--- a/src/HOL/Equiv_Relations.thy Wed Aug 10 22:03:58 2016 +0200
+++ b/src/HOL/Equiv_Relations.thy Wed Aug 10 22:05:00 2016 +0200
@@ -1,20 +1,19 @@
-(* Authors: Lawrence C Paulson, Cambridge University Computer Laboratory
- Copyright 1996 University of Cambridge
+(* Title: HOL/Equiv_Relations.thy
+ Author: Lawrence C Paulson, 1996 Cambridge University Computer Laboratory
*)
section \<open>Equivalence Relations in Higher-Order Set Theory\<close>
theory Equiv_Relations
-imports Groups_Big Relation
+ imports Groups_Big Relation
begin
subsection \<open>Equivalence relations -- set version\<close>
-definition equiv :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" where
- "equiv A r \<longleftrightarrow> refl_on A r \<and> sym r \<and> trans r"
+definition equiv :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool"
+ where "equiv A r \<longleftrightarrow> refl_on A r \<and> sym r \<and> trans r"
-lemma equivI:
- "refl_on A r \<Longrightarrow> sym r \<Longrightarrow> trans r \<Longrightarrow> equiv A r"
+lemma equivI: "refl_on A r \<Longrightarrow> sym r \<Longrightarrow> trans r \<Longrightarrow> equiv A r"
by (simp add: equiv_def)
lemma equivE:
@@ -23,20 +22,18 @@
using assms by (simp add: equiv_def)
text \<open>
- Suppes, Theorem 70: \<open>r\<close> is an equiv relation iff \<open>r\<inverse> O
- r = r\<close>.
+ Suppes, Theorem 70: \<open>r\<close> is an equiv relation iff \<open>r\<inverse> O r = r\<close>.
- First half: \<open>equiv A r ==> r\<inverse> O r = r\<close>.
+ First half: \<open>equiv A r \<Longrightarrow> r\<inverse> O r = r\<close>.
\<close>
-lemma sym_trans_comp_subset:
- "sym r ==> trans r ==> r\<inverse> O r \<subseteq> r"
- by (unfold trans_def sym_def converse_unfold) blast
+lemma sym_trans_comp_subset: "sym r \<Longrightarrow> trans r \<Longrightarrow> r\<inverse> O r \<subseteq> r"
+ unfolding trans_def sym_def converse_unfold by blast
-lemma refl_on_comp_subset: "refl_on A r ==> r \<subseteq> r\<inverse> O r"
- by (unfold refl_on_def) blast
+lemma refl_on_comp_subset: "refl_on A r \<Longrightarrow> r \<subseteq> r\<inverse> O r"
+ unfolding refl_on_def by blast
-lemma equiv_comp_eq: "equiv A r ==> r\<inverse> O r = r"
+lemma equiv_comp_eq: "equiv A r \<Longrightarrow> r\<inverse> O r = r"
apply (unfold equiv_def)
apply clarify
apply (rule equalityI)
@@ -45,11 +42,10 @@
text \<open>Second half.\<close>
-lemma comp_equivI:
- "r\<inverse> O r = r ==> Domain r = A ==> equiv A r"
+lemma comp_equivI: "r\<inverse> O r = r \<Longrightarrow> Domain r = A \<Longrightarrow> equiv A r"
apply (unfold equiv_def refl_on_def sym_def trans_def)
apply (erule equalityE)
- apply (subgoal_tac "\<forall>x y. (x, y) \<in> r --> (y, x) \<in> r")
+ apply (subgoal_tac "\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r")
apply fast
apply fast
done
@@ -57,62 +53,54 @@
subsection \<open>Equivalence classes\<close>
-lemma equiv_class_subset:
- "equiv A r ==> (a, b) \<in> r ==> r``{a} \<subseteq> r``{b}"
+lemma equiv_class_subset: "equiv A r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> r``{a} \<subseteq> r``{b}"
\<comment> \<open>lemma for the next result\<close>
- by (unfold equiv_def trans_def sym_def) blast
+ unfolding equiv_def trans_def sym_def by blast
-theorem equiv_class_eq: "equiv A r ==> (a, b) \<in> r ==> r``{a} = r``{b}"
+theorem equiv_class_eq: "equiv A r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> r``{a} = r``{b}"
apply (assumption | rule equalityI equiv_class_subset)+
apply (unfold equiv_def sym_def)
apply blast
done
-lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r``{a}"
