src/HOL/Equiv_Relations.thy
author hoelzl
Thu Jan 31 11:31:27 2013 +0100 (2013-01-31)
changeset 50999 3de230ed0547
parent 46752 e9e7209eb375
child 51112 da97167e03f7
permissions -rw-r--r--
introduce order topology
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(*  Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1996  University of Cambridge
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*)
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header {* Equivalence Relations in Higher-Order Set Theory *}
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theory Equiv_Relations
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imports Big_Operators Relation Plain
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begin
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subsection {* Equivalence relations -- set version *}
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definition equiv :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" where
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  "equiv A r \<longleftrightarrow> refl_on A r \<and> sym r \<and> trans r"
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lemma equivI:
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  "refl_on A r \<Longrightarrow> sym r \<Longrightarrow> trans r \<Longrightarrow> equiv A r"
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  by (simp add: equiv_def)
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lemma equivE:
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  assumes "equiv A r"
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  obtains "refl_on A r" and "sym r" and "trans r"
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  using assms by (simp add: equiv_def)
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text {*
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  Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O
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  r = r"}.
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  First half: @{text "equiv A r ==> r\<inverse> O r = r"}.
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*}
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lemma sym_trans_comp_subset:
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    "sym r ==> trans r ==> r\<inverse> O r \<subseteq> r"
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  by (unfold trans_def sym_def converse_unfold) blast
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lemma refl_on_comp_subset: "refl_on A r ==> r \<subseteq> r\<inverse> O r"
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  by (unfold refl_on_def) blast
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lemma equiv_comp_eq: "equiv A r ==> r\<inverse> O r = r"
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  apply (unfold equiv_def)
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  apply clarify
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  apply (rule equalityI)
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   apply (iprover intro: sym_trans_comp_subset refl_on_comp_subset)+
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  done
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text {* Second half. *}
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lemma comp_equivI:
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    "r\<inverse> O r = r ==> Domain r = A ==> equiv A r"
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  apply (unfold equiv_def refl_on_def sym_def trans_def)
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  apply (erule equalityE)
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  apply (subgoal_tac "\<forall>x y. (x, y) \<in> r --> (y, x) \<in> r")
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   apply fast
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  apply fast
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  done
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subsection {* Equivalence classes *}
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lemma equiv_class_subset:
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  "equiv A r ==> (a, b) \<in> r ==> r``{a} \<subseteq> r``{b}"
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  -- {* lemma for the next result *}
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  by (unfold equiv_def trans_def sym_def) blast
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theorem equiv_class_eq: "equiv A r ==> (a, b) \<in> r ==> r``{a} = r``{b}"
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  apply (assumption | rule equalityI equiv_class_subset)+
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  apply (unfold equiv_def sym_def)
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  apply blast
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  done
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lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r``{a}"
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  by (unfold equiv_def refl_on_def) blast
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lemma subset_equiv_class:
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    "equiv A r ==> r``{b} \<subseteq> r``{a} ==> b \<in> A ==> (a,b) \<in> r"
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  -- {* lemma for the next result *}
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  by (unfold equiv_def refl_on_def) blast
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lemma eq_equiv_class:
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    "r``{a} = r``{b} ==> equiv A r ==> b \<in> A ==> (a, b) \<in> r"
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  by (iprover intro: equalityD2 subset_equiv_class)
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lemma equiv_class_nondisjoint:
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    "equiv A r ==> x \<in> (r``{a} \<inter> r``{b}) ==> (a, b) \<in> r"
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  by (unfold equiv_def trans_def sym_def) blast
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lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"
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  by (unfold equiv_def refl_on_def) blast
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theorem equiv_class_eq_iff:
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  "equiv A r ==> ((x, y) \<in> r) = (r``{x} = r``{y} & x \<in> A & y \<in> A)"
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  by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
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theorem eq_equiv_class_iff:
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  "equiv A r ==> x \<in> A ==> y \<in> A ==> (r``{x} = r``{y}) = ((x, y) \<in> r)"
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  by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
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subsection {* Quotients *}
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definition quotient :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set"  (infixl "'/'/" 90) where
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  "A//r = (\<Union>x \<in> A. {r``{x}})"  -- {* set of equiv classes *}
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lemma quotientI: "x \<in> A ==> r``{x} \<in> A//r"
