Quellcode-Bibliothek Semilat.thy Sprache: Isabelle

(*  Title:      HOL/MicroJava/DFA/Semilat.thy(*  Title:      HOL/MicroJava/DFA/Semilat.thy '  ' <> ' \<Rightarrow> bool"
    Author:     Tobias Nipkow
    Copyright   2000 TUM
*)

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

section \<open>Semilattices\<close>

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

type_synonym 'a ord = "'a \<Rightarrow> 'a \<Rightarrow> bool"
type_synonym 'a binop = "'a \<Rightarrow> 'a \<Rightarrow> 'a"
type_synonym 'a sl = "'a set \<times> 'a ord \<times> 'a binop"

definition lesub :: "'a \ 'a ord \ 'a \ bool"
  where "lesub x r y \ r x y"

definition lesssub :: "'a \ 'a ord \ 'a \ bool"
  where "lesssub x r y \ lesub x r y \ x \ y"

definition plussub :: "'a \ ('a \ 'b \ 'c) \ 'b \ 'c"  where
  where "plussub x f y = f x y"

   "lesssub x ry\ lesub x r y \ x \ y"
  lesub
  "lesssub"  \open(_ /'__\close [0 00,5] 0)
  "plussub"  (\<open>(_ /+'__ _)\<close> [65, 1000, 66] 65)

notation"esub ((_ /<='__ _)\ [50, 1000, 51] 50) and
  "lesub"  (\<open>(_ /\<sqsubseteq>\<^bsub>_\<^esub> _)\<close> [50, 0, 51] 50) and
  \<open>(_ /\<sqsubset>\<^bsub>_\<^esub> _)\<close> [50, 0, 51] 50) and
  "plussub"  (

(* allow \<sub> instead of \<bsub>..\<esub> *)
abbreviation (input)
  lesub1"
  where

abbreviation (input)
  lesssub1 :: "'a \ 'a ord \ 'a \ bool" (\(_ /\\<^sub>_ _)\ [50, 1000, 51] 50)
  where " \\<^sub>r y == x \\<^bsub>r\<^esub> y"

abbreviation (input)
  java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 46
  where " where " <>\<^sub>r y == x \<sqsubset>\<^bsub>r\<^esub> y"

   "x \\<^sub>f y == x \\<^bsub>f\<^esub> y"
  ord

definition order :: "'a ord \ bool" where
    " r \ \x y. (x,y) \ r"

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

definition acc :: "'a ord \ bool" where
  "acc r \ wf {(y,x). x \\<^sub>r y}"

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

definition semilat  " r \ (\x. x \\<^sub>r x) \ (\x y. x \\<^sub>r y \ y \\<^sub>r x \ x=y) \ (\x y z. x \\<^sub>r y \ y \\<^sub>r z \ x \\<^sub>r z)"
" \<equiv> \<lambda>(A,r,f). order r \<and> closed A f \<and>   " r <{x.
                       (\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^sub>r x \<squnion>\<^sub>f y) \<and>

semilatsl

definition" \ \(A,r,f). order r \ closed A f \
  "is_ub r x y u \ (x,u)\r \ (y,u)\r"

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

definition some_lub :: "('a \ 'a) set \ 'a \ 'a \ 'a" where
  "some_lub r x y \ SOME z. is_lub r x y z"

locale Semilat =
  fixes:"asetjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
  fixes r :: "'a ord"
  fixes f :: "'java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumes : "semilat (A r,f"

lemma" r x y SOME z. is_lub r x y z"
  (*<*) by (unfold order_def) (simp (no_asm_simp)) (*>*)

lemma order_antisym: "\ order r; x \\<^sub>r y; y \\<^sub>r x \ \ x = y"
  (*<*) by (unfold order_def) (simp (no_asm_simp)) (*>*) A ::"a set"

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

lemma   semilat", f)java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
  (*<*) by (unfold order_def lesssub_def) blast (*>*)(*<*) by (unfold order_def) (simp (no_asm_simp)) (*>*)

lemma order_less_trans  (*<*) by (unfold order_def) (simp (no_asm_simp)) (*>*)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  (*<*) by (unfold order_def lesssub_def) blast (*>*)(<*)byunfoldorder_def (*>*)

lemma topD [simp, intro]: "top r T \ x \\<^sub>r T"
(

lemma top_le_conv [simp]: "\ order r; top r T \ \ (T \\<^sub>r x) = (x = T)"
  (*<*) by (blast intro: order_antisym) (*>*)

lemma semilat_Def
"semilat(A,rjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^sub>r x \<squnion>\<^sub>f y) \<and>
                 (lemma top_le_conv [simp]: "\<lbrakk> order r; top r T \<rbrakk> \<Longrightarrow> (T \<sqsubseteq>\<^sub>r x) = (x = T)"
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
  *<*) by (unfold semilat_def) clarsimp (*>*)"(A,r,f) \ order r \ closed A f \

