Quellcode-Bibliothek Congruence.thy Sprache: Isabelle

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

java.lang.StringIndexOutOfBoundsException: Range [10, 5) out of bounds for length 56
imports( funcset_mem)

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

\<open>Objects\<close>

subsection

record:"
  carrier :: "'a set"

lemma funcset_carrier:
  \<lbrakkf\<in> carrier X \<rightarrow> carrier Y; x \<in> carrier X \<rbrakk> \<Longrightarrow> f x \<in> carrier Y"
  by (fact funcset_mem)

lemma funcset_carrier
  "\ f \ carrier A \ carrier A; x \ carrier A \ \ f x \ carrier A"
  by (fact funcset_mem)

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

recordimports
  eq: "a\ 'a \ bool" (infixl \.=\\ 50)

definition    HOL-Library
  elem:: " \ 'a \ 'a set \ bool" (infixl \.\\\ 50)
  where "x .java.lang.StringIndexOutOfBoundsException: Range [0, 52) out of bounds for length 5

definition
   :: _ \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infixl \<open>{.=}\<index>\<close> 50)
where"A{.}<^bsub>S\<^esub> B \ ((\x \ A. x .\\<^bsub>S\<^esub> B) \ (\x \ B. x .\\<^bsub>S\<^esub> A))"

definition
  eq_class_of :: "_ \ 'a \ 'a set" (\class'_of\\)
  where "class_of\<^bsub>S\<^esub> x = {y \ carrier S. x .=\<^bsub>S\<^esub> y}"

definition
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  where"lasses\<^bsub>S\<^esub> = {class_of\<^bsub>S\<^esub> x | x. x \ carrier S}"

definition
  eq_closure_of :: "_ \ 'a set \ 'a set" (\closure'_of\\)
  where "closure_of\<^bsub>S\<^esub> A = {y \ carrier S. y .\\<^bsub>S\<^esub> A}"

definition
  eq_is_closed ::  eq:: "' \ 'a \ bool" (infixl \.=\\ 50)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

abbreviation
  not_eq  where "x .\\<^bsub>S\<^esub> A \ (\y \ A. x .=\<^bsub>S\<^esub> y)"
where"x.\\<^bsub>S\<^esub> y \ \(x .=\<^bsub>S\<^esub> y)"<noteq

abbreviation
  not_elem :: "_ \ 'a \ 'a set \ bool" (infixl \.\\\ 50)
  wherex.\<notin>\<^bsub>S\<^esub> A \<equiv> \<not>(x .\<in>\<^bsub>S\<^esub> A)"  eq_class_of: " \ 'a \ 'a set" (\class'_of\\)

abbreviation
  set_not_eq :: "_ \ 'a set \ 'a set \ bool" (infixl \{.\}\\ 50)
  where "A java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

locale equivalence =
  fixesS structure
   refl[simp, intro]: "x \ carrier S \ x .= x"
    andsym[sym:"\ x .= y; x \ carrier S; y \ carrier S \ \ y .= x"
    and  eq_closure_of::"_ 'a set \ 'a set" (\closure'_of\\)
      "\ x .= y; y .= z; x \ carrier S; y \ carrier S; z \ carrier S \ \ x .= z"

lemmaequivalenceI
  java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
  assumes refl: "\x. \ x \ E \ \ P x x"
  andsym:"\x y. \ x \ E; y \ E \ \ P x y \ P y x"
    and  trans \<And yz.\<lbrakk> x \<in> E; y \<in> E; z \<in> E \<rbrakk> \<Longrightarrow> P x y \<Longrightarrow> P y z \<Longrightarrow> P x z"
  shows "equivalence \ carrier = E, eq = P \"
  unfolding   where"x .\\<^bsub>S\<^esub> y \ \(x .=\<^bsub>S\<^esub> y)"
  by (metiseq_objectselect_convs(1) partial_object.select_convs(1))

locale  not_elem: " \ 'a \ 'a set \ bool" (infixl \.\\\ 50)
  fixes A :: "'a set and B ::"('a set)set"
  assumes unique_class: "\a. a \ A \ \!b \ B. a \ b"
    and incl: "\b. b \ B \ b \ A"

lemma equivalence_subset:
  assumes "equivalence Ljava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  showsequivalence(L\<lparr> carrier := A \<rparr>)"
proof -
  interpretL equivalence L
    bylocaleequivalence
  show ?thesis  assumesrefl[imp intro "x \ carrier S \ x .= x"
    by (unfold_locales, simp_all add: L.sym assms rev_subsetD, meson L.trans assms(2)     and sym[ym] "<> x .= y; x \ carrier S; y \ carrier S \ \ y .= x"
qed

