Quelle Semilat.thy Sprache: Isabelle

    Author:     Tobias Nipkow
    Copyright   2000 TUM
*)

chapter \<open>Bytecode Verifier \label{cha:bv}\<close>

section \<open>Semilattices\<close>

theory Semilat
imports Main "HOL-Library.While_Combinator"
begin

type_synonym' ord ="'\Rightarrowa\<Rightarrow> bool"
type_synonym 'a binop = "'a \<Rightarrow> 'a \<Rightarrow> 'a"
type_synonym    Copyright*)

lesub"java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
"lesubxry \ r x y"

definition lesssub :: "'a \ 'a ord \ 'a \ bool"
where

definition""  (\<open>(_ /<='__ _)\<close> [50, 1000, 51] 50) and(<>_ /' )<>5,100 15)and
  where

notation
  ""java.lang.NullPointerException
  "lesssub"  (\<open>(_ /<'__ _)\<close> [50, 1000, 51] 50) and
  "plussub"    "lesssub"  (

notation
  "lesub"  (\<open>(_ /\<sqsubseteq>\<^bsub>_\<^esub> _)\<close> [50, 0, 51] 50) and\<open>(_ /\<squnion>\<^bsub>_\<^esub> _)\<close> [65, 0, 66] 65)
  "esssub"(\<open>(_ /\<sqsubset>\<^bsub>_\<^esub> _)\<close> [50, 0, 51] 50) and "x \\<^sub>r y == x \\<^bsub>r\<^esub> y"
  "plussub"  (\<open>(_ /\<squnion>\<^bsub>_\<^esub> _)\<close> [65, 0, 66] 65) x

(* allow \<sub> instead of \<bsub>..\<esub> *)
abbreviation (input)
  lesub1 :: "'a \ 'a ord \ 'a \ bool" (\(_ /\\<^sub>_ _)\ [50, 1000, 51] 50)
  where "x \\<^sub>r y == x \\<^bsub>r\<^esub> y"

abbreviation (input)
  lesssub1 :: "'a \ 'a ord \ 'a \ bool" (\(_ /\\<^sub>_ _)\ [50, 1000, 51] 50)
x\sqsubset

abbreviation (input)
  plussub1
where

definition ord :: "" r \<equiv> \<lambda>x y. (x,y) \<in> r"
  ord

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

definition top :: "'a ord \ 'a \ bool" where
  "top r T \ \x. x \\<^sub>r T"

  semilat
  "cc\equiv> wf {y,x). x \\<^sub>r y}"

definition closed :: "'a set \ 'a binop \ bool" where
  "closed (\x\A. \y\A. y \\<^sub>r x \\<^sub>f y) \

definition :: "'a \ bool" where
  semilat
                       (\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^sub>r x \<squnion>\<^sub>f y) \<and>:'\java.lang.StringIndexOutOfBoundsException: Index 119 out of bounds for length 119

                        A : ' "

definition is_ub :: "('a \ 'a) set \ 'a \ 'a \ 'a \ bool" where
  "is_ub r x y u \ (x,u)\r \ (y,u)\r"

definition is_lub :: "('a \ 'a) set \ 'a \ 'a \ 'a \ bool" where
    assumessemilat,r )

definition some_lub
  some_lubjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50

locale Semilat
  fixes:' setjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
  fixes r :: "'a ord"
  fixes(*<*) by (unfold order_def) blast
assumes: "emilat (A,r, "

lemma order_refl [simp, intro]: "order r \ x \\<^sub>r x"


lemma order_antisym: "\<lbrakk> order r; x \<sqsubseteq>\<^sub>r y; y \<sqsubseteq>\<^sub>r x \<rbrakk> \<Longrightarrow> x = y"
  (*<*)

lemma order_trans: "\ order r; x \\<^sub>r y; y \\<^sub>r z \ \ x \\<^sub>r z"
  *<*  (unfold ) blast

lemma order_less_irrefl [intro, simp]: "order rjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  (*<*) by (unfold order_def lesssub_def) blast (*>*)  *<*) by (simp add: top_def) (*>*)

lemma order_less_trans: "\ order r; x \\<^sub>r y; y \\<^sub>r z \ \ x \\<^sub>r z"
  (*<*) by (unfold order_def lesssub_def) blast (*>*):

lemma topD [
  (*<*) by (simp add: top_def) (*>*)

java.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 120
  (*<*) by (blast intro: order_antisym) (*>*)

lemma semilat_Def(
semilat
                 (\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^sub>r x \<squnion>\<^sub>f y) \<and>
                 (\<forall>x\<in>A. \<forall>y\<in>A. y \<sqsubseteq>\<^sub>r x \<squnion>\<^sub>f y) \<and>

