Quelle SchorrWaite.thy Sprache: Isabelle

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

section

SchorrWaite
   HeapSyntax
begin SchorrWaite

java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

definition
  \<comment> \<open>Relations induced by a mapping\<close>rel(m y"
:( Rightarrow :: (ajava.lang.NullPointerException
where, x= "

definition
  relS java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
elS\>

=relS_def
   clarsimponly

definitionblast
  reachable :: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply

rel_defs

appl java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

apply(ruleequalityI erule ,assumption java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
apply
doneclarsimp :Image_iff

lemma   } }java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
donejava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

: addrs_def java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
applyclarsimprule
applyerule)
apply( add(imp: )
done
reachable_empty} "
(autointro
applyblast
done

lemma still_reachable_eq: ( classical
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
)
done

lemma reachable_null: "reachable mS {Null} =
( dd addrs_def
done

lemma reachable_empty
apply( add  addrs_def
done

lemma
apply
apply blast(
done ( add  addrs_def

java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply( addreachable_def)
apply blastwhere  =(y(x,)
done

lemma rel_upd1 (rule classical)
apply (rule (<>. \<not> m x) \<Longrightarrow> (R |m) = R"
apply(a:rel_defs
done simp :converse_rtranclE

[]: "\java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
  classical
apply (by(  add elim
done

definition
  \<comment> \<open>Restriction of a relation\<close>
  restr (auto simp rel_def)
    (    (\<open>(\<open>notation=\<open>mixfix relation restriction\<close>\<close>_/ | _)\<close> [50, 51] 50)xy
estr,)(,)

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

<>x.
by simp )

lemmarestr_rtrancl
(applysimpadd)

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by (auto simp

lemmarestr_upd rq: ))(( =))=(=)
apply (
( a b)
apply ( \forall>p.a\notin  stack
  auto
donejava.lang.NullPointerException

lemma apply stack ,]java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
   auto:)

lemma rel_upd3:apply add S_def:restr_def
apply java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
( add fun_upd_apply
done java.lang.StringIndexOutOfBoundsException: Range [0, 53) out of bounds for length 23

definition
  <> \<open>A short form for the stack mapping function for List\<close>)
  S :: java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58

text\foralljava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

lemma [|stkOk_cons    cltptkl     )()java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply stack
java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
done

lemma
simp
( text
applysimp:fun_upd_applyS_def( addfun_upd_apply)+
done

lemma java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
"
apply(induct_tacjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
done

lemmarule_format]:
  xs
( stack
apply       plemma]:"<
done

primrec
lemmac   : txs   L t java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
where
  stkOk_nil:  "stkOk stkOkc((x:=g)stkOkcl r(:g)iLRefx =stkOk l R( )xsjava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
|lemma]: "\x. x \ set xs \
    stkOkr t (xs
       applystkOk(:g)iR  stkOkiLxs
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

text []:"

lemma   ( )
  applyinduct
apply (induct
apply (auto \<open>The Schorr-Waite algorithm\<close>

[:"
stkOk (x:g iLtxs  l  iL t "
apply (induct xs)
apply (auto simp:eq_sym_conv=root=Null;

lemmasimpjava.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
  r=     l  txs
apply (    ( )
apply(auto)
one

R subsection

apply (induct SchorrWaiteAlgorithm
applyauto:java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
done

lemma simp\<And>x. x \<notin> set xs \<Longrightarrow>
stkOk  )   ( )xs      (Ref"
applyxs
apply (auto:eq_sym_conv
done

lemma [simp]: "\x. x \ set xs \
=)  r ( )xs\<in> R) \<and>                                         \<comment> \<open>\<open>i5\<close>\<close>(\<forall>x. x \<in> R \<and> \<not>m x \<longrightarrow>                                         \<comment> \<open>\<open>i4\<close>\<close>
                q =t  : ;p: ^.                 q := ;t:p^rp.  pl\<comment> \<open>\<open>swing\<close>\<close>
apply simp)p. =qpc=
done

open>The Schorr-Waite algorithm\<close>

SchorrWaiteAlgorithm
"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
c,   , )? l  }
t         eqapply java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
WHILEjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
INV
(Scjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
\forall in
R= (java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 26

                 x \<in> reachable (relS{l,r}|m) ({t}\<union>set(map r stack))) \<and>r  auto
<.<
          (fix c m l r t p q
("
DO = \<or> t^.m
      THEN  invInv blast
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
                q =; =prp^r= ^l
^java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
        :""and?"and i5 ? : "I6
p^:True=                   OD
{(<forall>x. (x \<in> R) = m x) \<and> (r = iR \<and> l = iL) }"     from tDisj i4
( " cml tpqojava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
c, ,,t    java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
     "<>. ?Inv "<
(Aseq?((
proof)     \<exists>stack. ?popInv stack"  =  "?inv c m l (r(p \<rightarrow> t)) p (p^.r)"apply (simp>. ?swInv=
   " c( \ False)) (m(t \ True)) (l(t \ p)) r (t^.l) t"
     c m    "ifB2 = ^cjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
    assume"Pre rule)
     ?java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
  next inv : "?" and: ?obtain :? "    byjava.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
    fix c m l r t p q
    let\<exists>stack. ?Inv stack"  =  "?inv c m l r t p"
assume ?clr  restr ::"('a \<times> 'a) set \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'a) set"

