Quelle Congruence.thy Sprache: Isabelle

(*  Title:      HOL/Algebra/Congruence.thy
    Author:     Clemens Ballarin, started 3 January 2008
    with thanks to Paulo Emílio de Vilhena
*)

theorywith to*    :     Clemens Ballarin, started 3 January200java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
   :'
    Main
".FuncSet" :_
begin

section \<open>Objects\<close>

subsection \<open>Structure with Carrier Set.\<close>

record 'a partial_object =
  carrier :: "'a set"

lemma funcset_carrier:
  "\ f \ carrier X \ carrier Y; x \ carrier X \ \ f x \ carrier Y"
  by fact )

lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10


subsection  "

record 'a eq_object = java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
a

definition

where

definition
    \>\<^bsub>S\<^esub> y \<equiv> \<not>(x .=\<^bsub>S\<^esub> y)"
"

definition
: _
  where "class_of\<^bsub>S\<^esub> x = {y \ carrier S. x .=\<^bsub>S\<^esub> y}"

definition
  eq_classesequivalence  )
assumes simp

definition  []
  java.lang.NullPointerException
  where :

definition
  eq_is_closed
      :

abbreviation
:">x java.lang.StringIndexOutOfBoundsException: Index 154 out of bounds for length 154
java.lang.NullPointerException

by .select_convsselect_convs
:_
  fixesa":a)java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45

abbreviation
  set_not_eq  " L
  where :

  =
  fixes
  [, ]:java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
  s:\lbrakk
    and trans [trans]:
      "\ x .= y; y .= z; x \ carrier S; y \ carrier S; z \ carrier S \ \ x .= z"

lemma equivalenceI
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
: "And.\lbrakk>
    and    sym: " \ A"
      transjava.lang.StringIndexOutOfBoundsException: Index 154 out of bounds for length 154
showsjava.lang.NullPointerException
  unfolding equivalence_def using   a
      using java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40

locale partition =
  fixes shows
  assumes )  ]:
    and"

:
  assumesjava.lang.StringIndexOutOfBoundsException: Range [13, 9) out of bounds for length 48

proof -
interpretequivalence
   "
    by elim)
    by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

(* Lemmas by Stephan Hohe *)

:
"{}"
  assumes unfolding
showsa.
  unfolding elem_def Rstructurejava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75

lemma (java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0
  assumes   "
  shows assms )
  unfolding

lemma elemE:
  fixes   Rstructure A'A"
   .
    and by add)
""
assms by autoauto

lemma (in =java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
  assumes "
    andRstructure)
  shows "a .in A"
  using assms by   assumes"

lemma (inshows

  shows "a .\ B"
usingby intro elim: subsetD

  equivalence ]:
  assumes java.lang.StringIndexOutOfBoundsException: Index 138 out of bounds for length 138
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
assms

lemma\
  fixes R  (  elem_cong_l)
"java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
    shows=java.lang.NullPointerException
"A {=} B"
  using assms unfolding set_eq_def by auto

lemma set_eqI2:
    java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
  assumesjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
      equivalence
  showsjava.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 87
  using    assms: )

lemma set_eqD1:
  fixes R (structure)
  assumes "A {.=} A'" and "a \ A"
  shows "\a'\A'. a .= a'"
  using assms by (simp add: set_eq_def elem_def)

lemma set_eqD2:
  fixes R (structure)
  assumes "A {.=} A'" and "a' \ A'"
  shows "java.lang.StringIndexOutOfBoundsException: Range [0, 33) out of bounds for length 30
  usingby(: )

lemma
  \<And> java.lang.NullPointerException
  assumes "A {.=} B"
    and "\ \a \ A. a .\ B; \b \ B. b .\ A \ \ P"
  shows
  using assms unfolding set_eq_def by blast

lemma set_eqE2:  assumes: java.lang.StringIndexOutOfBoundsException: Index 113 out of bounds for length 113
  fixesunfolding eq_closure_of_def
   " .=}Bjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
    and "
   Pjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  using assms unfolding set_eq_def  elem_cong_lunfolding by

lemma set_eqE':
  fixes R    " A" x
  =a
    and "\a' b'. \ a' \ A; b' \ B \ \ b .= a' \ a .= b' \ P"   [     byjava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
    [,
  by set_eqE2

