--- a/src/HOL/Equiv_Relations.thy Fri Mar 27 22:06:46 2020 +0100
+++ b/src/HOL/Equiv_Relations.thy Sat Mar 28 17:27:01 2020 +0000
@@ -34,21 +34,21 @@
unfolding refl_on_def by blast
lemma equiv_comp_eq: "equiv A r \<Longrightarrow> r\<inverse> O r = r"
- apply (unfold equiv_def)
- apply clarify
- apply (rule equalityI)
- apply (iprover intro: sym_trans_comp_subset refl_on_comp_subset)+
- done
+ unfolding equiv_def
+ by (iprover intro: sym_trans_comp_subset refl_on_comp_subset equalityI)
text \<open>Second half.\<close>
-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 \<longrightarrow> (y, x) \<in> r")
- apply fast
- apply fast
- done
+lemma comp_equivI:
+ assumes "r\<inverse> O r = r" "Domain r = A"
+ shows "equiv A r"
+proof -
+ have *: "\<And>x y. (x, y) \<in> r \<Longrightarrow> (y, x) \<in> r"
+ using assms by blast
+ show ?thesis
+ unfolding equiv_def refl_on_def sym_def trans_def
+ using assms by (auto intro: *)
+qed
subsection \<open>Equivalence classes\<close>
@@ -58,10 +58,7 @@
unfolding equiv_def trans_def sym_def by blast
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
+ by (intro equalityI equiv_class_subset; force simp add: equiv_def sym_def)
lemma equiv_class_self: "equiv A r \<Longrightarrow> a \<in> A \<Longrightarrow> a \<in> r``{a}"
unfolding equiv_def refl_on_def by blast
@@ -91,7 +88,7 @@
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"
+lemma quotientI: "x \<in> A \<Longrightarrow> r``{x} \<in> A//r"
unfolding quotient_def by blast
lemma quotientE: "X \<in> A//r \<Longrightarrow> (\<And>x. X = r``{x} \<Longrightarrow> x \<in> A \<Longrightarrow> P) \<Longrightarrow> P"
@@ -101,32 +98,31 @@
unfolding equiv_def refl_on_def quotient_def by blast
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)
- apply assumption
- apply (unfold equiv_def trans_def sym_def)
- apply blast
- done
+ unfolding quotient_def equiv_def trans_def sym_def by blast
lemma 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) \<in> r \<Longrightarrow> X = Y"
- apply (clarify elim!: quotientE)
- apply (rule equiv_class_eq)
- apply assumption
- apply (unfold equiv_def sym_def trans_def)
- apply blast
- done
+ assumes "equiv A r" "X \<in> A//r" "Y \<in> A//r" and xy: "x \<in> X" "y \<in> Y" "(x, y) \<in> r"
+ shows "X = Y"
+proof -
+ obtain a b where "a \<in> A" and a: "X = r `` {a}" and "b \<in> A" and b: "Y = r `` {b}"
+ using assms by (auto elim!: quotientE)
+ then have "(a,b) \<in> r"
+ using xy \<open>equiv A r\<close> unfolding equiv_def sym_def trans_def by blast
+ then show ?thesis
+ unfolding a b by (rule equiv_class_eq [OF \<open>equiv A r\<close>])
+qed
lemma quotient_eq_iff:
- "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)
- apply blast
- done
+ assumes "equiv A r" "X \<in> A//r" "Y \<in> A//r" and xy: "x \<in> X" "y \<in> Y"
+ shows "X = Y \<longleftrightarrow> (x, y) \<in> r"
+proof
+ assume L: "X = Y"
+ with assms show "(x, y) \<in> r"
+ unfolding equiv_def sym_def trans_def by (blast elim!: quotientE)
+next
+ assume \<section>: "(x, y) \<in> r" show "X = Y"
+ by (rule quotient_eqI) (use \<section> assms in \<open>blast+\<close>)
+qed
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)
@@ -189,22 +185,22 @@
\<comment> \<open>lemma required to prove \<open>UN_equiv_class\<close>\<close>
by auto
-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"
+lemma UN_equiv_class:
+ assumes "equiv A r" "f respects r" "a \<in> A"
+ shows "(\<Union>x \<in> r``{a}. f x) = f a"
\<comment> \<open>Conversion rule\<close>
- 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
+proof -
+ have \<section>: "\<forall>x\<in>r `` {a}. f x = f a"
+ using assms unfolding equiv_def congruent_def sym_def by blast
+ show ?thesis
+ by (iprover intro: assms UN_constant_eq [OF equiv_class_self \<section>])
+qed
lemma UN_equiv_class_type:
- "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)
- apply auto
- done
+ assumes r: "equiv A r" "f respects r" and X: "X \<in> A//r" and AB: "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B"
+ shows "(\<Union>x \<in> X. f x) \<in> B"
+ using assms unfolding quotient_def
+ by (auto simp: UN_equiv_class [OF r])
text \<open>
Sufficient conditions for injectiveness. Could weaken premises!
