1 (* Title: HOL/Relation.thy
2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
3 Copyright 1996 University of Cambridge
9 imports Datatype Finite_Set
12 subsection {* Definitions *}
15 converse :: "('a * 'b) set => ('b * 'a) set"
16 ("(_^-1)" [1000] 999) where
17 "r^-1 == {(y, x). (x, y) : r}"
20 converse ("(_\<inverse>)" [1000] 999)
23 rel_comp :: "[('a * 'b) set, ('b * 'c) set] => ('a * 'c) set"
25 "r O s == {(x,z). EX y. (x, y) : r & (y, z) : s}"
28 Image :: "[('a * 'b) set, 'a set] => 'b set"
29 (infixl "``" 90) where
30 "r `` s == {y. EX x:s. (x,y):r}"
33 Id :: "('a * 'a) set" where -- {* the identity relation *}
34 "Id == {p. EX x. p = (x,x)}"
37 Id_on :: "'a set => ('a * 'a) set" where -- {* diagonal: identity over a set *}
38 "Id_on A == \<Union>x\<in>A. {(x,x)}"
41 Domain :: "('a * 'b) set => 'a set" where
42 "Domain r == {x. EX y. (x,y):r}"
45 Range :: "('a * 'b) set => 'b set" where
46 "Range r == Domain(r^-1)"
49 Field :: "('a * 'a) set => 'a set" where
50 "Field r == Domain r \<union> Range r"
53 refl_on :: "['a set, ('a * 'a) set] => bool" where -- {* reflexivity over a set *}
54 "refl_on A r == r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)"
57 refl :: "('a * 'a) set => bool" where -- {* reflexivity over a type *}
58 "refl == refl_on UNIV"
61 sym :: "('a * 'a) set => bool" where -- {* symmetry predicate *}
62 "sym r == ALL x y. (x,y): r --> (y,x): r"
65 antisym :: "('a * 'a) set => bool" where -- {* antisymmetry predicate *}
66 "antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y"
69 trans :: "('a * 'a) set => bool" where -- {* transitivity predicate *}
70 "trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
73 irrefl :: "('a * 'a) set => bool" where
74 "irrefl r \<equiv> \<forall>x. (x,x) \<notin> r"
77 total_on :: "'a set => ('a * 'a) set => bool" where
78 "total_on A r \<equiv> \<forall>x\<in>A.\<forall>y\<in>A. x\<noteq>y \<longrightarrow> (x,y)\<in>r \<or> (y,x)\<in>r"
80 abbreviation "total \<equiv> total_on UNIV"
83 single_valued :: "('a * 'b) set => bool" where
84 "single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)"
87 inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set" where
88 "inv_image r f == {(x, y). (f x, f y) : r}"
91 subsection {* The identity relation *}
93 lemma IdI [intro]: "(a, a) : Id"
96 lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
97 by (unfold Id_def) (iprover elim: CollectE)
99 lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
100 by (unfold Id_def) blast
102 lemma refl_Id: "refl Id"
103 by (simp add: refl_on_def)
105 lemma antisym_Id: "antisym Id"
106 -- {* A strange result, since @{text Id} is also symmetric. *}
107 by (simp add: antisym_def)
109 lemma sym_Id: "sym Id"
110 by (simp add: sym_def)
112 lemma trans_Id: "trans Id"
113 by (simp add: trans_def)
116 subsection {* Diagonal: identity over a set *}
118 lemma Id_on_empty [simp]: "Id_on {} = {}"
119 by (simp add: Id_on_def)
121 lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A"
122 by (simp add: Id_on_def)
124 lemma Id_onI [intro!,no_atp]: "a : A ==> (a, a) : Id_on A"
125 by (rule Id_on_eqI) (rule refl)
127 lemma Id_onE [elim!]:
128 "c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
129 -- {* The general elimination rule. *}
130 by (unfold Id_on_def) (iprover elim!: UN_E singletonE)
132 lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)"
135 lemma Id_on_def' [nitpick_unfold, code]:
136 "Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)"
139 lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
143 subsection {* Composition of two relations *}
145 lemma rel_compI [intro]:
146 "(a, b) : r ==> (b, c) : s ==> (a, c) : r O s"
147 by (unfold rel_comp_def) blast
149 lemma rel_compE [elim!]: "xz : r O s ==>
150 (!!x y z. xz = (x, z) ==> (x, y) : r ==> (y, z) : s ==> P) ==> P"
151 by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE)
154 "(a, c) : r O s ==> (!!y. (a, y) : r ==> (y, c) : s ==> P) ==> P"
155 by (iprover elim: rel_compE Pair_inject ssubst)
157 lemma R_O_Id [simp]: "R O Id = R"
160 lemma Id_O_R [simp]: "Id O R = R"
163 lemma rel_comp_empty1[simp]: "{} O R = {}"
166 lemma rel_comp_empty2[simp]: "R O {} = {}"
169 lemma O_assoc: "(R O S) O T = R O (S O T)"
172 lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
173 by (unfold trans_def) blast
175 lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