- by (unfold equiv_def refl_on_def) blast
+lemma equiv_class_self: "equiv A r \<Longrightarrow> a \<in> A \<Longrightarrow> a \<in> r``{a}"
+ unfolding equiv_def refl_on_def by blast
-lemma subset_equiv_class:
- "equiv A r ==> r``{b} \<subseteq> r``{a} ==> b \<in> A ==> (a,b) \<in> r"
+lemma subset_equiv_class: "equiv A r \<Longrightarrow> r``{b} \<subseteq> r``{a} \<Longrightarrow> b \<in> A \<Longrightarrow> (a, b) \<in> r"
\<comment> \<open>lemma for the next result\<close>
- by (unfold equiv_def refl_on_def) blast
+ unfolding equiv_def refl_on_def by blast
-lemma eq_equiv_class:
- "r``{a} = r``{b} ==> equiv A r ==> b \<in> A ==> (a, b) \<in> r"
+lemma eq_equiv_class: "r``{a} = r``{b} \<Longrightarrow> equiv A r \<Longrightarrow> b \<in> A \<Longrightarrow> (a, b) \<in> r"
by (iprover intro: equalityD2 subset_equiv_class)
-lemma equiv_class_nondisjoint:
- "equiv A r ==> x \<in> (r``{a} \<inter> r``{b}) ==> (a, b) \<in> r"
- by (unfold equiv_def trans_def sym_def) blast
+lemma equiv_class_nondisjoint: "equiv A r \<Longrightarrow> x \<in> (r``{a} \<inter> r``{b}) \<Longrightarrow> (a, b) \<in> r"
+ unfolding equiv_def trans_def sym_def by blast
-lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"
- by (unfold equiv_def refl_on_def) blast
+lemma equiv_type: "equiv A r \<Longrightarrow> r \<subseteq> A \<times> A"
+ unfolding equiv_def refl_on_def by blast
-theorem equiv_class_eq_iff:
- "equiv A r ==> ((x, y) \<in> r) = (r``{x} = r``{y} & x \<in> A & y \<in> A)"
+lemma equiv_class_eq_iff: "equiv A r \<Longrightarrow> (x, y) \<in> r \<longleftrightarrow> r``{x} = r``{y} \<and> x \<in> A \<and> y \<in> A"
by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
-theorem eq_equiv_class_iff:
- "equiv A r ==> x \<in> A ==> y \<in> A ==> (r``{x} = r``{y}) = ((x, y) \<in> r)"
+lemma eq_equiv_class_iff: "equiv A r \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> r``{x} = r``{y} \<longleftrightarrow> (x, y) \<in> r"
by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
subsection \<open>Quotients\<close>
-definition quotient :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set" (infixl "'/'/" 90) where
- "A//r = (\<Union>x \<in> A. {r``{x}})" \<comment> \<open>set of equiv classes\<close>
+definition quotient :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set" (infixl "'/'/" 90)
+ where "A//r = (\<Union>x \<in> A. {r``{x}})" \<comment> \<open>set of equiv classes\<close>
lemma quotientI: "x \<in> A ==> r``{x} \<in> A//r"
- by (unfold quotient_def) blast
+ unfolding quotient_def by blast
-lemma quotientE:
- "X \<in> A//r ==> (!!x. X = r``{x} ==> x \<in> A ==> P) ==> P"
- by (unfold quotient_def) blast
+lemma quotientE: "X \<in> A//r \<Longrightarrow> (\<And>x. X = r``{x} \<Longrightarrow> x \<in> A \<Longrightarrow> P) \<Longrightarrow> P"
+ unfolding quotient_def by blast
-lemma Union_quotient: "equiv A r ==> \<Union>(A//r) = A"
- by (unfold equiv_def refl_on_def quotient_def) blast
+lemma Union_quotient: "equiv A r \<Longrightarrow> \<Union>(A//r) = A"
+ unfolding equiv_def refl_on_def quotient_def by blast
-lemma quotient_disj:
- "equiv A r ==> X \<in> A//r ==> Y \<in> A//r ==> X = Y | (X \<inter> Y = {})"
+lemma quotient_disj: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> Y \<in> A//r \<Longrightarrow> X = Y \<or> X \<inter> Y = {}"
apply (unfold quotient_def)
apply clarify
apply (rule equiv_class_eq)
@@ -122,108 +110,96 @@
done
lemma quotient_eqI:
- "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y; (x,y) \<in> r|] ==> X = Y"
