src/HOL/Equiv_Relations.thy
 author huffman Thu Feb 18 14:21:44 2010 -0800 (2010-02-18) changeset 35216 7641e8d831d2 parent 30198 922f944f03b2 child 35725 4d7e3cc9c52c permissions -rw-r--r--
get rid of many duplicate simp rule warnings
1 (*  Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
2     Copyright   1996  University of Cambridge
3 *)
5 header {* Equivalence Relations in Higher-Order Set Theory *}
7 theory Equiv_Relations
8 imports Finite_Set Relation Plain
9 begin
11 subsection {* Equivalence relations *}
13 locale equiv =
14   fixes A and r
15   assumes refl_on: "refl_on A r"
16     and sym: "sym r"
17     and trans: "trans r"
19 text {*
20   Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O
21   r = r"}.
23   First half: @{text "equiv A r ==> r\<inverse> O r = r"}.
24 *}
26 lemma sym_trans_comp_subset:
27     "sym r ==> trans r ==> r\<inverse> O r \<subseteq> r"
28   by (unfold trans_def sym_def converse_def) blast
30 lemma refl_on_comp_subset: "refl_on A r ==> r \<subseteq> r\<inverse> O r"
31   by (unfold refl_on_def) blast
33 lemma equiv_comp_eq: "equiv A r ==> r\<inverse> O r = r"
34   apply (unfold equiv_def)
35   apply clarify
36   apply (rule equalityI)
37    apply (iprover intro: sym_trans_comp_subset refl_on_comp_subset)+
38   done
40 text {* Second half. *}
42 lemma comp_equivI:
43     "r\<inverse> O r = r ==> Domain r = A ==> equiv A r"
44   apply (unfold equiv_def refl_on_def sym_def trans_def)
45   apply (erule equalityE)
46   apply (subgoal_tac "\<forall>x y. (x, y) \<in> r --> (y, x) \<in> r")
47    apply fast
48   apply fast
49   done
52 subsection {* Equivalence classes *}
54 lemma equiv_class_subset:
55   "equiv A r ==> (a, b) \<in> r ==> r{a} \<subseteq> r{b}"
56   -- {* lemma for the next result *}
57   by (unfold equiv_def trans_def sym_def) blast
59 theorem equiv_class_eq: "equiv A r ==> (a, b) \<in> r ==> r{a} = r{b}"
60   apply (assumption | rule equalityI equiv_class_subset)+
61   apply (unfold equiv_def sym_def)
62   apply blast
63   done
65 lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r{a}"
66   by (unfold equiv_def refl_on_def) blast
68 lemma subset_equiv_class:
69     "equiv A r ==> r{b} \<subseteq> r{a} ==> b \<in> A ==> (a,b) \<in> r"
70   -- {* lemma for the next result *}
71   by (unfold equiv_def refl_on_def) blast
73 lemma eq_equiv_class:
74     "r{a} = r{b} ==> equiv A r ==> b \<in> A ==> (a, b) \<in> r"
75   by (iprover intro: equalityD2 subset_equiv_class)
77 lemma equiv_class_nondisjoint:
78     "equiv A r ==> x \<in> (r{a} \<inter> r{b}) ==> (a, b) \<in> r"
79   by (unfold equiv_def trans_def sym_def) blast
81 lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"
82   by (unfold equiv_def refl_on_def) blast
84 theorem equiv_class_eq_iff:
85   "equiv A r ==> ((x, y) \<in> r) = (r{x} = r{y} & x \<in> A & y \<in> A)"
86   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
88 theorem eq_equiv_class_iff:
89   "equiv A r ==> x \<in> A ==> y \<in> A ==> (r{x} = r{y}) = ((x, y) \<in> r)"
90   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
93 subsection {* Quotients *}
