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" ("(_^-1)" [1000] 999)
17 "r^-1 == {(y, x). (x, y) : r}"
19 const_syntax (xsymbols)
20 converse ("(_\<inverse>)" [1000] 999)
23 rel_comp :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set" (infixr "O" 75)
24 "r O s == {(x,z). EX y. (x, y) : s & (y, z) : r}"
26 Image :: "[('a * 'b) set, 'a set] => 'b set" (infixl "``" 90)
27 "r `` s == {y. EX x:s. (x,y):r}"
29 Id :: "('a * 'a) set" -- {* the identity relation *}
30 "Id == {p. EX x. p = (x,x)}"
32 diag :: "'a set => ('a * 'a) set" -- {* diagonal: identity over a set *}
33 "diag A == \<Union>x\<in>A. {(x,x)}"
35 Domain :: "('a * 'b) set => 'a set"
36 "Domain r == {x. EX y. (x,y):r}"
38 Range :: "('a * 'b) set => 'b set"
39 "Range r == Domain(r^-1)"
41 Field :: "('a * 'a) set => 'a set"
42 "Field r == Domain r \<union> Range r"
44 refl :: "['a set, ('a * 'a) set] => bool" -- {* reflexivity over a set *}
45 "refl A r == r \<subseteq> A \<times> A & (ALL x: A. (x,x) : r)"
47 sym :: "('a * 'a) set => bool" -- {* symmetry predicate *}
48 "sym r == ALL x y. (x,y): r --> (y,x): r"
50 antisym:: "('a * 'a) set => bool" -- {* antisymmetry predicate *}
51 "antisym r == ALL x y. (x,y):r --> (y,x):r --> x=y"
53 trans :: "('a * 'a) set => bool" -- {* transitivity predicate *}
54 "trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
56 single_valued :: "('a * 'b) set => bool"
57 "single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)"
59 inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set"
60 "inv_image r f == {(x, y). (f x, f y) : r}"
63 reflexive :: "('a * 'a) set => bool" -- {* reflexivity over a type *}
64 "reflexive == refl UNIV"
67 subsection {* The identity relation *}
69 lemma IdI [intro]: "(a, a) : Id"
72 lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
73 by (unfold Id_def) (iprover elim: CollectE)
75 lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
76 by (unfold Id_def) blast
78 lemma reflexive_Id: "reflexive Id"
79 by (simp add: refl_def)
81 lemma antisym_Id: "antisym Id"
82 -- {* A strange result, since @{text Id} is also symmetric. *}
83 by (simp add: antisym_def)
85 lemma sym_Id: "sym Id"
86 by (simp add: sym_def)
88 lemma trans_Id: "trans Id"
89 by (simp add: trans_def)
92 subsection {* Diagonal: identity over a set *}
94 lemma diag_empty [simp]: "diag {} = {}"
95 by (simp add: diag_def)
97 lemma diag_eqI: "a = b ==> a : A ==> (a, b) : diag A"
98 by (simp add: diag_def)
100 lemma diagI [intro!]: "a : A ==> (a, a) : diag A"
101 by (rule diag_eqI) (rule refl)
104 "c : diag A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
105 -- {* The general elimination rule. *}
106 by (unfold diag_def) (iprover elim!: UN_E singletonE)
108 lemma diag_iff: "((x, y) : diag A) = (x = y & x : A)"
111 lemma diag_subset_Times: "diag A \<subseteq> A \<times> A"
115 subsection {* Composition of two relations *}
117 lemma rel_compI [intro]:
118 "(a, b) : s ==> (b, c) : r ==> (a, c) : r O s"
119 by (unfold rel_comp_def) blast
121 lemma rel_compE [elim!]: "xz : r O s ==>
122 (!!x y z. xz = (x, z) ==> (x, y) : s ==> (y, z) : r ==> P) ==> P"
123 by (unfold rel_comp_def) (iprover elim!: CollectE splitE exE conjE)
126 "(a, c) : r O s ==> (!!y. (a, y) : s ==> (y, c) : r ==> P) ==> P"
127 by (iprover elim: rel_compE Pair_inject ssubst)
129 lemma R_O_Id [simp]: "R O Id = R"
132 lemma Id_O_R [simp]: "Id O R = R"
135 lemma O_assoc: "(R O S) O T = R O (S O T)"
138 lemma trans_O_subset: "trans r ==> r O r \<subseteq> r"
139 by (unfold trans_def) blast
141 lemma rel_comp_mono: "r' \<subseteq> r ==> s' \<subseteq> s ==> (r' O s') \<subseteq> (r O s)"
144 lemma rel_comp_subset_Sigma:
145 "s \<subseteq> A \<times> B ==> r \<subseteq> B \<times> C ==> (r O s) \<subseteq> A \<times> C"
149 subsection {* Reflexivity *}
151 lemma reflI: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl A r"
152 by (unfold refl_def) (iprover intro!: ballI)
154 lemma reflD: "refl A r ==> a : A ==> (a, a) : r"
155 by (unfold refl_def) blast
157 lemma reflD1: "refl A r ==> (x, y) : r ==> x : A"
158 by (unfold refl_def) blast
160 lemma reflD2: "refl A r ==> (x, y) : r ==> y : A"
161 by (unfold refl_def) blast
163 lemma refl_Int: "refl A r ==> refl B s ==> refl (A \<inter> B) (r \<inter> s)"
164 by (unfold refl_def) blast
166 lemma refl_Un: "refl A r ==> refl B s ==> refl (A \<union> B) (r \<union> s)"