+ "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> Y \<in> A//r \<Longrightarrow> x \<in> X \<Longrightarrow> y \<in> Y \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> X = Y"
apply (clarify elim!: quotientE)
- apply (rule equiv_class_eq, assumption)
- apply (unfold equiv_def sym_def trans_def, blast)
+ apply (rule equiv_class_eq)
+ apply assumption
+ apply (unfold equiv_def sym_def trans_def)
+ apply blast
done
lemma quotient_eq_iff:
- "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y|] ==> (X = Y) = ((x,y) \<in> r)"
- apply (rule iffI)
- prefer 2 apply (blast del: equalityI intro: quotient_eqI)
+ "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> Y \<in> A//r \<Longrightarrow> x \<in> X \<Longrightarrow> y \<in> Y \<Longrightarrow> X = Y \<longleftrightarrow> (x, y) \<in> r"
+ apply (rule iffI)
+ prefer 2
+ apply (blast del: equalityI intro: quotient_eqI)
apply (clarify elim!: quotientE)
- apply (unfold equiv_def sym_def trans_def, blast)
+ apply (unfold equiv_def sym_def trans_def)
+ apply blast
done
-lemma eq_equiv_class_iff2:
- "\<lbrakk> equiv A r; x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> ({x}//r = {y}//r) = ((x,y) : r)"
-by(simp add:quotient_def eq_equiv_class_iff)
-
+lemma eq_equiv_class_iff2: "equiv A r \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> {x}//r = {y}//r \<longleftrightarrow> (x, y) \<in> r"
+ by (simp add: quotient_def eq_equiv_class_iff)
lemma quotient_empty [simp]: "{}//r = {}"
-by(simp add: quotient_def)
+ by (simp add: quotient_def)
-lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"
-by(simp add: quotient_def)
+lemma quotient_is_empty [iff]: "A//r = {} \<longleftrightarrow> A = {}"
+ by (simp add: quotient_def)
-lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"
-by(simp add: quotient_def)
-
+lemma quotient_is_empty2 [iff]: "{} = A//r \<longleftrightarrow> A = {}"
+ by (simp add: quotient_def)
lemma singleton_quotient: "{x}//r = {r `` {x}}"
-by(simp add:quotient_def)
+ by (simp add: quotient_def)
-lemma quotient_diff1:
- "\<lbrakk> inj_on (%a. {a}//r) A; a \<in> A \<rbrakk> \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
-apply(simp add:quotient_def inj_on_def)
-apply blast
-done
+lemma quotient_diff1: "inj_on (\<lambda>a. {a}//r) A \<Longrightarrow> a \<in> A \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
+ unfolding quotient_def inj_on_def by blast
+
subsection \<open>Refinement of one equivalence relation WRT another\<close>
-lemma refines_equiv_class_eq:
- "\<lbrakk>R \<subseteq> S; equiv A R; equiv A S\<rbrakk> \<Longrightarrow> R``(S``{a}) = S``{a}"
+lemma refines_equiv_class_eq: "R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> R``(S``{a}) = S``{a}"
by (auto simp: equiv_class_eq_iff)
-lemma refines_equiv_class_eq2:
- "\<lbrakk>R \<subseteq> S; equiv A R; equiv A S\<rbrakk> \<Longrightarrow> S``(R``{a}) = S``{a}"
+lemma refines_equiv_class_eq2: "R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> S``(R``{a}) = S``{a}"
by (auto simp: equiv_class_eq_iff)
-lemma refines_equiv_image_eq:
- "\<lbrakk>R \<subseteq> S; equiv A R; equiv A S\<rbrakk> \<Longrightarrow> (\<lambda>X. S``X) ` (A//R) = A//S"
+lemma refines_equiv_image_eq: "R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> (\<lambda>X. S``X) ` (A//R) = A//S"
by (auto simp: quotient_def image_UN refines_equiv_class_eq2)
lemma finite_refines_finite:
- "\<lbrakk>finite (A//R); R \<subseteq> S; equiv A R; equiv A S\<rbrakk> \<Longrightarrow> finite (A//S)"
- apply (erule finite_surj [where f = "\<lambda>X. S``X"])
- apply (simp add: refines_equiv_image_eq)
- done
