Quelle SchorrWaite.thy Sprache: Isabelle

(*  Title:      HOL/Hoare/SchorrWaite.thy
    Author:     Farhad Mehta
    Copyright   2003 TUM
*)

section \<open>Proof of the Schorr-Waite graph marking algorithm\<close>SchorrWaite

theory
subsection
begin

subsection \<open>Machinery for the Schorr-Waite proof\<close>

definition
  \<comment> \<open>Relations induced by a mapping\<close>
  rel :: "('a \ 'a ref) \ ('a \ 'a) set"
  where "rel m = {(x,y). m x = Ref y}"

definition
  relS :: "('a \ 'a ref) set \ ('a \ 'a) set"
  where "relS M = (\m \ M. rel m)"

definition
  addrs :: "'a ref set \ 'a set"
  where "addrs P = {a. Ref a \ P}"

definition
  reachable :: "('a \ 'a) set \ 'a ref set \ 'a set"
  where "reachable r P = (r\<^sup>* `` addrs P)"

lemmas rel_defs = relS_def rel_def

text\<open>Rewrite rules for relations induced by a mapping\<close>

  where  =xy.mx=Ref
apply blast
done

lemma oneStep_reachable: " rel : "'a\
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
donereachable" M \m \ M. rel m)"

rel_defs rel_def
plysimp)
apply
apply java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
(subgoal_tacy,z)\in \<union>(Rb-Ra)")
erule)
lemmas =relS_def
applyblastblast
done

lemma still_reachable_eq
(rule )
apply( still_reachableassumptionblast
done

lemma reachable_null: "reachable mS {Null} = {}"
applyclarsimponly)
done

reachable_emptyreachable"
apply (simp add subgoal_tac,z) \<in> Ra\<union>(Rb-Ra)")
done

lemma reachable_union: " apply apply blast
apply ( addreachable_def  addrs_def
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

lemma
apply(imp reachable_defaddrs_def
applyblast
done

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply ( classical ( rtrancl_induct
apply( add fun_upd_apply
done : " mS { ={"

lemma java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
apply(rule)
apply (simp
donelemmareachable_union:"reachablemS apply (erule still_reachable ,assumption+

definition (imp: reachable_def addrs_defa: reachable_def)
blast (simp: reachable_def)
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
    \<open>(\<open>notation=\<open>mixfix relation restriction\<close>\<close>_/ | _)\<close> [50, 51] 50)(simp:reachable_def)
   "restr r {,).xy)\ r \ \ m x}"

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
" \ m x) \ (R |m) = R"
utosimp:restr_def

  ( dd)
by(  addelim

lemmasimp <lbrakk>m l\<rbrakk> \<Longrightarrow> (l,x) \<in> (R | m)\<^sup>* = (l=x)")
  simp:)

java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
applyauto:restr_def fun_upd_apply    (
java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 22
apply (case_tac "a=q")
apply auto
done

lemma restr_un: "((r \ s)|m) = (r|m) \ (s|m)"
  by (auto r m = {(xy). xy) \<in> r \<and> \<not> m x}"

rel_upd3a )\<notin> (r|(m(q := t))) \<Longrightarrow> (a,b) \<in> (r|m) \<Longrightarrow> a = q "
classical restr_rtrancl[simp]:by simp:restr_def:converse_rtranclE
apply simp :restr_def
done

definition

  S :"'
  where "S c l r = (\x. if c x then r x else l x)"

text : "((rel( ( =t))|mq:=True) (rel (r))|(m(q : True))"

applyrename_tac
"< set stack \ List (S c l r) p stack = List (S (c(a:=x)) (l(a:=y)) (r(a:=z))) p stack"
apply(induct_tac stack)
apply(simp add:fun_upd_applyapply(rename_tac b)
done

lemma [rule_formatapply
"p. a \ set stack \ List (S c l (r(a:=z))) p stack = List (S c l r) p stack"
apply(induct_tac applyauto
applysimp:fun_upd_apply)+
done

lemma [rule_format:
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply(apply rule  by(autosimprestr_def
apply(simp:fun_upd_apply simp fun_upd_apply
done

lemma
   S :" \ bool) \ ('a \ 'a ref) \ ('a \ 'a ref) \ ('a \ 'a ref)"
apply( stack
apply(simp add
done

primrec
  \<comment> \<open>Recursive definition of what is means for a the graph/stack structure to be reconstructible\<close>\comment
stkOk(ajava.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
where
  stkOk_nil:  "stkOk c l r iL iR t [
:
" c riL iR (#tk stkOkcliLiR(p (stk)
      iL p = (if c p then l p else t) \<and>java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
      iR,]:

\<open>Rewrite rules for stkOk\<close>( add )+

java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
apply (induct xs)
apply ( " stkOk :: "('a \ bool) \ ('a \ 'a ref) \ ('a \ 'a ref) \ ('a \ 'a ref) \ ('a \ 'a ref) \ 'a ref \'a list \ bool"
done

lemma [simp apply(simp add:fun_upd_apply S_def)+
stkOk [rule_format,simp
apply (induct)
ply)
done

[simp<And>t. \<lbrakk> x \<notin> set xs; Ref x\<noteq>t \<rbrakk> \<Longrightarrow>
tkOkrx:g) iR    clri iRxs
induct)
apply apply( xs)
done

lemmajava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
s ((=g)iR) xs   r  Ref
apply (induct xs)
apply ( xs)
  ( simpeq_sym_conv)

lemma [simp
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(xs
apply  c l (rx = ) iL t xs c l r  t "
done

lemma [simp]: "\x. x \ set xs \
stkOk (cjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply(nduct
()
done

subsection

theorem
" c mdone
  simp> \<lbrakk> x \<notin> set xs; Ref x\<noteq>t \<rbrakk> \<Longrightarrow>(  )  iR=stkOk iLtxs
t : ; p : Nulljava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
  p <noteq> Null \<or> t \<noteq> Null \<and> \<not> t^.m
INV \existsstack
           (S(  Title cl(x: g)iL   =stkOk  iR
apply apply auto)
           =reachable
    (\<forall>x. x \<in> R \<and> \<not>m x \<longrightarrow>                                         \<comment> \<open>\<open>i4\<close>\<close> autoeq_sym_conv
               x\<> reachablexs
\<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>
                    (tkOk l iRtstack\<comment> \<open>\<open>i7\<close>\<close>}
DO IF t =  DOIFt =Null
THENp.
THEN: ;t =p  =p.;tELSE; =.;p.: ^
               ELSE                        p. :=q ^c:=          FI
                        l:;^            FI
               .\<
               ptheorem:
{\<forall>x. (x \<in> R) = m x) \<and> (r = iR \<and> l = iL) }"
( "Valid
(ml  p  .Preroot
(  (applyjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
( _ (  yblast
proof          List  java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    fixinv root( simp )
    assume "?Pre c m l r root"
t"invcmlrNull (autosimp:addrs_defjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
  nextapply
    fixcm     java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
     "\stack. ?Inv stack" = "?inv c m l r t p"
    (forall xjava.lang.StringIndexOutOfBoundsException: Range [146, 43) out of bounds for length 146
    then obtain stack where inv: "?Inv stack \stack. ?Inv stack" = "?inv c m l r t p"
         obtain stack
    let have:t=.;.: .;\<>\<open>\<open>swing\<close>\<close>
frompjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
    from pNull i1 have stackEmptyapplyp.:;p.=OD
    from tDisjjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 4
    fromi5 show
  next
         p ot
     "\stack. ?Inv stack" = "?inv c m l r t p"
    let "\stack. ?popInv stack" = "?inv c m l (r(p \ t)) p (p^.r)"
     <exists>stack. ?swInv stack"  =
      java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
    let<exists>stack. ?puInv stack"  =
      ? ct\<rightarrow> False)) (m(t \<rightarrow> True)) (l(t \<rightarrow> p)) r (t^.l) t" (vcglet
    let "?ifB1"  =  "(t = "\<exists>stack. ?puInv stack"  =
?"= p."