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  by (unfold quotient_def) blast
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lemma quotientE:
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  "X \<in> A//r ==> (!!x. X = r``{x} ==> x \<in> A ==> P) ==> P"
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  by (unfold quotient_def) blast
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lemma Union_quotient: "equiv A r ==> Union (A//r) = A"
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  by (unfold equiv_def refl_on_def quotient_def) blast
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lemma quotient_disj:
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  "equiv A r ==> X \<in> A//r ==> Y \<in> A//r ==> X = Y | (X \<inter> Y = {})"
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  apply (unfold quotient_def)
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  apply clarify
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  apply (rule equiv_class_eq)
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   apply assumption
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  apply (unfold equiv_def trans_def sym_def)
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  apply blast
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  done
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lemma quotient_eqI:
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  "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y; (x,y) \<in> r|] ==> X = Y" 
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  apply (clarify elim!: quotientE)
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  apply (rule equiv_class_eq, assumption)
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  apply (unfold equiv_def sym_def trans_def, blast)
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  done
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lemma quotient_eq_iff:
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  "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y|] ==> (X = Y) = ((x,y) \<in> r)" 
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  apply (rule iffI)  
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   prefer 2 apply (blast del: equalityI intro: quotient_eqI) 
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  apply (clarify elim!: quotientE)
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  apply (unfold equiv_def sym_def trans_def, blast)
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  done
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lemma eq_equiv_class_iff2:
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  "\<lbrakk> equiv A r; x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> ({x}//r = {y}//r) = ((x,y) : r)"
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by(simp add:quotient_def eq_equiv_class_iff)
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lemma quotient_empty [simp]: "{}//r = {}"
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by(simp add: quotient_def)
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lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"
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by(simp add: quotient_def)
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lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"
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by(simp add: quotient_def)
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lemma singleton_quotient: "{x}//r = {r `` {x}}"
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by(simp add:quotient_def)
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lemma quotient_diff1:
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  "\<lbrakk> inj_on (%a. {a}//r) A; a \<in> A \<rbrakk> \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
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apply(simp add:quotient_def inj_on_def)
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apply blast
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done
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subsection {* Defining unary operations upon equivalence classes *}
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text{*A congruence-preserving function*}
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definition congruent :: "('a \<times> 'a) set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"  where
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  "congruent r f \<longleftrightarrow> (\<forall>(y, z) \<in> r. f y = f z)"
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lemma congruentI:
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  "(\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z) \<Longrightarrow> congruent r f"
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  by (auto simp add: congruent_def)
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lemma congruentD:
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  "congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> f y = f z"
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  by (auto simp add: congruent_def)
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abbreviation
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  RESPECTS :: "('a => 'b) => ('a * 'a) set => bool"
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    (infixr "respects" 80) where
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  "f respects r == congruent r f"
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lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"
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  -- {* lemma required to prove @{text UN_equiv_class} *}
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  by auto
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lemma UN_equiv_class:
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  "equiv A r ==> f respects r ==> a \<in> A
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    ==> (\<Union>x \<in> r``{a}. f x) = f a"
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  -- {* Conversion rule *}
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  apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)
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  apply (unfold equiv_def congruent_def sym_def)
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  apply (blast del: equalityI)
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  done
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lemma UN_equiv_class_type:
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  "equiv A r ==> f respects r ==> X \<in> A//r ==>
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    (!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"
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  apply (unfold quotient_def)
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  apply clarify
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  apply (subst UN_equiv_class)
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     apply auto
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  done
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text {*
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  Sufficient conditions for injectiveness.  Could weaken premises!