+ "finite (A//R) \<Longrightarrow> R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> finite (A//S)"
+ by (erule finite_surj [where f = "\<lambda>X. S``X"]) (simp add: refines_equiv_image_eq)
lemma finite_refines_card_le:
- "\<lbrakk>finite (A//R); R \<subseteq> S; equiv A R; equiv A S\<rbrakk> \<Longrightarrow> card (A//S) \<le> card (A//R)"
- apply (subst refines_equiv_image_eq [of R S A, symmetric])
- apply (auto simp: card_image_le [where f = "\<lambda>X. S``X"])
- done
+ "finite (A//R) \<Longrightarrow> R \<subseteq> S \<Longrightarrow> equiv A R \<Longrightarrow> equiv A S \<Longrightarrow> card (A//S) \<le> card (A//R)"
+ by (subst refines_equiv_image_eq [of R S A, symmetric])
+ (auto simp: card_image_le [where f = "\<lambda>X. S``X"])
subsection \<open>Defining unary operations upon equivalence classes\<close>
-text\<open>A congruence-preserving function\<close>
+text \<open>A congruence-preserving function.\<close>
-definition congruent :: "('a \<times> 'a) set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" where
- "congruent r f \<longleftrightarrow> (\<forall>(y, z) \<in> r. f y = f z)"
+definition congruent :: "('a \<times> 'a) set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
+ where "congruent r f \<longleftrightarrow> (\<forall>(y, z) \<in> r. f y = f z)"
-lemma congruentI:
- "(\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z) \<Longrightarrow> congruent r f"
+lemma congruentI: "(\<And>y z. (y, z) \<in> r \<Longrightarrow> f y = f z) \<Longrightarrow> congruent r f"
by (auto simp add: congruent_def)
-lemma congruentD:
- "congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> f y = f z"
+lemma congruentD: "congruent r f \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> f y = f z"
by (auto simp add: congruent_def)
-abbreviation
- RESPECTS :: "('a => 'b) => ('a * 'a) set => bool"
- (infixr "respects" 80) where
- "f respects r == congruent r f"
+abbreviation RESPECTS :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" (infixr "respects" 80)
+ where "f respects r \<equiv> congruent r f"
-lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"
+lemma UN_constant_eq: "a \<in> A \<Longrightarrow> \<forall>y \<in> A. f y = c \<Longrightarrow> (\<Union>y \<in> A. f y) = c"
\<comment> \<open>lemma required to prove \<open>UN_equiv_class\<close>\<close>
by auto
-lemma UN_equiv_class:
- "equiv A r ==> f respects r ==> a \<in> A
- ==> (\<Union>x \<in> r``{a}. f x) = f a"
+lemma UN_equiv_class: "equiv A r \<Longrightarrow> f respects r \<Longrightarrow> a \<in> A \<Longrightarrow> (\<Union>x \<in> r``{a}. f x) = f a"
\<comment> \<open>Conversion rule\<close>
- apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)
+ apply (rule equiv_class_self [THEN UN_constant_eq])
+ apply assumption
+ apply assumption
apply (unfold equiv_def congruent_def sym_def)
apply (blast del: equalityI)
done
lemma UN_equiv_class_type:
- "equiv A r ==> f respects r ==> X \<in> A//r ==>
- (!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"
+ "equiv A r \<Longrightarrow> f respects r \<Longrightarrow> X \<in> A//r \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<in> B) \<Longrightarrow> (\<Union>x \<in> X. f x) \<in> B"
apply (unfold quotient_def)
apply clarify
apply (subst UN_equiv_class)
@@ -232,15 +208,15 @@
text \<open>
Sufficient conditions for injectiveness. Could weaken premises!
- major premise could be an inclusion; bcong could be \<open>!!y. y \<in>
- A ==> f y \<in> B\<close>.
+ major premise could be an inclusion; \<open>bcong\<close> could be
+ \<open>\<And>y. y \<in> A \<Longrightarrow> f y \<in> B\<close>.
