author | wenzelm |
Wed, 29 Sep 1999 14:38:03 +0200 | |
changeset 7648 | 8258b93cdd32 |
parent 7516 | a1d476251238 |
child 7713 | f4fe9d620107 |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: HOL/equalities |
923 | 2 |
ID: $Id$ |
1465 | 3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
923 | 4 |
Copyright 1994 University of Cambridge |
5 |
||
6 |
Equalities involving union, intersection, inclusion, etc. |
|
7 |
*) |
|
8 |
||
9 |
writeln"File HOL/equalities"; |
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10 |
||
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AddSIs [equalityI]; |
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1548 | 13 |
section "{}"; |
14 |
||
5069 | 15 |
Goal "{x. False} = {}"; |
2891 | 16 |
by (Blast_tac 1); |
1531 | 17 |
qed "Collect_False_empty"; |
18 |
Addsimps [Collect_False_empty]; |
|
19 |
||
5069 | 20 |
Goal "(A <= {}) = (A = {})"; |
2891 | 21 |
by (Blast_tac 1); |
1531 | 22 |
qed "subset_empty"; |
23 |
Addsimps [subset_empty]; |
|
24 |
||
5069 | 25 |
Goalw [psubset_def] "~ (A < {})"; |
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by (Blast_tac 1); |
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27 |
qed "not_psubset_empty"; |
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AddIffs [not_psubset_empty]; |
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29 |
|
5069 | 30 |
Goal "{x. P x | Q x} = {x. P x} Un {x. Q x}"; |
4748 | 31 |
by (Blast_tac 1); |
32 |
qed "Collect_disj_eq"; |
|
33 |
||
5069 | 34 |
Goal "{x. P x & Q x} = {x. P x} Int {x. Q x}"; |
4748 | 35 |
by (Blast_tac 1); |
36 |
qed "Collect_conj_eq"; |
|
37 |
||
38 |
||
1548 | 39 |
section "insert"; |
923 | 40 |
|
1531 | 41 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a {}*) |
5069 | 42 |
Goal "insert a A = {a} Un A"; |
2891 | 43 |
by (Blast_tac 1); |
1531 | 44 |
qed "insert_is_Un"; |
45 |
||
5069 | 46 |
Goal "insert a A ~= {}"; |
4089 | 47 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
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Added insert_not_empty, UN_empty and UN_insert (to set_ss).
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48 |
qed"insert_not_empty"; |
1531 | 49 |
Addsimps[insert_not_empty]; |
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50 |
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Added insert_not_empty, UN_empty and UN_insert (to set_ss).
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51 |
bind_thm("empty_not_insert",insert_not_empty RS not_sym); |
1531 | 52 |
Addsimps[empty_not_insert]; |
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53 |
|
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Goal "a:A ==> insert a A = A"; |
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by (Blast_tac 1); |
923 | 56 |
qed "insert_absorb"; |
6832 | 57 |
(* Addsimps [insert_absorb] causes recursive calls |
58 |
when there are nested inserts, with QUADRATIC running time |
|
4605 | 59 |
*) |
923 | 60 |
|
5069 | 61 |
Goal "insert x (insert x A) = insert x A"; |
2891 | 62 |
by (Blast_tac 1); |
1531 | 63 |
qed "insert_absorb2"; |
64 |
Addsimps [insert_absorb2]; |
|
65 |
||
5069 | 66 |
Goal "insert x (insert y A) = insert y (insert x A)"; |
2891 | 67 |
by (Blast_tac 1); |
1879 | 68 |
qed "insert_commute"; |
69 |
||
5069 | 70 |
Goal "(insert x A <= B) = (x:B & A <= B)"; |
2891 | 71 |
by (Blast_tac 1); |
923 | 72 |
qed "insert_subset"; |
1531 | 73 |
Addsimps[insert_subset]; |
74 |
||
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75 |
Goal "insert a A ~= insert a B ==> A ~= B"; |
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|
76 |
by (Blast_tac 1); |
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Distributed Psubset stuff to basic set theory files, incl Finite.
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|
77 |
qed "insert_lim"; |
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78 |
|
1531 | 79 |
(* use new B rather than (A-{a}) to avoid infinite unfolding *) |
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80 |
Goal "a:A ==> ? B. A = insert a B & a ~: B"; |
1553 | 81 |
by (res_inst_tac [("x","A-{a}")] exI 1); |
2891 | 82 |
by (Blast_tac 1); |
1531 | 83 |
qed "mk_disjoint_insert"; |
923 | 84 |
|
4882 | 85 |
bind_thm ("insert_Collect", prove_goal thy |
5590 | 86 |
"insert a (Collect P) = {u. u ~= a --> P u}" (K [Auto_tac])); |
4882 | 87 |
|
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88 |
Goal "u: A ==> (UN x:A. insert a (B x)) = insert a (UN x:A. B x)"; |
2891 | 89 |
by (Blast_tac 1); |
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90 |
qed "UN_insert_distrib"; |
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91 |
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section "``"; |
923 | 93 |
|
5069 | 94 |
Goal "f``{} = {}"; |
2891 | 95 |
by (Blast_tac 1); |
923 | 96 |
qed "image_empty"; |
1531 | 97 |
Addsimps[image_empty]; |
923 | 98 |
|
5069 | 99 |
Goal "f``insert a B = insert (f a) (f``B)"; |
2891 | 100 |
by (Blast_tac 1); |
923 | 101 |
qed "image_insert"; |
1531 | 102 |
Addsimps[image_insert]; |
923 | 103 |
|
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104 |
Goal "x:A ==> (%x. c) `` A = {c}"; |
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|
105 |
by (Blast_tac 1); |
