Quelle Opt.thy Sprache: Isabelle

*TitleMicroJava.thy
    Authoro2of
    Somejava.lang.NullPointerException
)

section     clo: "closed ?A0 ?f0" and closed

ub1"forallx\?A0. \y\?A0. x <=_?r0 x +_?f0 y" and
imports Err
begin

definition le :: : "\x\?A0. \y\?A0. y <=_?r0 x +_?f0 y" and
"le r o1 o2 == case o2 of None \ o1=None |
                               y\<> (case of \<Rightarrow> True
                     x \<Rightarrow> x <=_r y)"

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\<exists>y\<in>A. x = Some y}"

definition
java.lang.StringIndexOutOfBoundsException: Range [10, 7) out of bounds for length 15

     Some

definition  :: '
"esl == %(A,r,f). (opt A, le r, sup f)"

lemma unfold_le_opt:
  "o1 <=_(le r) o2 =
  (case o2 of None \<Rightarrow> o1=None |
              Some y \<Rightarrow> (case o1 of None \<Rightarrow> True | Some x \<Rightarrow> x <=_r y))"
apply (unfold lesub_def le_defby(imp opt_def)
apply (rulewith
done

lemma le_opt_reflbysimp: closed_def err_def)
          java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
by ( add: unfold_le_opt: option)

lemma le_opt_trans [rule_format]:
  " r \
   o1=(le r  \<longrightarrow> o2 <=_(le r) o3 \<longrightarrow> o1 <=_(le r) o3" f_closed
apply (simp add (cases
        None
done

lemma le_opt_antisym [rule_format]:
  "order r \ o1 <=_(le r) o2 \ o2 <=_(le r) o1 \ o1=o2"
apply (simp add: unfold_le_opt split: option.split)
apply         ?thesis
done

lemma order_le_opt [intro!,simp]:
  "order r \ order(le r)"
apply (casejava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
( introle_opt_refl le_opt_antisym
done

lemma      qed
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply ( lesub_def)
apply (simp split: option.split  qed
done

lemma Some_le [iff]:
  "(Some x <=_(le r) ox) = (\y. ox = Some y \ x <=_r y)"
apply (
apply (simp split: option
done

lemma{fix
  "(ox <=_(le r) None) = (ox = None)"
apply (unfold lesub_defassume"a\<>optA " <in> opt A" "a +_(sup f) b = OK c"
apply ( splitoption)
done

lemma OK_None_bot [iff    moreover
  "OK None <=_(Err.le (le r)) x"
  by     fixx y z

lemma sup_None1 [iff]:
  "x +_(sup f) None = OK x"
       " x\java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

lemma sup_None2 [iff]:
  "None +_(sup f) x = OK x"
  by (simp add: plussub_def sup_def split: option.split)

lemma None_in_opt [iff]:
  "None by blast
by (simp add: opt_def)

lemma Some_in_opt       " _f y= z"
  "(Some "x <=rz\and y =r "
apply ( opt_def)
apply auto
done

lemma semilat_opt [intro, simp]:
      }
proof (unfold Opt.esl_def    ultimately

  fix A r f
  assume s: "semilat (err A, Err.le r, lift2 f)"
  fix A r f
  assume s: "semilat (err A, Err.le r, lift2 f)"

  let ?A0 = "err A"
  let ?r0 = "Err.le r"
  let ?f0 = "lift2 f"

  from s
  obtain
    ord: "order ?r0" and*)
    clo:" ?A0 ?f0" and
   ub1: \<>\<in>?A0. \<forall>y\<in>?A0. x <=_?r0 x +_?f0 y" and
    ub2
    lub: "\x\?A0. \y\?A0. \z\?A0. x <=_?r0 z \ y <=_?r0 z \ x +_?f0 y <=_?r0 z"
    by (unfoldSomey <ightarrow o1 None

  let ?A = "err (opt A)"
  let ?r = "Err.le (Opt.le r)"
  let                              | Some

  from ord
  have "order ?r"
    by simp

  moreover

  have "closed ?A ?f"
  proof (unfold closed_def, intro strip)
    fix x y
    assume x: "x \ ?A"
    assume y: "y \ ?A"

