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products/Sources/formale Sprachen/PVS/ACCoRD/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 28.9.2014 mit Größe 96 kB

Quelle SchorrWaite.thy Sprache: Isabelle

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

section

theory
imports
begin

subsection

definition
\<open>Machinery for the Schorr-Waite proof\<close>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  rel :: "(
   "rel m={,) m }java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

definition
  relS"' 'a ref) set \ ('a \ 'a) set"
  where " "rel m = {(xy).m  =Refy}java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

definition
  addrs :apply blast
  where "addrs P = {a. Ref a \ P}"

definitiondone
   :: " where"elS=(Union
  where

lemmas=relS_def(  :Image_iffapply

apply "( z <>Ra

  ( UnE =rel_def
applyblast
done

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
done

lemma
( simpImage_iff

apply: "reachable mS{}={"
apply( "(y zjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
(blast
apply (auto(simp: reachable_defrel_defs)
applyblast
done

lemma still_reachable_eq: "
( dd rel_defs)
applyapplyblast
done

lemma reachable_nullapply ( simp( )
simp: (imprel_defs
done

emmareachable}
apply :rtrancl_into_rtranclapply java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
done

reachable_union:(  )java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
apply( add rel_defsapply(imp reachable_def
applyjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
done

java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
apply(simp: rel_defs
apply
done

lemma rel_upd1:apply simp:  rel_defs addrs_defblastwherem={,.xy)
apply (ruleclassicaljava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
by (a  add)
done

lemmaapplysimp fun_upd_applybyauto:restr_def)
apply (rule classical
apply (simpapply(rule)
one(uto:restr_defconverse_rtranclE

definition
  \<comment> \<open>Restriction of a relation\<close>
  restr ::"('a \ 'a) set \ ('a \ bool) \ ('a \ 'a) set"

  where "restr r m = {(x,).java.lang.StringIndexOutOfBoundsException: Range [29, 22) out of bounds for length 22

textjava.lang.NullPointerException

lemmalemma: "(,b
" (\x. \ m x) \ (R |m) = R"
by (auto simp add:restr_def)

lemma restr_rtrancl[simp]: " \m l\ \ (R | m)\<^sup>* `` {l} = {l}"
by (auto simp add

" (\forall\<not> m x) \<Longrightarrow> (R |m) = R" (autoapply(rule
(auto add elimconverse_rtranclE)

lemma by (uto(simp add fun_upd_apply
apply (auto  \<comment> \<open>A short form for the stack mapping function for List\<close>: (a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a ref) \<Rightarrow> ('a \<Rightarrow> 'a ref) \<Rightarrow> ('a \<Rightarrow> 'a ref)"
  ab)
apply (case_tac java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

done

lemma( add S_def
    by,imp

lemma rel_upd3: "(a, b) \ (r|(m(q := t))) \ (a,b) \ (r|m) \ a = q "
( classicalauto  addjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
apply( add)
done

definitiondone
   [rule_format,simp]:
S: ('a
  apply(imp:restr_def)

textapplyinduct_tac)

lemmajava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
apply(induct_tac stack
apply(simp add:fun_upd_apply S_def)+
done

lemma    :: "' where "S c l r = (\x. if c x then r x else l x)"
"java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 23
apply(stkOk   (#)=(  r Ref)java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
done

lemma [rule_format(induct_tac)
"apply(simp add:fun_upd_apply S_def)+
apply(induct_tac stack)
apply(simp add
done

lemma [rule_format

apply(induct_tacapplyinduct_tac
apply(simp: S_def
done

primrec
    stkOk (c(x := f)) l r iL iR t xs = stkOk c l r iL iR t xs"
java.lang.StringIndexOutOfBoundsException: Index 258 out of bounds for length 258
where
  stkOk_nil:  "stkOk c l r iL iR t [] = True"
| stkOk_cons[]: "
    "stkOk c l r iL iR t (p#java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
a(nduct_tac
      iRsimp java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95

text

   l((: )iLt =stkOkiLt"
  stkOk xs
induct
apply (auto simp
done

java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
   l  )     r  )   ( =c iLx "
applyinduct xs)
applyauto:eq_sym_convjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
done

lemma [simp" c l r iLiR applyinduct ))
stkOk) iR=stkOk  iR
apply (induct xs)
       java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
done

lemma stkOk_r_rewritesimp:"\x. x \ set xs \
stkOkxs
apply( xs
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
done

