src/HOL/Equiv_Relations.thy
 author berghofe Wed May 07 10:56:33 2008 +0200 (2008-05-07) changeset 26791 3581a9c71909 parent 25482 4ed49eccb1eb child 28229 4f06fae6a55e permissions -rw-r--r--
Instantiated subst rule to avoid problems with HO unification.
1 (*  ID:         $Id$
2     Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
3     Copyright   1996  University of Cambridge
4 *)
6 header {* Equivalence Relations in Higher-Order Set Theory *}
8 theory Equiv_Relations
9 imports Finite_Set Relation
10 begin
12 subsection {* Equivalence relations *}
14 locale equiv =
15   fixes A and r
16   assumes refl: "refl A r"
17     and sym: "sym r"
18     and trans: "trans r"
20 text {*
21   Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O
22   r = r"}.
24   First half: @{text "equiv A r ==> r\<inverse> O r = r"}.
25 *}
27 lemma sym_trans_comp_subset:
28     "sym r ==> trans r ==> r\<inverse> O r \<subseteq> r"
29   by (unfold trans_def sym_def converse_def) blast
31 lemma refl_comp_subset: "refl A r ==> r \<subseteq> r\<inverse> O r"
32   by (unfold refl_def) blast
34 lemma equiv_comp_eq: "equiv A r ==> r\<inverse> O r = r"
35   apply (unfold equiv_def)
36   apply clarify
37   apply (rule equalityI)
38    apply (iprover intro: sym_trans_comp_subset refl_comp_subset)+
39   done
41 text {* Second half. *}
43 lemma comp_equivI:
44     "r\<inverse> O r = r ==> Domain r = A ==> equiv A r"
45   apply (unfold equiv_def refl_def sym_def trans_def)
46   apply (erule equalityE)
47   apply (subgoal_tac "\<forall>x y. (x, y) \<in> r --> (y, x) \<in> r")
48    apply fast
49   apply fast
50   done
53 subsection {* Equivalence classes *}
55 lemma equiv_class_subset:
56   "equiv A r ==> (a, b) \<in> r ==> r{a} \<subseteq> r{b}"
57   -- {* lemma for the next result *}
58   by (unfold equiv_def trans_def sym_def) blast
60 theorem equiv_class_eq: "equiv A r ==> (a, b) \<in> r ==> r{a} = r{b}"
61   apply (assumption | rule equalityI equiv_class_subset)+
62   apply (unfold equiv_def sym_def)
63   apply blast
64   done
66 lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r{a}"
67   by (unfold equiv_def refl_def) blast
69 lemma subset_equiv_class:
70     "equiv A r ==> r{b} \<subseteq> r{a} ==> b \<in> A ==> (a,b) \<in> r"
71   -- {* lemma for the next result *}
72   by (unfold equiv_def refl_def) blast
74 lemma eq_equiv_class:
75     "r{a} = r{b} ==> equiv A r ==> b \<in> A ==> (a, b) \<in> r"
76   by (iprover intro: equalityD2 subset_equiv_class)
78 lemma equiv_class_nondisjoint:
79     "equiv A r ==> x \<in> (r{a} \<inter> r{b}) ==> (a, b) \<in> r"
80   by (unfold equiv_def trans_def sym_def) blast
82 lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"
83   by (unfold equiv_def refl_def) blast
85 theorem equiv_class_eq_iff:
86   "equiv A r ==> ((x, y) \<in> r) = (r{x} = r{y} & x \<in> A & y \<in> A)"
87   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
89 theorem eq_equiv_class_iff:
90   "equiv A r ==> x \<in> A ==> y \<in> A ==> (r{x} = r{y}) = ((x, y) \<in> r)"
91   by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
