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(* ID: $Id$


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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 HigherOrder Set Theory *}


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theory Equiv_Relations


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imports Relation Finite_Set


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begin


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subsection {* Equivalence relations *}


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locale equiv =


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fixes A and r


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assumes refl: "refl A r"


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and sym: "sym r"


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and trans: "trans r"


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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_def) blast


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lemma refl_comp_subset: "refl A r ==> r \<subseteq> r\<inverse> O r"


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by (unfold refl_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 (rules intro: sym_trans_comp_subset refl_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_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_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_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 (rules 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_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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constdefs


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quotient :: "['a set, ('a*'a) set] => 'a set set" (infixl "'/'/" 90)


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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_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 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 congruencepreserving function*}


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locale congruent =


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fixes r and f


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assumes congruent: "(y,z) \<in> r ==> f y = f z"


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syntax


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RESPECTS ::"['a => 'b, ('a * 'a) set] => bool" (infixr "respects" 80)


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translations


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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 congruencepreserving function of two arguments*}


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locale congruent2 =


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fixes r1 and r2 and f


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assumes congruent2:


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"(y1,z1) \<in> r1 ==> (y2,z2) \<in> r2 ==> f y1 y2 = f z1 z2"


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text{*Abbreviation for the common case where the relations are identical*}


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syntax


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RESPECTS2 ::"['a => 'b, ('a * 'a) set] => bool" (infixr "respects2 " 80)


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translations


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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_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_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_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 (assumption  rule congt 


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erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+


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done


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subsection {* Cardinality results *}


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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])


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apply assumption


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apply (rule dvd_partition)


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prefer 4 apply (blast dest: quotient_disj)


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apply (simp_all add: Union_quotient equiv_type finite_quotient)


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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(fastsimp simp add:inj_on_def)


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apply (simp add:setsum_constant_nat)


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done


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15300

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ML


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{*


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val UN_UN_split_split_eq = thm "UN_UN_split_split_eq";


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val UN_constant_eq = thm "UN_constant_eq";


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val UN_equiv_class = thm "UN_equiv_class";


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val UN_equiv_class2 = thm "UN_equiv_class2";


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val UN_equiv_class_inject = thm "UN_equiv_class_inject";


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val UN_equiv_class_type = thm "UN_equiv_class_type";


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val UN_equiv_class_type2 = thm "UN_equiv_class_type2";


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val Union_quotient = thm "Union_quotient";


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val comp_equivI = thm "comp_equivI";


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val congruent2I = thm "congruent2I";


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val congruent2_commuteI = thm "congruent2_commuteI";


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val congruent2_def = thm "congruent2_def";


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val congruent2_implies_congruent = thm "congruent2_implies_congruent";


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val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN";


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val congruent_def = thm "congruent_def";


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val eq_equiv_class = thm "eq_equiv_class";


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val eq_equiv_class_iff = thm "eq_equiv_class_iff";


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val equiv_class_eq = thm "equiv_class_eq";


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val equiv_class_eq_iff = thm "equiv_class_eq_iff";


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val equiv_class_nondisjoint = thm "equiv_class_nondisjoint";


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val equiv_class_self = thm "equiv_class_self";


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val equiv_comp_eq = thm "equiv_comp_eq";


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val equiv_def = thm "equiv_def";


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val equiv_imp_dvd_card = thm "equiv_imp_dvd_card";


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val equiv_type = thm "equiv_type";


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val finite_equiv_class = thm "finite_equiv_class";


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val finite_quotient = thm "finite_quotient";


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val quotientE = thm "quotientE";


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val quotientI = thm "quotientI";


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val quotient_def = thm "quotient_def";


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val quotient_disj = thm "quotient_disj";


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val refl_comp_subset = thm "refl_comp_subset";


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val subset_equiv_class = thm "subset_equiv_class";


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val sym_trans_comp_subset = thm "sym_trans_comp_subset";


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*}


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end