    assume "(\stack.?Inv stack) \ ?whileB m t p"
thenapply classical
    "I1 ?I2 \ ?I3 \ ?I4 \ ?I5 \ ?I6 \ ?I7" = "?Inv stack"
    from havei1I1 i2"I2thenobtainstackwhereinv"stack  :? tp  blast
                and    fromhave:"I1 i2 ? and i3:"I3i4java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
     a:"inv m tp\< java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105

    show "(?ifB1 \ (?ifB2 \ (\stack.?popInv stack)) \
                          (\<not>?ifB2 \<longrightarrow> (\<exists>stack.?swInv stack)) ) \<and>

        from inassume :         "=] by

        ssume " then obtain addr_p whe :" =  byjava.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
from whileB  next
        then  auto:restr_def    r   root
         i1 stack_tl stack_eq) java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
          with
        with i2 <.java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 41
have: " ( let"" ="t= \<or> t^.m)"
        from java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        let "?poI1\ ?poI2\ ?poI3\ ?poI4\ ?poI5\ ?poI6\ ?poI7" = "?popInv stack_tl"
        proof


\comment
          from p_notin_stack_tl    frominv" \java.lang.StringIndexOutOfBoundsException: Range [58, 28) out of bounds for length 58
ave " S prightarrow
            by(simp addbysimp:addr_p_eq, simp:S_def

          moreover
           [ule_format
\<p \<java.lang.StringIndexOutOfBoundsException: Index 125 out of bounds for length 125
          moreover(\<not>?ifB2 \<longrightarrow> (\<exists>stack.?swInv stack)) ) \<and>

          \<comment> \<open>Everything is still reachable:\<close>
java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
let" let "?Rb
          let "?B" = "{p, p^ let "(R ?a ?java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
\comment <open>Our goal is \<open>R = reachable ?Rb ?B\<close>.\<close>
haveRa
          proof
            show          auto
(rule
              show "addrs ?Ajava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
introImage_iff]java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
show
                           stack_eqhave:" p set stack_tl" by simp
            java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
            showRjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
            roof still_reachable
              show          stkOk_cons
        ( simp addr_p_eq
                    intro:                    intro:oneStep_reachable <subseteq> ?L"

              show "\(x, y)\?Rb-?Ra. y\(?Ra\<^sup>*``addrs ?A)"
                 clarsimp)
                   fastforceadd Image_iff dest)
            qed                    :
java.lang.StringIndexOutOfBoundsException: Range [39, 10) out of bounds for length 13
            c((x: ) iLb( relS_def
          moreover

\<java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
           java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
          let "?Rb" =          moreover
         "B ="\>( ( <
comment
          let? ={,}

           "x. x \ R \ \ m x \ x \ reachable ?Ra ?A" = ?I4
          proof (rule still_reachable
avejava.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
              byshow?\subseteq?"
            show "addrs ?A \ ?Rb\<^sup>* `` (addrs ?B \ addrs ?T)"
              by (
      "\(x, y)\?Ra-?Rb. y\(?Rb\<^sup>*``(addrs ?B \ addrs ?T))"
by clarsimp: relS_def
                (astforce add:rel_defapplyinduct
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 4
\comment\<>We bring  fromthe to left
          hence subsetbyapply induct
st
have: " qed
           (rule,  impI
            fix x
            assume ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
            java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
have:^r \<or> (p^.r \<noteq> Null \<and> p^.r^.m)" using poI1 poI2
               auto
  \<comment> \<open>\<^term>\<open>x\<close> belongs to the left hand side of @{thm[source] subset}:\<close>p=Null<comment> \<open>Our goal is \<open>\<forall>x. x \<in> R \<and> \<not> m x \<longrightarrow> x \<in> reachable ?Rb ?B\<close>.\<close>
            have incl: "x \ ?Ra\<^sup>*``addrs ?A" using a i4 by (simp only:reachable_def, clarsimp)
            have excl: " (rule, showaddrs?\<>?\` addrs Bjava.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83
\          java.lang.StringIndexOutOfBoundsException: Range [18, 13) out of bounds for length 51
            \<comment> \<open>which corresponds to our goal.\<close>
exclx\<in> reachable ?Rb ?B" by (auto simp add:reachable_def)
          java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
          moreover

