Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: LBVCorrect.thy Sprache: Isabelle

Original von: Isabelle^©

(*  Author:     Gerwin Klein
    Copyright   1999 Technische Universitaet Muenchen
*)

section \<open>Correctness of the LBV\<close>

theory LBVCorrect
imports LBVSpec Typing_Framework
begin

locale lbvs = lbv +
  fixes s0  :: 'a ("s\<^sub>0")
  fixes c   :: "'a list"
  fixes ins :: "'b list"
  fixes phi :: "'a list" ("\")
  defines phi_def:
  "\ \ map (\pc. if c!pc = \ then wtl (take pc ins) c 0 s0 else c!pc)
       [0..<length ins]"

  assumes bounded: "bounded step (length ins)"
  assumes cert: "cert_ok c (length ins) \ \ A"
  assumes pres: "pres_type step (length ins) A"

lemma (in lbvs) phi_None [intro?]:
  "\ pc < length ins; c!pc = \ \ \ \ ! pc = wtl (take pc ins) c 0 s0"
  by (simp add: phi_def)

lemma (in lbvs) phi_Some [intro?]:
  "\ pc < length ins; c!pc \ \ \ \ \ ! pc = c ! pc"
  by (simp add: phi_def)

lemma (in lbvs) phi_len [simp]:
  "length \ = length ins"
  by (simp add: phi_def)

lemma (in lbvs) wtl_suc_pc:
  assumes all: "wtl ins c 0 s\<^sub>0 \ \"
  assumes pc:  "pc+1 < length ins"
  shows "wtl (take (pc+1) ins) c 0 s0 \\<^sub>r \!(pc+1)"
proof -
  from all pc
  have "wtc c (pc+1) (wtl (take (pc+1) ins) c 0 s0) \ T" by (rule wtl_all)
  with pc show ?thesis by (simp add: phi_def wtc split: if_split_asm)
qed

lemma (in lbvs) wtl_stable:
  assumes wtl: "wtl ins c 0 s0 \ \"
  assumes s0:  "s0 \ A"
  assumes pc:  "pc < length ins"
  shows "stable r step \ pc"
proof (unfold stable_def, clarify)
  fix pc' s' assume step: "(pc',s') \ set (step pc (\ ! pc))"
                      (is "(pc',s') \ set (?step pc)")

  from bounded pc step have pc': "pc' < length ins" by (rule boundedD)

  from wtl have tkpc: "wtl (take pc ins) c 0 s0 \ \" (is "?s1 \ _") by (rule wtl_take)
  from wtl have s2: "wtl (take (pc+1) ins) c 0 s0 \ \" (is "?s2 \ _") by (rule wtl_take)

  from wtl pc have wt_s1: "wtc c pc ?s1 \ \" by (rule wtl_all)

  have c_Some: "\pc t. pc < length ins \ c!pc \ \ \ \!pc = c!pc"
    by (simp add: phi_def)
  from pc have c_None: "c!pc = \ \ \!pc = ?s1" ..

  from wt_s1 pc c_None c_Some
  have inst: "wtc c pc ?s1 = wti c pc (\!pc)"
    by (simp add: wtc split: if_split_asm)

  from pres cert s0 wtl pc have "?s1 \ A" by (rule wtl_pres)
  with pc c_Some cert c_None
  have "\!pc \ A" by (cases "c!pc = \") (auto dest: cert_okD1)
  with pc pres
  have step_in_A: "snd`set (?step pc) \ A" by (auto dest: pres_typeD2)

  show "s' <=_r \!pc'"
  proof (cases "pc' = pc+1")
    case True
    with pc' cert
    have cert_in_A: "c!(pc+1) \ A" by (auto dest: cert_okD1)
    from True pc' have pc1: "pc+1 < length ins" by simp
    with tkpc have "?s2 = wtc c pc ?s1" by - (rule wtl_Suc)
    with inst
    have merge: "?s2 = merge c pc (?step pc) (c!(pc+1))" by (simp add: wti)
    also
    from s2 merge have "\ \ \" (is "?merge \ _") by simp
    with cert_in_A step_in_A
    have "?merge = (map snd [(p',t') \ ?step pc. p'=pc+1] ++_f (c!(pc+1)))"
      by (rule merge_not_top_s)
    finally
    have "s' <=_r ?s2" using step_in_A cert_in_A True step
      by (auto intro: pp_ub1')
    also
    from wtl pc1 have "?s2 <=_r \!(pc+1)" by (rule wtl_suc_pc)
    also note True [symmetric]
    finally show ?thesis by simp
  next
    case False
    from wt_s1 inst
    have "merge c pc (?step pc) (c!(pc+1)) \ \" by (simp add: wti)
    with step_in_A
    have "\(pc', s')\set (?step pc). pc'\pc+1 \ s' <=_r c!pc'"
      by - (rule merge_not_top)
    with step False
    have ok: "s' <=_r c!pc'" by blast
    moreover
    from ok
    have "c!pc' = \ \ s' = \" by simp
    moreover
    from c_Some pc'
    have "c!pc' \ \ \ \!pc' = c!pc'" by auto
    ultimately
    show ?thesis by (cases "c!pc' = \") auto
  qed
qed