  (*<*) by (unfold semilat_def) clarsimp (*>*)

lemma (in Semilat) orderI [simp, intro]: "order r"
  (*<*) using semilat by (simp add: semilat_Def) (*>*)

lemma (in


lemma closedD: "\<lbrakk> closed A f; x\<in>A; y\<in>A \<rbrakk> \<Longrightarrow> x \<squnion>\<^sub>f y \<in> A"
  (*<*)

lemma
  (*<*) by (simp add: closed_def) (*>*)

java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
(

lemma (in Semilat) refl_r [java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma (in Semilat) antisym_r [intro?]: "\ x \\<^sub>r y; y \\<^sub>r x \ \ x = y"
(<)( order_antisym

lemma (
  (*<*)

lemma lemma( Semilat [, intro
  (*<*) using semilat by (simp add: semilat_Def) (*>*)

in),?]
  (*<*) using semilat by (simp add: semilat_Def) (*>*)

in [simp:
"lbrakk
  (*<*) using semilat by (simp add: semilat_Def) (*>*)

lemma (in Semilat) plus_le_conv [simp]:
  " (*<*) using semilat by (simp add: semilat_Def) (*>*)
(*<*) by (blast intro: ub1 ub2 lub order_trans) (*>*)

lemma (in Semilat) le_iff_plus_unchanged: "lemma (in Semilat) le_iff_plus_unchanged: "\ x \ A; y \ A \ \ (x \\<^sub>r y) = (x \\<^sub>f y = y)"
(*<*)
apply (rule iffI)(*<*)
apply(blast intro: antisym_r)
apply e subst
apply simperule)
done
(*>*)

lemma
(*<*)
applyrule)
apply (blast intro: order_antisym lub ub1)
apply(rule)
apply simp )
done
(*>*)

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
:
  shows "a \\<^sub>f (b \\<^sub>f c) = a \\<^sub>f b \\<^sub>f c"
(*<*)
proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  from a b have ab: "a \\<^sub>f b \ A" ..
  from this haveabc have': a \\<^sub>f (b \\<^sub>f c) \ A" ..
from  bcbjava.lang.NullPointerException
  from a this

  show ?thesis
  proof
    show " proof -
    proof -
      from a b have "a \\<^sub>r a \\<^sub>f b" ..
      also from ab c have "\ \\<^sub>r \ \\<^sub>f c" ..
      finallyhavea<" " \<sqsubseteq>\<^sub>r (a \<squnion>\<^sub>f b) \<squnion>\<^sub>f c" .
a bhave" \\<^sub>r a \\<^sub>f b" ..
      also from ab c have "\ \\<^sub>r \ \\<^sub>f c" ..
      finally have "b<": "b \\<^sub>r (a \\<^sub>f b) \\<^sub>f c" .
      fromchavec<": "c \<sqsubseteq>\<^sub>r (a \<squnion>\<^sub>f b) \<squnion>\<^sub>f c" ..
      from "b<" "c<" b c abc       from abhave" \\<^sub>r a \\<^sub>f b" ..
      from "a<" this
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
    show
    proof -
      from b c have "b \\<^sub>r b \\<^sub>f c" ..
      also from      from"a" this a bc show ?thesis ..
      finally haveb<:" \\<^sub>r a \\<^sub>f (b \\<^sub>f c)" .
      fromproof -
      also froma bc have"dots>\\<^sub>r a \\<^sub>f \" ..
       have"<" " \\<^sub>r a \\<^sub>f (b \\<^sub>f c)" .
      from a bc have "a<": "a \\<^sub>r a \\<^sub>f (b \\<^sub>f c)" ..
      from "a<" "b<" a b abc' have "a \\<^sub>f b \\<^sub>r a \\<^sub>f (b \\<^sub>f c)" ..
      from this "c<"        a  have "
    qed
  qed
qed
(*>*)

lemma (in Semilat) plus_com_lemma:
  "\a \ A; b \ A\ \ a \\<^sub>f b \\<^sub>r b \\<^sub>f a"
(*<*)
proof -
  assume a: "a \ A" and b: "b \ A"
from ahavea <sqsubseteq>\<^sub>r b \<squnion>\<^sub>f a" ..
  moreover from b a have "b \\<^sub>r b \\<^sub>f a" ..
  moreoverab
  moreover from b a have "b \\<^sub>f a \ A" ..
.
qed
(*>*)

lemmain) :
  "\a \ A; b \ A\ \ a \\<^sub>f b = b \\<^sub>f a"
  (*<*) by(blast intro: order_antisym plus_com_lemma) (*>*) in) :

lemma is_lubD:
  "is_lubproof -
  (*<*) by (simp add: is_lub_def) (*>*) : ajava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44

lemma is_ubI:
  "\ (x,u) \ r; (y,u) \ r \ \ is_ub r x y u"
  (*<*) by (simp add: is_ub_def) (*>*)

lemma is_ubD:
  "is_ub r x ultimately ?thesis..
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

is_lub_bigger1]:
  "is_lub (r\<^sup>*) x y y = ((x,y)\r\<^sup>*)"
(*<*)
apply (unfold   (*<*) by(blast intro: order_antisym plus_com_lemma) (*>*)
apply blast
done
(*>*)