lemma [VARS
stkOklemma] \And.\<lbrakk> x \<notin> set xs; Ref x\<noteq>t \<rbrakk> \<Longrightarrow> c (x=)r     clr iR"
done
apply (auto [simpWHILE
done

lemma [simp] INV {<>.
stkOkList ( stkOk  (=g iLtxsclriL"
applyinduct
apply(simp

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

theorem:
"" ( simp)
{R = reachable ( {l, r) {} \<and> (\<forall>x. \<not> m x) \<and> iR = r \<and> iL = l}
t: root     lx=)r iR x xsstkOkriL x "
( )
INV( simp)

          java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
          :g)l   iR java.lang.NullPointerException

xjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
(foralljava.lang.NullPointerException
          (
( c r iL  )
  t= \<or> t^.m
      THEN IF p^.c
                      IF ^.
ELSEt  ^. ^r=p^.;
                        ^l : ;p. = True
      ELSE q := p; p := t; t := t apply (auto:eq_sym_conv^l  q .:True
p^subsection
   "reachable r java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  (is "Valid
    ({,,r,p,,)  cml  }
         (_(applyblast
         (Aseq
proof ()
  {
    fix c m l r t          (seqAseqjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
    assume           (<>x\<>   {
s" c l Null by(simp add: addrs_def
   (\<forall>x. x \<in> R \<and> \<not>m x \<longrightarrow>                                         \<comment> \<open>\<open>i4\<close>\<close>    ?    rootbyadd)
    fix c mlrt  java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
let
    assume  DO IFt  Null
then where: "? stack" byjava.lang.StringIndexOutOfBoundsException: Range [54, 55) out of bounds for length 54
    froma  pNullELSE: t : ^r ^ pl                    comment

aveI1 ? and""and ?  +
    apply^  ;p.c =FI
    from tDisj i4\<
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  next  isjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 4
    fix(c,,  ,letjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
let\existsstack     java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
"
existsstackstack
      "?inv (c(p \ True)) m (l(p \ t)) (r(p \ p^.l)) (p^.r) p"
let
  ?(t\\<rightarrow> False)) (m(t \<rightarrow> True)) (l(t \<rightarrow> p)) r (t^.l) t"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    let "?ifB2"  =  " ?Precobtainstackapply()

    ssume\<exists>stack.?Inv stack) \<and> ?whileB m t p"
         :Inv"andwhileB whileBmt "by
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
   i1  :""andI3:"I4java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
                and java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105

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

            \<not>?ifB1 \<longrightarrow> (\<exists>stack.?puInv stack))"
      \<forall>x. \<not> m x) \<Longrightarrow> (R |m) = R"
      {
ifB1pNull tack
        fromby( simp:restr_def:converse_rtranclE
eaddr_p_eq" auto
        with (utofrom i6 (<forall>x.(x \<in> R) = m x) \<and> r = iR \<and> l = iL"  by(auto simp: stackEmpty fun_eq_iff intro:RisMarked)
          yautowith obtain where: "stack = (addr p #stack_tl"
                with   java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
stackDist    rule
                 stackDistdistinct?  (  Null
"<>poI2and> ?poI3\ ?poI4\ ?poI5\ ?poI6\ ?poI7" = "?popInv stack_tl"
        have "?popInv stack_tl"
proof

          \<comment> \<open>List property is maintained:\<close>
          from i1 i1 ifB2    where llambda>x. if c x then r x else l x)"
          poI1poI1List r > t))) (p^.r) stack_tl"
            ( add  stack_eq add S_def)

          moreover
          \<comment> \<open>Everything on the stack is marked:\<close>lemma[,]:
          fromapply )
          moreover

          \<comment> \<open>Everything is still reachable:\<close> rule_format(java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 0
=reachableAjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
         obtain where<>java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
let
          apply add S_defproof )
[,simp:
          proof
                               neStep_reachable iffD2

              (from  p_notin_stack_tljava.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
                   intro:oneStep_reachable    :         ? stack_tl "? ?L"
              show
    fromp_notin_stack_tl
            qediL :addrs_def
            showjava.lang.NullPointerException
             (rule)
              show "addrsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                by(fastforcefrom havepoI2 simpjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
intro
            next
show          "R =reachable? ?Ajava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 13
byclarsimp:)
                    simp:rel_def

          qed
          with have: "R =reachable let"
          moreover

          \<comment> \<open>If it is reachable and not marked, it is still reachable using...\<close>"}unionsetmap (\
letjava.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
                   h :java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
          \<comment> \<open>Our goal is \<open>\<forall>x. x \<in> R \<and> \<not> m x \<longrightarrow> x \<in> reachable ?Rb ?B\<close>.\<close> rx:( simp:rel_def addrs_def
T  .java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30

java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 4
          proof
            ( simp java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 17
              by (autonext (( : )    iR x) xs c l<> <open now aterm  right the of
show
              by (           poI4            java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
            java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