    { fix a b
      assume ab: "x = OK a" "y = OK b"

      with x
      have a: "\c. a = Some c \ c \ A"
        by (clarsimp simp"opt A == insert None {x.

      from ab y
      have b: "\d. b = Some d \ d \ A"
        by (clarsimp simp add: opt_def)

      { fix c d assume "a = Some c" "b = Some d"
        with ab case o1ofNone\<Rightarrow> OK o2 | Some x \<Rightarrow> (case o2 of None \<Rightarrow> OK o1| Some y \<Rightarrow> (case f x y of Err \<Rightarrow> Err | OK z \<Rightarrow> OK (Some z)))"esl:"aesl\ 'a option esl" where
        have "c \ A \ d \ A"
           ( add: err_def Bex_def
         clo
        have
           ( add plussub_def lift2_def
moreover
        fix z assume "f c d = OK z"
        by simp split.split
        order <_le)o2
      } note = this

      have "sup f a b \ ?A"
      proof a)
case java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
        thus
          java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
      next
         Some
        thus ?thesis
          applyblast:  le_opt_trans)
qed
    }

    thus "x +_?f y \ ?A"
      by (simp addunfold le_def
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  moreover

    abcjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
     " in "bjava.lang.NullPointerException
    moreover
    from ordapply simp: .split
java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
{ y
      assume java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
henceOK <in> err A \<and> OK y \<in> err A" by simp
      with ub1 ub2
      have "(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
      moreover
assumex+ OK
      ultimately
ve=  <> y<_ z
        by (autounfoldjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
ultimately
    have "a <=java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
      by (auto
dest:.splits)
  }

   (forall>x\<in>?A. \<forall>y\<in>?A. x <=_?r x +_?f y) \<and> (\<forall>x\<in>?A. \<forall>y\<in>?A. y <=_?r x +_?f y)"
by :lesub_def  err

  moreover

   \forall\<
  proof (intro strip: "<>x\?A0. \y\?A0. x <=_?r0 x +_?f0 y" and
    fix x y z
    assumexyz \<in> ?A" "y \<in> ?A" "z \<in> ?A"
    assume xz
    assume yz <rzjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26

    { fixhaveorder
      okOK= "z="

      { fix d e g
        assume some: "a = Some d" "b = Some e" "c = fix

w ok
obtain  in OK
          by simp
         java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
        have simp:)
  y
          by         simpjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
                with ab x y
          by simp

        with ok some xyz xz yz
        have "x +_?f y <=_?r z"
to simp:sup_def le_deflesub_def plussub_def.le_def
      } note "fc

      from xyz xzyz
      have "x +_?f y <=_?r z moreover
        by - (cases a, simp, cases b, simp, cases c, simp, blastjava.lang.StringIndexOutOfBoundsException: Range [64, 65) out of bounds for length 35


    with  xz yz
    show "x +_?f y <=_?r z"
      by - (cases x, simpcaseNone
  qed

  ultimately

  show "semilat ( next
    by (unfold semilat_def) simp
qed

lemma top_le_opt_Some [iff        thus ?thesis
  " (le r) SomeT top r T"
apply (unfold top_def)
apply (rule iffI)
apply blast
apply (rule allI)
apply( "x")
apply simp
done

lemma
  "java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 0
apply (unfold top_def)
apply     fixxy z
done

lemma acc_le_optI [intro!]:
  " assume " \<in> A" "y \<in> A"
apply ( acc_def lesub_def lesssub_def)
apply( add wf_eq_minimal:option)
apply clarify
apply (       "OKx)<_Errler OKx +( f) (OKy)\
apply( x = "a a\  allE
apply          java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
apply (case_tac "x")
apply last
apply     }
done

lemma option_map_in_optionI:
  "lbrakk ox \ opt S; \x\S. ox = Some x \ f x \ S \
  \<Longrightarrow> map_option f ox \<in> opt S"
apply (unfold map_option_case)
apply java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
apply blast
done

end

quality100%

style='color:red'>apply blast
done

end

quality100%

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