\<close>
lemma UN_equiv_class_inject:
- "equiv A r ==> f respects r ==>
- (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r
- ==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)
- ==> X = Y"
+ "equiv A r \<Longrightarrow> f respects r \<Longrightarrow>
+ (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) \<Longrightarrow> X \<in> A//r ==> Y \<in> A//r
+ \<Longrightarrow> (\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<Longrightarrow> (x, y) \<in> r)
+ \<Longrightarrow> X = Y"
apply (unfold quotient_def)
apply clarify
apply (rule equiv_class_eq)
@@ -254,33 +230,30 @@
subsection \<open>Defining binary operations upon equivalence classes\<close>
-text\<open>A congruence-preserving function of two arguments\<close>
+text \<open>A congruence-preserving function of two arguments.\<close>
-definition congruent2 :: "('a \<times> 'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool" where
- "congruent2 r1 r2 f \<longleftrightarrow> (\<forall>(y1, z1) \<in> r1. \<forall>(y2, z2) \<in> r2. f y1 y2 = f z1 z2)"
+definition congruent2 :: "('a \<times> 'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool"
+ where "congruent2 r1 r2 f \<longleftrightarrow> (\<forall>(y1, z1) \<in> r1. \<forall>(y2, z2) \<in> r2. f y1 y2 = f z1 z2)"
lemma congruent2I':
assumes "\<And>y1 z1 y2 z2. (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
shows "congruent2 r1 r2 f"
using assms by (auto simp add: congruent2_def)
-lemma congruent2D:
- "congruent2 r1 r2 f \<Longrightarrow> (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
+lemma congruent2D: "congruent2 r1 r2 f \<Longrightarrow> (y1, z1) \<in> r1 \<Longrightarrow> (y2, z2) \<in> r2 \<Longrightarrow> f y1 y2 = f z1 z2"
by (auto simp add: congruent2_def)
-text\<open>Abbreviation for the common case where the relations are identical\<close>
-abbreviation
- RESPECTS2:: "['a => 'a => 'b, ('a * 'a) set] => bool"
- (infixr "respects2" 80) where
- "f respects2 r == congruent2 r r f"
+text \<open>Abbreviation for the common case where the relations are identical.\<close>
+abbreviation RESPECTS2:: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> bool" (infixr "respects2" 80)
+ where "f respects2 r \<equiv> congruent2 r r f"
lemma congruent2_implies_congruent:
- "equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"
- by (unfold congruent_def congruent2_def equiv_def refl_on_def) blast
+ "equiv A r1 \<Longrightarrow> congruent2 r1 r2 f \<Longrightarrow> a \<in> A \<Longrightarrow> congruent r2 (f a)"
+ unfolding congruent_def congruent2_def equiv_def refl_on_def by blast
lemma congruent2_implies_congruent_UN:
- "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>
+ "equiv A1 r1 \<Longrightarrow> equiv A2 r2 \<Longrightarrow> congruent2 r1 r2 f \<Longrightarrow> a \<in> A2 \<Longrightarrow>
congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2``{a}. f x1 x2)"
apply (unfold congruent_def)
apply clarify
@@ -291,20 +264,19 @@
done
lemma UN_equiv_class2:
- "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2
- ==> (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2"
- by (simp add: UN_equiv_class congruent2_implies_congruent
- congruent2_implies_congruent_UN)
+ "equiv A1 r1 \<Longrightarrow> equiv A2 r2 \<Longrightarrow> congruent2 r1 r2 f \<Longrightarrow> a1 \<in> A1 \<Longrightarrow> a2 \<in> A2 \<Longrightarrow>
+ (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. f x1 x2) = f a1 a2"
+ by (simp add: UN_equiv_class congruent2_implies_congruent congruent2_implies_congruent_UN)
lemma UN_equiv_class_type2:
- "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f
- ==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2
- ==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)
- ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
+ "equiv A1 r1 \<Longrightarrow> equiv A2 r2 \<Longrightarrow> congruent2 r1 r2 f
+ \<Longrightarrow> X1 \<in> A1//r1 \<Longrightarrow> X2 \<in> A2//r2
+ \<Longrightarrow> (\<And>x1 x2. x1 \<in> A1 \<Longrightarrow> x2 \<in> A2 \<Longrightarrow> f x1 x2 \<in> B)
