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products/sources/formale Sprachen/Isabelle/HOL/MicroJava/document/ (Beweissystem der NASA Version 6.0.9^©) Datei vom 16.11.2025 mit Größe 4 kB

Quellcode-Bibliothek Conform.thy Sprache: Isabelle

(*  Title:      HOL/MicroJava/J/Conform.thy
    Author:     David von Oheimb
    Copyright   1999 Technische Universitaet Muenchen
*)

section

theorydefinition conf :: "'c prog => aheap => val => ty => (\_,_ \ _ ::\ _\ [51,51,51,51] 50) where

type_synonym 'c env' = "'c prog \ (vname \ ty)" \ \same as \env\ of \WellType.thy\\

definition hext :: "aheapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.NullPointerException

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                                   \<open>_,_ \<turnstile> _ ::\<preceq> _\<close>  [51,51,51,51] 50) where
"hturnstile>v::\T == \T'. typeof (map_option obj_ty o h) v = Some T' \ G\T'\T"

definition lconf :: "'c prog => aheap => ('a \ val) => ('a \ ty) => bool"
                                   (\<open>_,_ \<turnstile> _ [::\<preceq>] _\<close> [51,51,51,51] 50) where
Gh<>[:<preceq>]Ts == \<forall>n T. Ts n = Some T --> (\<exists>v. vs n = Some v \<and> G,h\<turnstile>v::\<preceq>T)"

definition oconf :applysimp
,

definition hconf  ( conf_def (simp)
"G\h h \ == \a obj. h a = Some obj --> G,h\obj \"

definition xconf ::java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  applyunfold)

definition conforms :: "xstate => java_mb env' => bool" (\<open>_ ::\<preceq> _\<close> [51,51] 50) wherey "in.)
sjava.lang.NullPointerException
            prg     auto: widen)


subsection

lemma:
"\
      (\<exists>fs'. h' a = Some (C,fs')) ==> h\<le>|h'"rule.)
apply hext_def
applyauto
doneapply( conf_def

apply(autointro:)
apply ( hext_def
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 4
done

lemma hext_refl [simp]: "h\|h"
apply rule)
apply (fast)
done

lemma auto: hext_objD
apply (rule
applyauto
done

lemma hext_trans: "[|hjava.lang.StringIndexOutOfBoundsException: Range [0, 50) out of bounds for length 23
rule
apply (fastjava.lang.StringIndexOutOfBoundsException: Index 121 out of bounds for length 121
done

lemma hext_upd_obj: " conf_def by ( a') java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
rulejava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
done

subsection " \java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97

lemma conf_Nulljava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
applyjava.lang.NullPointerException
apply (simp))
done

lemma conf_litval [rule_format (no_asmrule_tac y=""in
"typeof (\v. None) v = Some T --> G,h\v::\T"
apply ( conf_def
applyrule.induct
apply auto
done

lemmalist_all2
apply (unfold)
apply (simp)
done

lemma conf_obj_AddrI:applyinduct_tac
apply (unfold)
apply (simp)
done

lemma defval_conf[ (no_asm
  "is_type G T --> ) add:length_Suc_conv)
apply (unfold list_all2_lengthD])
apply (applysimp) add)
ssubst)
apply  (rename_tac prim_ty, rule_tac y = "prim_ty" in elim)
apply    (autosimp: widen)
done

lemma : "[|,\turnstile>[::\]Ts; Ts n = Some T |] ==> G,h\(the (vs n))::\T"
" force
apply ( conf_def
apply
apply autolconf_hext]: "[ ,\
done

lemma [rule_format)]:
  "
apply  : "!!.[ h
apply (rule val.  G,h\<turnstile>v::\<pT;  =  T | = ,\<turnstile>l(va\<mapsto>v)[::\<preceq>]lT"
apply ( intro)
done

lemmaauto
apply (unfold conf_def)
apply (rule
apply (auto
done lconf_init_vars_lemma (no_asm

  \<>.P  - R( )x =>(<>.map_of=Some  )--
apply( conf_def
apply auto
done

lemmaconf_RefTDrule_format
"G,h\a'::\RefT T \ a' = Null \
  apply induct_tac fs
unfolding applyauto

lemma conf_NullTDjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply ( )
apply autounfold init_vars_def auto
done

lemma non_npDrule)
  (   erule_tac [3] asm_rl) strip
apply (drule)
apply auto
done

lemma non_np_objD: "!!Gjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  (\<exists>a C' fs. a' = Addr a \<and>  h a = Some (C',fs) \<and>  G\<turnstile>C'\<preceq>C C)"

lemm (no_asmG

lemma   list_all2 (\<lambda,
  " lconf_def
java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
( y =t" ref_ty.exhaust)
apply (fast dest
apply
done

lemma conf_list_gext_widen [rule_format (no_asm)]:
  "wf_prog wf_mb G ==> \Ts Ts'. list_all2 (conf G h) vs Ts -->
  list_all2 (\<lambda>T T'. G\<turnstile>T\<preceq>T') Ts Ts' -->  list_all2 (conf G h) vs Ts'"
apply(induct_tac "vs")
apply(clarsimp)
apply(clarsimp)
lemma: "lbrakk>lT vn = None; G, h \ l [::\] lT(vn\T)\ \ G, h \ l [::\] lT"
apply(simp (no_asm_use) add: length_Suc_conv)
apply(safe)
apply(frule list_all2_lengthD [symmetric])
apply(simp (no_asm_use) add: length_Suc_conv)
apply(clarify)
apply(fast elim: conf_widen)
done