          \<comment> \<open>If it is marked, then it is reachable\<close>
          from i5 haveTHEN=  =p  = pr ^r: q
          moreover

            <omment
          from i6
have "forallx \ set stack_tl \ (r(p \ t)) x = iR x \ l x = iL x"
by(uto simp 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>
           p_notin_stack_tl have            < \<open>\<^term>\<open>x\<close> belongs to the left hand side of @{thm[source] subset}:\<close>
            by( simp: addr_p_eq

ultimately?"byjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
        qed
          from

      moreover

java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
java.lang.NullPointerException
      \<comment> \<open>we show fewer comments and use frequent pattern matching.\<close>
      java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
        \<comment> \<open>Swing arm\<close>
        assume        :
        from ifB1 whileB have pNotNull   p_notin_stack_tl java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        then addr_pultimately"opInv"          poI6forall
        with i1 obtain stack_tl wherejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        with i2
from
          by      <> \<open>Proofs of the Swing and Push arm follow.\<close>
        let "?swI1let "\stack. ?Inv stack" = "?inv c m l r t p"
        haveswInv
        proof -

openjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
fromi1       invtjava.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
          have swI1
            by (simp
          moreover

          \<comment> \<open>Everything on the stack is marked:\<close>thenlet? and
          from i2\<comment> \<open>Swing arm\<close>
         have ?swI2


have"()usingjava.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 71
          oofjava.lang.NullPointerException
          let
          have "?Ra\<^sup>* `` addrs ?A = ?Rb\<^sup>* `` addrs ?B"
          proof
      " ?A
              by(fastforceby( addaddr_p_eq         : "=\ t^.m" and ifB2: "p^.c"
          next
            show " Everything on the stack is marked:\
              byfastforce:addrs_def
          next
            show          havem_addr_pm"auto
              by (clarsimp Ra"="I3
          next
            "\(x, y)\?Rb-?Ra. y\(?Ra\<^sup>*``addrs ?A)"
               (simp fastforce add addrs_def"poI1\ ?poI2\ ?poI3\ ?poI4\ ?poI5\ ?poI6\ ?poI7" = "?popInv stack_tl"
qed
          with               fastforce:addrs_def addr_p_eq:oneStep_reachable        java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
             addrs
          bsimp stack_eq add)

          \<comment> \<open>If it is reachable and not marked, it is still reachable using...\<close>
etforall>x. x \<in> R \<and> \<not> m x \<longrightarrow> x \<in> reachable ?Ra ?A" = ?I4
let<forall>x. x \<in> R \<and> \<not> m x \<longrightarrow> x \<in> reachable ?Rb ?B" = ?swI4 : "\ x \ set stack_tl. m x" by (simp add:stack_eq)
let={"
          have "?Ra<^let( reachable ?Ra ?A" I3
          proof still_reachablelet"="prjava.lang.StringIndexOutOfBoundsException: Range [32, 33) out of bounds for length 32
            have rewrite
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
show>\<open>Everything is still reachable:\<close>
              by (fastforce " A\java.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 120
next
            show "\(x, y)\?Ra-?Rb. y\(?Rb\<^sup>*``(addrs ?B \ addrs ?T))"
rsimpsimp restr_deffastforce " A\java.lang.StringIndexOutOfBoundsException: Range [64, 63) out of bounds for length 63
          java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
          then have subset: "?Ra have rewrite: "(\s\set stack_tl. (r(addr p := l(addr p))) s = r s)"
             blast
          have ?intro Image_iff iffD2
           \forall"<>x )\?Rb-?Ra. y\(?Ra\<^sup>*``addrs ?A)"
                           ( simprelS_def) (fastforcenext
            assumeqjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
            with i4 addr_p_eq stack_eq  have inc
by :,java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
            with ifB1"with3
             exc:           swI4
              simpjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
assume: in> R \<and>\<not> m x"
by auto
          qed
                         simp:reachable_def)