(java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
letjava.lang.NullPointerException
v
have" [ byjava.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
a :" :p=Ref"by
    i5"x.(x \ R) = m x) \ r = iR \ l = iL" by(auto simp: stackEmpty fun_eq_iff intro:RisMarked)
  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    fixl t
\<exists. b java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
    let  "a=q)
ists
      java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    let"stack puInvstack =
      "?inv :" (stackby )
    ifB1t  \<or> t^.m)"
letifB2c

poI1java.lang.StringIndexOutOfBoundsException: Range [112, 35) out of bounds for length 112
    then obtain java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    letjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
inv Sc   java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
                and : "?I5"andhaveh : ( cl(java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
    have stackDistjava.lang.StringIndexOutOfBoundsException: Range [70, 64) out of bounds for length 0

    induct_tac

java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
    proof stack
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 7
assume  ?
  whileB: "
then addr_p          >
withobtain           "\
java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
        with simp
        haveinduct_tac: [THENjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
  stackDist addrjava.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
      let?\>poI2qed
"popInvstack_tl"showRjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
        proof(ule)

|:
i1
          java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
                    "java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36

          moreover next
java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
           i2 :by(:relS_def
          moreover( simprel_defaddrs_defrel_upd2

          \<comment> \<open>Everything is still reachable:\<close>
let(   Ra)" let "(R = reachable ?Ra ?A)"
          let "? ((x=g) iLiR clarsimp simp:relS_def)simpjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
          comment> \<open>If it is reachable and not marked, it is still reachable using...\<close>
          \<comment> \<open>Our goal is \<open>R = reachable ?Rb ?B\<close>.\<close>
          have
          let?" =" java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 77
            show T={ ^."
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
              show "addrs ?A \ ?Rb\<^sup>* `` addrs ?B" by(fastforce simp:addrs_def relS_def rel_def addr_p_eq
intro  "java.lang.StringIndexOutOfBoundsException: Range [87, 35) out of bounds for length 87
              show "\(x,y) \ ?Ra-?Rb. y \ (?Rb\<^sup>* `` addrs ?B)" by (clarsimp simp:relS_def)
fastforceaddImage_iff :rel_upd1
            qed
             "R L"
      proof           ?  t pr"
              show "addrs ?B \ ?Ra\<^sup>* `` addrs ?A"
                by(fastforce simpjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                    ( simp
            java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
               "\(x, y)\?Rb-?Ra. y\(?Ra\<^sup>*``addrs ?A)"
                 )
(fastforce            "st
qed
          qedproof allIrule)
          with i3 java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
          moreover

\comment
let""RSpDisj
           "Rb relS{,rp\rightarrow> by auto

java.lang.StringIndexOutOfBoundsException: Index 146 out of bounds for length 146
let={,p."

java.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
proof still_reachable
            have rewrite: "\s\set stack_tl. (r(p \ t)) s = r s"
              by java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
A <*addrs
( :map_cong addrs_def:
            show            (assume:x
              by (clarsimp simp:restr_def relS_def)
                (fastforce simp(<>.m           fromincl  show
          java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
          \<comment> \<open>We now bring a term from the right to the left of the subset relation.\<close>
           subset\<^sup>* `` addrs ?A - ?Rb\<^sup>* `` addrs ?T \<subseteq> ?Rb\<^sup>* `` addrs ?B"
            blast
                      <> <>And also belongs tojava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
proofrulepoI6
            fix x
"
moreover
            java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
               auto
\>\<open>\<^term>\<open>x\<close> belongs to the left hand side of @{thm[source] subset}:\<close>
            have incl: "x \ ?Ra\<^sup>*``addrs ?A" using a i4 by (simp only:reachable_def, clarsimp)
            have excl:{
mment>\<>And also belongs the hand
            \<comment> \<open>which corresponds to our goal.\<close>

          qed
reovernext

java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
from  poI5\forallx      java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

        clarsimp)
          from
:\<xx\<
            (auto simp: addr_p_eq stack_eq)

          moreoverhence\<exists>stack. ?popInv stack" ..