                (fastforce          <> \<open>If it is reachable and not marked, it is still reachable using...\<close>           ""             "^
          qed
                    have "?Ra\<^sup>* `` addrs ?A \<subseteq> ?Rb\<^sup>* `` (addrs ?B \<union> addrs ?T)"
          hence subset"?\<^sup>* `` addrs ?A - ?Rb\<^sup>* `` addrs ?T \ ?Rb\<^sup>* `` addrs ?B"
            byblast
          have poI4: "\x. x \ R \ \ m x \ x \ reachable ?Rb ?B"
proof allI" subseteq?Rb^sup>*`(addrs ? \ addrs ?T)"
              by ( cong :stack_eq intro)
             a  \<in> R \<and> \<not> m x"
            \<comment> \<open>First, a disjunction on \<^term>\<open>p^.r\<close> used later in the proof\<close>
            incl
              by autoqed
            \<comment> \<open>\<^term>\<open>x\<close> belongs to the left hand side of @{thm[source] subset}:\<close> iL
              IF  \<or> t^.m

\comment\open therefore the
\<omment
i7ifB2
           :\<>.x
            (uto: addr_p_eqfun_upd_apply q := p p java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

\<comment> \<open>If it is marked, then it is reachable\<close>byjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
from have poI5: "x. m x \ x \ R" .
          moreover clarsimpstack_eq)

                      "popInvstack_tl java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
i7
          have poI6thus excl


          letchable\<close>

          \<comment> \<open>If it is on the stack, then its \<^term>\<open>l\<close> and \<^term>\<open>r\<close> fields can be reconstructed\<close>           i5have: ".{
p_notin_stack_tl
by:stack_eq)

  ?stack_tl "forall.
        qedbyautosimpaddr_p_eq fun_upd_apply
         "\stack. ?popInv stack" ..
      }
      moreover

\comment

                from p_notin_stack_tl i7 have poI7: "stkOk c l (r(p \<rightarrow> t)) iL iR p stack_tl"java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
      {
        \<comment> \<open>Swing arm\<close>let    (=Null

java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
        then ?\>
with  stack_tl       \<comment> \<open>we show fewer comments and use frequent pattern matching.\<close>
          :"swI2
        frommoreover
          bysimp
                 obtain     have:" tack i1by( List_distinct)
"?swInv stack"
java.lang.StringIndexOutOfBoundsException: Range [10, 8) out of bounds for length 15

          \<comment> \<open>List property is maintained:\<close>
          from i1 p_notin_stack_tl nifB2
          have swI1
             simp: stack_eq t  \<or> t^.m" and ifB2: "p^.c"
          moreover

\>
          from i2

          moreover

          \<comment> \<open>Everything is still reachable:\<close>with  "."by
le? ?A  ?"
          let "R = reachable ?Rb simp
           show
          byclarsimp)( simp: Image_iff poI1
            show "addrs ?A \ ?Rb\<^sup>* `` addrs ?B"
by( simp rel_defs intro Image_iffjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
          next
" ?B ?Ra\<^sup>* `` addrs ?A"
              by(fastforce simp            (addaddr_p_eq, simp S_def

            show
               ( simp fastforce \<
          next
            show"(x, y)\?Rb-?Ra. y\(?Ra\<^sup>*``addrs ?A)"           java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
              by (clarsimp? "}java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24

          with i3 (rule           ?  p,.}
          have: "?swI3 by (addreachable_def)
          moreover

                                < \<open>Everything is still reachable:\<close>
          letl "R=reachable ?Rb ?" "
          let
           "Ra^sup>*``addrs ?A \ ?Rb\<^sup>*``(addrs ?B \ addrs ?T)"
          proof (rule :relS_def)                                addrsjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 63

