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_def, code]:
136 "(Id_on (A :: 'a => bool)) = (%(x, y). x = y \<and> A x)"
137 by (auto simp add: fun_eq_iff
138 elim: Id_onE[unfolded mem_def] intro: Id_onI[unfolded mem_def])
140 lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
144 subsection {* Composition of two relations *}
146 lemma rel_compI [intro]:
147 "(a, b) : r ==> (b, c) : s ==> (a, c) : r O s"
148 by (unfold rel_comp_def) blast
150 lemma rel_compE [elim!]: "xz : r O s ==>
151 (!!x y z. xz = (x, z) ==> (x, y) : r ==> (y, z) : s ==> P) ==> P"
152 by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE)
155 "(a, c) : r O s ==> (!!y. (a, y) : r ==> (y, c) : s ==> P) ==> P"
156 by (iprover elim: rel_compE Pair_inject ssubst)
158 lemma R_O_Id [simp]: "R O Id = R"
161 lemma Id_O_R [simp]: "Id O R = R"
164 lemma rel_comp_empty1[simp]: "{} O R = {}"
167 lemma rel_comp_empty2[simp]: "R O {} = {}"
170 lemma O_assoc: "(R O S) O T = R O (S O T)"
173 lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
174 by (unfold trans_def) blast
176 lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
179 lemma rel_comp_subset_Sigma:
180 "r \<subseteq> A \<times> B ==> s \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
183 lemma rel_comp_distrib[simp]: "R O (S \<union> T) = (R O S) \<union> (R O T)"
186 lemma rel_comp_distrib2[simp]: "(S \<union> T) O R = (S O R) \<union> (T O R)"
189 lemma rel_comp_UNION_distrib: "s O UNION I r = UNION I (%i. s O r i)"
192 lemma rel_comp_UNION_distrib2: "UNION I r O s = UNION I (%i. r i O s)"
196 subsection {* Reflexivity *}
198 lemma refl_onI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r"
199 by (unfold refl_on_def) (iprover intro!: ballI)
201 lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r"
202 by (unfold refl_on_def) blast
204 lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A"
205 by (unfold refl_on_def) blast
207 lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A"
208 by (unfold refl_on_def) blast
210 lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)"
211 by (unfold refl_on_def) blast
213 lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)"
214 by (unfold refl_on_def) blast
217 "ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)"
218 by (unfold refl_on_def) fast
221 "ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)"
222 by (unfold refl_on_def) blast
224 lemma refl_on_empty[simp]: "refl_on {} {}"
225 by(simp add:refl_on_def)
227 lemma refl_on_Id_on: "refl_on A (Id_on A)"
228 by (rule refl_onI [OF Id_on_subset_Times Id_onI])
231 subsection {* Antisymmetry *}
234 "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
235 by (unfold antisym_def) iprover
237 lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
238 by (unfold antisym_def) iprover
240 lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
241 by (unfold antisym_def) blast
243 lemma antisym_empty [simp]: "antisym {}"
244 by (unfold antisym_def) blast
246 lemma antisym_Id_on [simp]: "antisym (Id_on A)"
247 by (unfold antisym_def) blast
250 subsection {* Symmetry *}
252 lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
253 by (unfold sym_def) iprover
255 lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r"
256 by (unfold sym_def, blast)
258 lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)"
259 by (fast intro: symI dest: symD)
261 lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)"
262 by (fast intro: symI dest: symD)
264 lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)"
265 by (fast intro: symI dest: symD)
267 lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)"
268 by (fast intro: symI dest: symD)
270 lemma sym_Id_on [simp]: "sym (Id_on A)"
271 by (rule symI) clarify
274 subsection {* Transitivity *}
277 "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
278 by (unfold trans_def) iprover
280 lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
281 by (unfold trans_def) iprover
283 lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)"
284 by (fast intro: transI elim: transD)
286 lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)"
287 by (fast intro: transI elim: transD)
289 lemma trans_Id_on [simp]: "trans (Id_on A)"
290 by (fast intro: transI elim: transD)
292 lemma trans_diff_Id: " trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r-Id)"
293 unfolding antisym_def trans_def by blast
295 subsection {* Irreflexivity *}
297 lemma irrefl_diff_Id[simp]: "irrefl(r-Id)"
298 by(simp add:irrefl_def)
300 subsection {* Totality *}
302 lemma total_on_empty[simp]: "total_on {} r"
303 by(simp add:total_on_def)