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106 |
qed "image_constant"; |
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107 |
|
5069 | 108 |
Goal "f``(g``A) = (%x. f (g x)) `` A"; |
3457 | 109 |
by (Blast_tac 1); |
4059 | 110 |
qed "image_image"; |
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|
111 |
|
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|
112 |
Goal "x:A ==> insert (f x) (f``A) = f``A"; |
2891 | 113 |
by (Blast_tac 1); |
1884 | 114 |
qed "insert_image"; |
115 |
Addsimps [insert_image]; |
|
116 |
||
5069 | 117 |
Goal "(f``A = {}) = (A = {})"; |
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118 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
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Moved image_is_empty from Finite.ML to equalities.ML
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|
119 |
qed "image_is_empty"; |
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120 |
AddIffs [image_is_empty]; |
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121 |
|
5281 | 122 |
Goal "f `` {x. P x} = {f x | x. P x}"; |
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123 |
by (Blast_tac 1); |
5281 | 124 |
qed "image_Collect"; |
125 |
Addsimps [image_Collect]; |
|
126 |
||
5590 | 127 |
Goalw [image_def] "(%x. if P x then f x else g x) `` S \ |
128 |
\ = (f `` (S Int {x. P x})) Un (g `` (S Int {x. ~(P x)}))"; |
|
4686 | 129 |
by (Simp_tac 1); |
2891 | 130 |
by (Blast_tac 1); |
1748 | 131 |
qed "if_image_distrib"; |
132 |
Addsimps[if_image_distrib]; |
|
133 |
||
5590 | 134 |
val prems = Goal "[|M = N; !!x. x:N ==> f x = g x|] ==> f``M = g``N"; |
4136 | 135 |
by (simp_tac (simpset() addsimps image_def::prems) 1); |
136 |
qed "image_cong"; |
|
137 |
||
1748 | 138 |
|
1548 | 139 |
section "Int"; |
923 | 140 |
|
5069 | 141 |
Goal "A Int A = A"; |
2891 | 142 |
by (Blast_tac 1); |
923 | 143 |
qed "Int_absorb"; |
1531 | 144 |
Addsimps[Int_absorb]; |
923 | 145 |
|
5590 | 146 |
Goal "A Int (A Int B) = A Int B"; |
4609 | 147 |
by (Blast_tac 1); |
148 |
qed "Int_left_absorb"; |
|
149 |
||
5590 | 150 |
Goal "A Int B = B Int A"; |
2891 | 151 |
by (Blast_tac 1); |
923 | 152 |
qed "Int_commute"; |
153 |
||
5069 | 154 |
Goal "A Int (B Int C) = B Int (A Int C)"; |
4609 | 155 |
by (Blast_tac 1); |
156 |
qed "Int_left_commute"; |
|
157 |
||
5590 | 158 |
Goal "(A Int B) Int C = A Int (B Int C)"; |
2891 | 159 |
by (Blast_tac 1); |
923 | 160 |
qed "Int_assoc"; |
161 |
||
4609 | 162 |
(*Intersection is an AC-operator*) |
7648 | 163 |
bind_thms ("Int_ac", [Int_assoc, Int_left_absorb, Int_commute, Int_left_commute]); |
4609 | 164 |
|
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165 |
Goal "B<=A ==> A Int B = B"; |
4662 | 166 |
by (Blast_tac 1); |
167 |
qed "Int_absorb1"; |
|
168 |
||
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169 |
Goal "A<=B ==> A Int B = A"; |
4662 | 170 |
by (Blast_tac 1); |
171 |
qed "Int_absorb2"; |
|
172 |
||
5069 | 173 |
Goal "{} Int B = {}"; |
2891 | 174 |
by (Blast_tac 1); |
923 | 175 |
qed "Int_empty_left"; |
1531 | 176 |
Addsimps[Int_empty_left]; |
923 | 177 |
|
5069 | 178 |
Goal "A Int {} = {}"; |
2891 | 179 |
by (Blast_tac 1); |
923 | 180 |
qed "Int_empty_right"; |
1531 | 181 |
Addsimps[Int_empty_right]; |
182 |
||
5490 | 183 |
Goal "(A Int B = {}) = (A <= -B)"; |
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184 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
3356 | 185 |
qed "disjoint_eq_subset_Compl"; |
186 |
||
5069 | 187 |
Goal "UNIV Int B = B"; |
2891 | 188 |
by (Blast_tac 1); |
1531 | 189 |
qed "Int_UNIV_left"; |
190 |
Addsimps[Int_UNIV_left]; |
|
191 |
||
5069 | 192 |
Goal "A Int UNIV = A"; |
2891 | 193 |
by (Blast_tac 1); |
1531 | 194 |
qed "Int_UNIV_right"; |
195 |
Addsimps[Int_UNIV_right]; |
|
923 | 196 |
|
5069 | 197 |
Goal "A Int B = Inter{A,B}"; |
4634 | 198 |
by (Blast_tac 1); |
199 |
qed "Int_eq_Inter"; |
|
200 |
||
5590 | 201 |
Goal "A Int (B Un C) = (A Int B) Un (A Int C)"; |
2891 | 202 |
by (Blast_tac 1); |
923 | 203 |
qed "Int_Un_distrib"; |
204 |
||
5590 | 205 |
Goal "(B Un C) Int A = (B Int A) Un (C Int A)"; |
2891 | 206 |
by (Blast_tac 1); |
1618 | 207 |
qed "Int_Un_distrib2"; |
208 |
||
5069 | 209 |
Goal "(A Int B = UNIV) = (A = UNIV & B = UNIV)"; |
4089 | 210 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
1531 | 211 |
qed "Int_UNIV"; |
212 |
Addsimps[Int_UNIV]; |
|
213 |
||
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214 |
Goal "(C <= A Int B) = (C <= A & C <= B)"; |
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215 |
by (Blast_tac 1); |
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216 |
qed "Int_subset_iff"; |
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217 |
|
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218 |
|
1548 | 219 |
section "Un"; |
923 | 220 |
|
5069 | 221 |
Goal "A Un A = A"; |
2891 | 222 |
by (Blast_tac 1); |
923 | 223 |
qed "Un_absorb"; |
1531 | 224 |
Addsimps[Un_absorb]; |
923 | 225 |
|
5069 | 226 |
Goal " A Un (A Un B) = A Un B"; |
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227 |
by (Blast_tac 1); |
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228 |
qed "Un_left_absorb"; |
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229 |
|
5590 | 230 |
