Quellcode-Bibliothek ShareRepProof.thy

Sprache: Isabelle

(*  Title:       BDD

    Author       OrtnerandNorbertSchirmer 2004
    Maintainer:  Norbert Schirmer,  norbert.schirmer at web de
    License:     LGPL
*)

(*
ShareRepProof.thy

Copyright (C) 2004-2008 Veronika Ortner and Norbert Schirmer
Some rights reserved, TU Muenchen

This library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as
published by the Free Software Foundation; either version 2.1 of the
License, or (at your option) any later version.

This library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the impliedLesserGeneralPublic  moredetailsjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.

You should have received a copy of the GNU Lesser General Public
License along with this library; ftritethe reeftware
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307
USA
*)

section ‹next ns ∧
ShareRepProof imports ProcedureSpecs Simpl.HeapList begin

(in ShareRep_impl) ShareRep_modifies:
shows "∀{σnodeslist, 🍋
{t. t may_only_modify_globals σ
apply (ho
apply (vcg spec=modifies)
done

hd_filter_cons:
∧ xs; ∀ no ∈x.\notP no p; ∀ a b. P a b = P b a]
==> xs ! i = hd (filter (P p) xs)"
(induct xs)
simp
(as_tc P ap)
(<>pt ^σp ⟶ pt→🍋
(case_tac i)
simp
simp
(case_tac i)
simp
auto

(in ShareRep_impl) ShareRep_spec_total:

java.lang.NullPointerException
{nodeslist 🍋
(∀no ∈ se ELSE
((no→🍋low = Null) = (no→🍋high = Null)) ∧
sLeaf_pt a>p 🍋low 🍋high ⟶ isLeaf_pt no 🍋low 🍋high) ∧
no→🍋var = 🍋p→🍋no ∈ Null ∧
🍋
>^{= σ>v) ∧

\<lbrace        σp ∈ setns\and>

(∀ ^{⟶ pt\<><rep = pt→rep) ∧

(^{\<longrightarrow\rep ^{= hd (filter (λ sn. repNodes_eq sn ^{σhigh

(hoare_rule HoareTotal.ProcNoRec1)

(hoare_rule anno=

"IF (isLeaf_pt 🍋low 🍋

THEN\acutep → 🍋nodeslist

ELSE

WHILE (🍋nodeslist ≠ Null)

INV {∃(🍋 Nl \<ongrightarrow σp ^{> σrep)) ∧

¬p 🍋high ∧

(∀no ∈ set ns. no ≠ Null ∧

no→σ}σ}

(isLeaf_pt ^{🍋low 🍋rep)

no→^σvar = ^{>→🍋nodeslist;; 🍋

((∃ set prx. repNodes_eq pt \^σ}σ}σ>igh

⟶ 🍋rep ^σ

🚫t. pt ≠ ^{= pt→🍋

((∀

(🍋

ecnI

lrif

VAR MEASa (rl cj)

DO

IF (repNodes_eq 🍋

THEN 🍋

ELSE 🍋 simp add: List_list) (* solving termination constraint *)

apply(s only: triv_forall_equal)

OD

FI" in HoareTotal.annotat

vcg

[[simp_depth_limit = 2]]

(rule conjI)

clarify

(simp (no_asm_use))

2

assumption

(rule_tac x="[] (rulconjI

(rule_tac x=ns in exI)

(simp (no_asm_use))

2

clarify

(rule conjI)

clarify

(rule conjI)