305 lemma total_on_diff_Id[simp]: "total_on A (r-Id) = total_on A r"
306 by(simp add: total_on_def)
308 subsection {* Converse *}
310 lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
311 by (simp add: converse_def)
313 lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
314 by (simp add: converse_def)
316 lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
317 by (simp add: converse_def)
319 lemma converseE [elim!]:
320 "yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
321 -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
322 by (unfold converse_def) (iprover elim!: CollectE splitE bexE)
324 lemma converse_converse [simp]: "(r^-1)^-1 = r"
325 by (unfold converse_def) blast
327 lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
330 lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
333 lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
336 lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
339 lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
342 lemma converse_Id [simp]: "Id^-1 = Id"
345 lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A"
348 lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
349 by (unfold refl_on_def) auto
351 lemma sym_converse [simp]: "sym (converse r) = sym r"
352 by (unfold sym_def) blast
354 lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
355 by (unfold antisym_def) blast
357 lemma trans_converse [simp]: "trans (converse r) = trans r"
358 by (unfold trans_def) blast
360 lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
361 by (unfold sym_def) fast
363 lemma sym_Un_converse: "sym (r \<union> r^-1)"
364 by (unfold sym_def) blast
366 lemma sym_Int_converse: "sym (r \<inter> r^-1)"
367 by (unfold sym_def) blast
369 lemma total_on_converse[simp]: "total_on A (r^-1) = total_on A r"
370 by (auto simp: total_on_def)
373 subsection {* Domain *}
375 declare Domain_def [no_atp]
377 lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
378 by (unfold Domain_def) blast
380 lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
381 by (iprover intro!: iffD2 [OF Domain_iff])
383 lemma DomainE [elim!]:
384 "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
385 by (iprover dest!: iffD1 [OF Domain_iff])
387 lemma Domain_empty [simp]: "Domain {} = {}"
390 lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"
393 lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
396 lemma Domain_Id [simp]: "Domain Id = UNIV"
399 lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
402 lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
405 lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
408 lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
411 lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
414 lemma Domain_converse[simp]: "Domain(r^-1) = Range r"
415 by(auto simp:Range_def)
417 lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
420 lemma fst_eq_Domain: "fst ` R = Domain R"
421 by (auto intro!:image_eqI)
423 lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
426 lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
430 subsection {* Range *}
432 lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
433 by (simp add: Domain_def Range_def)
435 lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
436 by (unfold Range_def) (iprover intro!: converseI DomainI)
438 lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
439 by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
441 lemma Range_empty [simp]: "Range {} = {}"
444 lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}"
447 lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
450 lemma Range_Id [simp]: "Range Id = UNIV"
453 lemma Range_Id_on [simp]: "Range (Id_on A) = A"
456 lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
459 lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
462 lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
465 lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
468 lemma Range_converse[simp]: "Range(r^-1) = Domain r"
471 lemma snd_eq_Range: "snd ` R = Range R"
472 by (auto intro!:image_eqI)
475 subsection {* Field *}
477 lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
478 by(auto simp:Field_def Domain_def Range_def)
480 lemma Field_empty[simp]: "Field {} = {}"
481 by(auto simp:Field_def)
483 lemma Field_insert[simp]: "Field (insert (a,b) r) = {a,b} \<union> Field r"
484 by(auto simp:Field_def)
486 lemma Field_Un[simp]: "Field (r \<union> s) = Field r \<union> Field s"
487 by(auto simp:Field_def)
489 lemma Field_Union[simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