Goal "A Un B = B Un A"; |
2891 | 231 |
by (Blast_tac 1); |
923 | 232 |
qed "Un_commute"; |
233 |
||
5069 | 234 |
Goal "A Un (B Un C) = B Un (A Un C)"; |
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|
235 |
by (Blast_tac 1); |
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|
236 |
qed "Un_left_commute"; |
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|
237 |
|
5590 | 238 |
Goal "(A Un B) Un C = A Un (B Un C)"; |
2891 | 239 |
by (Blast_tac 1); |
923 | 240 |
qed "Un_assoc"; |
241 |
||
4609 | 242 |
(*Union is an AC-operator*) |
7648 | 243 |
bind_thms ("Un_ac", [Un_assoc, Un_left_absorb, Un_commute, Un_left_commute]); |
4609 | 244 |
|
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245 |
Goal "A<=B ==> A Un B = B"; |
4662 | 246 |
by (Blast_tac 1); |
247 |
qed "Un_absorb1"; |
|
248 |
||
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|
249 |
Goal "B<=A ==> A Un B = A"; |
4662 | 250 |
by (Blast_tac 1); |
251 |
qed "Un_absorb2"; |
|
252 |
||
5069 | 253 |
Goal "{} Un B = B"; |
2891 | 254 |
by (Blast_tac 1); |
923 | 255 |
qed "Un_empty_left"; |
1531 | 256 |
Addsimps[Un_empty_left]; |
923 | 257 |
|
5069 | 258 |
Goal "A Un {} = A"; |
2891 | 259 |
by (Blast_tac 1); |
923 | 260 |
qed "Un_empty_right"; |
1531 | 261 |
Addsimps[Un_empty_right]; |
262 |
||
5069 | 263 |
Goal "UNIV Un B = UNIV"; |
2891 | 264 |
by (Blast_tac 1); |
1531 | 265 |
qed "Un_UNIV_left"; |
266 |
Addsimps[Un_UNIV_left]; |
|
267 |
||
5069 | 268 |
Goal "A Un UNIV = UNIV"; |
2891 | 269 |
by (Blast_tac 1); |
1531 | 270 |
qed "Un_UNIV_right"; |
271 |
Addsimps[Un_UNIV_right]; |
|
923 | 272 |
|
5069 | 273 |
Goal "A Un B = Union{A,B}"; |
4634 | 274 |
by (Blast_tac 1); |
275 |
qed "Un_eq_Union"; |
|
276 |
||
5069 | 277 |
Goal "(insert a B) Un C = insert a (B Un C)"; |
2891 | 278 |
by (Blast_tac 1); |
923 | 279 |
qed "Un_insert_left"; |
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|
280 |
Addsimps[Un_insert_left]; |
923 | 281 |
|
5069 | 282 |
Goal "A Un (insert a B) = insert a (A Un B)"; |
2891 | 283 |
by (Blast_tac 1); |
1917 | 284 |
qed "Un_insert_right"; |
3384
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|
285 |
Addsimps[Un_insert_right]; |
1917 | 286 |
|
5069 | 287 |
Goal "(insert a B) Int C = (if a:C then insert a (B Int C) \ |
5590 | 288 |
\ else B Int C)"; |
4686 | 289 |
by (Simp_tac 1); |
3356 | 290 |
by (Blast_tac 1); |
291 |
qed "Int_insert_left"; |
|
292 |
||
5069 | 293 |
Goal "A Int (insert a B) = (if a:A then insert a (A Int B) \ |
5590 | 294 |
\ else A Int B)"; |
4686 | 295 |
by (Simp_tac 1); |
3356 | 296 |
by (Blast_tac 1); |
297 |
qed "Int_insert_right"; |
|
298 |
||
5590 | 299 |
Goal "A Un (B Int C) = (A Un B) Int (A Un C)"; |
2891 | 300 |
by (Blast_tac 1); |
923 | 301 |
qed "Un_Int_distrib"; |
302 |
||
5590 | 303 |
Goal "(B Int C) Un A = (B Un A) Int (C Un A)"; |
4609 | 304 |
by (Blast_tac 1); |
305 |
qed "Un_Int_distrib2"; |
|
306 |
||
5590 | 307 |
Goal "(A Int B) Un (B Int C) Un (C Int A) = \ |
308 |
\ (A Un B) Int (B Un C) Int (C Un A)"; |
|
2891 | 309 |
by (Blast_tac 1); |
923 | 310 |
qed "Un_Int_crazy"; |
311 |
||
5069 | 312 |
Goal "(A<=B) = (A Un B = B)"; |
4306
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
313 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
923 | 314 |
qed "subset_Un_eq"; |
315 |
||
5069 | 316 |
Goal "(A <= insert b C) = (A <= C | b:A & A-{b} <= C)"; |
2891 | 317 |
by (Blast_tac 1); |
923 | 318 |
qed "subset_insert_iff"; |
319 |
||
5069 | 320 |
Goal "(A Un B = {}) = (A = {} & B = {})"; |
4089 | 321 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
923 | 322 |
qed "Un_empty"; |
1531 | 323 |
Addsimps[Un_empty]; |
923 | 324 |
|
5319
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
325 |
Goal "(A Un B <= C) = (A <= C & B <= C)"; |
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
326 |
by (Blast_tac 1); |
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
327 |
qed "Un_subset_iff"; |
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
328 |
|
5331 | 329 |
Goal "(A - B) Un (A Int B) = A"; |
330 |
by (Blast_tac 1); |
|
331 |
qed "Un_Diff_Int"; |
|
332 |
||
5319
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
333 |
|
5931 | 334 |
section "Set complement"; |
923 | 335 |
|
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
336 |
Goal "A Int (-A) = {}"; |
2891 | 337 |
by (Blast_tac 1); |
923 | 338 |
qed "Compl_disjoint"; |
1531 | 339 |
Addsimps[Compl_disjoint]; |
923 | 340 |
|
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
341 |
Goal "A Un (-A) = UNIV"; |
2891 | 342 |
by (Blast_tac 1); |
923 | 343 |
qed "Compl_partition"; |
344 |
||
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
345 |
Goal "- (-A) = (A:: 'a set)"; |
2891 | 346 |
by (Blast_tac 1); |
923 | 347 |
qed "double_complement"; |
1531 | 348 |
Addsimps[double_complement]; |
923 | 349 |
|
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
350 |
Goal "-(A Un B) = (-A) Int (-B)"; |
2891 | 351 |
by (Blast_tac 1); |
923 | 352 |
qed "Compl_Un"; |
353 |
||
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
354 |
Goal "-(A Int B) = (-A) Un (-B)"; |
2891 | 355 |
by (Blast_tac 1); |
923 | 356 |
qed "Compl_Int"; |
357 |
||
5490 | 358 |
Goal "-(UN x:A. B(x)) = (INT x:A. -B(x))"; |
2891 | 359 |
by (Blast_tac 1); |
923 | 360 |