94 subsection {* Quotients *}
96 constdefs
97   quotient :: "['a set, ('a*'a) set] => 'a set set"  (infixl "'/'/" 90)
98   "A//r == \<Union>x \<in> A. {r{x}}"  -- {* set of equiv classes *}
100 lemma quotientI: "x \<in> A ==> r{x} \<in> A//r"
101   by (unfold quotient_def) blast
103 lemma quotientE:
104   "X \<in> A//r ==> (!!x. X = r{x} ==> x \<in> A ==> P) ==> P"
105   by (unfold quotient_def) blast
107 lemma Union_quotient: "equiv A r ==> Union (A//r) = A"
108   by (unfold equiv_def refl_def quotient_def) blast
110 lemma quotient_disj:
111   "equiv A r ==> X \<in> A//r ==> Y \<in> A//r ==> X = Y | (X \<inter> Y = {})"
112   apply (unfold quotient_def)
113   apply clarify
114   apply (rule equiv_class_eq)
115    apply assumption
116   apply (unfold equiv_def trans_def sym_def)
117   apply blast
118   done
120 lemma quotient_eqI:
121   "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y; (x,y) \<in> r|] ==> X = Y"
122   apply (clarify elim!: quotientE)
123   apply (rule equiv_class_eq, assumption)
124   apply (unfold equiv_def sym_def trans_def, blast)
125   done
127 lemma quotient_eq_iff:
128   "[|equiv A r; X \<in> A//r; Y \<in> A//r; x \<in> X; y \<in> Y|] ==> (X = Y) = ((x,y) \<in> r)"
129   apply (rule iffI)
130    prefer 2 apply (blast del: equalityI intro: quotient_eqI)
131   apply (clarify elim!: quotientE)
132   apply (unfold equiv_def sym_def trans_def, blast)
133   done
135 lemma eq_equiv_class_iff2:
136   "\<lbrakk> equiv A r; x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> ({x}//r = {y}//r) = ((x,y) : r)"
140 lemma quotient_empty [simp]: "{}//r = {}"
143 lemma quotient_is_empty [iff]: "(A//r = {}) = (A = {})"
146 lemma quotient_is_empty2 [iff]: "({} = A//r) = (A = {})"
150 lemma singleton_quotient: "{x}//r = {r  {x}}"
153 lemma quotient_diff1:
154   "\<lbrakk> inj_on (%a. {a}//r) A; a \<in> A \<rbrakk> \<Longrightarrow> (A - {a})//r = A//r - {a}//r"
156 apply blast
157 done
159 subsection {* Defining unary operations upon equivalence classes *}
161 text{*A congruence-preserving function*}
162 locale congruent =
163   fixes r and f
164   assumes congruent: "(y,z) \<in> r ==> f y = f z"
166 abbreviation
167   RESPECTS :: "('a => 'b) => ('a * 'a) set => bool"
168     (infixr "respects" 80) where
169   "f respects r == congruent r f"
172 lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. f y = c ==> (\<Union>y \<in> A. f(y))=c"
173   -- {* lemma required to prove @{text UN_equiv_class} *}
174   by auto
176 lemma UN_equiv_class:
177   "equiv A r ==> f respects r ==> a \<in> A
178     ==> (\<Union>x \<in> r{a}. f x) = f a"
179   -- {* Conversion rule *}
180   apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)
181   apply (unfold equiv_def congruent_def sym_def)
182   apply (blast del: equalityI)
183   done
185 lemma UN_equiv_class_type:
186   "equiv A r ==> f respects r ==> X \<in> A//r ==>
187     (!!x. x \<in> A ==> f x \<in> B) ==> (\<Union>x \<in> X. f x) \<in> B"
188   apply (unfold quotient_def)
189   apply clarify
190   apply (subst UN_equiv_class)