490 by(auto simp:Field_def)
492 lemma Field_converse[simp]: "Field(r^-1) = Field r"
493 by(auto simp:Field_def)
496 subsection {* Image of a set under a relation *}
498 declare Image_def [no_atp]
500 lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
501 by (simp add: Image_def)
503 lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
504 by (simp add: Image_def)
506 lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
507 by (rule Image_iff [THEN trans]) simp
509 lemma ImageI [intro,no_atp]: "(a, b) : r ==> a : A ==> b : r``A"
510 by (unfold Image_def) blast
512 lemma ImageE [elim!]:
513 "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
514 by (unfold Image_def) (iprover elim!: CollectE bexE)
516 lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
517 -- {* This version's more effective when we already have the required @{text a} *}
520 lemma Image_empty [simp]: "R``{} = {}"
523 lemma Image_Id [simp]: "Id `` A = A"
526 lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
529 lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
533 "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
534 by (simp add: single_valued_def, blast)
536 lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
539 lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
542 lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
543 by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
545 lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
546 -- {* NOT suitable for rewriting *}
549 lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
552 lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
555 lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
558 text{*Converse inclusion requires some assumptions*}
560 "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
561 apply (rule equalityI)
562 apply (rule Image_INT_subset)
563 apply (simp add: single_valued_def, blast)
566 lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
570 subsection {* Single valued relations *}
572 lemma single_valuedI:
573 "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
574 by (unfold single_valued_def)
576 lemma single_valuedD:
577 "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
578 by (simp add: single_valued_def)
580 lemma single_valued_rel_comp:
581 "single_valued r ==> single_valued s ==> single_valued (r O s)"
582 by (unfold single_valued_def) blast
584 lemma single_valued_subset:
585 "r \<subseteq> s ==> single_valued s ==> single_valued r"
586 by (unfold single_valued_def) blast
588 lemma single_valued_Id [simp]: "single_valued Id"
589 by (unfold single_valued_def) blast
591 lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
592 by (unfold single_valued_def) blast
595 subsection {* Graphs given by @{text Collect} *}
597 lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
600 lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
603 lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
607 subsection {* Inverse image *}
609 lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
610 by (unfold sym_def inv_image_def) blast
612 lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
613 apply (unfold trans_def inv_image_def)
614 apply (simp (no_asm))
618 lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
619 by (auto simp:inv_image_def)
621 lemma converse_inv_image[simp]: "(inv_image R f)^-1 = inv_image (R^-1) f"
622 unfolding inv_image_def converse_def by auto
625 subsection {* Finiteness *}
627 lemma finite_converse [iff]: "finite (r^-1) = finite r"
628 apply (subgoal_tac "r^-1 = (%(x,y). (y,x))`r")
631 apply (erule finite_imageD [unfolded inj_on_def])
632 apply (simp split add: split_split)
633 apply (erule finite_imageI)
634 apply (simp add: converse_def image_def, auto)
636 prefer 2 apply assumption
640 lemma finite_Domain: "finite r ==> finite (Domain r)"
641 by (induct set: finite) (auto simp add: Domain_insert)
643 lemma finite_Range: "finite r ==> finite (Range r)"
644 by (induct set: finite) (auto simp add: Range_insert)
646 lemma finite_Field: "finite r ==> finite (Field r)"
647 -- {* A finite relation has a finite field (@{text "= domain \<union> range"}. *}
648 apply (induct set: finite)
649 apply (auto simp add: Field_def Domain_insert Range_insert)
653 subsection {* Miscellaneous *}
655 text {* Version of @{thm[source] lfp_induct} for binary relations *}
658 lfp_induct_set [of "(a, b)", split_format (complete)]
660 text {* Version of @{thm[source] subsetI} for binary relations *}
662 lemma subrelI: "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s"