qed "Compl_UN"; |
361 |
||
5490 | 362 |
Goal "-(INT x:A. B(x)) = (UN x:A. -B(x))"; |
2891 | 363 |
by (Blast_tac 1); |
923 | 364 |
qed "Compl_INT"; |
365 |
||
4615 | 366 |
Addsimps [Compl_Un, Compl_Int, Compl_UN, Compl_INT]; |
367 |
||
923 | 368 |
(*Halmos, Naive Set Theory, page 16.*) |
369 |
||
5069 | 370 |
Goal "((A Int B) Un C = A Int (B Un C)) = (C<=A)"; |
4306
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
371 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
923 | 372 |
qed "Un_Int_assoc_eq"; |
373 |
||
374 |
||
1548 | 375 |
section "Union"; |
923 | 376 |
|
5069 | 377 |
Goal "Union({}) = {}"; |
2891 | 378 |
by (Blast_tac 1); |
923 | 379 |
qed "Union_empty"; |
1531 | 380 |
Addsimps[Union_empty]; |
381 |
||
5069 | 382 |
Goal "Union(UNIV) = UNIV"; |
2891 | 383 |
by (Blast_tac 1); |
1531 | 384 |
qed "Union_UNIV"; |
385 |
Addsimps[Union_UNIV]; |
|
923 | 386 |
|
5069 | 387 |
Goal "Union(insert a B) = a Un Union(B)"; |
2891 | 388 |
by (Blast_tac 1); |
923 | 389 |
qed "Union_insert"; |
1531 | 390 |
Addsimps[Union_insert]; |
923 | 391 |
|
5069 | 392 |
Goal "Union(A Un B) = Union(A) Un Union(B)"; |
2891 | 393 |
by (Blast_tac 1); |
923 | 394 |
qed "Union_Un_distrib"; |
1531 | 395 |
Addsimps[Union_Un_distrib]; |
923 | 396 |
|
5069 | 397 |
Goal "Union(A Int B) <= Union(A) Int Union(B)"; |
2891 | 398 |
by (Blast_tac 1); |
923 | 399 |
qed "Union_Int_subset"; |
400 |
||
5069 | 401 |
Goal "(Union M = {}) = (! A : M. A = {})"; |
4306
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
402 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
403 |
qed "Union_empty_conv"; |
4003 | 404 |
AddIffs [Union_empty_conv]; |
405 |
||
5069 | 406 |
Goal "(Union(C) Int A = {}) = (! B:C. B Int A = {})"; |
4306
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
407 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
923 | 408 |
qed "Union_disjoint"; |
409 |
||
1548 | 410 |
section "Inter"; |
411 |
||
5069 | 412 |
Goal "Inter({}) = UNIV"; |
2891 | 413 |
by (Blast_tac 1); |
1531 | 414 |
qed "Inter_empty"; |
415 |
Addsimps[Inter_empty]; |
|
416 |
||
5069 | 417 |
Goal "Inter(UNIV) = {}"; |
2891 | 418 |
by (Blast_tac 1); |
1531 | 419 |
qed "Inter_UNIV"; |
420 |
Addsimps[Inter_UNIV]; |
|
421 |
||
5069 | 422 |
Goal "Inter(insert a B) = a Int Inter(B)"; |
2891 | 423 |
by (Blast_tac 1); |
1531 | 424 |
qed "Inter_insert"; |
425 |
Addsimps[Inter_insert]; |
|
426 |
||
5069 | 427 |
Goal "Inter(A) Un Inter(B) <= Inter(A Int B)"; |
2891 | 428 |
by (Blast_tac 1); |
1564
822575c737bd
Deleted faulty comment; proved new rule Inter_Un_subset
paulson
parents:
1553
diff
changeset
|
429 |
qed "Inter_Un_subset"; |
1531 | 430 |
|
5069 | 431 |
Goal "Inter(A Un B) = Inter(A) Int Inter(B)"; |
2891 | 432 |
by (Blast_tac 1); |
923 | 433 |
qed "Inter_Un_distrib"; |
434 |
||
1548 | 435 |
section "UN and INT"; |
923 | 436 |
|
437 |
(*Basic identities*) |
|
438 |
||
7648 | 439 |
bind_thm ("not_empty", prove_goal Set.thy "(A ~= {}) = (? x. x:A)" (K [Blast_tac 1])); |
4136 | 440 |
|
5069 | 441 |
Goal "(UN x:{}. B x) = {}"; |
2891 | 442 |
by (Blast_tac 1); |
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
changeset
|
443 |
qed "UN_empty"; |
1531 | 444 |
Addsimps[UN_empty]; |
445 |
||
5069 | 446 |
Goal "(UN x:A. {}) = {}"; |
3457 | 447 |
by (Blast_tac 1); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
448 |
qed "UN_empty2"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
449 |
Addsimps[UN_empty2]; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
450 |
|
5069 | 451 |
Goal "(UN x:A. {x}) = A"; |
4645 | 452 |
by (Blast_tac 1); |
453 |
qed "UN_singleton"; |
|
454 |
Addsimps [UN_singleton]; |
|
455 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
456 |
Goal "k:I ==> A k Un (UN i:I. A i) = (UN i:I. A i)"; |
4634 | 457 |
by (Blast_tac 1); |
458 |
qed "UN_absorb"; |
|
459 |
||
5069 | 460 |
Goal "(INT x:{}. B x) = UNIV"; |
2891 | 461 |
by (Blast_tac 1); |
1531 | 462 |
qed "INT_empty"; |
463 |
Addsimps[INT_empty]; |
|
464 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
465 |
Goal "k:I ==> A k Int (INT i:I. A i) = (INT i:I. A i)"; |
4634 | 466 |
by (Blast_tac 1); |
467 |
qed "INT_absorb"; |
|
468 |
||
5069 | 469 |
Goal "(UN x:insert a A. B x) = B a Un UNION A B"; |
2891 | 470 |
by (Blast_tac 1); |
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
changeset
|
471 |
qed "UN_insert"; |
1531 | 472 |
Addsimps[UN_insert]; |
473 |
||
5999 | 474 |
Goal "(UN i: A Un B. M i) = (UN i: A. M i) Un (UN i:B. M i)"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
475 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
476 |
qed "UN_Un"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
477 |
|
5069 | 478 |
Goal "(UN x : (UN y:A. B y). C x) = (UN y:A. UN x: B y. C x)"; |
4771 | 479 |
by (Blast_tac 1); |
480 |
qed "UN_UN_flatten"; |
|
481 |
||
6292
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
482 |
Goal "((UN i:I. A i) <= B) = (ALL i:I. A i <= B)"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
483 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
484 |
qed "UN_subset_iff"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
485 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
486 |
Goal "(B <= (INT i:I. A i)) = (ALL i:I. B <= A i)"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