+ \<Longrightarrow> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
apply (unfold quotient_def)
apply clarify
apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
- congruent2_implies_congruent quotientI)
+ congruent2_implies_congruent quotientI)
done
lemma UN_UN_split_split_eq:
@@ -315,12 +287,12 @@
by auto
lemma congruent2I:
- "equiv A1 r1 ==> equiv A2 r2
- ==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)
- ==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)
- ==> congruent2 r1 r2 f"
+ "equiv A1 r1 \<Longrightarrow> equiv A2 r2
+ \<Longrightarrow> (\<And>y z w. w \<in> A2 \<Longrightarrow> (y,z) \<in> r1 \<Longrightarrow> f y w = f z w)
+ \<Longrightarrow> (\<And>y z w. w \<in> A1 \<Longrightarrow> (y,z) \<in> r2 \<Longrightarrow> f w y = f w z)
+ \<Longrightarrow> congruent2 r1 r2 f"
\<comment> \<open>Suggested by John Harrison -- the two subproofs may be\<close>
- \<comment> \<open>\emph{much} simpler than the direct proof.\<close>
+ \<comment> \<open>\<^emph>\<open>much\<close> simpler than the direct proof.\<close>
apply (unfold congruent2_def equiv_def refl_on_def)
apply clarify
apply (blast intro: trans)
@@ -328,8 +300,8 @@
lemma congruent2_commuteI:
assumes equivA: "equiv A r"
- and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"
- and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"
+ and commute: "\<And>y z. y \<in> A \<Longrightarrow> z \<in> A \<Longrightarrow> f y z = f z y"
+ and congt: "\<And>y z w. w \<in> A \<Longrightarrow> (y,z) \<in> r \<Longrightarrow> f w y = f w z"
shows "f respects2 r"
apply (rule congruent2I [OF equivA equivA])
apply (rule commute [THEN trans])
@@ -344,7 +316,7 @@
text \<open>Suggested by Florian Kammüller\<close>
-lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
+lemma finite_quotient: "finite A \<Longrightarrow> r \<subseteq> A \<times> A \<Longrightarrow> finite (A//r)"
\<comment> \<open>recall @{thm equiv_type}\<close>
apply (rule finite_subset)
apply (erule_tac [2] finite_Pow_iff [THEN iffD2])
@@ -352,93 +324,94 @@
apply blast
done
-lemma finite_equiv_class:
- "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
+lemma finite_equiv_class: "finite A \<Longrightarrow> r \<subseteq> A \<times> A \<Longrightarrow> X \<in> A//r \<Longrightarrow> finite X"
apply (unfold quotient_def)
apply (rule finite_subset)
prefer 2 apply assumption
apply blast
done
-lemma equiv_imp_dvd_card:
- "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
- ==> k dvd card A"
+lemma equiv_imp_dvd_card: "finite A \<Longrightarrow> equiv A r \<Longrightarrow> \<forall>X \<in> A//r. k dvd card X \<Longrightarrow> k dvd card A"
apply (rule Union_quotient [THEN subst [where P="\<lambda>A. k dvd card A"]])
apply assumption
apply (rule dvd_partition)
- prefer 3 apply (blast dest: quotient_disj)
- apply (simp_all add: Union_quotient equiv_type)
+ prefer 3 apply (blast dest: quotient_disj)
+ apply (simp_all add: Union_quotient equiv_type)
done
-lemma card_quotient_disjoint:
- "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"
-apply(simp add:quotient_def)
-apply(subst card_UN_disjoint)
- apply assumption
+lemma card_quotient_disjoint: "finite A \<Longrightarrow> inj_on (\<lambda>x. {x} // r) A \<Longrightarrow> card (A//r) = card A"
+ apply (simp add:quotient_def)
+ apply (subst card_UN_disjoint)
+ apply assumption
+ apply simp
+ apply (fastforce simp add:inj_on_def)
apply simp
- apply(fastforce simp add:inj_on_def)
-apply simp
-done
+ done
subsection \<open>Projection\<close>
-definition proj where "proj r x = r `` {x}"
+definition proj :: "('b \<times> 'a) set \<Rightarrow> 'b \<Rightarrow> 'a set"
+ where "proj r x = r `` {x}"
-lemma proj_preserves:
-"x \<in> A \<Longrightarrow> proj r x \<in> A//r"
-unfolding proj_def by (rule quotientI)
+lemma proj_preserves: "x \<in> A \<Longrightarrow> proj r x \<in> A//r"
+ unfolding proj_def by (rule quotientI)
lemma proj_in_iff:
-assumes "equiv A r"
-shows "(proj r x \<in> A//r) = (x \<in> A)"
-apply(rule iffI, auto simp add: proj_preserves)