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  major premise could be an inclusion; bcong could be @{text "!!y. y \<in>
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  A ==> f y \<in> B"}.
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*}
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lemma UN_equiv_class_inject:
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  "equiv A r ==> f respects r ==>
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    (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r
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    ==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)
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    ==> X = Y"
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  apply (unfold quotient_def)
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  apply clarify
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  apply (rule equiv_class_eq)
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   apply assumption
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  apply (subgoal_tac "f x = f xa")
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   apply blast
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  apply (erule box_equals)
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   apply (assumption | rule UN_equiv_class)+
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  done
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subsection {* Defining binary operations upon equivalence classes *}
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text{*A congruence-preserving function of two arguments*}
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definition congruent2 :: "('a \<times> 'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool" where
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  "congruent2 r1 r2 f \<longleftrightarrow> (\<forall>(y1, z1) \<in> r1. \<forall>(y2, z2) \<in> r2. f y1 y2 = f z1 z2)"
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lemma congruent2I':
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  assumes "\<And>y1 z1 y2 z2. (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
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  shows "congruent2 r1 r2 f"
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  using assms by (auto simp add: congruent2_def)
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lemma congruent2D:
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  "congruent2 r1 r2 f \<Longrightarrow> (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
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  using assms by (auto simp add: congruent2_def)
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text{*Abbreviation for the common case where the relations are identical*}
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abbreviation
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  RESPECTS2:: "['a => 'a => 'b, ('a * 'a) set] => bool"
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    (infixr "respects2" 80) where
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  "f respects2 r == congruent2 r r f"
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lemma congruent2_implies_congruent:
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    "equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"
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  by (unfold congruent_def congruent2_def equiv_def refl_on_def) blast
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lemma congruent2_implies_congruent_UN:
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  "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>
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    congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2``{a}. f x1 x2)"
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  apply (unfold congruent_def)
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  apply clarify
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  apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
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  apply (simp add: UN_equiv_class congruent2_implies_congruent)
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  apply (unfold congruent2_def equiv_def refl_on_def)
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  apply (blast del: equalityI)
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  done
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lemma UN_equiv_class2:
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  "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2
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    ==> (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2"
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  by (simp add: UN_equiv_class congruent2_implies_congruent
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    congruent2_implies_congruent_UN)
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lemma UN_equiv_class_type2:
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  "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f
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    ==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2
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    ==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)
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    ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
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  apply (unfold quotient_def)
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  apply clarify
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  apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
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    congruent2_implies_congruent quotientI)
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  done
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lemma UN_UN_split_split_eq:
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  "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
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    (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
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  -- {* Allows a natural expression of binary operators, *}
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  -- {* without explicit calls to @{text split} *}
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  by auto
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lemma congruent2I:
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  "equiv A1 r1 ==> equiv A2 r2
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    ==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)
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    ==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)
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    ==> congruent2 r1 r2 f"
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  -- {* Suggested by John Harrison -- the two subproofs may be *}
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  -- {* \emph{much} simpler than the direct proof. *}
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  apply (unfold congruent2_def equiv_def refl_on_def)
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  apply clarify
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  apply (blast intro: trans)
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  done
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lemma congruent2_commuteI:
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  assumes equivA: "equiv A r"
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    and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"
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    and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"
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  shows "f respects2 r"
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  apply (rule congruent2I [OF equivA equivA])
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   apply (rule commute [THEN trans])
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     apply (rule_tac [3] commute [THEN trans, symmetric])
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       apply (rule_tac [5] sym)
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       apply (rule congt | assumption |
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         erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
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  done
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subsection {* Quotients and finiteness *}
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text {*Suggested by Florian Kammüller*}
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lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
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  -- {* recall @{thm equiv_type} *}
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  apply (rule finite_subset)
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   apply (erule_tac [2] finite_Pow_iff [THEN iffD2])
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  apply (unfold quotient_def)