487 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
488 |
qed "INT_subset_iff"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
489 |
|
5069 | 490 |
Goal "(INT x:insert a A. B x) = B a Int INTER A B"; |
2891 | 491 |
by (Blast_tac 1); |
1531 | 492 |
qed "INT_insert"; |
493 |
Addsimps[INT_insert]; |
|
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
changeset
|
494 |
|
5999 | 495 |
Goal "(INT i: A Un B. M i) = (INT i: A. M i) Int (INT i:B. M i)"; |
496 |
by (Blast_tac 1); |
|
497 |
qed "INT_Un"; |
|
498 |
||
5941
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
499 |
Goal "u: A ==> (INT x:A. insert a (B x)) = insert a (INT x:A. B x)"; |
2891 | 500 |
by (Blast_tac 1); |
2021 | 501 |
qed "INT_insert_distrib"; |
502 |
||
5069 | 503 |
Goal "Union(B``A) = (UN x:A. B(x))"; |
2891 | 504 |
by (Blast_tac 1); |
923 | 505 |
qed "Union_image_eq"; |
6292
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
506 |
Addsimps [Union_image_eq]; |
923 | 507 |
|
6283 | 508 |
Goal "f `` Union S = (UN x:S. f `` x)"; |
509 |
by (Blast_tac 1); |
|
510 |
qed "image_Union_eq"; |
|
511 |
||
5069 | 512 |
Goal "Inter(B``A) = (INT x:A. B(x))"; |
2891 | 513 |
by (Blast_tac 1); |
923 | 514 |
qed "Inter_image_eq"; |
6292
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
515 |
Addsimps [Inter_image_eq]; |
923 | 516 |
|
5941
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
517 |
Goal "u: A ==> (UN y:A. c) = c"; |
2891 | 518 |
by (Blast_tac 1); |
923 | 519 |
qed "UN_constant"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
520 |
Addsimps[UN_constant]; |
923 | 521 |
|
5941
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
522 |
Goal "u: A ==> (INT y:A. c) = c"; |
2891 | 523 |
by (Blast_tac 1); |
923 | 524 |
qed "INT_constant"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
525 |
Addsimps[INT_constant]; |
923 | 526 |
|
5069 | 527 |
Goal "(UN x:A. B(x)) = Union({Y. ? x:A. Y=B(x)})"; |
2891 | 528 |
by (Blast_tac 1); |
923 | 529 |
qed "UN_eq"; |
530 |
||
531 |
(*Look: it has an EXISTENTIAL quantifier*) |
|
5069 | 532 |
Goal "(INT x:A. B(x)) = Inter({Y. ? x:A. Y=B(x)})"; |
2891 | 533 |
by (Blast_tac 1); |
923 | 534 |
qed "INT_eq"; |
535 |
||
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
536 |
|
923 | 537 |
(*Distributive laws...*) |
538 |
||
5069 | 539 |
Goal "A Int Union(B) = (UN C:B. A Int C)"; |
2891 | 540 |
by (Blast_tac 1); |
923 | 541 |
qed "Int_Union"; |
542 |
||
5069 | 543 |
Goal "Union(B) Int A = (UN C:B. C Int A)"; |
4674 | 544 |
by (Blast_tac 1); |
545 |
qed "Int_Union2"; |
|
546 |
||
4306
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
547 |
(* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: |
923 | 548 |
Union of a family of unions **) |
5069 | 549 |
Goal "(UN x:C. A(x) Un B(x)) = Union(A``C) Un Union(B``C)"; |
2891 | 550 |
by (Blast_tac 1); |
923 | 551 |
qed "Un_Union_image"; |
552 |
||
553 |
(*Equivalent version*) |
|
5069 | 554 |
Goal "(UN i:I. A(i) Un B(i)) = (UN i:I. A(i)) Un (UN i:I. B(i))"; |
2891 | 555 |
by (Blast_tac 1); |
923 | 556 |
qed "UN_Un_distrib"; |
557 |
||
5069 | 558 |
Goal "A Un Inter(B) = (INT C:B. A Un C)"; |
2891 | 559 |
by (Blast_tac 1); |
923 | 560 |
qed "Un_Inter"; |
561 |
||
5069 | 562 |
Goal "(INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C)"; |
2891 | 563 |
by (Blast_tac 1); |
923 | 564 |
qed "Int_Inter_image"; |
565 |
||
566 |
(*Equivalent version*) |
|
5069 | 567 |
Goal "(INT i:I. A(i) Int B(i)) = (INT i:I. A(i)) Int (INT i:I. B(i))"; |
2891 | 568 |
by (Blast_tac 1); |
923 | 569 |
qed "INT_Int_distrib"; |
570 |
||
571 |
(*Halmos, Naive Set Theory, page 35.*) |
|
5069 | 572 |
Goal "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))"; |
2891 | 573 |
by (Blast_tac 1); |
923 | 574 |
qed "Int_UN_distrib"; |
575 |
||
5069 | 576 |
Goal "B Un (INT i:I. A(i)) = (INT i:I. B Un A(i))"; |
2891 | 577 |
by (Blast_tac 1); |
923 | 578 |
qed "Un_INT_distrib"; |
579 |
||
5278 | 580 |
Goal "(UN i:I. A(i)) Int (UN j:J. B(j)) = (UN i:I. UN j:J. A(i) Int B(j))"; |
2891 | 581 |
by (Blast_tac 1); |
923 | 582 |
qed "Int_UN_distrib2"; |
583 |
||
5278 | 584 |
Goal "(INT i:I. A(i)) Un (INT j:J. B(j)) = (INT i:I. INT j:J. A(i) Un B(j))"; |
2891 | 585 |
by (Blast_tac 1); |
923 | 586 |
qed "Un_INT_distrib2"; |
587 |
||
2512 | 588 |
|
589 |
section"Bounded quantifiers"; |
|
590 |
||
3860 | 591 |
(** The following are not added to the default simpset because |
592 |
(a) they duplicate the body and (b) there are no similar rules for Int. **) |
|
2512 | 593 |
|
5069 | 594 |
Goal "(ALL x:A Un B. P x) = ((ALL x:A. P x) & (ALL x:B. P x))"; |
2891 | 595 |
by (Blast_tac 1); |
2519 | 596 |
qed "ball_Un"; |
597 |
||
5069 | 598 |
Goal "(EX x:A Un B. P x) = ((EX x:A. P x) | (EX x:B. P x))"; |
2891 | 599 |
by (Blast_tac 1); |
2519 | 600 |
qed "bex_Un"; |
2512 | 601 |
|
5069 | 602 |
Goal "(ALL z: UNION A B. P z) = (ALL x:A. ALL z:B x. P z)"; |
4771 | 603 |
by (Blast_tac 1); |
604 |
qed "ball_UN"; |
|
605 |
||
5069 | 606 |
Goal "(EX z: UNION A B. P z) = (EX x:A. EX z:B x. P z)"; |
4771 | 607 |
by (Blast_tac 1); |
608 |
qed "bex_UN"; |
|
609 |
||
2512 | 610 |
|
1548 | 611 |
section "-"; |
923 | 612 |
|
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
613 |
Goal "A-B = A Int (-B)"; |
4609 | 614 |