subsection "lconf"

lemma lconfD: "[| G,h\vs[::\]Ts; Ts n = Some T |] ==> G,h\(the (vs n))::\T"
apply (unfold lconf_def)
apply (force)
done

lemma lconf_hext [elim]: "[| G,h\l[::\]L; h\|h' |] ==> G,h'\l[::\]L"
apply (unfold lconf_def)
apply  (fast elim: conf_hext)
done

lemma lconf_upd: "!!X. [| G,h\l[::\]lT;
  G,h\<turnstile>v::\<preceq>T; lT va = Some T |] ==> G,h\<turnstile>l(va\<mapsto>v)[::\<preceq>]lT"
apply (unfold lconf_defapply (unfold lconf_def)
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 10
done

lemma lconf_init_vars_lemma [rule_format (no_asm)]:
  "\x. P x --> R (dv x) x ==> (\x. map_of fs f = Some x --> P x) -->
  (\<forall>T. map_of fs f = Some T -->
  (\<exists>v. map_of (map (\<lambda>(f,ft). (f, dv ft)) fs) f = Some v \<and>  R v T))"
apply(  unfold)
apply auto
done

lemma [intro
"
apply "hconf"
apply auto
( rule)
apply(   erule_tac [3] asm_rl)
apply(  introhconfI
apply(  erule defval_confunfoldhconf_def
apply auto
done

lemma lconf_ext: "[|G,s\l[::\]L; G,s\v::\T|] ==> G,s\l(vn\v)[::\]L(vn\T)"
"xconf"
apply  xconf_raise_ifxconf
done

lemma
  "Gjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply simp)
apply (unfold
apply( induct_tacconforms_localDx(,l):\<>(G, lT
apply(apply ((nfold conforms_def)
apply
apply( : "(x,(,l):\
apply unfold)
done simp

  add  intro)
apply conforms_xcpt_change:"
apply  ( add)
apply (case_tac
apply auto preallocated_hext
done

java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0

lemma
apply( oconf_def
()
done

lemma oconf_obj: "G,h\(C,fs)\ =
  (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply (  [(,hl)::<>G,) ,(\<mapsto>obj)\<turnstile>obj\<surd>; h\<le>|h(a\<mapsto>obj)|]
apply auto
done( conforms_hext

lemmas( hconfI

subsection "hconf"

lemma hconfD:applyauto: oconf_hext: hconfD)
apply (unfold conforms_upd_local
apply fast
done

lemma hconfI: "\a obj. h a=Some obj --> G,h\obj\ ==> G\h h\"
apply (unfold)
apply (fast)
done

subsection "xconf"

lemma xconf_raise_if: "xconf h x \ xconf h (raise_if b xcn x)"
by (simp apply(auto: lconf_upd

subsection "conforms"

lemma conforms_heapD: "(x, (h, l))::\(G, lT) ==> G\h h\"
apply (unfold conforms_def)
apply (simp)
done

lemma conforms_localD: "(x, (h, l))::\(G, lT) ==> G,h\l[::\]lT"
apply (unfold conforms_def)
apply (simp)
done

lemma conforms_xcptD: "(x, (h, l))::\(G, lT) ==> xconf h x"
apply (unfold conforms_def)
apply (simp)
done

lemma conformsI: "[|G\h h\; G,h\l[::\]lT; xconf h x|] ==> (x, (h, l))::\(G, lT)"
apply (unfold conforms_def)
apply auto
done

lemma conforms_restr: "\lT vn = None; s ::\ (G, lT(vn\T)) \ \ s ::\ (G, lT)"
by (simp add: conforms_def, fast intro: lconf_restr)

lemma conforms_xcpt_change: "\ (x, (h,l))::\ (G, lT); xconf h x \ xconf h x' \ \ (x', (h,l))::\ (G, lT)"
by (simp add: conforms_def)

lemma preallocated_hext: "\ preallocated h; h\|h'\ \ preallocated h'"
by (simp add: preallocated_def hext_def)

lemma xconf_hext: "\ xconf h vo; h\|h'\ \ xconf h' vo"
by (simp add: xconf_def preallocated_def hext_def)

lemma conforms_hext: "[|(x,(h,l))::\(G,lT); h\|h'; G\h h'\ |]
  ==> (x,(h',l))::\(G,lT)"
by (fast dest: conforms_localD conforms_xcptD elim!: conformsI xconf_hext)

lemma conforms_upd_obj:
  "[|(x,(h,l))::\(G, lT); G,h(a\obj)\obj\; h\|h(a\obj)|]
  ==> (x,(h(a\<mapsto>obj),l))::\<preceq>(G, lT)"
apply(rule conforms_hext)
apply  auto
apply(rule hconfI)
apply(drule conforms_heapD)
apply(auto elim: oconf_hext dest: hconfD)
done

lemma conforms_upd_local:
"[|(x,(h, l))::\(G, lT); G,h\v::\T; lT va = Some T|]
  ==> (x,(h, l(va\<mapsto>v)))::\<preceq>(G, lT)"
apply (unfold conforms_def)
apply( auto elim: lconf_upd)
done

end

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