178 lemma rel_comp_subset_Sigma:
179 "r \<subseteq> A \<times> B ==> s \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
182 lemma rel_comp_distrib[simp]: "R O (S \<union> T) = (R O S) \<union> (R O T)"
185 lemma rel_comp_distrib2[simp]: "(S \<union> T) O R = (S O R) \<union> (T O R)"
188 lemma rel_comp_UNION_distrib: "s O UNION I r = UNION I (%i. s O r i)"
191 lemma rel_comp_UNION_distrib2: "UNION I r O s = UNION I (%i. r i O s)"
195 subsection {* Reflexivity *}
197 lemma refl_onI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r"
198 by (unfold refl_on_def) (iprover intro!: ballI)
200 lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r"
201 by (unfold refl_on_def) blast
203 lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A"
204 by (unfold refl_on_def) blast
206 lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A"
207 by (unfold refl_on_def) blast
209 lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)"
210 by (unfold refl_on_def) blast
212 lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)"
213 by (unfold refl_on_def) blast
216 "ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)"
217 by (unfold refl_on_def) fast
220 "ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)"
221 by (unfold refl_on_def) blast
223 lemma refl_on_empty[simp]: "refl_on {} {}"
224 by(simp add:refl_on_def)
226 lemma refl_on_Id_on: "refl_on A (Id_on A)"
227 by (rule refl_onI [OF Id_on_subset_Times Id_onI])
229 lemma refl_on_def' [nitpick_unfold, code]:
230 "refl_on A r = ((\<forall>(x, y) \<in> r. x : A \<and> y : A) \<and> (\<forall>x \<in> A. (x, x) : r))"
231 by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2)
233 subsection {* Antisymmetry *}
236 "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
237 by (unfold antisym_def) iprover
239 lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
240 by (unfold antisym_def) iprover
242 lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
243 by (unfold antisym_def) blast
245 lemma antisym_empty [simp]: "antisym {}"
246 by (unfold antisym_def) blast
248 lemma antisym_Id_on [simp]: "antisym (Id_on A)"
249 by (unfold antisym_def) blast
252 subsection {* Symmetry *}
254 lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
255 by (unfold sym_def) iprover
257 lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r"
258 by (unfold sym_def, blast)
260 lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)"
261 by (fast intro: symI dest: symD)
263 lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)"
264 by (fast intro: symI dest: symD)
266 lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)"
267 by (fast intro: symI dest: symD)
269 lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)"
270 by (fast intro: symI dest: symD)
272 lemma sym_Id_on [simp]: "sym (Id_on A)"
273 by (rule symI) clarify
276 subsection {* Transitivity *}
279 "trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
280 by (auto simp add: trans_def)
283 "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
284 by (unfold trans_def) iprover
286 lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
287 by (unfold trans_def) iprover
289 lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)"
290 by (fast intro: transI elim: transD)
292 lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)"
293 by (fast intro: transI elim: transD)
295 lemma trans_Id_on [simp]: "trans (Id_on A)"
296 by (fast intro: transI elim: transD)
298 lemma trans_diff_Id: " trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r-Id)"
299 unfolding antisym_def trans_def by blast
301 subsection {* Irreflexivity *}
303 lemma irrefl_distinct:
304 "irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)"
305 by (auto simp add: irrefl_def)
307 lemma irrefl_diff_Id[simp]: "irrefl(r-Id)"
308 by(simp add:irrefl_def)
310 subsection {* Totality *}
312 lemma total_on_empty[simp]: "total_on {} r"
313 by(simp add:total_on_def)
315 lemma total_on_diff_Id[simp]: "total_on A (r-Id) = total_on A r"
316 by(simp add: total_on_def)
318 subsection {* Converse *}
320 lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
321 by (simp add: converse_def)
323 lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
324 by (simp add: converse_def)
326 lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
327 by (simp add: converse_def)
329 lemma converseE [elim!]:
330 "yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
331 -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
332 by (unfold converse_def) (iprover elim!: CollectE splitE bexE)
334 lemma converse_converse [simp]: "(r^-1)^-1 = r"
335 by (unfold converse_def) blast
337 lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
340 lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
343 lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
346 lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
349 lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
352 lemma converse_Id [simp]: "Id^-1 = Id"
355 lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A"
358 lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
359 by (unfold refl_on_def) auto
361 lemma sym_converse [simp]: "sym (converse r) = sym r"
362 by (unfold sym_def) blast
364 lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
365 by (unfold antisym_def) blast
367 lemma trans_converse [simp]: "trans (converse r) = trans r"
368 by (unfold trans_def) blast
370 lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
371 by (unfold sym_def) fast
373 lemma sym_Un_converse: "sym (r \<union> r^-1)"
374 by (unfold sym_def) blast
376 lemma sym_Int_converse: "sym (r \<inter> r^-1)"
377 by (unfold sym_def) blast
379 lemma total_on_converse[simp]: "total_on A (r^-1) = total_on A r"
380 by (auto simp: total_on_def)
383 subsection {* Domain *}
385 declare Domain_def [no_atp]
387 lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
388 by (unfold Domain_def) blast
390 lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
391 by (iprover intro!: iffD2 [OF Domain_iff])
393 lemma DomainE [elim!]:
394 "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
395 by (iprover dest!: iffD1 [OF Domain_iff])
399 by (auto simp add: image_def Bex_def)
401 lemma Domain_empty [simp]: "Domain {} = {}"
404 lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"
407 lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
410 lemma Domain_Id [simp]: "Domain Id = UNIV"
413 lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
416 lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
419 lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
422 lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
425 lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
428 lemma Domain_converse[simp]: "Domain(r^-1) = Range r"
429 by(auto simp:Range_def)
431 lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
434 lemma fst_eq_Domain: "fst ` R = Domain R"
437 lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
440 lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
444 subsection {* Range *}
446 lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
447 by (simp add: Domain_def Range_def)
449 lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
450 by (unfold Range_def) (iprover intro!: converseI DomainI)
452 lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
453 by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
457 by (auto simp add: image_def Bex_def)
459 lemma Range_empty [simp]: "Range {} = {}"
462 lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}"
465 lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
468 lemma Range_Id [simp]: "Range Id = UNIV"
471 lemma Range_Id_on [simp]: "Range (Id_on A) = A"
474 lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
477 lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
480 lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
483 lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
486 lemma Range_converse[simp]: "Range(r^-1) = Domain r"
489 lemma snd_eq_Range: "snd ` R = Range R"
493 subsection {* Field *}
495 lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
496 by(auto simp:Field_def Domain_def Range_def)
498 lemma Field_empty[simp]: "Field {} = {}"
499 by(auto simp:Field_def)
501 lemma Field_insert[simp]: "Field (insert (a,b) r) = {a,b} \<union> Field r"
502 by(auto simp:Field_def)
504 lemma Field_Un[simp]: "Field (r \<union> s) = Field r \<union> Field s"
505 by(auto simp:Field_def)
507 lemma Field_Union[simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
508 by(auto simp:Field_def)
510 lemma Field_converse[simp]: "Field(r^-1) = Field r"
511 by(auto simp:Field_def)
514 subsection {* Image of a set under a relation *}
516 declare Image_def [no_atp]