by (Blast_tac 1); |
4662 | 615 |
qed "Diff_eq"; |
4609 | 616 |
|
7516 | 617 |
Goal "(A-B = {}) = (A<=B)"; |
618 |
by (Blast_tac 1); |
|
619 |
qed "Diff_eq_empty_iff"; |
|
620 |
Addsimps[Diff_eq_empty_iff]; |
|
621 |
||
5069 | 622 |
Goal "A-A = {}"; |
2891 | 623 |
by (Blast_tac 1); |
923 | 624 |
qed "Diff_cancel"; |
1531 | 625 |
Addsimps[Diff_cancel]; |
923 | 626 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
627 |
Goal "A Int B = {} ==> A-B = A"; |
4674 | 628 |
by (blast_tac (claset() addEs [equalityE]) 1); |
629 |
qed "Diff_triv"; |
|
630 |
||
5069 | 631 |
Goal "{}-A = {}"; |
2891 | 632 |
by (Blast_tac 1); |
923 | 633 |
qed "empty_Diff"; |
1531 | 634 |
Addsimps[empty_Diff]; |
923 | 635 |
|
5069 | 636 |
Goal "A-{} = A"; |
2891 | 637 |
by (Blast_tac 1); |
923 | 638 |
qed "Diff_empty"; |
1531 | 639 |
Addsimps[Diff_empty]; |
640 |
||
5069 | 641 |
Goal "A-UNIV = {}"; |
2891 | 642 |
by (Blast_tac 1); |
1531 | 643 |
qed "Diff_UNIV"; |
644 |
Addsimps[Diff_UNIV]; |
|
645 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
646 |
Goal "x~:A ==> A - insert x B = A-B"; |
2891 | 647 |
by (Blast_tac 1); |
1531 | 648 |
qed "Diff_insert0"; |
649 |
Addsimps [Diff_insert0]; |
|
923 | 650 |
|
651 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a 0*) |
|
5069 | 652 |
Goal "A - insert a B = A - B - {a}"; |
2891 | 653 |
by (Blast_tac 1); |
923 | 654 |
qed "Diff_insert"; |
655 |
||
656 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a 0*) |
|
5069 | 657 |
Goal "A - insert a B = A - {a} - B"; |
2891 | 658 |
by (Blast_tac 1); |
923 | 659 |
qed "Diff_insert2"; |
660 |
||
5069 | 661 |
Goal "insert x A - B = (if x:B then A-B else insert x (A-B))"; |
4686 | 662 |
by (Simp_tac 1); |
2891 | 663 |
by (Blast_tac 1); |
1531 | 664 |
qed "insert_Diff_if"; |
665 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
666 |
Goal "x:B ==> insert x A - B = A-B"; |
2891 | 667 |
by (Blast_tac 1); |
1531 | 668 |
qed "insert_Diff1"; |
669 |
Addsimps [insert_Diff1]; |
|
670 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
671 |
Goal "a:A ==> insert a (A-{a}) = A"; |
2922 | 672 |
by (Blast_tac 1); |
923 | 673 |
qed "insert_Diff"; |
674 |
||
5069 | 675 |
Goal "A Int (B-A) = {}"; |
2891 | 676 |
by (Blast_tac 1); |
923 | 677 |
qed "Diff_disjoint"; |
1531 | 678 |
Addsimps[Diff_disjoint]; |
923 | 679 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
680 |
Goal "A<=B ==> A Un (B-A) = B"; |
2891 | 681 |
by (Blast_tac 1); |
923 | 682 |
qed "Diff_partition"; |
683 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
684 |
Goal "[| A<=B; B<= C |] ==> (B - (C - A)) = (A :: 'a set)"; |
2891 | 685 |
by (Blast_tac 1); |
923 | 686 |
qed "double_diff"; |
687 |
||
5069 | 688 |
Goal "A Un (B-A) = A Un B"; |
4645 | 689 |
by (Blast_tac 1); |
690 |
qed "Un_Diff_cancel"; |
|
691 |
||
5069 | 692 |
Goal "(B-A) Un A = B Un A"; |
4645 | 693 |
by (Blast_tac 1); |
694 |
qed "Un_Diff_cancel2"; |
|
695 |
||
696 |
Addsimps [Un_Diff_cancel, Un_Diff_cancel2]; |
|
697 |
||
5069 | 698 |
Goal "A - (B Un C) = (A-B) Int (A-C)"; |
2891 | 699 |
by (Blast_tac 1); |
923 | 700 |
qed "Diff_Un"; |
701 |
||
5069 | 702 |
Goal "A - (B Int C) = (A-B) Un (A-C)"; |
2891 | 703 |
by (Blast_tac 1); |
923 | 704 |
qed "Diff_Int"; |
705 |
||
5069 | 706 |
Goal "(A Un B) - C = (A - C) Un (B - C)"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
707 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
708 |
qed "Un_Diff"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
709 |
|
5069 | 710 |
Goal "(A Int B) - C = A Int (B - C)"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
711 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
712 |
qed "Int_Diff"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
713 |
|
5069 | 714 |
Goal "C Int (A-B) = (C Int A) - (C Int B)"; |
4748 | 715 |
by (Blast_tac 1); |
716 |
qed "Diff_Int_distrib"; |
|
717 |
||
5069 | 718 |
Goal "(A-B) Int C = (A Int C) - (B Int C)"; |
4645 | 719 |
by (Blast_tac 1); |
4748 | 720 |
qed "Diff_Int_distrib2"; |
4645 | 721 |
|
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
722 |
Goal "A - (- B) = A Int B"; |
5632 | 723 |
by Auto_tac; |
724 |
qed "Diff_Compl"; |
|
725 |
Addsimps [Diff_Compl]; |
|
726 |
||
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
727 |
|
5238 | 728 |
section "Quantification over type \"bool\""; |
729 |
||
730 |
Goal "(ALL b::bool. P b) = (P True & P False)"; |
|
731 |
by Auto_tac; |
|
732 |
by (case_tac "b" 1); |
|
733 |
by Auto_tac; |
|
734 |
qed "all_bool_eq"; |
|
735 |
||
5762 | 736 |
bind_thm ("bool_induct", conjI RS (all_bool_eq RS iffD2) RS spec); |
737 |
||
5238 | 738 |
Goal "(EX b::bool. P b) = (P True | P False)"; |
739 |
by Auto_tac; |
|
740 |
by (case_tac "b" 1); |
|
741 |
by Auto_tac; |
|
742 |
qed "ex_bool_eq"; |
|
743 |
||
744 |
Goal "A Un B = (UN b. if b then A else B)"; |
|
6301 | 745 |
by (auto_tac(claset()delWrapper"bspec",simpset()addsimps [split_if_mem2])); |
5238 | 746 |
qed "Un_eq_UN"; |
747 |
||
748 |
Goal "(UN b::bool. A b) = (A True Un A False)"; |
|
749 |
by Auto_tac; |
|
750 |
by (case_tac "b" 1); |
|
751 |
by Auto_tac; |
|
752 |
qed "UN_bool_eq"; |
|
753 |
||
754 |
Goal "(INT b::bool. A b) = (A True Int A False)"; |
|
755 |
by Auto_tac; |
|
756 |
by (case_tac "b" 1); |
|
757 |
by Auto_tac; |
