_tl ifB2
           poI1
y : stack_eq:java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

          moreover
\<by clarsimprelS_def          "java.lang.StringIndexOutOfBoundsException: Range [104, 23) out of bounds for length 104
          from i2 havepoI2java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
          moreover

           T {java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
Rreachable) ""
                    qed
          "B={p,^r"
          \<comment> \<open>Our goal is \<open>R = reachable ?Rb ?B\<close>.\<close>
          have "?Ra\<^sup>* `` addrs ?A = ?Rb\<^sup>* `` addrs ?B" (is "?L = ?R")
          proof

            proof (          \<comment> \<open>If it is reachable and not marked, it is still reachable using...\<close>
              show"addrs?A ?Rb\<^sup>* `` addrs ?B" by(fastforce simp:addrs_def relS_def rel_def addr_p_eq
                   intro:oneStep_reachable           "Ra
                        have?\<
( show? <subseteq> ?Rb\<^sup>* `` addrs ?B"
            qed
            show "?qed
            proof (ruleshow"addrs?B \ ?Ra\<^sup>* `` addrs ?A"
               "addrs ?B \ ?Ra\<^sup>* `` addrs ?A"
                by(fastforcebyblast
    :oneStep_reachable[THEN])
            nextnext
showorall)
                by (clarsimp simp:relS_def)
                   next
            qed
          ed
byclarsimp )(fastforce:rel_def addrs_def:rel_upd1
          moreover

\<>                simp clarsimp
let"<> x \ m x \ x \ reachable ?Ra ?A" = ?I4
          let ?          ijava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
          let "?B" =           proof( allI(  add)
          \<comment> \<open>Our goal is \<open>\<forall>x. x \<in> R \<and> \<not> m x \<longrightarrow> x \<in> reachable ?Rb ?B\<close>.\<close>"
           ?  "t,^.}java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

          have "?Ra\<^sup>* `` addrs ?A \ ?Rb\<^sup>* `` (addrs ?B \ addrs ?T)"
          proof (rule           fromi5
            have          have "swI5 .
              by (autojava.lang.StringIndexOutOfBoundsException: Range [14, 10) out of bounds for length 18
            java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  : : i5
\
              by (clarsimpjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
                >\<
qed
          \<comment> \<open>We now bring a term from the right to the left of the subset relation.\<close>qed
           ?"
            by java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
have: "\x. x \ R \ \ m x \ x \ reachable ?Rb ?B"
          prooff x
            fixjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
ssume \<in> R \<and> \<not> m x"
            \<comment> \<open>First, a disjunction on \<^term>\<open>p^.r\<close> used later in the proof\<close>:reachable_deffrom i7
            have pDisj:"p^.r }
              by auto
            \<comment> \<open>\<^term>\<open>x\<close> belongs to the left hand side of @{thm[source] subset}:\<close>
   by(  addreachable_def
            have         obtainaddr_t: t"by
            \<comment> \<open>And therefore also belongs to the right hand side of @{thm[source]subset},\<close>
with i2 have t_notin_stack: "(addr t) \<notin> set stack" by blastswI5let
          <comment> \<open>If it is not on the stack, then its \<^term>\<open>l\<close> and \<^term>\<open>r\<close> fields are unchanged\<close>
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 13
moreover

          \<comment> \<open>If it is marked, then it is reachable\<close>
f i7
moreover

          \<comment> \<open>If it is not on the stack, then its \<^term>\<open>l\<close> and \<^term>\<open>r\<close> fields are unchanged\<close>          fromi2
          m
have
            by(auto

moreover

          let   Rb" java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 0
            i7 poI7 cl           rule)
             clarsimp:stack_eq)

ultimately fun_upd_apply
        qed
        hence "\stack. ?popInv stack" ..
      }
      moreover

      \<comment> \<open>Since they are in principle simmilar to the Pop arm proof,\<close>
<
      {
        \<comment> \<open>Swing arm\<close>
assume: "ifB1 nifB2: \
        from           "\<^>*` ?A=? :relS_def)(simp: addrs_defdest)
        then obtain addr_p where addr_p_eq: "p = Ref addr_p" by clarsimp( still_reachable_eq
        with i1 obtain stack_tlby( simpaddrs_def addr_t_eq:oneStep_reachable[THEN "\(x, y)\?Rb-?Ra. y\(?Ra\<^sup>*``addrs ?A)"
        with i2 have" ?B \ ?Ra\<^sup>* `` addrs ?A"
        from          i3
          ysimp
        let "?swI1\?swI2\?swI3\?swI4\?swI5\?swI6\?swI7" = "?swInv stack"
        have "?swInv stack"
        proof -by clarsimp:relS_deffastforcesimprel_defImage_iff :)