-unfolding proj_def quotient_def proof clarsimp
- fix y assume y: "y \<in> A" and "r `` {x} = r `` {y}"
- moreover have "y \<in> r `` {y}" using assms y unfolding equiv_def refl_on_def by blast
- ultimately have "(x,y) \<in> r" by blast
- thus "x \<in> A" using assms unfolding equiv_def refl_on_def by blast
+ assumes "equiv A r"
+ shows "proj r x \<in> A//r \<longleftrightarrow> x \<in> A"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ assume ?rhs
+ then show ?lhs by (simp add: proj_preserves)
+next
+ assume ?lhs
+ then show ?rhs
+ unfolding proj_def quotient_def
+ proof clarsimp
+ fix y
+ assume y: "y \<in> A" and "r `` {x} = r `` {y}"
+ moreover have "y \<in> r `` {y}"
+ using assms y unfolding equiv_def refl_on_def by blast
+ ultimately have "(x, y) \<in> r" by blast
+ then show "x \<in> A"
+ using assms unfolding equiv_def refl_on_def by blast
+ qed
qed
-lemma proj_iff:
-"\<lbrakk>equiv A r; {x,y} \<subseteq> A\<rbrakk> \<Longrightarrow> (proj r x = proj r y) = ((x,y) \<in> r)"
-by (simp add: proj_def eq_equiv_class_iff)
+lemma proj_iff: "equiv A r \<Longrightarrow> {x, y} \<subseteq> A \<Longrightarrow> proj r x = proj r y \<longleftrightarrow> (x, y) \<in> r"
+ by (simp add: proj_def eq_equiv_class_iff)
(*
lemma in_proj: "\<lbrakk>equiv A r; x \<in> A\<rbrakk> \<Longrightarrow> x \<in> proj r x"
unfolding proj_def equiv_def refl_on_def by blast
*)
-lemma proj_image: "(proj r) ` A = A//r"
-unfolding proj_def[abs_def] quotient_def by blast
+lemma proj_image: "proj r ` A = A//r"
+ unfolding proj_def[abs_def] quotient_def by blast
-lemma in_quotient_imp_non_empty:
-"\<lbrakk>equiv A r; X \<in> A//r\<rbrakk> \<Longrightarrow> X \<noteq> {}"
-unfolding quotient_def using equiv_class_self by fast
+lemma in_quotient_imp_non_empty: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> X \<noteq> {}"
+ unfolding quotient_def using equiv_class_self by fast
-lemma in_quotient_imp_in_rel:
-"\<lbrakk>equiv A r; X \<in> A//r; {x,y} \<subseteq> X\<rbrakk> \<Longrightarrow> (x,y) \<in> r"
-using quotient_eq_iff[THEN iffD1] by fastforce
+lemma in_quotient_imp_in_rel: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> {x, y} \<subseteq> X \<Longrightarrow> (x, y) \<in> r"
+ using quotient_eq_iff[THEN iffD1] by fastforce
-lemma in_quotient_imp_closed:
-"\<lbrakk>equiv A r; X \<in> A//r; x \<in> X; (x,y) \<in> r\<rbrakk> \<Longrightarrow> y \<in> X"
-unfolding quotient_def equiv_def trans_def by blast
+lemma in_quotient_imp_closed: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> x \<in> X \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> y \<in> X"
+ unfolding quotient_def equiv_def trans_def by blast
-lemma in_quotient_imp_subset:
-"\<lbrakk>equiv A r; X \<in> A//r\<rbrakk> \<Longrightarrow> X \<subseteq> A"
-using in_quotient_imp_in_rel equiv_type by fastforce
+lemma in_quotient_imp_subset: "equiv A r \<Longrightarrow> X \<in> A//r \<Longrightarrow> X \<subseteq> A"
+ using in_quotient_imp_in_rel equiv_type by fastforce
subsection \<open>Equivalence relations -- predicate version\<close>
-text \<open>Partial equivalences\<close>
+text \<open>Partial equivalences.\<close>
-definition part_equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
- "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)"
+definition part_equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
+ where "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)"
\<comment> \<open>John-Harrison-style characterization\<close>
-lemma part_equivpI:
- "(\<exists>x. R x x) \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> part_equivp R"
+lemma part_equivpI: "\<exists>x. R x x \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> part_equivp R"
by (auto simp add: part_equivp_def) (auto elim: sympE transpE)
lemma part_equivpE:
@@ -447,7 +420,7 @@
proof -
from assms have 1: "\<exists>x. R x x"
and 2: "\<And>x y. R x y \<longleftrightarrow> R x x \<and> R y y \<and> R x = R y"
- by (unfold part_equivp_def) blast+
+ unfolding part_equivp_def by blast+
from 1 obtain x where "R x x" ..