|
758 |
qed "INT_bool_eq"; |
|
759 |
||
760 |
||
6292
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
761 |
section "Pow"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
762 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
763 |
Goalw [Pow_def] "Pow {} = {{}}"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
764 |
by Auto_tac; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
765 |
qed "Pow_empty"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
766 |
Addsimps [Pow_empty]; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
767 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
768 |
Goal "Pow (insert a A) = Pow A Un (insert a `` Pow A)"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
769 |
by Safe_tac; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
770 |
by (etac swap 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
771 |
by (res_inst_tac [("x", "x-{a}")] image_eqI 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
772 |
by (ALLGOALS Blast_tac); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
773 |
qed "Pow_insert"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
774 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
775 |
Goal "Pow (- A) = {-B |B. A: Pow B}"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
776 |
by Safe_tac; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
777 |
by (Blast_tac 2); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
778 |
by (res_inst_tac [("x", "-x")] exI 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
779 |
by (ALLGOALS Blast_tac); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
780 |
qed "Pow_Compl"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
781 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
782 |
Goal "Pow UNIV = UNIV"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
783 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
784 |
qed "Pow_UNIV"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
785 |
Addsimps [Pow_UNIV]; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
786 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
787 |
Goal "Pow(A) Un Pow(B) <= Pow(A Un B)"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
788 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
789 |
qed "Un_Pow_subset"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
790 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
791 |
Goal "(UN x:A. Pow(B(x))) <= Pow(UN x:A. B(x))"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
792 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
793 |
qed "UN_Pow_subset"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
794 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
795 |
Goal "A <= Pow(Union(A))"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
796 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
797 |
qed "subset_Pow_Union"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
798 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
799 |
Goal "Union(Pow(A)) = A"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
800 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
801 |
qed "Union_Pow_eq"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
802 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
803 |
Goal "Pow(A Int B) = Pow(A) Int Pow(B)"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
804 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
805 |
qed "Pow_Int_eq"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
806 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
807 |
Goal "Pow(INT x:A. B(x)) = (INT x:A. Pow(B(x)))"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
808 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
809 |
qed "Pow_INT_eq"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
810 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
811 |
Addsimps [Union_Pow_eq, Pow_Int_eq]; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
812 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
813 |
|
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
814 |
section "Miscellany"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
815 |
|
5069 | 816 |
Goal "(A = B) = ((A <= (B::'a set)) & (B<=A))"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
817 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
818 |
qed "set_eq_subset"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
819 |
|
5069 | 820 |
Goal "A <= B = (! t. t:A --> t:B)"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
821 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
822 |
qed "subset_iff"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
823 |
|
5069 | 824 |
Goalw [psubset_def] "((A::'a set) <= B) = ((A < B) | (A=B))"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
825 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
826 |
qed "subset_iff_psubset_eq"; |
2021 | 827 |
|
5069 | 828 |
Goal "(!x. x ~: A) = (A={})"; |
4423 | 829 |
by (Blast_tac 1); |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
830 |
qed "all_not_in_conv"; |
3907 | 831 |
AddIffs [all_not_in_conv]; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
832 |
|
6007 | 833 |
|
5189
362e4d6213c5
Added theorem distinct_lemma (needed for datatypes).