@@ -213,19 +209,23 @@
\<close>
lemma UN_equiv_class_inject:
- "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)
- apply assumption
- apply (subgoal_tac "f x = f xa")
- apply blast
- apply (erule box_equals)
- apply (assumption | rule UN_equiv_class)+
- done
+ assumes "equiv A r" "f respects r"
+ and eq: "(\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y)"
+ and X: "X \<in> A//r" and Y: "Y \<in> A//r"
+ and fr: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x = f y \<Longrightarrow> (x, y) \<in> r"
+ shows "X = Y"
+proof -
+ obtain a b where "a \<in> A" and a: "X = r `` {a}" and "b \<in> A" and b: "Y = r `` {b}"
+ using assms by (auto elim!: quotientE)
+ then have "\<Union> (f ` r `` {a}) = f a" "\<Union> (f ` r `` {b}) = f b"
+ by (iprover intro: UN_equiv_class [OF \<open>equiv A r\<close>] assms)+
+ then have "f a = f b"
+ using eq unfolding a b by (iprover intro: trans sym)
+ then have "(a,b) \<in> r"
+ using fr \<open>a \<in> A\<close> \<open>b \<in> A\<close> by blast
+ then show ?thesis
+ unfolding a b by (rule equiv_class_eq [OF \<open>equiv A r\<close>])
+qed
subsection \<open>Defining binary operations upon equivalence classes\<close>
@@ -253,15 +253,20 @@
unfolding congruent_def congruent2_def equiv_def refl_on_def by blast
lemma congruent2_implies_congruent_UN:
- "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
- apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
- apply (simp add: UN_equiv_class congruent2_implies_congruent)
- apply (unfold congruent2_def equiv_def refl_on_def)
- apply (blast del: equalityI)
- done
+ assumes "equiv A1 r1" "equiv A2 r2" "congruent2 r1 r2 f" "a \<in> A2"
+ shows "congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2``{a}. f x1 x2)"
+ unfolding congruent_def
+proof clarify
+ fix c d
+ assume cd: "(c,d) \<in> r1"
+ then have "c \<in> A1" "d \<in> A1"
+ using \<open>equiv A1 r1\<close> by (auto elim!: equiv_type [THEN subsetD, THEN SigmaE2])
+ with assms show "\<Union> (f c ` r2 `` {a}) = \<Union> (f d ` r2 `` {a})"
+ proof (simp add: UN_equiv_class congruent2_implies_congruent)
+ show "f c a = f d a"
+ using assms cd unfolding congruent2_def equiv_def refl_on_def by blast
+ qed
+qed
lemma UN_equiv_class2:
"equiv A1 r1 \<Longrightarrow> equiv A2 r2 \<Longrightarrow> congruent2 r1 r2 f \<Longrightarrow> a1 \<in> A1 \<Longrightarrow> a2 \<in> A2 \<Longrightarrow>
@@ -273,11 +278,10 @@
\<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)
- done
+ unfolding quotient_def
+ by (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
+ congruent2_implies_congruent quotientI)
+
lemma UN_UN_split_split_eq:
"(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
@@ -293,60 +297,63 @@
\<Longrightarrow> congruent2 r1 r2 f"
\<comment> \<open>Suggested by John Harrison -- the two subproofs may be\<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)
- done
+ unfolding congruent2_def equiv_def refl_on_def
+ by (blast intro: trans)
lemma congruent2_commuteI:
assumes equivA: "equiv A r"
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])
- apply (rule_tac [3] commute [THEN trans, symmetric])
- apply (rule_tac [5] sym)
- apply (rule congt | assumption |
- erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
- done
+proof (rule congruent2I [OF equivA equivA])
+ note eqv = equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2]
+ show "\<And>y z w. \<lbrakk>w \<in> A; (y, z) \<in> r\<rbrakk> \<Longrightarrow> f y w = f z w"
+ by (iprover intro: commute [THEN trans] sym congt elim: eqv)
+ show "\<And>y z w. \<lbrakk>w \<in> A; (y, z) \<in> r\<rbrakk> \<Longrightarrow> f w y = f w z"
+ by (iprover intro: congt elim: eqv)
+qed
subsection \<open>Quotients and finiteness\<close>
text \<open>Suggested by Florian Kammüller\<close>
-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])
- apply (unfold quotient_def)
- apply blast
- done
+lemma finite_quotient:
+ assumes "finite A" "r \<subseteq> A \<times> A"
+ shows "finite (A//r)"
+ \<comment> \<open>recall @{thm equiv_type}\<close>
+proof -
+ have "A//r \<subseteq> Pow A"
+ using assms unfolding quotient_def by blast
+ moreover have "finite (Pow A)"
+ using assms by simp
+ ultimately show ?thesis
+ by (iprover intro: finite_subset)
+qed
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
+ unfolding quotient_def
+ by (erule rev_finite_subset) blast
-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)
- done
+lemma equiv_imp_dvd_card:
+ assumes "finite A" "equiv A r" "\<And>X. X \<in> A//r \<Longrightarrow> k dvd card X"
+ shows "k dvd card A"
+proof (rule Union_quotient [THEN subst])
+ show "k dvd card (\<Union> (A // r))"
+ apply (rule dvd_partition)
+ using assms
+ by (auto simp: Union_quotient dest: quotient_disj)
+qed (use assms in blast)
-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
- done
+lemma card_quotient_disjoint:
+ assumes "finite A" "inj_on (\<lambda>x. {x} // r) A"
+ shows "card (A//r) = card A"
+proof -
+ have "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> r `` {j} \<noteq> r `` {i}"
+ using assms by (fastforce simp add: quotient_def inj_on_def)
+ with assms show ?thesis
+ by (simp add: quotient_def card_UN_disjoint)
+qed
subsection \<open>Projection\<close>