lemma (in lbvs) phi_not_top:
  assumes wtl: "wtl ins c 0 s0 \ \"
  assumes pc:  "pc < length ins"
  shows "\!pc \ \"
proof (cases "c!pc = \")
  case False with pc
  have "\!pc = c!pc" ..
  also from cert pc have "\ \ \" by (rule cert_okD4)
  finally show ?thesis .
next
  case True with pc
  have "\!pc = wtl (take pc ins) c 0 s0" ..
  also from wtl have "\ \ \" by (rule wtl_take)
  finally show ?thesis .
qed

lemma (in lbvs) phi_in_A:
  assumes wtl: "wtl ins c 0 s0 \ \"
  assumes s0:  "s0 \ A"
  shows "\ \ list (length ins) A"
proof -
  { fix x assume "x \ set \"
    then obtain xs ys where "\ = xs @ x # ys"
      by (auto simp add: in_set_conv_decomp)
    then obtain pc where pc: "pc < length \" and x: "\!pc = x"
      by (simp add: that [of "length xs"] nth_append)

    from pres cert wtl s0 pc
    have "wtl (take pc ins) c 0 s0 \ A" by (auto intro!: wtl_pres)
    moreover
    from pc have "pc < length ins" by simp
    with cert have "c!pc \ A" ..
    ultimately
    have "\!pc \ A" using pc by (simp add: phi_def)
    hence "x \ A" using x by simp
  }
  hence "set \ \ A" ..
  thus ?thesis by (unfold list_def) simp
qed

lemma (in lbvs) phi0:
  assumes wtl: "wtl ins c 0 s0 \ \"
  assumes 0:   "0 < length ins"
  shows "s0 <=_r \!0"
proof (cases "c!0 = \")
  case True
  with 0 have "\!0 = wtl (take 0 ins) c 0 s0" ..
  moreover have "wtl (take 0 ins) c 0 s0 = s0" by simp
  ultimately have "\!0 = s0" by simp
  thus ?thesis by simp
next
  case False
  with 0 have "phi!0 = c!0" ..
  moreover
  from wtl have "wtl (take 1 ins) c 0 s0 \ \" by (rule wtl_take)
  with 0 False
  have "s0 <=_r c!0" by (auto simp add: neq_Nil_conv wtc split: if_split_asm)
  ultimately
  show ?thesis by simp
qed

theorem (in lbvs) wtl_sound:
  assumes wtl: "wtl ins c 0 s0 \ \"
  assumes s0: "s0 \ A"
  shows "\ts. wt_step r \ step ts"
proof -
  have "wt_step r \ step \"
  proof (unfold wt_step_def, intro strip conjI)
    fix pc assume "pc < length \"
    then have pc: "pc < length ins" by simp
    with wtl show "\!pc \ \" by (rule phi_not_top)
    from wtl s0 pc show "stable r step \ pc" by (rule wtl_stable)
  qed
  thus ?thesis ..
qed

theorem (in lbvs) wtl_sound_strong:
  assumes wtl: "wtl ins c 0 s0 \ \"
  assumes s0: "s0 \ A"
  assumes nz: "0 < length ins"
  shows "\ts \ list (length ins) A. wt_step r \ step ts \ s0 <=_r ts!0"
proof -
  from wtl s0 have "\ \ list (length ins) A" by (rule phi_in_A)
  moreover
  have "wt_step r \ step \"
  proof (unfold wt_step_def, intro strip conjI)
    fix pc assume "pc < length \"
    then have pc: "pc < length ins" by simp
    with wtl show "\!pc \ \" by (rule phi_not_top)
    from wtl s0 pc show "stable r step \ pc" by (rule wtl_stable)
  qed
  moreover
  from wtl nz have "s0 <=_r \!0" by (rule phi0)
  ultimately
  show ?thesis by fast
qed

end

¤ Dauer der Verarbeitung: 0.4 Sekunden (vorverarbeitet) ¤

Download des Quellennavigators
Download des sprechenden Kalenders
in der Quellcodebibliothek suchen

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.