lemma is_lub_bigger2 [iff]:
  "is_lub (*<*) by (simp add: is_lub_def) (*>*)
(*<*)
( is_lub_def)
apply blast(*<*) by (simp add: is_ub_def) (*>*)
lemm:
(*>*)

lemma extend_lub:
  "\ single_valued r; is_lub (r\<^sup>*) x y u; (x',x) \ r \
  \<Longrightarrow> \<exists>v. is_lub (r\<^sup>*) x' y v"
(*<*)
apply (unfold is_lub_def
apply (case_tac  [iff:
apply (case_tac "(yjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  apply is_lub_def
  :  dest
apply (rule
apply (rule conjI)
apply (blast intro: converse_rtrancl_into_rtrancl(*>*)
apply (blast introlemma extend_lub
             elim: converse_rtranclE
done
(*>*)

lemma single_valued_has_lubs [rule_format]:
  "\single_valued r; (x,u) \ r\<^sup>*\ \ (\y. (y,u) \ r\<^sup>* \
  (\<exists>z. is_lub (r\<^sup>*) x y z))"
(*<*)
apply java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
apply apply (last: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
  apply blast
apply (blast intro:
applyjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
done
(*>*)

lemma some_lub_conv:
  "\acyclic r; is_lub (r\<^sup>*) x y u\ \ some_lub (r\<^sup>*) x y = u"
(*<*)
java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
apply (rule someI2
java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
apply  apply( intro)
done
(*>*)

lemma is_lub_some_lub:
  "\single_valued r; acyclic r; (x,u)\r\<^sup>*; (y,u)\r\<^sup>*\
  <Longrightarrow> is_lub (r\<^sup>*) x y (some_lub (r\<^sup>*) x y)"
  (*<*) by (fastforce dest: single_valued_has_lubs simp add: some_lub_conv) (*>*)

\<open>An executable lub-finder\<close>

definition exec_lub :: "('a * 'a) set \ ('a \ 'a) \ 'a binop" where
"exec_lub r f x y \ while (\z. (x,z) \ r\<^sup>*) f y"

lemma exec_lub_reflapply( intro antisymD!: )
by (simp add

lemma acyclic_single_valued_finite is_lub_some_lub:
"\acyclic r; single_valued r; (x,y) \ r\<^sup>*\
  <Longrightarrow> is_lub (r\<^sup>*) x y (some_lub (r\<^sup>*) x y)"
(*<*)
apply(erule converse_rtrancl_induct
subsection
  apply(simp only:java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
  apply(blast intro:rtrancl_into_trancl2 rtrancl_trancl_trancl)
apply simp
apply(rename_tac exec_lub_refl     "
apply(subgoal_tac "r \ {a. (x,a) \ r\<^sup>*} \ {b. (b,y) \ r\<^sup>*} =
                    (x,' r\ {a. (x', a) \ r\<^sup>*} \ {b. (b, y) \ r\<^sup>*})")
apply simp
apply(blast intro:converse_rtrancl_into_rtrancl
            elim"r;single_valued ;(x,)\ r\<^sup>*\
done
(*>*)

lemma exec_lub_conv:
  "\ acyclic r; \x y. (x,y) \ r \ f x = y; is_lub (r\<^sup>*) x y u \ \
  exec_lub   x y =u"
(*<*)
apply(unfold exec_lub_def)
apply(rule_tac P = "\z. (y,z) \ r\<^sup>* \ (z,u) \ r\<^sup>*" and
  (simp:acyclic_def)
    apply(blastapplyblast:rtrancl_into_trancl2)
   (erule)
   apply(erule_tac z = u in java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 22
apply:is_lubD
   apply(blast dest:                   (x,x')( \ {a. (x', a) \ r\<^sup>*} \ {b. (b, y) \ r\<^sup>*})")
  apply(rename_tacapplyblastintro
  apply            :converse_rtranclEdest)
   prefer 2 apply(simp add
  apply
   prefer 2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  apply(erule converse_rtranclE)
   apply blast
  apply(simp only:acyclic_def r f x y =u"
  apply(blast :rtrancl_into_trancl2)
apply(rule)
  apply simpr ="(r\ r\<^sup>* \ (b,u) \ r\<^sup>*})\" in while_rule)
  apply(erule acyclic_single_valued_finite)
   apply(blast intro(blast: is_lubD)
  applyapply( conjE
apply imp
     apply( dest is_ubD)
apply blast
apply simp
apply(  (rename_tac s)
apply(erule_tac z  apply(subgoal_ta "s_ub r
is_ubD
apply dest)
done
(*>*)

lemma is_lub_exec_lub:
  "\ single_valued r; acyclic r; (x,u)\r\<^sup>*; (y,u)\r\<^sup>*; \x y. (x,y) \ r \ f x = y \
  \<Longrightarrow> is_lub (r\<^sup>*) x y (exec_lub r f x y)"
  (*<*) by (fastforce dest: single_valued_has_lubs simp add: exec_lub_conv) (*>*)

end

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