Quelle Congruence.thy Sprache: Isabelle

     thanks
AuthorBallarin 0
    with thanks to Paulo Emílio de Vilhena
*)

theory Congruence
  imports
    Main
    "HOL-Library.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 funcset_mem)

lemma funcset_carrier':
    imports
  by factfuncset_mem

subsection \<open>Structure with Carrier and Equivalence Relation \<open>eq\<close>\<close>

record apartial_object
  eqsection\<open>Objects\<close>

definition
: _\<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixl \<open>.\<in>\<index>\<close> 50)funcset_carrier"> \ carrier X \ carrier Y; x \ carrier X \ \ f x \ carrier Y"
    by (factfuncset_mem ':

definition
fact

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
  eq_classes :: "_ \ ('a set) set" (\classes\\)
  where "classes\<^bsub>S\<^esub> = {class_of\<^bsub>S\<^esub> x | x. x \ carrier S}"

definition
  eq_closure_of :: "_ \ 'a set \ 'a set" (\closure'_of\\)
(funcset_memset_eq" \ 'a set \ 'a set \ bool" (infixl \{.=}\\ 50)

definition
   "A .=\
  where

abbreviation
  not_eq :: " .remove_elemof insert b ' ].prems
  where "x .\\<^bsub>S\<^esub> y \ \(x .=\<^bsub>S\<^esub> y)"

abbreviation
  not_elem :: "_ \ 'a \ 'a set \ bool" (infixl \.\\\ 50)
  where "x .\\<^bsub>S\<^esub> A \ \(x .\\<^bsub>S\<^esub> A)"

abbreviation
  set_not_eq :: "_ by (metis add.commute insert_iff partition.incl sum.subset_diff)
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

locale equivalence java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
Sstructure
     java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
      thesis assms
    and trans [trans
       ): \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>

equivalenceI
  a java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
  assumesmetis)(4 (5) elemI)
    and
      equivalencejava.lang.NullPointerException
  shows "equivalence \ carrier = E, eq = P \"
  unfolding equivalence_def\>  S"
  by (metis . A

locale partition set_eqI )
  fixes
    equivalence
    and incl: "\b. b \ B \ b \ A"

    and carr: "x \ carrier S" "x' \ carrier S" "y \ carrier S"
  assumes "shows",  ,
L
proof -
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

  show ? " A
    by (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed  structure

(* Lemmas by Stephan Hohe *)

    "
  fixes R bymetis closure_inclusion
  assumesequivalencejava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
ajava.lang.NullPointerException
ldingjava.lang.StringIndexOutOfBoundsException: Range [26, 21) out of bounds for length 40

Sjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
in"\Ajava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

  unfolding elem_def "java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11

lemma elemE

  assumes(closure_of}java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
    and  java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 0
   java.lang.NullPointerException
auto by (using complete_classesauto

()  []:
s   "java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
simp)
  shows:
  lemmajava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31

lemma
):
   "a\>Bjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
inter

lemma  java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
  java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
s     x
  using assms byblastthus  thus

:
  fixes R (structure)

    and
"A\
  using

java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 27
   x inclass1
  assumes "\a. a \ A \ \b \ B. a .= b"
java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
A=
  using assms   w z "java.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94

lemma set_eqD1
(java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
assumes}anda\<in> A"
   eq_class_of_def
   thesis

lemma b java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
RstructureSjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
' "
  using assms byclass1

:
f by
  assumes java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    and "\ \a \ A. a .\ B; \b \ B. b .\ A \ \ P"
( arbitrary
  using unfolding set_eq_def

lemma set_eqE2 eq_classes_def byauto
  fixes Rcase udisjoint_sumpartition_from_equivalence
  assumes
and
  shows " simpa: .hyps .hyps2java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
using    simp:)

lemma set_eqE':
   (   insert_iff
fs
     partition[  " ']insert.prems
  shows "P"
  using( add insert_iff.incl.)

lemma (java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
assumesA\<subseteq> carrier S" "A' \<subseteq> carrier S" "A {.=} A'"
  shows "finiteAjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
  using   finite_UnionD)

lemma  using assmsblast
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shows "A {.=} B \ B {.=} A"
  using assmsjava.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58

lemma (in

  by simp

lemma   "java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
meson   sym
  by simp

lemma (in equivalence) set_eq_trans_aux:
  assumesxxx."
    and "A {.=} B" "B {.=} C"
showsjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
   assms(simp add subset_iff

corollary (in
  assumes is_closed

  shows " Ajava.lang.NullPointerException
  set_eqI
  show
next
show
qed

lemma   assms :  : )
  assumes closed
    and\<subseteq
  shows   " java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
    unfolding
  using S
  by (blast dest: closed sym)

lemma in) closure_of_eq
  assumes "A \ carrier S" "x \ closure_of A"
  shows "java.lang.StringIndexOutOfBoundsException: Range [0, 81) out of bounds for length 47
  equivalence

Ajava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
  assumes assms is_closedI
  shows
using   AA  eq_is_closed_def

corollary (
  assumes    " closure_of)"
  shows "\ x .= x'; x \ carrier S \ \ x \ A"
  using symusing  ( add)