  ) eq_elem_cong_r
  assumes

  assmselemEset_eqE

lemma structure
  assumes "A \ carrier S" "B \ carrier S"
  =java.lang.NullPointerException
  using assms unfolding set_eq_defsimp assms)

lemma ()

  by simp

equivalence]:
  "\ A {.=} B; B = C \ \ A {.=} C"
  by simp

( ) set_eq_trans_aux

.  ="
  shows a
  using assms by "P"

corollary

"{}B" { java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
proofjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  set_eqI
java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
next
">java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
qed

) :
meson unique_class )
      ) disjoint_union
  shows "is_closed A"
  unfolding eq_is_closed_def elem_def
  using S
blast  sym

lemma " java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
  assumes      assms  java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
shows
  using

    :" : ('"
">=
  showsjava.lang.NullPointerException
  usingshows

corollary  rule
"
  shows "\ x .= x'; x \ carrier S \ \ x \ A"
sym bymeson)

lemma closure_of_closed [and
  fixes S ( False assms  blast
  shows "closure_of A \ carrier S"
  unfolding eq_closure_of_def by

:
  fixesjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  assumes "a .\ A" "a \ carrier S"
   "a \ closure_of A"
  by (simp addpartition "

lemma closure_ofI2:
  )

  shows "a assms unique_class partition_axioms by fastforce
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma closure_of_memE:
  fixesS()
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
lemmashowsjava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 36
  shows A\<carrier
shows.

lemma lding elem_def using   assmsblast: set_eqI closure_of_memIelim)
  fixesS (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   "a <>carrier "
    and " "a .\ A"
s java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  using

lemma (inclosure_of).= losure_of
  equivalence
    unfolding partition_defequivalence_def
proofauto
  let ?f = "\x. THE b. b \ B \ x \ b"
  show"x. x \ A \ x \ ?f x"
    using unique_classby (metis(mono_tags  closure_of_eqis_closedI by java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
  how"\x y. \ x \ A; y \ A \ \ y \ ?f x \ x \ ?f y" "
usingbymetis,)the_equality
  show closure_idem_strongsimp add eq_is_closed_def
    using
lem ( equivalenceclasses_coverture

lemmajava.lang.NullPointerException
  by( Sup_le_iff unique_class subsetIsubset_antisym

(npartition: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close> "
  assumes "b1 \ B" "b2 \ B"
    and <inter> b2 \<noteq> {}"
  shows"b1= java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
proof (rule ccontr)
  assume "b1 \ b2"
\in "a\ b1" "a \ b2"
    using  hows"x \ {y \ carrier S. x .= y}" using refl by simp
   False  x
qed

java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  fixes  assumes
  assumes
    and " showsjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 55
   class1
proof
   "a. a \ A \ \!b. b \ B \ a \ b"
  proof (rule y  x:"x>carrierS"" = class_of xjava.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
               y "y \ carrier S" "class2 = class_of y"
    using (1-)s ". java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
                        and2\<in> B" "a \<in> b2" "b1 \<noteq> b2" using assms(1) by blast
    thus False using assms
  qed
(structure
  show " z \ y .= z" using x y unfolding eq_class_of_def by blast
qed  assumes "A {.= A'" and "

  ? using
umes"bjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  shows   fixes (java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 33
proof
  assumes
    usingshow"<And>class1 class2. \<lbrakk> class1 \<in> classes; class2 \<in> classes \<rbrakk> \<Longrightarrow> ina=a"
nextlemmaset_eqE
    ixesdisjoint_union
    usingassms
qed

: <^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  "\ finite B; finite A; partition A B\ \ (\b\B. \a\b. f a) = (\a\A. f a)"
proof(nduct. \<Sum>x\<in>c. f x) = (\<Sum>x\<in>(carrier S). f x)"using assms java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
   t ?  disjoint_sum partition_from_equivalence
next
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 3
    by( ddinsert(1)insert(2))
  also have "... = (\a\b. f a) + (\a\(A - b). f a)"
    using partition.remove_elem[of A   using assms unfoldingset_eq_def by( add elem_def
   by metisDiff_insert_absorbfinite_Diff)
  finally how "(\b\(insert b B'). \a\b. f a) = (\a\A. f a)"
    usingpartition.remove_elemof A "insert bB" b insert
    by metis.commute partition sumsubset_diff
qed

lemma (in partition) disjoint_sum:  assumes " \ carrier S" "A' \ carrier S" "A {.=} A'"
  assumes "
  shows "(\b\B. \a\b. f a) = (\a\A. f a)"
proof -
  have "finite B"
    by (simp add: assmsfinite_UnionD partition_coverture)
  thus?thesis disjoint_sum partition_axioms by blast
qed