               (auto add:p_notin_stack_tl:fun_upd_other)
            show "addrs ?A \ ?Rb\<^sup>* `` (addrs ?B \ addrs ?T)"
              by (fastforce cong:map_cong            "<> \<>(,y
          java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
ed
              (simprestr_def  add Image_ifffun_upd_applyrel_upd1
          qed
          thencomment(only)
                       \forallx.x\<>Rjava.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
          have?
          proof ruleby auto:addrs_def
            fix x
             :"in> R \\ m x"
            with i4              by ( simp
              by( only, clarsimp
            with
have "x\java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
              by (auto simp add:addrs_def)
            from inc exc subset (rule)
               (utosimp:reachable_def
          qed
moreover

          \<comment> \<open>If it is marked, then it is reachable\<close>
          fromi5
          have "?swI5" .            show"(x, y)\?Ra-?Rb. y\(?Rb\<^sup>*``(addrs ?B \ addrs ?T))"
          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>
          then  subsetRa
have"swI6java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
            by clarsimp "?wI7java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
          moreover

          a :x\<in> R \<and> \<not> m x"
           java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 33
          have "?swI7"
            by (haveexc x\notin?java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

          hbyuto)
        qed
"
      }
      moreover

      {
\comment>
                            with
        from          java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
        then \<>\<open>List property is maintained:\<close>
        with i1 obtain ""
          java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
        with i2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        let " moreover
        have "?puInv new_stack"
        proof -

          \<comment> \<open>List property is maintained:\<close>
          from i1 t_notin_stack
          have           poI6
            by (simp add:addr_t_eq
          moreover

          \<comment> \<open>Everything on the stack is marked:\<close>
           i2
          havefromp_notin_stack_tl have: "stkOkc proof(java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
new_stack_eq)
          moreover

          \<comment> \<open>Everything is still reachable:\<close>             "addrs? \Proofs of the Swing and Push arm follow.\
let = Ra =      <comment> \<open>we show fewer comments and use frequent pattern matching.\<close>
         ifB1"ifB1"and"
          have?Ra^sup* `addrs  ?byclarsimprelS_deffastforcerel_def :java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
          proof rule)
            show "addrs ?A \ ?Rb\<^sup>* `` addrs ?B"
              by(astforce: rel_defs intro Image_iff show
          next
show?
              by(fastforce simp  with
          java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
            show "\(x, y)\?Ra-?Rb. y\(?Rb\<^sup>*``addrs ?B)"
              by( simp) (fastforcesimp:rel_def  addrs_defdest
          next
            show "\(x, y)\?Rb-?Ra. y\(?Ra\<^sup>*``addrs ?A)"
              by (clarsimp            "?\<^sup*`addrs ? java.lang.StringIndexOutOfBoundsException: Range [0, 90) out of bounds for length 18
          qed
           swI2swI2
           java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
          java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

\>
          let "\x. x \ R \ \ m x \ x \ reachable ?Ra ?A" = ?I4
          let"x. x \ R \ \ ?new_m x \ x \ reachable ?Rb ?B" = ?puI4
          let ( still_reachable_eq
          have "?Ra\<^sup>*``addrs ?A \ ?Rb\<^sup>*``(addrs ?B \ addrs ?T)"
proof (rule)
show?java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
by(simp intro)
          next
            "(x, y)\?Ra-?Rb. y\(?Rb\<^sup>*``(addrs ?B \ addrs ?T))"
restr_upd
                 (fastforce
         java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
          then have subset "\(x, y)\?Ra-?Rb. y\(?Rb\<^sup>*``addrs ?B)"
            byblast
java.lang.StringIndexOutOfBoundsException: Range [70, 20) out of bounds for length 20
)
            fixwith  have t_notin_stackby(larsimp     "\java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
            assume a: "x \ R \ \ ?new_m x"
            have xDisj
            with
              by (fastforce simp:java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 31
            have : "x\<> ?Rb\<^sup>*`` addrs ?T"
usingan_m_addr_t
              by (clarsimp simp
            from inc           i6
              by (auto java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 20
qed
          moreover

          from i5
          have " by simponly: )
            byby( simp rel_defs intro Image_iff iffD2
          moreover


          i6
          have "?puI6"
            by (simp add:new_stack_eq)
                   moreover          next

          java.lang.StringIndexOutOfBoundsException: Range [0, 136) out of bounds for length 18
          stackDist i7
          have "?puI7" by (clarsimp simp:addr_t_eq new_stack_eq)

          ultimately show ?thesis
        java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
        then have "\stack. ?puInv stack" by blast

      }
      ultimately<
    qed
  }
qed

end

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