95 definition quotient :: "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set"  (infixl "'/'/" 90) where
96   [code del]: "A//r = (\<Union>x \<in> A. {r{x}})"  -- {* set of equiv classes *}
98 lemma quotientI: "x \<in> A ==> r{x} \<in> A//r"
99   by (unfold quotient_def) blast
101 lemma quotientE:
102   "X \<in> A//r ==> (!!x. X = r{x} ==> x \<in> A ==> P) ==> P"
103   by (unfold quotient_def) blast
105 lemma Union_quotient: "equiv A r ==> Union (A//r) = A"
106   by (unfold equiv_def refl_on_def quotient_def) blast
108 lemma quotient_disj:
109   "equiv A r ==> X \<in> A//r ==> Y \<in> A//r ==> X = Y | (X \<inter> Y = {})"
110   apply (unfold quotient_def)
111   apply clarify
112   apply (rule equiv_class_eq)
113    apply assumption
114   apply (unfold equiv_def trans_def sym_def)
115   apply blast
116   done
118 lemma quotient_eqI:
119   "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y; (x,y) \<in> r|] ==> X = Y"
120   apply (clarify elim!: quotientE)
121   apply (rule equiv_class_eq, assumption)
122   apply (unfold equiv_def sym_def trans_def, blast)
123   done
125 lemma quotient_eq_iff:
126   "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y|] ==> (X = Y) = ((x,y) \<in> r)"
127   apply (rule iffI)
128    prefer 2 apply (blast del: equalityI intro: quotient_eqI)
129   apply (clarify elim!: quotientE)
130   apply (unfold equiv_def sym_def trans_def, blast)
131   done
133 lemma eq_equiv_class_iff2:
134   "\<lbrakk> equiv A r; x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> ({x}//r = {y}//r) = ((x,y) : r)"
135 by(simp add:quotient_def eq_equiv_class_iff)
138 lemma quotient_empty [simp]: "{}//r = {}"
139 by(simp add: quotient_def)
141 lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"
142 by(simp add: quotient_def)
144 lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"
145 by(simp add: quotient_def)
148 lemma singleton_quotient: "{x}//r = {r  {x}}"
151 lemma quotient_diff1:
152   "\<lbrakk> inj_on (%a. {a}//r) A; a \<in> A \<rbrakk> \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
153 apply(simp add:quotient_def inj_on_def)
154 apply blast
155 done
157 subsection {* Defining unary operations upon equivalence classes *}
159 text{*A congruence-preserving function*}
160 locale congruent =
161   fixes r and f
162   assumes congruent: "(y,z) \<in> r ==> f y = f z"
164 abbreviation
165   RESPECTS :: "('a => 'b) => ('a * 'a) set => bool"
166     (infixr "respects" 80) where
167   "f respects r == congruent r f"
170 lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"
171   -- {* lemma required to prove @{text UN_equiv_class} *}
172   by auto
174 lemma UN_equiv_class:
175   "equiv A r ==> f respects r ==> a \<in> A
176     ==> (\<Union>x \<in> r{a}. f x) = f a"
177   -- {* Conversion rule *}
178   apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)
179   apply (unfold equiv_def congruent_def sym_def)
180   apply (blast del: equalityI)
181   done
183 lemma UN_equiv_class_type:
184   "equiv A r ==> f respects r ==> X \<in> A//r ==>
185     (!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"
186   apply (unfold quotient_def)
187   apply clarify
188   apply (subst UN_equiv_class)