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  apply blast
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  done
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lemma finite_equiv_class:
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  "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
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  apply (unfold quotient_def)
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  apply (rule finite_subset)
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   prefer 2 apply assumption
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  apply blast
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  done
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lemma equiv_imp_dvd_card:
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  "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
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    ==> k dvd card A"
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  apply (rule Union_quotient [THEN subst [where P="\<lambda>A. k dvd card A"]])
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   apply assumption
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  apply (rule dvd_partition)
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     prefer 3 apply (blast dest: quotient_disj)
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    apply (simp_all add: Union_quotient equiv_type)
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  done
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lemma card_quotient_disjoint:
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 "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"
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apply(simp add:quotient_def)
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apply(subst card_UN_disjoint)
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   apply assumption
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  apply simp
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 apply(fastforce simp add:inj_on_def)
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apply simp
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done
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subsection {* Equivalence relations -- predicate version *}
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text {* Partial equivalences *}
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definition part_equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
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  "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> (\<forall>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y)"
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    -- {* John-Harrison-style characterization *}
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lemma part_equivpI:
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  "(\<exists>x. R x x) \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> part_equivp R"
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  by (auto simp add: part_equivp_def) (auto elim: sympE transpE)
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lemma part_equivpE:
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  assumes "part_equivp R"
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  obtains x where "R x x" and "symp R" and "transp R"
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proof -
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  from assms have 1: "\<exists>x. R x x"
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    and 2: "\<And>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y"
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    by (unfold part_equivp_def) blast+
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  from 1 obtain x where "R x x" ..
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  moreover have "symp R"
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  proof (rule sympI)
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    fix x y
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    assume "R x y"
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    with 2 [of x y] show "R y x" by auto
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  qed
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  moreover have "transp R"
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  proof (rule transpI)
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    fix x y z
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    assume "R x y" and "R y z"
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    with 2 [of x y] 2 [of y z] show "R x z" by auto
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  qed
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  ultimately show thesis by (rule that)
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qed
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lemma part_equivp_refl_symp_transp:
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  "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> symp R \<and> transp R"
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  by (auto intro: part_equivpI elim: part_equivpE)
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lemma part_equivp_symp:
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  "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
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  by (erule part_equivpE, erule sympE)
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lemma part_equivp_transp:
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  "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
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  by (erule part_equivpE, erule transpE)
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lemma part_equivp_typedef:
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  "part_equivp R \<Longrightarrow> \<exists>d. d \<in> {c. \<exists>x. R x x \<and> c = Collect (R x)}"
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  by (auto elim: part_equivpE)
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text {* Total equivalences *}
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definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
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  "equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" -- {* John-Harrison-style characterization *}
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lemma equivpI:
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  "reflp R \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> equivp R"
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  by (auto elim: reflpE sympE transpE simp add: equivp_def)
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lemma equivpE:
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  assumes "equivp R"
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  obtains "reflp R" and "symp R" and "transp R"
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  using assms by (auto intro!: that reflpI sympI transpI simp add: equivp_def)
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lemma equivp_implies_part_equivp:
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  "equivp R \<Longrightarrow> part_equivp R"
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  by (auto intro: part_equivpI elim: equivpE reflpE)
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lemma equivp_equiv:
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  "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"
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  by (auto intro!: equivI equivpI [to_set] elim!: equivE equivpE [to_set])
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lemma equivp_reflp_symp_transp:
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  shows "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"
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  by (auto intro: equivpI elim: equivpE)
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lemma identity_equivp:
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  "equivp (op =)"
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  by (auto intro: equivpI reflpI sympI transpI)
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lemma equivp_reflp:
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  "equivp R \<Longrightarrow> R x x"
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  by (erule equivpE, erule reflpE)
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lemma equivp_symp:
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  "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
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  by (erule equivpE, erule sympE)
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lemma equivp_transp:
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  "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
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  by (erule equivpE, erule transpE)
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end
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