1 (* Title: HOL/Relation.thy
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
4 Copyright 1996 University of Cambridge
13 subsection {* Definitions *}
16 converse :: "('a * 'b) set => ('b * 'a) set"
17 ("(_^-1)" [1000] 999) where
18 "r^-1 == {(y, x). (x, y) : r}"
21 converse ("(_\<inverse>)" [1000] 999)
24 rel_comp :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set"
26 "r O s == {(x,z). EX y. (x, y) : s & (y, z) : r}"
29 Image :: "[('a * 'b) set, 'a set] => 'b set"
30 (infixl "``" 90) where
31 "r `` s == {y. EX x:s. (x,y):r}"
34 Id :: "('a * 'a) set" where -- {* the identity relation *}
35 "Id == {p. EX x. p = (x,x)}"
38 diag :: "'a set => ('a * 'a) set" where -- {* diagonal: identity over a set *}
39 "diag A == \<Union>x\<in>A. {(x,x)}"
42 Domain :: "('a * 'b) set => 'a set" where
43 "Domain r == {x. EX y. (x,y):r}"
46 Range :: "('a * 'b) set => 'b set" where
47 "Range r == Domain(r^-1)"
50 Field :: "('a * 'a) set => 'a set" where
51 "Field r == Domain r \<union> Range r"
54 refl :: "['a set, ('a * 'a) set] => bool" where -- {* reflexivity over a set *}
55 "refl A r == r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)"
58 sym :: "('a * 'a) set => bool" where -- {* symmetry predicate *}
59 "sym r == ALL x y. (x,y): r --> (y,x): r"
62 antisym :: "('a * 'a) set => bool" where -- {* antisymmetry predicate *}
63 "antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y"
66 trans :: "('a * 'a) set => bool" where -- {* transitivity predicate *}
67 "trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
70 single_valued :: "('a * 'b) set => bool" where
71 "single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)"
74 inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set" where
75 "inv_image r f == {(x, y). (f x, f y) : r}"
78 reflexive :: "('a * 'a) set => bool" where -- {* reflexivity over a type *}
79 "reflexive == refl UNIV"
82 subsection {* The identity relation *}
84 lemma IdI [intro]: "(a, a) : Id"
87 lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
88 by (unfold Id_def) (iprover elim: CollectE)
90 lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
91 by (unfold Id_def) blast
93 lemma reflexive_Id: "reflexive Id"
94 by (simp add: refl_def)
96 lemma antisym_Id: "antisym Id"
97 -- {* A strange result, since @{text Id} is also symmetric. *}
98 by (simp add: antisym_def)
100 lemma sym_Id: "sym Id"
101 by (simp add: sym_def)
103 lemma trans_Id: "trans Id"
104 by (simp add: trans_def)
107 subsection {* Diagonal: identity over a set *}
109 lemma diag_empty [simp]: "diag {} = {}"
110 by (simp add: diag_def)
112 lemma diag_eqI: "a = b ==> a : A ==> (a, b) : diag A"
113 by (simp add: diag_def)
115 lemma diagI [intro!]: "a : A ==> (a, a) : diag A"
116 by (rule diag_eqI) (rule refl)
119 "c : diag A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
120 -- {* The general elimination rule. *}
121 by (unfold diag_def) (iprover elim!: UN_E singletonE)
123 lemma diag_iff: "((x, y) : diag A) = (x = y & x : A)"
126 lemma diag_subset_Times: "diag A \<subseteq> A \<times> A"
130 subsection {* Composition of two relations *}
132 lemma rel_compI [intro]:
133 "(a, b) : s ==> (b, c) : r ==> (a, c) : r O s"
134 by (unfold rel_comp_def) blast
136 lemma rel_compE [elim!]: "xz : r O s ==>
137 (!!x y z. xz = (x, z) ==> (x, y) : s ==> (y, z) : r ==> P) ==> P"
138 by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE)
141 "(a, c) : r O s ==> (!!y. (a, y) : s ==> (y, c) : r ==> P) ==> P"
142 by (iprover elim: rel_compE Pair_inject ssubst)
144 lemma R_O_Id [simp]: "R O Id = R"
147 lemma Id_O_R [simp]: "Id O R = R"
150 lemma O_assoc: "(R O S) O T = R O (S O T)"
153 lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
154 by (unfold trans_def) blast
156 lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
159 lemma rel_comp_subset_Sigma:
160 "s \<subseteq> A \<times> B ==> r \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
164 subsection {* Reflexivity *}
166 lemma reflI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl A r"
167 by (unfold refl_def) (iprover intro!: ballI)
169 lemma reflD: "refl A r ==> a : A ==> (a, a) : r"
170 by (unfold refl_def) blast
172 lemma reflD1: "refl A r ==> (x, y) : r ==> x : A"
173 by (unfold refl_def) blast
175 lemma reflD2: "refl A r ==> (x, y) : r ==> y : A"
176 by (unfold refl_def) blast
178 lemma refl_Int: "refl A r ==> refl B s ==> refl (A \<inter> B) (r \<inter> s)"
179 by (unfold refl_def) blast
181 lemma refl_Un: "refl A r ==> refl B s ==> refl (A \<union> B) (r \<union> s)"
182 by (unfold refl_def) blast
185 "ALL x:S. refl (A x) (r x) ==> refl (INTER S A) (INTER S r)"
186 by (unfold refl_def) fast
189 "ALL x:S. refl (A x) (r x) \<Longrightarrow> refl (UNION S A) (UNION S r)"
190 by (unfold refl_def) blast
192 lemma refl_diag: "refl A (diag A)"
193 by (rule reflI [OF diag_subset_Times diagI])
196 subsection {* Antisymmetry *}
199 "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
200 by (unfold antisym_def) iprover
202 lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
203 by (unfold antisym_def) iprover
205 lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
206 by (unfold antisym_def) blast
208 lemma antisym_empty [simp]: "antisym {}"
209 by (unfold antisym_def) blast
211 lemma antisym_diag [simp]: "antisym (diag A)"
212 by (unfold antisym_def) blast
215 subsection {* Symmetry *}
217 lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
218 by (unfold sym_def) iprover
220 lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r"
221 by (unfold sym_def, blast)
223 lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)"
224 by (fast intro: symI dest: symD)
226 lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)"
227 by (fast intro: symI dest: symD)
229 lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)"
230 by (fast intro: symI dest: symD)
232 lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)"
233 by (fast intro: symI dest: symD)
235 lemma sym_diag [simp]: "sym (diag A)"
236 by (rule symI) clarify
239 subsection {* Transitivity *}
242 "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
243 by (unfold trans_def) iprover
245 lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
246 by (unfold trans_def) iprover
248 lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)"
249 by (fast intro: transI elim: transD)
251 lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)"
252 by (fast intro: transI elim: transD)