          \<comment> \<open>List property is maintained:\<close>
           i1 nifB2
          have swI1let
            by           et T ={}"
          moreover

          \<comment> \<open>Everything on the stack is marked:\<close>
          from i2qed
          have: "?swI2" .
ver

\<oreover
          let "R = reachable ?Ranext
          letR=reachable ?\commentjava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
          have "?Ra\<^sup>* `` addrs ?A = ?Rb\<^sup>* `` addrs ?B"
frulejava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
            showjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
              by(             "addrs A qed
          thenhavesubset:ew_stack_eq:self_reachable
            show "addrs}
              by(fastforce simp:addrs_def rel_defs               by (clarsimp simp:relS_def new_stack_eq restr_un          havepuI4

            show        from nifB1 whileB have tNotNull: "t \<noteq> Null" by clarsimp
               ( simp fastforcerel_def
          next
           have n_m_addr_t
              by(larsimp            ahavex<
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 13
          with
          have swI3: "?swI3" by             excx\notin
          moreover

          \<comment> \<open>If it is reachable and not marked, it is still reachable using...\<close>
<
          lethaveexc notin
           ? usingan_m_addr_t
          have "?Ra\<^sup>*``addrs ?A \ ?Rb\<^sup>*``(addrs ?B \ addrs ?T)"
          proof (rule still_reachable)
            have
              by (auto simp add:p_notin_stack_tl intro
            show "addrs ( simp:addrs_def i3 addr_t_eq fun_upd_apply introself_reachablejava.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98
               (fastforce:map_cong simp:stack_eq addrs_def rewrite intro:self_reachable)
          next
            show "\(x, y)\?Ra-?Rb. y\(?Rb\<^sup>*``(addrs ?B \ addrs ?T))"
               (clarsimprelS_def) (fastforce:rel_def  fun_upd_apply:rel_upd1
          qed
            :"Ra\<^sup>*``addrs ?A - ?Rb\<^sup>*``addrs ?T \ ?Rb\<^sup>*``addrs ?B"
            by blast
          have ?swI4
          proof (rule allI, rule impI)
             by(simp
            assumea "
            with i4           \<comment> \<open>If it is marked, then it is reachable\<close>
by eachable_def
            with ifB1 a
            have exc: "x \ ?Rb\<^sup>*`` addrs ?T"
byauto add:)
            from inc\<comment> \<open>If it is not on the stack, then its \<^term>\<open>l\<close> and \<^term>\<open>r\<close> fields are unchanged\<close>:from
              by (auto simp
          qed
          moreover

          \<comment> \<open>If it is marked, then it is reachable\<close>   t_notin_stack
          from               ( simp:relS_defjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
          have "?swI5" .           puI3"puI3 byreachable_def)
          moreover

          \<comment> \<open>If it is not on the stack, then its \<^term>\<open>l\<close> and \<^term>\<open>r\<close> fields are unchanged\<close>          comment
          from java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
          have "?swI6"
            by clarsimp
          java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

          \<comment> \<open>If it is on the stack, then its \<^term>\<open>l\<close> and \<^term>\<open>r\<close> fields can be reconstructed\<close>rule)
          from stackDist i7 nifB2
          have"swI7
            bynext

          ultimately show ?thesis byauto

        then have "\stack. ?swInv stack" by blast
      }
      moreover

      java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
        \<comment> \<open>Push arm\<close>
        assume nifB1: "\?ifB1"
        from nifB1 whileB have tNotNull: "t \ Null" by clarsimp
        then obtain             (imp:new_stack_eq
        with i1 obtain new_stack where new_stack_eq

        with i2 have t_notin_stack: "(addr t) \ set stack" by blast
                  fromstackDist i6 t_notin_stack
? new_stack
        proof -

          \<comment> \<open>List property is maintained:\<close>

          have puI1: "?puI1"
            by add:addr_t_eq, simp add:S_def
          moreover