167 by (unfold refl_def) blast
170 "ALL x:S. refl (A x) (r x) ==> refl (INTER S A) (INTER S r)"
171 by (unfold refl_def) fast
174 "ALL x:S. refl (A x) (r x) \<Longrightarrow> refl (UNION S A) (UNION S r)"
175 by (unfold refl_def) blast
177 lemma refl_diag: "refl A (diag A)"
178 by (rule reflI [OF diag_subset_Times diagI])
181 subsection {* Antisymmetry *}
184 "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
185 by (unfold antisym_def) iprover
187 lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
188 by (unfold antisym_def) iprover
190 lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
191 by (unfold antisym_def) blast
193 lemma antisym_empty [simp]: "antisym {}"
194 by (unfold antisym_def) blast
196 lemma antisym_diag [simp]: "antisym (diag A)"
197 by (unfold antisym_def) blast
200 subsection {* Symmetry *}
202 lemma symI: "(!!a b. (a, b) : r ==> (b, a) : r) ==> sym r"
203 by (unfold sym_def) iprover
205 lemma symD: "sym r ==> (a, b) : r ==> (b, a) : r"
206 by (unfold sym_def, blast)
208 lemma sym_Int: "sym r ==> sym s ==> sym (r \<inter> s)"
209 by (fast intro: symI dest: symD)
211 lemma sym_Un: "sym r ==> sym s ==> sym (r \<union> s)"
212 by (fast intro: symI dest: symD)
214 lemma sym_INTER: "ALL x:S. sym (r x) ==> sym (INTER S r)"
215 by (fast intro: symI dest: symD)
217 lemma sym_UNION: "ALL x:S. sym (r x) ==> sym (UNION S r)"
218 by (fast intro: symI dest: symD)
220 lemma sym_diag [simp]: "sym (diag A)"
221 by (rule symI) clarify
224 subsection {* Transitivity *}
227 "(!!x y z. (x, y) : r ==> (y, z) : r ==> (x, z) : r) ==> trans r"
228 by (unfold trans_def) iprover
230 lemma transD: "trans r ==> (a, b) : r ==> (b, c) : r ==> (a, c) : r"
231 by (unfold trans_def) iprover
233 lemma trans_Int: "trans r ==> trans s ==> trans (r \<inter> s)"
234 by (fast intro: transI elim: transD)
236 lemma trans_INTER: "ALL x:S. trans (r x) ==> trans (INTER S r)"
237 by (fast intro: transI elim: transD)
239 lemma trans_diag [simp]: "trans (diag A)"
240 by (fast intro: transI elim: transD)
243 subsection {* Converse *}
245 lemma converse_iff [iff]: "((a,b): r^-1) = ((b,a) : r)"
246 by (simp add: converse_def)
248 lemma converseI[sym]: "(a, b) : r ==> (b, a) : r^-1"
249 by (simp add: converse_def)
251 lemma converseD[sym]: "(a,b) : r^-1 ==> (b, a) : r"
252 by (simp add: converse_def)
254 lemma converseE [elim!]:
255 "yx : r^-1 ==> (!!x y. yx = (y, x) ==> (x, y) : r ==> P) ==> P"
256 -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
257 by (unfold converse_def) (iprover elim!: CollectE splitE bexE)
259 lemma converse_converse [simp]: "(r^-1)^-1 = r"
260 by (unfold converse_def) blast
262 lemma converse_rel_comp: "(r O s)^-1 = s^-1 O r^-1"
265 lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
268 lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
271 lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
274 lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
277 lemma converse_Id [simp]: "Id^-1 = Id"
280 lemma converse_diag [simp]: "(diag A)^-1 = diag A"
283 lemma refl_converse [simp]: "refl A (converse r) = refl A r"
284 by (unfold refl_def) auto
286 lemma sym_converse [simp]: "sym (converse r) = sym r"
287 by (unfold sym_def) blast
289 lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
290 by (unfold antisym_def) blast
292 lemma trans_converse [simp]: "trans (converse r) = trans r"
293 by (unfold trans_def) blast
295 lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
296 by (unfold sym_def) fast
298 lemma sym_Un_converse: "sym (r \<union> r^-1)"
299 by (unfold sym_def) blast
301 lemma sym_Int_converse: "sym (r \<inter> r^-1)"
302 by (unfold sym_def) blast
305 subsection {* Domain *}
307 lemma Domain_iff: "(a : Domain r) = (EX y. (a, y) : r)"
308 by (unfold Domain_def) blast
310 lemma DomainI [intro]: "(a, b) : r ==> a : Domain r"
311 by (iprover intro!: iffD2 [OF Domain_iff])
313 lemma DomainE [elim!]:
314 "a : Domain r ==> (!!y. (a, y) : r ==> P) ==> P"
315 by (iprover dest!: iffD1 [OF Domain_iff])
317 lemma Domain_empty [simp]: "Domain {} = {}"
320 lemma Domain_insert: "Domain (insert (a, b) r) = insert a (Domain r)"
323 lemma Domain_Id [simp]: "Domain Id = UNIV"
326 lemma Domain_diag [simp]: "Domain (diag A) = A"
329 lemma Domain_Un_eq: "Domain(A \<union> B) = Domain(A) \<union> Domain(B)"