lemma (in equivalence) set_eq_insert_aux: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  assumes "A \ carrier S"
    and "x \ carrier S" "x' \ carrier S" "x .= x'"
    and "y \ insert x A"
  shows "y .\ insert x' A"
  by (metis assms(1) assms(4) assms(5) contra_subsetD elemI elem_exact insert_iff

corollary (in"\<lbrakk> A = B; B {.=} C \<rbrakk> \<Longrightarrow> A {.=} C"
   assumes" \ carrier S"
    and "x \ carrier S" "x' \ carrier S" "x .= x'"
  shows "insert x A {.=} insert x' A"
  by (meson set_eqIassmsset_eq_insert_aux equivalence_axioms)

lemma (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumes ': " .= x'java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
    and  shows "\a. a \ A \ a .\ C"
  shows "{x, y} {.=} {x', y}"
  using assms set_eq_insertusingassms by add: eq_elem_cong_rsubset_iff)

lemma (in equivalence) closure_inclusion:
  assumes "A \ B"
  shows "closure_of A \ closure_of B"
  unfolding eq_closure_of_def using assms elem_subsetD by auto

lemma (in equivalence) classes_small:
  assumes"is_closed B"
    and"A\<subseteq> B"
  shows closure_of <subseteq> B"
  by (metis assmsproof(intro)

lemma
  assumes   "\b. b \ C \ b .\ A" using set_eq_trans_aux set_eq_sym assms by blast
  shows
usingassms by (blast intro set_eqI elem_exactclosure_of_memIelimclosure_of_memE

     S: "A carrier S"
assumes"is_closedA"
  shows "A = closure_of A"
  using assms by (simp add: eq_is_closed_def)

lemma ( equivalenceclosure_of_eq:
  "closure_of (closure_of A) {.=} closure_of A"
  by (simp add: classes_eq set_eq_sym)

lemma(inequivalence) closure_idem_strong:
  assumes "A \ carrier S"
  shows "closure_of (closure_of A) = closure_of A"
  using closure_of_eq complete_classes by auto

  using assmsclosure_of_eq[where = ]unfolding eq_is_closed_def by simp
   assumes"A \ carrier S"
  shows ""is_closed( A)java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
  using closure_idem_strong by (imp add: assms eq_is_closed_def

lemmalemmaclosure_of_closed[simp intro]:
  "classes carrierSjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
proof
  how "\classes \ carrier S"
    unfolding eq_classes_def eq_class_of_def by blast
next
  mma closure_of_memI
  proof
    fix x assume "x\<> carrier S"
    hence   assumes"a .\ A" "a \ carrier S"
    thus "x \ \{{y \ carrier S. x .= y} |x. x \ carrier S}" by blast
  qed
qed

lemma (in equivalence) disjoint_union:
  assumes "class1 \ classes" "class2 \ classes"
    and class1
    shows "class1 = class2"
proof -
  obtain x y   showsa <> closure_of"
               and y: "y \ carrier S" "class2 = class_of y"
    usingassms(1-2) unfolding eq_classes_def by blast
  obtain    where z: "z \ carrier S" "z \ class1 \ class2"
    using assms classes_coverture by fastforce
  hence "x .=z \ y .= z" using x y unfolding eq_class_of_def by blast
  hence "x .= y" using x y z trans sym by meson
  hence "class_of x = class_of y"
    unfolding eq_class_of_def using local  shows"P"
  thus ?thesis using x y by simp
qed

lemma(inequivalence) partition_from_equivalence:
  "partition (carrier S) classes"
proof (intro partitionI)  fixes (structure
  show " "\<And>a'. \<lbrakk>a \<in> carrier S; a' \<in> A; a .= a'\<rbrakk> \<Longrightarrow> P"
next  using by (eson elemE
  show "\class1 class2. \ class1 \ classes; class2 \ classes \ \lemma(in partition) equivalence_from_partition: \<^marker>\contributor \Paulo Emílio de Vilhena\\
                          class1 \<inter> class2 \<noteq> {} \<Longrightarrow> class1 = class2"
     roof)
qed

lemma (in equivalence) disjoint_sum:
  assumes "finite (carrier S)"
shows\<>c<in>classes. \<Sum>x\<in>c. f x) = (\<Sum>x\<in>(carrier S). f x)"
proof    using  by ( (mono_tags) theI'
have "finite classes"
    unfolding    usingunique_classby (metismono_tagslifting) the_equality)
?thesis disjoint_sumassmspartition_from_equivalencejava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
qed

end

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