189      apply auto
190   done
192 text {*
193   Sufficient conditions for injectiveness.  Could weaken premises!
194   major premise could be an inclusion; bcong could be @{text "!!y. y \<in>
195   A ==> f y \<in> B"}.
196 *}
198 lemma UN_equiv_class_inject:
199   "equiv A r ==> f respects r ==>
200     (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r
201     ==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)
202     ==> X = Y"
203   apply (unfold quotient_def)
204   apply clarify
205   apply (rule equiv_class_eq)
206    apply assumption
207   apply (subgoal_tac "f x = f xa")
208    apply blast
209   apply (erule box_equals)
210    apply (assumption | rule UN_equiv_class)+
211   done
214 subsection {* Defining binary operations upon equivalence classes *}
216 text{*A congruence-preserving function of two arguments*}
217 locale congruent2 =
218   fixes r1 and r2 and f
219   assumes congruent2:
220     "(y1,z1) \<in> r1 ==> (y2,z2) \<in> r2 ==> f y1 y2 = f z1 z2"
222 text{*Abbreviation for the common case where the relations are identical*}
223 abbreviation
224   RESPECTS2:: "['a => 'a => 'b, ('a * 'a) set] => bool"
225     (infixr "respects2" 80) where
226   "f respects2 r == congruent2 r r f"
229 lemma congruent2_implies_congruent:
230     "equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"
231   by (unfold congruent_def congruent2_def equiv_def refl_on_def) blast
233 lemma congruent2_implies_congruent_UN:
234   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>
235     congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2{a}. f x1 x2)"
236   apply (unfold congruent_def)
237   apply clarify
238   apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
239   apply (simp add: UN_equiv_class congruent2_implies_congruent)
240   apply (unfold congruent2_def equiv_def refl_on_def)
241   apply (blast del: equalityI)
242   done
244 lemma UN_equiv_class2:
245   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2
246     ==> (\<Union>x1 \<in> r1{a1}. \<Union>x2 \<in> r2{a2}. f x1 x2) = f a1 a2"
247   by (simp add: UN_equiv_class congruent2_implies_congruent
248     congruent2_implies_congruent_UN)
250 lemma UN_equiv_class_type2:
251   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f
252     ==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2
253     ==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)
254     ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
255   apply (unfold quotient_def)
256   apply clarify
257   apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
258     congruent2_implies_congruent quotientI)
259   done
261 lemma UN_UN_split_split_eq:
262   "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
263     (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
264   -- {* Allows a natural expression of binary operators, *}
265   -- {* without explicit calls to @{text split} *}
266   by auto
268 lemma congruent2I:
269   "equiv A1 r1 ==> equiv A2 r2
270     ==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)
271     ==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)
272     ==> congruent2 r1 r2 f"
273   -- {* Suggested by John Harrison -- the two subproofs may be *}
274   -- {* \emph{much} simpler than the direct proof. *}
275   apply (unfold congruent2_def equiv_def refl_on_def)
276   apply clarify
277   apply (blast intro: trans)
278   done
280 lemma congruent2_commuteI:
281   assumes equivA: "equiv A r"
282     and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"
283     and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"
284   shows "f respects2 r"
285   apply (rule congruent2I [OF equivA equivA])
286    apply (rule commute [THEN trans])
287      apply (rule_tac  commute [THEN trans, symmetric])
288        apply (rule_tac  sym)
289        apply (rule congt | assumption |
290          erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
291   done
294 subsection {* Quotients and finiteness *}
296 text {*Suggested by Florian Kamm�ller*}
298 lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
299   -- {* recall @{thm equiv_type} *}
300   apply (rule finite_subset)
301    apply (erule_tac  finite_Pow_iff [THEN iffD2])
302   apply (unfold quotient_def)
303   apply blast
304   done
306 lemma finite_equiv_class:
307   "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
308   apply (unfold quotient_def)
309   apply (rule finite_subset)
310    prefer 2 apply assumption
311   apply blast
312   done
314 lemma equiv_imp_dvd_card:
315   "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
316     ==> k dvd card A"
317   apply (rule Union_quotient [THEN subst [where P="\<lambda>A. k dvd card A"]])
318    apply assumption
319   apply (rule dvd_partition)
320      prefer 3 apply (blast dest: quotient_disj)
321     apply (simp_all add: Union_quotient equiv_type)
322   done
324 lemma card_quotient_disjoint:
325  "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"