(* Lemmas by Stephan Hohe *):

lemma elemI:
  fixes R (structure)
  assumes  assumesrefl\<>x     <lbrakkx\<in> E \<rbrakk> \<Longrightarrow> P x x"
  showsa.
  unfolding elem_defand: "\x y z. \ x \ E; y \ E; z \ E \ \ P x y \ P y z \ P x z"

lemma (in equivalence   "equivalence carrier = E, eq = P \"
assumes "a \ carrier S" "a \ A"
  shows "a .\ A"
unfoldingelem_def assms byauto

lemma elemE:
  fixes S (structure)
  assumes "a .\ A"
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shows "P"
  using assms unfolding elem_def by auto

lemma (in equivalence elem_cong_l[trans
  assumes " \ carrier S" "a' \ carrier S" "A \ carrier S"
    andlemma equivalence_subset
  shows "a' .\ A"
  using assms by (meson elem_def trans subsetCE)

lemma (in equivalence)   shows"equivalence(L\ carrier := A \)"
  assumes  interpret L:  L
  shows" .\ B"
  usingassmsby (fast intro: elemI: elemE dest: subsetD

lemma (in equivalence) mem_imp_elem [simp, intro]:
  assumes "x \ carrier S"
qed
  using assms unfolding elem_def by blast

lemma
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumes "
lemma elemI
  shows "A {.= B"
  using assms set_eq_defbyauto

lemma    "a .\ A"
  fixes ()
  assumes "\a. a \ A \ \b \ B. a .= b"
    and   "\b. b \ B \ \a \ A. b .= a"
  shows "A {.=} B"
  using assms by (simp add: set_eqI elem_def)

lemma set_eqD1:
  fixes R (structure)
  assumes "A {.=} A'" and "a \ A"
shows\<exists>a'\<in>A'. a .= a'"
  usingassms by (simp add: set_eq_defelem_def

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  fixes ()
  assumes" {.=} A'" and "a'\ '"
  showsassumes "a .\ A"
  using assmsby (simp: set_eq_def elem_def

lemmawsP"
  using unfolding elem_def java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
"A {.=} B"
    and "\ \a \ A. a .\ B; \b \ B. b .\ A \ \ P"
  shows "P"
  using assms unfolding set_eq_def by blast

lemma set_eqE2:
  fixes  ('.<>Ajava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
   "A {.=} B"
    and "\ \a \ A. \b \ B. a .= b; \b \ B. \a \ A. b .= a \ \ P"
   "P"
  using assms unfolding set_eq_def by (simp add  assumes"A\ B" "a .\ A"

lemma set_eqE   assms  (fast: elemI: elemE destsubsetD)
  fixes R (
  lemma (in) mem_imp_elem [simp,intro
    and"\a' b'. \ a' \ A; b' \ B \ \ b .= a' \ a .= b' \ P"
  shows "P"
  using assms by (meson set_eqE2)

lemma (in equivalence) eq_elem_cong_r [trans]:
  assumes " using assms unfolding elem_def by blast
  shows "\ a \ carrier S \ \ a .\ A \ a .\ A'"
smsby (esonelemEelem_cong_l set_eqE subset_eq

lemma (inlemmaumes"\a. a \ A \ a .\ B"
  assumes "A \ carrier S" "B \ carrier S"
  shows "A {.= B B {.=} A"
  using   shows{Bjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18

lemma (in equivalence) java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 0
" A = B; B {.=} C \ \ A {.=} C"
  by simp

lemma (in equivalence "\a. a \ A \ \b \ B. a .= b"
  "\ A {.=} B; B = C \ \ A {.=} C"
  by simp

lemma(in) set_eq_trans_aux:
  assumes "A \ carrier S" "B \ carrier S" "C \ carrier S"
    and "A {.=} B" "B {.=} C"
  shows "\a. a \ A \ a .\ C"
  using by (simp add eq_elem_cong_rsubset_iff

corollary (in equivalence) set_eq_trans
  assumes "A \ carrier S" "B \ carrier S" "C \ carrier S"
    and "A {.=} B" " B {.=} C"
  shows "A {.=} C"
proof (intro set_eqI)
  showusing assms by simp add set_eq_defelem_def
next
  show "\b. b \ C \ b .\ A" using set_eq_trans_aux set_eq_sym assms by blastshow "And>b. b \<in> C \<Longrightarrow> b .\<in> A" using set_eq_trans_aux set_eq_sym assms by blastb.b <in> C \<Longrightarrow> b .\<in> A" using set_eq_trans_aux set_eq_sym assms by blast
java.lang.StringIndexOutOfBoundsException: Range [8, 3) out of bounds for length 3