254 lemma trans_diag [simp]: "trans (diag A)"
255 by (fast intro: transI elim: transD)
258 subsection {* Converse *}
260 lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
261 by (simp add: converse_def)
263 lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
264 by (simp add: converse_def)
266 lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
267 by (simp add: converse_def)
269 lemma converseE [elim!]:
270 "yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
271 -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
272 by (unfold converse_def) (iprover elim!: CollectE splitE bexE)
274 lemma converse_converse [simp]: "(r^-1)^-1 = r"
275 by (unfold converse_def) blast
277 lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
280 lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
283 lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
286 lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
289 lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
292 lemma converse_Id [simp]: "Id^-1 = Id"
295 lemma converse_diag [simp]: "(diag A)^-1 = diag A"
298 lemma refl_converse [simp]: "refl A (converse r) = refl A r"
299 by (unfold refl_def) auto
301 lemma sym_converse [simp]: "sym (converse r) = sym r"
302 by (unfold sym_def) blast
304 lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
305 by (unfold antisym_def) blast
307 lemma trans_converse [simp]: "trans (converse r) = trans r"
308 by (unfold trans_def) blast
310 lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
311 by (unfold sym_def) fast
313 lemma sym_Un_converse: "sym (r \<union> r^-1)"
314 by (unfold sym_def) blast
316 lemma sym_Int_converse: "sym (r \<inter> r^-1)"
317 by (unfold sym_def) blast
320 subsection {* Domain *}
322 lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
323 by (unfold Domain_def) blast
325 lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
326 by (iprover intro!: iffD2 [OF Domain_iff])
328 lemma DomainE [elim!]:
329 "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
330 by (iprover dest!: iffD1 [OF Domain_iff])
332 lemma Domain_empty [simp]: "Domain {} = {}"
335 lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
338 lemma Domain_Id [simp]: "Domain Id = UNIV"
341 lemma Domain_diag [simp]: "Domain (diag A) = A"
344 lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
347 lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
350 lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
353 lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
356 lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
360 subsection {* Range *}
362 lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
363 by (simp add: Domain_def Range_def)
365 lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
366 by (unfold Range_def) (iprover intro!: converseI DomainI)
368 lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
369 by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
371 lemma Range_empty [simp]: "Range {} = {}"
374 lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
377 lemma Range_Id [simp]: "Range Id = UNIV"
380 lemma Range_diag [simp]: "Range (diag A) = A"
383 lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
386 lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
389 lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
392 lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
396 subsection {* Image of a set under a relation *}
398 lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
399 by (simp add: Image_def)
401 lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
402 by (simp add: Image_def)
404 lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
405 by (rule Image_iff [THEN trans]) simp
407 lemma ImageI [intro]: "(a, b) : r ==> a : A ==> b : r``A"
408 by (unfold Image_def) blast
410 lemma ImageE [elim!]:
411 "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
412 by (unfold Image_def) (iprover elim!: CollectE bexE)
414 lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
415 -- {* This version's more effective when we already have the required @{text a} *}
418 lemma Image_empty [simp]: "R``{} = {}"
421 lemma Image_Id [simp]: "Id `` A = A"
424 lemma Image_diag [simp]: "diag A `` B = A \<inter> B"
427 lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
431 "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
432 by (simp add: single_valued_def, blast)
434 lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
437 lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
440 lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
441 by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
443 lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
444 -- {* NOT suitable for rewriting *}
447 lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
450 lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
453 lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
456 text{*Converse inclusion requires some assumptions*}
458 "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
459 apply (rule equalityI)
460 apply (rule Image_INT_subset)
461 apply (simp add: single_valued_def, blast)
464 lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
468 subsection {* Single valued relations *}
470 lemma single_valuedI:
471 "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
472 by (unfold single_valued_def)
474 lemma single_valuedD:
475 "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
476 by (simp add: single_valued_def)
478 lemma single_valued_rel_comp:
479 "single_valued r ==> single_valued s ==> single_valued (r O s)"
480 by (unfold single_valued_def) blast
482 lemma single_valued_subset:
483 "r \<subseteq> s ==> single_valued s ==> single_valued r"
484 by (unfold single_valued_def) blast
486 lemma single_valued_Id [simp]: "single_valued Id"
487 by (unfold single_valued_def) blast
489 lemma single_valued_diag [simp]: "single_valued (diag A)"
490 by (unfold single_valued_def) blast
493 subsection {* Graphs given by @{text Collect} *}
495 lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
498 lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
501 lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
505 subsection {* Inverse image *}
507 lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
508 by (unfold sym_def inv_image_def) blast
510 lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
511 apply (unfold trans_def inv_image_def)
512 apply (simp (no_asm))