berghofe
parents:
5148
diff
changeset
|
834 |
(** for datatypes **) |
362e4d6213c5
Added theorem distinct_lemma (needed for datatypes).
berghofe
parents:
5148
diff
changeset
|
835 |
Goal "f x ~= f y ==> x ~= y"; |
362e4d6213c5
Added theorem distinct_lemma (needed for datatypes).
berghofe
parents:
5148
diff
changeset
|
836 |
by (Fast_tac 1); |
362e4d6213c5
Added theorem distinct_lemma (needed for datatypes).
berghofe
parents:
5148
diff
changeset
|
837 |
qed "distinct_lemma"; |
362e4d6213c5
Added theorem distinct_lemma (needed for datatypes).
berghofe
parents:
5148
diff
changeset
|
838 |
|
2021 | 839 |
|
840 |
(** Miniscoping: pushing in big Unions and Intersections **) |
|
841 |
local |
|
4059 | 842 |
fun prover s = prove_goal thy s (fn _ => [Blast_tac 1]) |
2021 | 843 |
in |
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
844 |
val UN_simps = map prover |
5941
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
845 |
["!!C. c: C ==> (UN x:C. insert a (B x)) = insert a (UN x:C. B x)", |
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
846 |
"!!C. c: C ==> (UN x:C. A x Un B) = ((UN x:C. A x) Un B)", |
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
847 |
"!!C. c: C ==> (UN x:C. A Un B x) = (A Un (UN x:C. B x))", |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
848 |
"(UN x:C. A x Int B) = ((UN x:C. A x) Int B)", |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
849 |
"(UN x:C. A Int B x) = (A Int (UN x:C. B x))", |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
850 |
"(UN x:C. A x - B) = ((UN x:C. A x) - B)", |
4231 | 851 |
"(UN x:C. A - B x) = (A - (INT x:C. B x))", |
852 |
"(UN x:f``A. B x) = (UN a:A. B(f a))"]; |
|
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
853 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
854 |
val INT_simps = map prover |
5941
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
855 |
["!!C. c: C ==> (INT x:C. A x Int B) = ((INT x:C. A x) Int B)", |
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
856 |
"!!C. c: C ==> (INT x:C. A Int B x) = (A Int (INT x:C. B x))", |
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
857 |
"!!C. c: C ==> (INT x:C. A x - B) = ((INT x:C. A x) - B)", |
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
858 |
"!!C. c: C ==> (INT x:C. A - B x) = (A - (UN x:C. B x))", |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
859 |
"(INT x:C. insert a (B x)) = insert a (INT x:C. B x)", |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
860 |
"(INT x:C. A x Un B) = ((INT x:C. A x) Un B)", |
4231 | 861 |
"(INT x:C. A Un B x) = (A Un (INT x:C. B x))", |
862 |
"(INT x:f``A. B x) = (INT a:A. B(f a))"]; |
|
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
863 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
864 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
865 |
val ball_simps = map prover |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
866 |
["(ALL x:A. P x | Q) = ((ALL x:A. P x) | Q)", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
867 |
"(ALL x:A. P | Q x) = (P | (ALL x:A. Q x))", |
3422 | 868 |
"(ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))", |
869 |
"(ALL x:A. P x --> Q) = ((EX x:A. P x) --> Q)", |
|
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
870 |
"(ALL x:{}. P x) = True", |
4136 | 871 |
"(ALL x:UNIV. P x) = (ALL x. P x)", |
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
872 |
"(ALL x:insert a B. P x) = (P(a) & (ALL x:B. P x))", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
873 |
"(ALL x:Union(A). P x) = (ALL y:A. ALL x:y. P x)", |
5233
3571ff68ceda
New rewrite rules for quantification over bounded UNIONs
paulson
parents:
5189
diff
changeset
|
874 |
"(ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)", |
3860 | 875 |
"(ALL x:Collect Q. P x) = (ALL x. Q x --> P x)", |
876 |
"(ALL x:f``A. P x) = (ALL x:A. P(f x))", |
|
877 |
"(~(ALL x:A. P x)) = (EX x:A. ~P x)"]; |
|
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
878 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
879 |
val ball_conj_distrib = |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
880 |
prover "(ALL x:A. P x & Q x) = ((ALL x:A. P x) & (ALL x:A. Q x))"; |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
881 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
882 |
val bex_simps = map prover |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
883 |
["(EX x:A. P x & Q) = ((EX x:A. P x) & Q)", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
884 |
"(EX x:A. P & Q x) = (P & (EX x:A. Q x))", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
885 |
"(EX x:{}. P x) = False", |
4136 | 886 |
"(EX x:UNIV. P x) = (EX x. P x)", |
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
887 |
"(EX x:insert a B. P x) = (P(a) | (EX x:B. P x))", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
888 |
"(EX x:Union(A). P x) = (EX y:A. EX x:y. P x)", |
5233
3571ff68ceda
New rewrite rules for quantification over bounded UNIONs
paulson
parents:
5189
diff
changeset
|
889 |
"(EX x: UNION A B. P x) = (EX a:A. EX x: B a. P x)", |
3860 | 890 |
"(EX x:Collect Q. P x) = (EX x. Q x & P x)", |
891 |
"(EX x:f``A. P x) = (EX x:A. P(f x))", |
|
892 |
"(~(EX x:A. P x)) = (ALL x:A. ~P x)"]; |
|
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
893 |
|
3426 | 894 |
val bex_disj_distrib = |
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
895 |
prover "(EX x:A. P x | Q x) = ((EX x:A. P x) | (EX x:A. Q x))"; |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
896 |
|
2021 | 897 |
end; |
898 |
||
7648 | 899 |
bind_thms ("UN_simps", UN_simps); |
900 |
bind_thms ("INT_simps", INT_simps); |
|
901 |
bind_thms ("ball_simps", ball_simps); |
|
902 |
bind_thms ("bex_simps", bex_simps); |
|
903 |
bind_thm ("ball_conj_distrib", ball_conj_distrib); |
|
904 |
bind_thm ("bex_disj_distrib", bex_disj_distrib); |
|
905 |
||
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
906 |
Addsimps (UN_simps @ INT_simps @ ball_simps @ bex_simps); |