  ,intro
  "Union= "
  shows "closure_of A \ carrier S"
   "java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46

mma:
  fixesassume  in
ajava.lang.NullPointerException
  showsx


lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  fixes S ( " \ class2 \ {}"
  assumes-
"\inclosure_of Ajava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
   assms eq_classes_def

lemma zwhere
  fixes S
hence java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
    and "\a \ carrier S; a .\ A\ \ P"
   java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
  using eq_closure_of_def

  equivalencejava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
S)
  assumes "a \ closure_of A"
and
  shows "P"
assms( closure_of_memE)

)java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 124
  "equivalence \ carrier = A, eq = (\x y. y \ (THE b. b \ B \ x \ b))\"
    unfolding partition_def equivalence_def
(auto
  letjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  show    "(Sum\java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
    usingunique_classmetis, lifting'
   finite
  by (, lifting)
  show  using   by blast
    using unique_class by (
qed

lemma (in partition) partition_coverture: "\B = A" \<^marker>\contributor \Paulo Emílio de Vilhena\\
  by (meson Sup_le_iff UnionI unique_class incl subsetI subset_antisym)

lemma (in partition) disjoint_union: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  assumes "b1 \ B" "b2 \ B"
    and "b1 \ b2 \ {}"
  shows "b1 = b2"
proof (rule ccontr)
  assume "b1 \ b2"
  obtain a where "a \ A" "a \ b1" "a \ b2"
    using assms(2-3) incl by blast
  thus False using unique_class \<open>b1 \<noteq> b2\<close> assms(1) assms(2) by blast
qed

lemma partitionI: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  fixes A :: "'a set" and B :: "('a set) set"
  assumes "\B = A"
    and "\b1 b2. \ b1 \ B; b2 \ B \ \ b1 \ b2 \ {} \ b1 = b2"
  shows "partition A B"
proof
  show "\a. a \ A \ \!b. b \ B \ a \ b"
  proof (rule ccontr)
    fix a assume "a \ A" "\!b. b \ B \ a \ b"
    then obtain b1 b2 where "b1 \ B" "a \ b1"
                        and "b2 \ B" "a \ b2" "b1 \ b2" using assms(1) by blast
    thus False using assms(2) by blast
  qed
next
  show "\b. b \ B \ b \ A" using assms(1) by blast
qed

lemma (in partition) remove_elem: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  assumes "b \ B"
  shows "partition (A - b) (B - {b})"
proof
  show "\a. a \ A - b \ \!b'. b' \ B - {b} \ a \ b'"
    using unique_class by fastforce
next
  show "\b'. b' \ B - {b} \ b' \ A - b"
    using assms unique_class incl partition_axioms partition_coverture by 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)
  finally show "(\b\(insert b B'). \a\b. f a) = (\a\A. f a)"
    using partition.remove_elem[of A "insert b B'" b] insert.prems
    by (metis add.commute insert_iff partition.incl sum.subset_diff)
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 -
  have "finite B"
    by (simp add: assms finite_UnionD partition_coverture)
  thus ?thesis using disjoint_sum assms 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 equivalence) 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'"
  shows "insert x A {.=} insert x' A"
  by (meson set_eqI assms set_eq_insert_aux sym equivalence_axioms)

lemma (in equivalence) set_eq_pairI: \<^marker>\<open>contributor \<open>Paulo Emílio de Vilhena\<close>\<close>
  assumes xx': "x .= x'"
    and carr: "x \ carrier S" "x' \ carrier S" "y \ carrier S"
  shows "{x, y} {.=} {x', y}"
  using assms set_eq_insert by simp

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 \ B"
  shows "closure_of A \ B"
  by (metis assms closure_inclusion eq_is_closed_def)

lemma (in equivalence) classes_eq:
  assumes "A \ carrier S"
  shows "A {.=} closure_of A"
  using assms by (blast intro: set_eqI elem_exact closure_of_memI elim: closure_of_memE)

lemma (in equivalence) complete_classes:
  assumes "is_closed A"
  shows "A = closure_of A"
  using assms by (simp add: eq_is_closed_def)

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

lemma (in equivalence) closure_idem_strong:
  assumes "A \ carrier S"
  shows "closure_of (closure_of A) = closure_of A"
  using assms closure_of_eq complete_classes is_closedI by auto

lemma (in equivalence) classes_consistent:
  assumes "A \ carrier S"
  shows "is_closed (closure_of A)"
  using closure_idem_strong by (simp add: assms eq_is_closed_def)

lemma (in equivalence) classes_coverture:
  "\classes = carrier S"
proof
  show "\classes \ carrier S"
    unfolding eq_classes_def eq_class_of_def by blast
next
  show "carrier S \ \classes" unfolding eq_classes_def eq_class_of_def
  proof
    fix x assume "x \ carrier S"
    hence "x \ {y \ carrier S. x .= y}" using refl by simp
    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 \ class2 \ {}"
    shows "class1 = class2"
proof -
  obtain x y where x: "x \ carrier S" "class1 = class_of x"
               and y: "y \ carrier S" "class2 = class_of y"
    using assms(1-2) unfolding eq_classes_def by blast
  obtain z   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.sym trans x y by blast
  thus ?thesis using x y by simp
qed

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

lemma (in equivalence) disjoint_sum:
  assumes "finite (carrier S)"
  shows "(\c\classes. \x\c. f x) = (\x\(carrier S). f x)"
proof -
  have "finite classes"
    unfolding eq_classes_def using assms by auto
  thus ?thesis using disjoint_sum assms partition_from_equivalence by blast
qed

end

quality97%

olor:red'>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

quality97%

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