          \<comment> \<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 97 out of bounds for length 97
            by             ? "

          ultimately
        java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
                   i1? ct<java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99
      }
      moreover

              from ifB1 whileB have pNotNull: "p \<noteq> Null" by clarsimp
      \<comment> \<open>Since they are in principle simmilar to the Pop arm proof,\<close>I1

      {

        assume ifB1: "?ifB1" andwithswI2".
         ifB1 bysimp
then addr_p :istinct java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 71
        with        have
        with havem_addr_p p.m" by clarsimp
        from stack_eqhave: " (stack) using i1 by ruleList_distinct)
byjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
          by :relS_defsimprel_def         ?\<and> ?poI2\<and> ?poI3\<and> ?poI4\<and> ?poI5\<and> ?poI6\<and> ?poI7" = "?popInv stack_tl"
hjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        proof -

          \<comment> \<open>List property is maintained:\<close>haveshowjava.lang.NullPointerException
          from i1
          have          comment:)(l
            by \"java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
          moreover

          \<comment> \<open>Everything on the stack is marked:\<close>"\sup>* "=reachable"=?java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
           swI3"bysimp :java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
          have swI2: "?swI2"
          moreover

\<omment
          let "R = reachable ?Ra ?A" = "?I3"
          et  reachableB ="swI3"
have?\<^sup>* `` addrs ?A = ?Rb\<^sup>* `` addrs ?B" ?
          proof (rule still_reachable_eq)
showAjava.lang.NullPointerException
              by(fastforce simp:addrs_def
          next

showby simp introjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69

show              \
byclarsimp:)
          next
            show "\(x, y)\?Rb-?Ra. y\(?Ra\<^sup>*``addrs ?A)"  :relS_def simprel_def  dest)
              by (clarsimp simp           >\(reachable_def
          qed
3
          have swI3: "?swI3" by (simphaveexc:           have?java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
          moreovera \java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

          \<comment> \<open>If it is reachable and not marked, it is still reachable using...\<close>
          letletT={,p."
             exc notin> ?Rb\<^sup>*`` addrs ?T"
          let ?T = "{t}"
have          from
          proof still_reachable
by(uto  add)

            show "addrs java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
              by (fastforce cong:map_cong simp              by(fastforcecongmap_congsimp          from
          java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
            show "\(x, y)\?Ra-?Rb. y\(?Rb\<^sup>*``(addrs ?B \ addrs ?T))"
              by (clarsimp simprelS_defrestr_def
          qed
thenhave: "?Ra\<^sup>*``addrs ?A - ?Rb\<^sup>*``addrs ?T \ ?Rb\<^sup>*``addrs ?B"
            by blast
          have ?swI4
          proof (           poI4
            ix
                         x
            with addr_p_eq  \a " \ R \ \ m x"
              by (simp only, clarsimpstackDist nifB2
            with ifB1then "stack. ?swInv stack" by blast
             exc:" <> Rb<^sup>*`` addrs ?T"
              by (auto
            from inc
( simp: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
          have "?" .
          moreover

java.lang.StringIndexOutOfBoundsException: Index 146 out of bounds for length 146
          from i6 stack_eq obtain< \<open>List property is maintained:\<close>
haveswI6
            by clarsimp
          moreover

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
          romi7nifB2
          have "?swI7"
            by (clarsimp simp java.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 111

                     still_reachable
        java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
                   have  : addrs_defself_reachable
      }byblast java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
      moreover

      {

        assume nifB1: "\?ifB1"
java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
whereaddr_t_eq addr_t
        withbyclarsimprelS_def  :java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 20
fromnifB1n_m_addr_t
        withi2t_notin_stack ( simpi4incin> ?Ra\<^sup>*``addrs ?A"
        let "?puI1
           java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
        proof

          from i1 t_notin_stack
          have              letjava.lang.NullPointerException
             simp:addr_t_eq, simp:S_defT=using
          moreover

          \<comment> \<open>Everything on the stack is marked:\<close>moreover
          from i2 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
f i5
            by            "puI5
          moreover

          \<comment> \<open>Everything is still reachable:\<close>
          by( congby : restr_defsimp addrs_defdest)
          let "R = reachable ?Rb ? thenhavesubset ?\<^sup>*``addrs ?A - ?Rb\<^sup>*``addrs ?T \ ?Rb\<^sup>*``addrs ?B"
          have "?Ra\<^sup>* `` addrs ?A = ?Rb\<^sup>* `` addrs ?B"
          proof "?"
            show
              by :x\<in> R \<and>\<not> m x"

have              (:,clarsimp
              fastforce:addrs_def addr_t_eq:oneStep_reachable[THEN])
          next
               ( simpaddaddrs_def
              by (clarsimp simp java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
          java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
            show  fromi6 java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
byclarsimp)
qed
          with i3qed
have: ?"by (simp add:reachable_defjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56

       show\><open>If it is on the stack, then its \<^term>\<open>l\<close> and \<^term>\<open>r\<close> fields can be reconstructed\<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 =moreover
          have "?Ra\<^sup>*``addrs ?A \ ?Rb\<^sup>*``(addrs ?B \ addrs ?T)"
          proof rulejava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
showaddrs ?A <subseteq> ?Rb\<^sup>* `` (addrs ?B \<union> addrs ?T)"
              by (fastforce ?"

            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
          qed

            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
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
            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( add)
          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>
stackDist i7
          have "? have "?uInv"

          ultimately show ?thesis by auto
        qed (simp new_stack_eqadd)
        then have

      java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
      ultimately            by (s addnew_stack_eq)
    qed
  }
qed

end

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