518 lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
519 by (simp add: Image_def)
521 lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
522 by (simp add: Image_def)
524 lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
525 by (rule Image_iff [THEN trans]) simp
527 lemma ImageI [intro,no_atp]: "(a, b) : r ==> a : A ==> b : r``A"
528 by (unfold Image_def) blast
530 lemma ImageE [elim!]:
531 "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
532 by (unfold Image_def) (iprover elim!: CollectE bexE)
534 lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
535 -- {* This version's more effective when we already have the required @{text a} *}
538 lemma Image_empty [simp]: "R``{} = {}"
541 lemma Image_Id [simp]: "Id `` A = A"
544 lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
547 lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
551 "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
552 by (simp add: single_valued_def, blast)
554 lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
557 lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
560 lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
561 by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
563 lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
564 -- {* NOT suitable for rewriting *}
567 lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
570 lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
573 lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
576 text{*Converse inclusion requires some assumptions*}
578 "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
579 apply (rule equalityI)
580 apply (rule Image_INT_subset)
581 apply (simp add: single_valued_def, blast)
584 lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
588 subsection {* Single valued relations *}
590 lemma single_valuedI:
591 "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
592 by (unfold single_valued_def)
594 lemma single_valuedD:
595 "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
596 by (simp add: single_valued_def)
598 lemma single_valued_rel_comp:
599 "single_valued r ==> single_valued s ==> single_valued (r O s)"
600 by (unfold single_valued_def) blast
602 lemma single_valued_subset:
603 "r \<subseteq> s ==> single_valued s ==> single_valued r"
604 by (unfold single_valued_def) blast
606 lemma single_valued_Id [simp]: "single_valued Id"
607 by (unfold single_valued_def) blast
609 lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
610 by (unfold single_valued_def) blast
613 subsection {* Graphs given by @{text Collect} *}
615 lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
618 lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
621 lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
625 subsection {* Inverse image *}
627 lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
628 by (unfold sym_def inv_image_def) blast
630 lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
631 apply (unfold trans_def inv_image_def)
632 apply (simp (no_asm))
636 lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
637 by (auto simp:inv_image_def)
639 lemma converse_inv_image[simp]: "(inv_image R f)^-1 = inv_image (R^-1) f"
640 unfolding inv_image_def converse_def by auto
643 subsection {* Finiteness *}
645 lemma finite_converse [iff]: "finite (r^-1) = finite r"
646 apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
649 apply (erule finite_imageD [unfolded inj_on_def])
650 apply (simp split add: split_split)
651 apply (erule finite_imageI)
652 apply (simp add: converse_def image_def, auto)
654 prefer 2 apply assumption
658 lemma finite_Domain: "finite r ==> finite (Domain r)"
659 by (induct set: finite) (auto simp add: Domain_insert)
661 lemma finite_Range: "finite r ==> finite (Range r)"
662 by (induct set: finite) (auto simp add: Range_insert)
664 lemma finite_Field: "finite r ==> finite (Field r)"
665 -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
666 apply (induct set: finite)
667 apply (auto simp add: Field_def Domain_insert Range_insert)
671 subsection {* Miscellaneous *}
673 text {* Version of @{thm[source] lfp_induct} for binary relations *}
676 lfp_induct_set [of "(a, b)", split_format (complete)]
678 text {* Version of @{thm[source] subsetI} for binary relations *}
680 lemma subrelI: "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s"