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products/sources/formale Sprachen/Isabelle/HOL/IMP/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 5 kB

Quelle VCG_Total_EX2.thy Sprache: Isabelle

(* Author: Tobias Nipkow *)

subsubsectionTotal

theory
importsHoare_Total_EX2 Hoare_Total_EX2
begin

text \<open>Theory
Theory \<open>VCG_Total_EX\<close> conatins a VCG built on top of a Hoare logic without logical variables.
Asaresultthe completeness proof runs into a problem. theory uses logic
with variablesproves andcompleteness
\<close>

text\<close>
invariants

datatype acom =
  Askip.\<close>
  Aassignvname     (\<open>(_ ::= _)\<close> [1000, 61] 61) |
  Aseq    acom       (<>_;/ _<close>  [60, 61] 60) |
  Aifbexp acom     \open( _/ THEN_/ELSE\<close>  [0, 0, 61] 61) |
  Awhile assn2 bexp acom
    (<open>({_'/_}/ WHILE _/ DO _)\<close>  [0, 0, 0, 61] 61)

notation com.SKIP (\<open>SKIP\<close>)

text\<open>Strip annotations:\<close>

fun strip :: "acom \ com" where
"strip SKIP = SKIP" |
"strip (x ::= a) = (x ::= a)" |
"strip (C\<^sub>1;; C\<^sub>2) = (strip C\<^sub>1;; strip C\<^sub>2)" |
"strip (IF b THEN C\<^sub>1 ELSE C\<^sub>2) = (IF b THEN strip C\<^sub>1 ELSE strip C\<^sub>2)" |
"strip ({_/_} WHILE b DO C) = (WHILE b DO strip C)"

text\<open>Weakest precondition from annotated commands:\<close>

fun pre :: "acom \ assn2 \ assn2" where
"pre SKIP Q = Q" |
"pre (x ::= a) Q = (\l s. Q l (s(x := aval a s)))" |
"pre (C\<^sub>1;; C\<^sub>2) Q = pre C\<^sub>1 (pre C\<^sub>2 Q)" |
"pre (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q =
  (\<lambda>l s. if bval b s then pre C\<^sub>1 Q l s else pre C\<^sub>2 Q l s)" |
"pre ({I/x} WHILE b DO C) Q = (\l s. \n. I (l(x:=n)) s)"

text\<open>Verification condition:\<close>

fun vc :: "acom \ assn2 \ bool" where
"vc SKIP Q = True" |
"vc (x ::= a) Q = True" |
"vc (C\<^sub>1;; C\<^sub>2) Q = (vc C\<^sub>1 (pre C\<^sub>2 Q) \ vc C\<^sub>2 Q)" |
"vc (IF b THEN C Aif bexp acom acom (\(IF _/ THEN _/ ELSE _)\ [0, 0, 61] 61) |
"vc ({I/x} WHILE b DO C) Q =
  (\<forall>l s. (I (l(x:=Suc(l x))) s \<longrightarrow> pre C I l s) \<and>
       (l x > 0 \<and> I l s \<longrightarrow> bval b s) \<and>
       (I     (<open({'_}/WHILE / DO _)\ [0, 0, 0, 61] 61)
       vc )"

lemma vc_sound: "vc C Q \ \\<^sub>t {pre C Q} strip C {Q}"
proof( C arbitrary
  case (Awhile I x b Cjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   ?case
pre"java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
  case
      using Awhile.IH[of  java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
" ({I/x} WHILE b DOQ \lambda .\<exists>n. I (l(x:=n)) s)"
    case showcase
      using Awhile
   insert,auto"c (x :=java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
qedauto: Seq  simp Assign

text\<open>Completeness:\<close>

lemma pre_mono:
       l x  0 <and> I l s \<longrightarrow> bval b s) \<and>
proof (induction C       I(l(x : 0)  \<longrightarrow> \<not> bval b s \<and> Q l s) \<and> C I)"
  case Aseq thus ?case by simp metis
qed simp_all

lemma vc_mono:
  "\l s. P l s \ P' l s \ vc C P \ vc C P'"
proof( C arbitrary  'java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
  case  thus ?bysimp (metis)
qed simp_all(simp  weaken_postOF[of I x]],goal_cases

lemma vc_complete:
"\\<^sub>t {P}c{Q} \ \C. strip C = c \ vc C Q \ (\l s. P l s \ pre C Q l s)"
  (is" \ \C. ?G P c Q C")
proof (induction rule: hoaret.induct)
  case Skip
  showusing.prems by (impmetis)
proof "?C Askip"bysimp
next
  case Assigna)
  show?case( "\C. ?C C")
  proof
next

SeqIH C1where: "?G c1Q C1 byblast
  from Seq.IH  \<java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 109
  show?aseis "\C. ?C C")
  proof
    show thus ?casebysimp
lemma vc_mono:
  qed
next
  case (If P b c1 Q c2)
  from If.IH obtain C1 where ih1: "?G (\l s. P l s \ bval b s) c1 Q C1"
    by blast(induction : P P'
from. obtainC2 ih2? \<
    by blast
   ?case( "\C. ?C C")
  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
     ( rule.induct
  qed
next
  case (While P x c b)
  from While.IH obtain C where
    ih:    show "bysimp java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
    by blast  proof "?C( x a)" by simp
?case (s "\C. ?C C")
  proof
    have "vc ({P/x} WHILE b DO C) (\l. P (l(x := 0)))"
      using ih While(2,3)
      by  (metisfun_upd_same)
    thus "?C( java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
qedusing ih2byfastforce!: pre_mono)
next
  casejava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
qed

text \<open>Two examples:\<close>from.IHobtain where ih1 \<lambda>l s. P l s \<and> bval b s) c1 Q C1"

lemma vc1: "vc
({by blast
(\<lambda>l s. s ''x'' \<le> 0)"
by auto

thm vc_sound[OF vc1 simplified

lemmavc2vc
  case( P x c b)
DOfromWhile.IHobtain where
    ('y'':: V '''';;
     {\<lambda>l s. l ''x'' = nat(s ''x'') \<and> l ''y'' = nat(s ''y'') / ''y''}
     WHILE Less (N 0) (V     by blast
(\<lambda>l s. s ''x'' \<le> 0)"
by auto

thm vc_sound vc2, simplified

      using While.(2,3)

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