191      apply auto
192   done
194 text {*
195   Sufficient conditions for injectiveness.  Could weaken premises!
196   major premise could be an inclusion; bcong could be @{text "!!y. y \<in>
197   A ==> f y \<in> B"}.
198 *}
200 lemma UN_equiv_class_inject:
201   "equiv A r ==> f respects r ==>
202     (\<Union>x \<in> X. f x) = (\<Union>y \<in> Y. f y) ==> X \<in> A//r ==> Y \<in> A//r
203     ==> (!!x y. x \<in> A ==> y \<in> A ==> f x = f y ==> (x, y) \<in> r)
204     ==> X = Y"
205   apply (unfold quotient_def)
206   apply clarify
207   apply (rule equiv_class_eq)
208    apply assumption
209   apply (subgoal_tac "f x = f xa")
210    apply blast
211   apply (erule box_equals)
212    apply (assumption | rule UN_equiv_class)+
213   done
216 subsection {* Defining binary operations upon equivalence classes *}
218 text{*A congruence-preserving function of two arguments*}
219 locale congruent2 =
220   fixes r1 and r2 and f
221   assumes congruent2:
222     "(y1,z1) \<in> r1 ==> (y2,z2) \<in> r2 ==> f y1 y2 = f z1 z2"
224 text{*Abbreviation for the common case where the relations are identical*}
225 abbreviation
226   RESPECTS2:: "['a => 'a => 'b, ('a * 'a) set] => bool"
227     (infixr "respects2" 80) where
228   "f respects2 r == congruent2 r r f"
231 lemma congruent2_implies_congruent:
232     "equiv A r1 ==> congruent2 r1 r2 f ==> a \<in> A ==> congruent r2 (f a)"
233   by (unfold congruent_def congruent2_def equiv_def refl_def) blast
235 lemma congruent2_implies_congruent_UN:
236   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a \<in> A2 ==>
237     congruent r1 (\<lambda>x1. \<Union>x2 \<in> r2{a}. f x1 x2)"
238   apply (unfold congruent_def)
239   apply clarify
240   apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
241   apply (simp add: UN_equiv_class congruent2_implies_congruent)
242   apply (unfold congruent2_def equiv_def refl_def)
243   apply (blast del: equalityI)
244   done
246 lemma UN_equiv_class2:
247   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f ==> a1 \<in> A1 ==> a2 \<in> A2
248     ==> (\<Union>x1 \<in> r1{a1}. \<Union>x2 \<in> r2{a2}. f x1 x2) = f a1 a2"
249   by (simp add: UN_equiv_class congruent2_implies_congruent
250     congruent2_implies_congruent_UN)
252 lemma UN_equiv_class_type2:
253   "equiv A1 r1 ==> equiv A2 r2 ==> congruent2 r1 r2 f
254     ==> X1 \<in> A1//r1 ==> X2 \<in> A2//r2
255     ==> (!!x1 x2. x1 \<in> A1 ==> x2 \<in> A2 ==> f x1 x2 \<in> B)
256     ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. f x1 x2) \<in> B"
257   apply (unfold quotient_def)
258   apply clarify
259   apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
260     congruent2_implies_congruent quotientI)
261   done
263 lemma UN_UN_split_split_eq:
264   "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
265     (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
266   -- {* Allows a natural expression of binary operators, *}
267   -- {* without explicit calls to @{text split} *}
268   by auto
270 lemma congruent2I:
271   "equiv A1 r1 ==> equiv A2 r2
272     ==> (!!y z w. w \<in> A2 ==> (y,z) \<in> r1 ==> f y w = f z w)
273     ==> (!!y z w. w \<in> A1 ==> (y,z) \<in> r2 ==> f w y = f w z)
274     ==> congruent2 r1 r2 f"
275   -- {* Suggested by John Harrison -- the two subproofs may be *}
276   -- {* \emph{much} simpler than the direct proof. *}
277   apply (unfold congruent2_def equiv_def refl_def)
278   apply clarify
279   apply (blast intro: trans)
280   done
282 lemma congruent2_commuteI:
283   assumes equivA: "equiv A r"
284     and commute: "!!y z. y \<in> A ==> z \<in> A ==> f y z = f z y"
285     and congt: "!!y z w. w \<in> A ==> (y,z) \<in> r ==> f w y = f w z"
286   shows "f respects2 r"
287   apply (rule congruent2I [OF equivA equivA])
288    apply (rule commute [THEN trans])
289      apply (rule_tac [3] commute [THEN trans, symmetric])
290        apply (rule_tac [5] sym)
291        apply (rule congt | assumption |
292          erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
293   done
296 subsection {* Quotients and finiteness *}
298 text {*Suggested by Florian Kammüller*}
300 lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
301   -- {* recall @{thm equiv_type} *}
302   apply (rule finite_subset)
303    apply (erule_tac [2] finite_Pow_iff [THEN iffD2])
304   apply (unfold quotient_def)
305   apply blast
306   done
308 lemma finite_equiv_class:
309   "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
310   apply (unfold quotient_def)
311   apply (rule finite_subset)
312    prefer 2 apply assumption
313   apply blast
314   done
316 lemma equiv_imp_dvd_card:
317   "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
318     ==> k dvd card A"
319   apply (rule Union_quotient [THEN subst [where P="\<lambda>A. k dvd card A"]])
320    apply assumption
321   apply (rule dvd_partition)
322      prefer 3 apply (blast dest: quotient_disj)
323     apply (simp_all add: Union_quotient equiv_type)
324   done
326 lemma card_quotient_disjoint:
327  "\<lbrakk> finite A; inj_on (\<lambda>x. {x} // r) A \<rbrakk> \<Longrightarrow> card(A//r) = card A"