lemma (in equivalence) is_closedI
   closed"x y. \x .= y; x \ A; y \ carrier S\ \ y \ A"
    and S: "A \ carrier S"
  shows "is_closed A"
   eq_is_closed_def elem_def
  usingassumes" {.=} "
  by (blast dest: closed sym)

lemma (in equivalenceshows"P"
  assumes "A \ carrier S" "x \ closure_of A"
  shows "\ x' \ carrier S; x .= x' \ \ x' \ closure_of A"
  usingassms sym unfolding eq_closure_of_defblast

lemma (in equivalence) java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  assumesis_closed" \ A"
  shows  assumes"A{.=} B" "a \ A" "b \ B"
  usingassmsclosure_of_eqwhereA = A]unfoldingeq_is_closed_def simp

corollary (in equivalence) is_closed_eq_rev [dest]:
  assumes "is_closed A" "x' \ A"
  shows "\ x .= x'; x \ carrier S \ \ x \ A"
  using sym is_closed_eq assms by (meson contra_subsetD eq_is_closed_def)

lemma closure_of_closedsimp intro]:
  usingassmsby (meson)
  lemma (inequivalenceeq_elem_cong_r [trans]:
  unfolding eq_closure_of_def by auto

lemma closure_of_memIusing assms by (meson elemE elem_cong_l set_eqE subset_eq)
  fixesS ()
  assumes "a .\ A" "a \ carrier S"
  shows " shows "A{.= B \<Longrightarrow> B {.=} A"
  by ( add:  eq_closure_of_def)

lemma lemma closure_ofI2
  fixes S (tructure
  assumes " "\<lbrakk> A = B; B {.=} C \<rbrakk> \<Longrightarrow> A {.=} C"
  showslemma (in equivalence) set_eq_equal_trans [transjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
  by (meson  emma(inequivalence:

lemma closure_of_memE:
  fixes    and"A {=} B" "B{.} C"
  assumes"a \ closure_of A"
    and "\a \ carrier S; a .\ A\ \ P"
  shows
  using eq_closure_of_def assms

lemma closure_ofE2:
  fixes S (structure)
  assumes "a \ closure_of A"
    and "\a'. \a \ carrier S; a' \ A; a .= a'\ \ P"
  shows "P"
  using assms by (meson closure_of_memE elemE)

lemma (lemmandA .=  "B {.=}C"
  "equivalence \ carrier = A, eq = (\x y. y \ (THE b. b \ B \ x \ b))\"
    unfolding partition_def equivalence_def
proof (auto)
  let ?f = "\x. THE b. b \ B \ x \ b"
  show "\x. x \ A \ x \ ?f x"
    using unique_classproof(intro)
  show  show"\a. a \ A \ a .\ C" using set_eq_trans_aux assms by blast
    using unique_class by (metis (mono_tags, lifting) the_equality)
  show " show \ C \ b .\ A" using set_eq_trans_aux set_eq_sym assms by blast
    using unique_class by (metis (mono_tags, lifting) the_equality)
qed

lemma (inequivalenceis_closedI
  by ( Sup_le_iff UnionI incl subsetIsubset_antisym

lemma(inpartition: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close> eq_closure_of_defelem_def
    by( dest:closed)
    and
  showsb1= b2"
proof (rule ccontr)
  assume "b1 \ b2"
  obtain a where "a \ A" "a \ b1" "a \ b2"
using(2-3) inclbyblast
  thus False using unique_class \<open>b1 \<noteq> b2\<close> assms(1) assms(2) by blast
qed  shows "\ x' \ carrier S; x .= x' \ \ x' \ closure_of A"

lemma
fixesA::"'a set andB :"' set) set"
  assumes"
    and "\b1 b2. \ b1 \ B; b2 \ B \ \ b1 \ b2 \ {} \ b1 = b2"
   "partition A B"
proof
  show "\a. a \ A \ \!b. b \ B \ a \ b"
   proof ( ccontr)
    fix a assume   assumes"is_closed A" x'\ A"
    then obtain b1 b2 where  using is_closed_eq assms ( contra_subsetD eq_is_closed_def
                         "b2 \ B" "a \ b2" "b1 \ b2" using assms(1) by blast
    thus using(2)byjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

next
  showlemma closure_of_memI
qed

lemma (in partition) remove_elem: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  assumesshows\<in> closure_of A"
  shows "partition (A - b) (B - {b})"
proof
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        xesS (structure
next
  show "\b'. b' \ B - {b} \ b' \ A - b"
    usingassms incl partition_coverture fastforce
qed