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

lemma (in Semilat) closedI [simp(\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. x \<sqsubseteq>\<^sub>r z \<and> y \<sqsubseteq>\<^sub>r z \<longrightarrow> x \<squnion>\<^sub>f y \<sqsubseteq>\<^sub>r z)"
  (*<*) using semilat by (simp add: semilat_Def) (*>*)

  (*<*) using semilat by (simp add: semilat_Def) (*>*)
  (*<*) by (unfold closed_def) blast (*>*)

lemma closed_UNIV [simp]: "closed UNIV f"
  (*<*) by (simp add: closed_def) (*>*)

  (*<*) by (simp add: closedD [OF closedI]) (*>*)

lemma (  *<*) by (simp add: closedD [OF closedI]) (*>*)

lemma (in Semilat) antisym_r [introjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  (*<*) by (rule order_antisym) auto (*>*)

lemma   ** by rule) auto (*>*)
  (*<*) by (auto intro: order_trans) (*>*)

in) ub1simp?]: "\ x \ A; y \ A \ \ x \\<^sub>r x \\<^sub>f y"
  (*<*) using semilat by (simp add: semilat_Def) (*>*)

lemmalemma( Semilat ub2 [simp intro: "\ x \ A; y \ A \ \ y \\<^sub>r x \\<^sub>f y"
  (*<*) using semilat by (simp add: semilat_Def) (*>*)

lemma( Semilat) lub, intro?]
  \<> x\<sqsubseteq>\<^sub>r z; y \<sqsubseteq>\<^sub>r z; x \<in> A; y \<in> A; z \<in> A \<rbrakk> \<Longrightarrow> x \<squnion>\<^sub>f y \<sqsubseteq>\<^sub>r z"


lemma (in
  "\ x \ A; y \ A; z \ A \ \ (x \\<^sub>f y \\<^sub>r z) = (x \\<^sub>r z \ y \\<^sub>r z)"
  (*<*) by (blast intro: ub1 ub2 lub order_trans) (*>*)

java.lang.StringIndexOutOfBoundsException: Index 161 out of bounds for length 161

apply ( blast lub ub2
apply (blastapply (rule)
apply (erule subst
apply simp
done
(*>*)

(
(*<*) ( iffIjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
apply (rule iffI)
apply (rule subst
apply (erulesubst
apply simp
done
(*>*)

lemma (in Semilat) plus_assoc [simp]:
  assumes a:  assumesa:"a \ A" and b: "b \ A" and c: "c \ A"
  shows "a \\<^sub>f (b \\<^sub>f c) = a \\<^sub>f b \\<^sub>f c"
(*<*)
proof -
  from a b have ab: "a \\<^sub>f b \ A" ..
  from this c have abc: from c : "(a \\<^sub>f b) \\<^sub>f c \ A" ..
  from b c have bc: "b \\<^sub>f c \ A" ..
  from a this abc"a \\<^sub>f (b \\<^sub>f c) \ A" ..

  show   b chave: " \<^sub>f c \ A" ..
  proof
    show "a \\<^sub>f (b \\<^sub>f c) \\<^sub>r (a \\<^sub>f b) \\<^sub>f c"
java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
      from a b have         ":"
      also      from   "\\<^sub>r a \\<^sub>f b" ..
      finally have "a<": "a \\<^sub>r (a \\<^sub>f b) \\<^sub>f c" . ab "cc
   "
      also from ab c have "\ \\<^sub>r \ \\<^sub>f c" ..
      finally have "b<": "b \\<^sub>r (a \\<^sub>f b) \\<^sub>f c" .
      from ab c have "c< qed
      from "b<" "c<" b c abc have "(a \\<^sub>f b) \\<^sub>f c \\<^sub>r a \\<^sub>f (b \\<^sub>f c)"
<aabc
    qed
    show  "" b
    proof
      from a bc "<>
      also from a bc have "\ \\<^sub>r a \\<^sub>f \" ..finally ":c
      finally have "b<": "b \\<^sub>r a \\<^sub>f (b \\<^sub>f c)" .
      from b c have "c \\<^sub>r b \\<^sub>f c" ..
      also from bc\<dots> \<sqsubseteq>\<^sub>r a \<squnion>\<^sub>f \<dots>" ..
      java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 5
      from   b  "\java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
      from "a<" "b<" a b abc' have moreover note java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
      from this "c<" ab c abc' show ?thesis ultimately show ?thesis .
    qed
  qed ( Semilatplus_commutative
qed
(*>*)