332 lemma Domain_Int_subset: "Domain(A \<inter> B) \<subseteq> Domain(A) \<inter> Domain(B)"
335 lemma Domain_Diff_subset: "Domain(A) - Domain(B) \<subseteq> Domain(A - B)"
338 lemma Domain_Union: "Domain (Union S) = (\<Union>A\<in>S. Domain A)"
341 lemma Domain_mono: "r \<subseteq> s ==> Domain r \<subseteq> Domain s"
345 subsection {* Range *}
347 lemma Range_iff: "(a : Range r) = (EX y. (y, a) : r)"
348 by (simp add: Domain_def Range_def)
350 lemma RangeI [intro]: "(a, b) : r ==> b : Range r"
351 by (unfold Range_def) (iprover intro!: converseI DomainI)
353 lemma RangeE [elim!]: "b : Range r ==> (!!x. (x, b) : r ==> P) ==> P"
354 by (unfold Range_def) (iprover elim!: DomainE dest!: converseD)
356 lemma Range_empty [simp]: "Range {} = {}"
359 lemma Range_insert: "Range (insert (a, b) r) = insert b (Range r)"
362 lemma Range_Id [simp]: "Range Id = UNIV"
365 lemma Range_diag [simp]: "Range (diag A) = A"
368 lemma Range_Un_eq: "Range(A \<union> B) = Range(A) \<union> Range(B)"
371 lemma Range_Int_subset: "Range(A \<inter> B) \<subseteq> Range(A) \<inter> Range(B)"
374 lemma Range_Diff_subset: "Range(A) - Range(B) \<subseteq> Range(A - B)"
377 lemma Range_Union: "Range (Union S) = (\<Union>A\<in>S. Range A)"
381 subsection {* Image of a set under a relation *}
383 lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
384 by (simp add: Image_def)
386 lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
387 by (simp add: Image_def)
389 lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
390 by (rule Image_iff [THEN trans]) simp
392 lemma ImageI [intro]: "(a, b) : r ==> a : A ==> b : r``A"
393 by (unfold Image_def) blast
395 lemma ImageE [elim!]:
396 "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
397 by (unfold Image_def) (iprover elim!: CollectE bexE)
399 lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
400 -- {* This version's more effective when we already have the required @{text a} *}
403 lemma Image_empty [simp]: "R``{} = {}"
406 lemma Image_Id [simp]: "Id `` A = A"
409 lemma Image_diag [simp]: "diag A `` B = A \<inter> B"
412 lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
416 "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
417 by (simp add: single_valued_def, blast)
419 lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
422 lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
425 lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
426 by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
428 lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
429 -- {* NOT suitable for rewriting *}
432 lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
435 lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
438 lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
441 text{*Converse inclusion requires some assumptions*}
443 "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
444 apply (rule equalityI)
445 apply (rule Image_INT_subset)
446 apply (simp add: single_valued_def, blast)
449 lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
453 subsection {* Single valued relations *}
455 lemma single_valuedI:
456 "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
457 by (unfold single_valued_def)
459 lemma single_valuedD:
460 "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
461 by (simp add: single_valued_def)
463 lemma single_valued_rel_comp:
464 "single_valued r ==> single_valued s ==> single_valued (r O s)"
465 by (unfold single_valued_def) blast
467 lemma single_valued_subset:
468 "r \<subseteq> s ==> single_valued s ==> single_valued r"
469 by (unfold single_valued_def) blast
471 lemma single_valued_Id [simp]: "single_valued Id"
472 by (unfold single_valued_def) blast
474 lemma single_valued_diag [simp]: "single_valued (diag A)"
475 by (unfold single_valued_def) blast
478 subsection {* Graphs given by @{text Collect} *}
480 lemma Domain_Collect_split [simp]: "Domain{(x,y). P x y} = {x. EX y. P x y}"
483 lemma Range_Collect_split [simp]: "Range{(x,y). P x y} = {y. EX x. P x y}"
486 lemma Image_Collect_split [simp]: "{(x,y). P x y} `` A = {y. EX x:A. P x y}"
490 subsection {* Inverse image *}
492 lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
493 by (unfold sym_def inv_image_def) blast
495 lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
496 apply (unfold trans_def inv_image_def)
497 apply (simp (no_asm))