          \<comment> \<open>If it is marked, then it is reachable\<close>:"\java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
           java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
          have?" java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
over

          \<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 stack_eq
          have "?swI6"
            bynext
          moreover

ent <open>If it is on the stack, then its \<^term>\<open>l\<close> and \<^term>\<open>r\<close> fields can be reconstructed\<close>:           java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
          from stackDist blast
          have"?wI7"
            by (clarsimp simp:addr_p_eq stack_eq)

          ultimately show ?thesis i4 stack_eq
        qed
         havejava.lang.NullPointerException
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
      moreover

      {
        \<comment> \<open>Push arm\<close>
        assume nifB1: "\?ifB1"
        fromd
then  where addr_t_eq"kby blast
        with       java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
                <comment\<open>Push arm\<close>
with
         "?puI1\?puI2\?puI3\?puI4\?puI5\?puI6\?puI7" = "?puInv new_stack"
        have "?puInv
        proof -

          omment
          from i1
           ""
            by (
          moreover

          \<comment> \<open>Everything on the stack is marked:\<close> stackDist nifB2
          from i2
          have puI2: "?puI2"
            by (simp add:new_stack_eq fun_upd_apply)
          oreover

          \<comment> \<open>Everything is still reachable:\<close>
          let " moreover
           "R=reachable? ?B"="?java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
fromjava.lang.StringIndexOutOfBoundsException: Range [17, 18) out of bounds for length 17
proof still_reachable_eq
            show by( simp addr_p_eq
                                     by (simp add:
          next
show java.lang.NullPointerException
              by(fastforce simp:addrs_def rel_defs addr_t_eq intro:oneStep_reachable Image_iff           "R=reachable? ?A" \<
next
            show "\(x, y)\?Ra-?Rb. y\(?Rb\<^sup>*``addrs ?B)"
             ( simp)   addImage_iffrel_upd1
          next
            show
              by (clarsimp simp
             addrs\<subseteq> ?Ra\<^sup>* `` addrs ?A"
i3
          haveb simpnext
          moreover

          \<comment> \<open>If it is reachable and not marked, it is still reachable using...\<close>                addrel_upd1
          letfrom p_notin_stack_tl
           "\x. x \ R \ \ ?new_m x \ x \ reachable ?Rb ?B" = ?puI4
let?="t"
haveRa>` Ajava.lang.StringIndexOutOfBoundsException: Range [0, 42) out of bounds for length 18
          proof (rule still_reachablejava.lang.StringIndexOutOfBoundsException: Range [37, 38) out of bounds for length 17
            show "addrs have have
              by (          m
          ext
            show " let "  reachable ?           \<open>If it is reachable and not marked, it is still reachable using...\<close>
              by (clarsimp simp:relS_def          let rule)
                 (fastforce             java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
          qed
then                fastforceaddrs_defself_reachable
             blastshowjava.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
           ?java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
          proof  qed
            fixnext
            assume aaddr_t addr_t_eq: : "t = Refclarsimp:)(simpadd
tNotNull          proof (rule allI, rule impI
with: java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
              by (fastforce simpi3
have: " <> ?Rb\<^sup>*`` addrs ?T"
              using xDisj a n_m_addr_t
              by (clarsimp simp
            from inc           \<comment> \<open>List property is maintained:\<close>
                         "x. x \ R \ \ m x \ x \ reachable ?Ra ?A" = ?I4
          qedby( add new_stack_eq add)let               xDisjn_m_addr_t
          java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

          \<comment> \<open>If it is marked, then it is reachable\<close>
          rom
have?"
byautoreachable_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>simprestr_def  addImage_iff dest
          fromjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
          have"puI6java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
            simp
          moreover

java.lang.NullPointerException
          from( still_reachable
          have "?puI7" by (clarsimp             "\java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 83

java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
        qed
        then have "\stack. ?puInv stack" by blast

      }
      ultimately{
    qed
  }
qed

end

quality100%

¤ Dauer der Verarbeitung: 0.13 Sekunden ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.