lemma( Semilatplus_com_lemma
  "\a \ A; b \ A\ \ a \\<^sub>f b \\<^sub>r b \\<^sub>f a"
(*<*)
proof
  assumea: " A" and b: "b \ A"
  from b a have "a \\<^sub>r b \\<^sub>f a" ..
  moreover from bjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  moreover note a b
  moreoverjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  show .
qed
(*>*)

lemma lemma [iff
  "\a \ A; b \ A\ \ a \\<^sub>f b = b \\<^sub>f a"
(

lemma is_lubD:
  "is_lub r x y u \<Longrightarrow> is_ub r x y u \<and> (\<forall>z. is_ub r x y z \<longrightarrow> (u,z) \<in> r)"
  (*<*)

lemma is_ubI:
applyunfold is_ub_def


a is_ubD
  "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  (*<*) by (simp add: is_ub_def) (*>*)

lemmais_lub_bigger1]
  "is_lub (r\<^sup>*) x y y = ((x,y)\r\<^sup>*)"
(*<*)
apply (unfold is_lub_def is_ub_def)
apply blast
done
(*>*)

lemma is_lub_bigger2 [iff]:
  "is_lub (r\<^sup>*) x y x = ((y,x)\r\<^sup>*)"
(*<*)
apply (unfoldis_lub_def is_ub_def)
apply blast apply(blastelimconverse_rtranclE: single_valuedD)
java.lang.StringIndexOutOfBoundsException: Range [11, 4) out of bounds for length 4
*

lemma:
  "\ single_valued r; is_lub (r\<^sup>*) x y u; (x',x) \ r \
  \<Longrightarrow> \<exists>v. is_lub (r\<^sup>*) x' y v"
(*<*)
applyjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply (case_tac "(y,x) \ r\<^sup>*")
apply (case_tac "(y,x') \ r\<^sup>*")
  apply blast
apply (blast elim: converse_rtranclE dest:
apply (rule exI)
apply (rule conjI)
apply (blast intro: converse_rtrancl_into_rtrancl dest applyclarify
( intro
             elim: converse_rtranclE

(*>*)

lemma single_valued_has_lubs
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  (apply (unfold some_lub_def is_lub_def)
(*<*)
apply (erule converse_rtrancl_induct)
apply clarify
apply  apply assumption
  apply blast
apply blast: converse_rtrancl_into_rtrancl
apply (blast intro
done
(*>*)

java.lang.NullPointerException
  "\acyclic r; is_lub (r\<^sup>*) x y u\ \ some_lub (r\<^sup>*) x y = u"
(*<*)
applysubsection
java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
apply
blast: destacyclic_impl_antisym_rtrancl
done
(*>*)

lemmais_lub_some_lub
  "\single_valued r; acyclic r; (x,u)\r\<^sup>*; (y,u)\r\<^sup>*\
java.lang.NullPointerException
  (*<*) 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 rfTT=Tjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
by (simpinsertx)(

lemma acyclic_single_valued_finite:
"lbrakkacyclic r r xy
  \<Longrightarrow> finite (r \<inter> {a. (x, a) \<in> r\<^sup>*} \<times> {b. (b, y) \<in> r\<^sup>*})"
(*<*)
apply(erulerf
apply(java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 5
apply only
  ( intro rtrancl_trancl_trancl
applyapply conjE
apply(rename_tac x x')
apply(subgoal_tac " pply(blast dest is_ubD)
insert (
apply simp
( :converse_rtrancl_into_rtrancl
elim :single_valuedD
done
(*>*)

lemma exec_lub_conv:
  "\ acyclic r; \x y. (x,y) \ r \ f x = y; is_lub (r\<^sup>*) x y u \ \
  exec_lub  u
(*<*)
apply(unfoldblastintro rtrancl_trancl_trancl
apply(rule_tac finite_acyclic_wf
                ( <>{ab) ya \<in> r\<^sup>* \<and> (b,u) \<in> r\<^sup>*})\<inverse>" in while_rule)
    apply dest is_ubD
   applyerule)
   apply(erule_tacs
    applyblast: is_lubD
   apply(blast
applyrename_tac
c"s_ub(\<^sup>*) x y s")
   prefer 2 apply(simp add:is_ub_def)
  apply(subgoal_tac "(u, s) \ r\<^sup>*")
   prefer 2 apply(blast dest:is_lubD)
  apply(erule converse_rtranclE)
   apply blast
  apply(simp only:acyclic_def)
  apply(blast intro:s_lubD)
applyapply(blast:rtrancl_into_rtrancl
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  apply(erule acyclic_single_valued_finite)
   apply(blast intro:single_valuedI)
  apply(simp add:is_lub_def is_ub_def)
apply simp
apply(erule acyclic_subset)
apply blast
apply simp
apply(erule conjE)
apply(erule_tac z = u in converse_rtranclE)
apply(blast dest: is_lubD is_ubD)
apply(blast dest:rtrancl_into_rtrancl)
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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