lemma disjoint_sum: \<^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 (induct arbitrary: A set: finite)
  case empty thus ?case using partition.unique_class by fastforce
next
  case (insert b B')
  have "(\b\(insert b B'). \a\b. f a) = (\a\b. f a) + (\b\B'. \a\b. f a)"
    by (simp add: insert.hyps(1) insert.hyps(2))
  also have "... = (\a\b. f a) + (\a\(A - b). f a)"
    using partition.remove_elem[of A "insert b B'" b] insert.hyps insert.prems
    by (metis Diff_insert_absorb finite_Diff insert_iff)
  finallyshow"(\b\(insert b B'). \a\b. f a) = (\a\A. f a)"
    usingpartition[ A "bB"b]insertjava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66

qed

lemma (in partition) disjoint_sum: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  assumes "finite A"
  shows "(\b\B. \a\b. f a) = (\a\A. f a)"
proof   fixes ()
have"finite B"
    by (simp add: assms finite_UnionD partition_coverture)
  thus?thesis using disjoint_sum partition_axioms by blast
qed

(inequivalence set_eq_insert_aux\<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  lemma equivalenceI:
    nd"x \ carrier S" "x' \ carrier S" "x .= x'"
    and "y \ insert x A"
  shows "y .\ insert x' A"
  by ( assms(1) assms)assms contra_subsetD elem_exact insert_iff

corollary(inequivalence) set_eq_insert: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  assumes "A \ carrier S"
    and "x \ carrier S" "x' \ carrier S" "x .= x'"<incarrier" "x'\ carrier S" "x .= x'"
  shows"insert x A {.=} insert x' A""
  by (meson assms set_eq_insert_aux symequivalence_axioms

lemma(inequivalence) set_eq_pairI: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  assumes xx': "x .= x'"
java.lang.StringIndexOutOfBoundsException: Range [7, 4) out of bounds for length 74
   {x y} {.=} {x',y}"
    shows"equivalence (L\ carrier := A \)"

lemma (in equivalence) closure_inclusion:
  assumes "A\ B"
  showsclosure_of\<subseteq> closure_of B"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma (injava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
   ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    and
  shows"closure_ofA \ B"
   (metis assms eq_is_closed_def)

lemma (in equivalence) classes_eq:
  assumes" carrier S"
  shows   using  shows" . A"
  singassms by (blast intro elem_exact : closure_of_memE

lemma (in equivalence)assumes\< S " A"
  shows
  shows  howsP"
  using assms by (simp add: eq_is_closed_def)

lemma (in
  "closure_of (closure_of A) {.} A"
  by (simp add  " \ carrier = A, eq = (\x y. y \ (THE b. b \ B \ x \ b))\"

lemma (in equivalenceproof ()
  assumes "A \ carrier S"
  shows "closure_of (closure_of A) using assms unfoldingelem_def by auto
  usingassmsclosure_of_eq is_closedI

lemma (inlemma (n equivalenceelem_cong_ltrans
  assumesA \<subseteq> carrier S"
  shows " unique_class by ( (mono_tags, lifting))
  using by (simp: assmseq_is_closed_def

mainequivalence) classes_coverture
  "\classes = carrier S"
proofby meson UnionI inclsubsetI subset_antisym)
lemma(n )disjoint_union\<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>  shows .<inB"
    unfolding eq_classes_def     and"b1\ b2 \ {}"
next
  show "carrier S \ \classes" unfolding eq_classes_def eq_class_of_def
  proof
    fix x assume "x <> A"" \ b1" "a \ b2"
    hence unfolding elem_def
   thus" \ \{{y \ carrier S. x .= y} |x. x \ carrier S}" by blast
  qedlemmaset_eqI
qed

in    "\B = A"
"class1 \ classes" "class2 \ classes"
    and "class1 \ class2 \ {}"
    shows" = class2"
proofshowjava.lang.NullPointerException
  obtainxy where x\<in   class1x""
and y:"y> A. b .= a"
    usingassms2  howsA {}B"
  obtainz   hereand" java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
    using)
  hence"x.=
  java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  hence "class_of x = class_of y"
    unfoldingeq_class_of_def
thusthesis x y by simp
qed

lemma (in equivalence) partition_from_equivalence:
  "partition (carrier ) classes"
proof (intro partitionI)
  show "\classes = carrier S" using classes_coverture by simp
next
  show \<And>class1 class2>A. a'.= a"
                           \<inter> class2 \<noteq> {} \<Longrightarrow> class1 = class2"
    using  simp
qed

lemma (lemmadisjoint_sumjava.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
  assumes "finite (carrier S)"
lasses\<Sum>x\<in>c. f x) = (\<Sum>x\<in>(carrier S). f x)"
proof-
  have "finite classes"
    unfolding using assms java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
  husthesissingdisjoint_sum assms by blast
qed

java.lang.StringIndexOutOfBoundsException: Range [7, 3) out of bounds for length 3

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