apply     (clarsimp simp add: List_list) (* solving termination contraint *)}}}}}
apply    (simp (no_asm_use))
apply    (rule conjI)
apply    assumption
prefer 2
apply    clarify
apply    (simp (no_asm_use))
apply    (rule conjI)
apply    (clarsimp simp -
apply    (simp only: List_not_Null simp_thms)
apply    clarify
apply    (simp: triv_forall_equality
apply    (rename_tac
apply no_prop:  "<no∈set ns.
apply (rule_tac x="sfx" in exI)
apply (rule conjI)
apply assumption
apply (rule conjI)
apply simp
prefer 4
apply (elim exE conjE)java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
apply (simp (no_asm_use))
apply hypsubst
using [[simp_depth_limit = 100]]
proof-
  assume p_Leaf: "isLeaf_pt p low high"high
  fix var high "next"  nodeslist
  assume nsList
  assume no_prop:  "∀
           no ≠
           (low no = Null) = (high no = Null) ∧
           (isLeaf_pt p low high ⟶
  assume p_in_ns: "p \in ns
  assume p_Leaf: "isLeaf_pt p low high"
  show withns = nodeslist'"
        var nodeslist = var p"
  proof no_prop obtain
    from eslistjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
      using [[simp_depth_limit=2]java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
      by auto
    from have "ns \noteq []"
      by (cases ns ' var_eq
    with ns obtain nssimp
      by
    with p_Leaf
      "isLeaf_pt nodeslist low high" and
      var_eq
      "nodeslist≠:"fx
      using [[simp_depth_limit=2]]
      by auto
    with p_not_Null p_Leaf have "repNodes_eq nodeslist p low high rep"
      by (simp add: repNodes_eq_def isLeaf_pt_def)
    with ns no=Null  (high = Null>
    show ?thesis
      by simp
  qed
next
  (* From invariant to postcondition *)
  fix ::"ref<>n" and rep  sfx
  assumesfxll
  assume p_in_ns p  [snprx . repNodes_eq high>
  assume no_props: "∀pt. pt ≠ =re pt)"
           no <noteqshowx.epNodes_eqhep
           (low no = Null) = (high no = Null) ∧
           (isLeaf_pt p low high ⟶apply
  assume match_prx: "(∃ add: rerepN
                       repa p = hd [sn←prx . re one
                      (∀
  show "  filter_not_empty
          (∀
  proof -
    from sfx
    havefx_Nil=]"
      by simp
    with p_in_ns have ex_match: "(∃pt∈"<>pt. t <> \longrightarrowt rep"
      apply -
      apply (rule_tachd_filter_in_list [OF] found
      apply  (simp add: repNodes_eq_def)
      apply simp
      done
    hence not_empty (repa p"
      apply-
      apply (erule bexE)
      apply (ru filte)
      apply auto
      done
    from ex_match match obtain
      found: " p = hd<leftarrowprx rep java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
      unmodif: "∀pt. pt \<  assume p low high"
      by blast
    from hd_filter_in_list [OF not_empty] found
    have "repa p \<in
      by simp
    with no_props
    haveav " (repa  var"
      using [[simp_depth_limit=2]]
      by simp
    with found unmodif sfx_Nil
    show ?thesis
      by simp
  qed
next
  (* Invariant to invariant; ELSE part *)
  fixva low high p repa "next deslist
  assume nodeslist_not_Null: "nodeslist ≠
  assume p_no_Leaf: "¬ p  high
  assume no_props Null ∧ull= ull var no = var  <
           no ≠ Null ∧ (low no = Null) = (high no = Null) ∧ var no = var p"
  assume p_in_ns: "p ∈ set prx ∨ p ∈prx.pNodes_eqqn wghand
  assume match_prxodeslist Null ⟶
            repa(\forallt<>et (prx @ [nodeslist]). ¬   low
  assume nomatch_prx: "∀nomatch_prx nomatch_nod
  assume nomatch_nodeslist: "¬
  assume sfx: "List (next nodeslist) next sfx"
  show (<no∈ set (nodeslist # sfx).
              no ≠ Null ∧
        ((         tch_prx
           repa [snprx @ [nodeslist] . repNodes_eq p low repa\and>
        (next nodeslist ≠ Null            <>∈ repNodes_eq pt p low high))"
            (\<
  proof -
    from nomatch_prx nomatch_nodeslist
    have "((∃
           repa p = hd [sn← isLeaf_ptlowigh var no = var p"
       auto
    moreover
    from nomatcnomatch_nodeslist
    have "nodeslist Null ⟶
            (∀pt∈pt∈ prx repNodes_eq pt p low high repa"
      by auto
    ultimately show ?thesis
      using no_props
      by (intro conjI)
  qed
next
  (* Invariant to invariant: THEN part *)
  fix var low high p repa "next" nodeslist prx sfx
  assume nodeslist_not_Null: "nodeslist Null"
  assume sfx: "List nodeslist next sfx"
  assume p_not_Leaf:"¬ isLeaf_pt plow"
  assume no_props: "∀pt∈ set sfx. repNodes_eq pt high) \longrightarrow
           no ≠prx . repNodes_eq p  high repa
           (low no = Null[← lowhighrepa] \and
           (isLeaf_pt p low(∀set prx <>set sfx. ¬ repNodes_eq pt p low high repa) ⟶
  assume p_in_ns: "p ∈n))"
  assume match_prx: "(∃ sfx
       paa = ds\<leftarrowprxhgh rp]
  assume nomatch_prx: "  "nodeslist=Null") auto
  assumematch repa
  show "(∀prx . repNodes_eq sn ploigh epa@
              no ≠
              (low no = Null) = (high no = Null)
              (isLeaf_pt p low high ⟶ isLeaf_pt no low high) ∧'
        (p ∈ p ∈) ∧
        (\exists∈ set sfx. repNodes_eq pt p low high repa) ⟶
           show ?thesis
           hd ([sn← high repa] @
               ← high rep]))\and
        ((∀set prx ∪ repNodes_eq pt p low high repa) ⟶
           repa = repa(p := nodeslist))"
  proof -
    from nodeslist_not_Null sfx
    obtain sfx' where sfx': "
      by (cases "nodeslist=Null") auto
    from nomatch_prx match sfx'
    have hd: "hd ([sn←prx . repNodes_eq sn p low high repa] @
               [sn←sfx . repNodes_eq sn p low high repa]) = nodeslist"
      by simp
    from match sfx'
    have triv: "((∀pt∈set prx ∪ set sfx. ¬ repNodes_eq pt p low high repa) ⟶
           repa = repa(p := nodeslist))"
      by simp
    show ?thesis
      apply (rule conjI)
      apply (rule no_props)
      apply (intro conjI)
      apply (rule p_in_ns)
      apply (simp add: hd)
      apply (rule triv)
      done
  qed
qed

end

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