          \<comment> \<open>Everything on the stack is marked:\<close>}
          from i2
          have puI2: "?puI2"
imp: fun_upd_apply
          moreover

          \<comment> \<open>Everything is still reachable:\<close>
          let "R = reachable ?Ra ?A" = "?I3"
          let "R = reachable ?Rb ?B" = "?puI3"
          have "?Ra\<^sup>* `` addrs ?A = ?Rb\<^sup>* `` addrs ?B"
          proof (rule still_reachable_eq)
            show "addrs ?A \ ?Rb\<^sup>* `` addrs ?B"
              by(fastforce simp:addrs_def rel_defs addr_t_eq intro:oneStep_reachable Image_iff[THEN iffD2])
          next
            show "addrs ?B \ ?Ra\<^sup>* `` addrs ?A"
              by(fastforce simp:addrs_def rel_defs addr_t_eq intro:oneStep_reachable Image_iff[THEN iffD2])
          next
            show "\(x, y)\?Ra-?Rb. y\(?Rb\<^sup>*``addrs ?B)"
              by (clarsimp simp:relS_def) (fastforce simp add:rel_def Image_iff addrs_def dest:rel_upd1)
          next
            show "\(x, y)\?Rb-?Ra. y\(?Ra\<^sup>*``addrs ?A)"
              by (clarsimp simp:relS_def) (fastforce simp add:rel_def Image_iff addrs_def fun_upd_apply dest:rel_upd2)
          qed
          with i3
          have puI3: "?puI3" by (simp add:reachable_def)
          moreover

          \<comment> \<open>If it is reachable and not marked, it is still reachable using...\<close>
          let "\x. x \ R \ \ m x \ x \ reachable ?Ra ?A" = ?I4
          let "\x. x \ R \ \ ?new_m x \ x \ reachable ?Rb ?B" = ?puI4
          let ?T = "{t}"
          have "?Ra\<^sup>*``addrs ?A \ ?Rb\<^sup>*``(addrs ?B \ addrs ?T)"
          proof (rule still_reachable)
            show "addrs ?A \ ?Rb\<^sup>* `` (addrs ?B \ addrs ?T)"
              by (fastforce simp:new_stack_eq addrs_def intro:self_reachable)
          next
            show "\(x, y)\?Ra-?Rb. y\(?Rb\<^sup>*``(addrs ?B \ addrs ?T))"
              by (clarsimp simp:relS_def new_stack_eq restr_un restr_upd)
                 (fastforce simp add:rel_def Image_iff restr_def addrs_def fun_upd_apply addr_t_eq dest:rel_upd3)
          qed
          then have subset: "?Ra\<^sup>*``addrs ?A - ?Rb\<^sup>*``addrs ?T \ ?Rb\<^sup>*``addrs ?B"
            by blast
          have ?puI4
          proof (rule allI, rule impI)
            fix x
            assume a: "x \ R \ \ ?new_m x"
            have xDisj: "x=(addr t) \ x\(addr t)" by simp
            with i4 a have inc: "x \ ?Ra\<^sup>*``addrs ?A"
              by (fastforce simp:addr_t_eq addrs_def reachable_def intro:self_reachable)
            have exc: "x \ ?Rb\<^sup>*`` addrs ?T"
              using xDisj a n_m_addr_t
              by (clarsimp simp add:addrs_def addr_t_eq)
            from inc exc subset  show "x \ reachable ?Rb ?B"
              by (auto simp add:reachable_def)
          qed
          moreover

          \<comment> \<open>If it is marked, then it is reachable\<close>
          from i5
          have "?puI5"
            by (auto simp:addrs_def i3 reachable_def addr_t_eq fun_upd_apply intro:self_reachable)
          moreover

          \<comment> \<open>If it is not on the stack, then its \<^term>\<open>l\<close> and \<^term>\<open>r\<close> fields are unchanged\<close>
          from i6
          have "?puI6"
            by (simp add:new_stack_eq)
          moreover

          \<comment> \<open>If it is on the stack, then its \<^term>\<open>l\<close> and \<^term>\<open>r\<close> fields can be reconstructed\<close>
          from stackDist i6 t_notin_stack i7
          have "?puI7" by (clarsimp simp:addr_t_eq new_stack_eq)

          ultimately show ?thesis by auto
        qed
        then have "\stack. ?puInv stack" by blast

      }
      ultimately show ?thesis by blast
    qed
  }
qed

end

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