moreover have "symp R"
proof (rule sympI)
@@ -464,30 +437,25 @@
ultimately show thesis by (rule that)
qed
-lemma part_equivp_refl_symp_transp:
- "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> symp R \<and> transp R"
+lemma part_equivp_refl_symp_transp: "part_equivp R \<longleftrightarrow> (\<exists>x. R x x) \<and> symp R \<and> transp R"
by (auto intro: part_equivpI elim: part_equivpE)
-lemma part_equivp_symp:
- "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
+lemma part_equivp_symp: "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
by (erule part_equivpE, erule sympE)
-lemma part_equivp_transp:
- "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
+lemma part_equivp_transp: "part_equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
by (erule part_equivpE, erule transpE)
-lemma part_equivp_typedef:
- "part_equivp R \<Longrightarrow> \<exists>d. d \<in> {c. \<exists>x. R x x \<and> c = Collect (R x)}"
+lemma part_equivp_typedef: "part_equivp R \<Longrightarrow> \<exists>d. d \<in> {c. \<exists>x. R x x \<and> c = Collect (R x)}"
by (auto elim: part_equivpE)
-text \<open>Total equivalences\<close>
+text \<open>Total equivalences.\<close>
-definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
- "equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" \<comment> \<open>John-Harrison-style characterization\<close>
+definition equivp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
+ where "equivp R \<longleftrightarrow> (\<forall>x y. R x y = (R x = R y))" \<comment> \<open>John-Harrison-style characterization\<close>
-lemma equivpI:
- "reflp R \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> equivp R"
+lemma equivpI: "reflp R \<Longrightarrow> symp R \<Longrightarrow> transp R \<Longrightarrow> equivp R"
by (auto elim: reflpE sympE transpE simp add: equivp_def)
lemma equivpE:
@@ -495,32 +463,25 @@
obtains "reflp R" and "symp R" and "transp R"
using assms by (auto intro!: that reflpI sympI transpI simp add: equivp_def)
-lemma equivp_implies_part_equivp:
- "equivp R \<Longrightarrow> part_equivp R"
+lemma equivp_implies_part_equivp: "equivp R \<Longrightarrow> part_equivp R"
by (auto intro: part_equivpI elim: equivpE reflpE)
-lemma equivp_equiv:
- "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"
+lemma equivp_equiv: "equiv UNIV A \<longleftrightarrow> equivp (\<lambda>x y. (x, y) \<in> A)"
by (auto intro!: equivI equivpI [to_set] elim!: equivE equivpE [to_set])
-lemma equivp_reflp_symp_transp:
- shows "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"
+lemma equivp_reflp_symp_transp: "equivp R \<longleftrightarrow> reflp R \<and> symp R \<and> transp R"
by (auto intro: equivpI elim: equivpE)
-lemma identity_equivp:
- "equivp (op =)"
+lemma identity_equivp: "equivp (op =)"
by (auto intro: equivpI reflpI sympI transpI)
-lemma equivp_reflp:
- "equivp R \<Longrightarrow> R x x"
+lemma equivp_reflp: "equivp R \<Longrightarrow> R x x"
by (erule equivpE, erule reflpE)
-lemma equivp_symp:
- "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
+lemma equivp_symp: "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y x"
by (erule equivpE, erule sympE)
-lemma equivp_transp:
- "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
+lemma equivp_transp: "equivp R \<Longrightarrow> R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
by (erule equivpE, erule transpE)
hide_const (open) proj