author | wenzelm |
Sat, 01 Jul 2000 19:55:22 +0200 | |
changeset 9230 | 17ae63f82ad8 |
parent 9206 | eaaee6bd74ba |
child 9312 | a93a7b6bb654 |
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"; |
|
10 |
||
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AddSIs [equalityI]; |
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|
1548 | 13 |
section "{}"; |
14 |
||
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15 |
(*supersedes Collect_False_empty*) |
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16 |
Goal "{s. P} = (if P then UNIV else {})"; |
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17 |
by (Force_tac 1); |
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18 |
qed "Collect_const"; |
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19 |
Addsimps [Collect_const]; |
1531 | 20 |
|
9077 | 21 |
(*If added as a simprule it can cause looping when equalityE is also present: |
22 |
A={} goes to {}<=A and A<={} and then back to A={} !*) |
|
5069 | 23 |
Goal "(A <= {}) = (A = {})"; |
2891 | 24 |
by (Blast_tac 1); |
1531 | 25 |
qed "subset_empty"; |
26 |
||
5069 | 27 |
Goalw [psubset_def] "~ (A < {})"; |
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by (Blast_tac 1); |
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29 |
qed "not_psubset_empty"; |
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AddIffs [not_psubset_empty]; |
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|
9206 | 32 |
Goal "(Collect P = {}) = (ALL x. ~ P x)"; |
7958 | 33 |
by Auto_tac; |
34 |
qed "Collect_empty_eq"; |
|
35 |
Addsimps[Collect_empty_eq]; |
|
36 |
||
8993 | 37 |
Goal "{x. ~ (P x)} = - {x. P x}"; |
38 |
by (Blast_tac 1); |
|
39 |
qed "Collect_neg_eq"; |
|
40 |
||
5069 | 41 |
Goal "{x. P x | Q x} = {x. P x} Un {x. Q x}"; |
4748 | 42 |
by (Blast_tac 1); |
43 |
qed "Collect_disj_eq"; |
|
44 |
||
5069 | 45 |
Goal "{x. P x & Q x} = {x. P x} Int {x. Q x}"; |
4748 | 46 |
by (Blast_tac 1); |
47 |
qed "Collect_conj_eq"; |
|
48 |
||
7845 | 49 |
Goal "{x. ALL y. P x y} = (INT y. {x. P x y})"; |
50 |
by (Blast_tac 1); |
|
51 |
qed "Collect_all_eq"; |
|
52 |
||
53 |
Goal "{x. ALL y: A. P x y} = (INT y: A. {x. P x y})"; |
|
54 |
by (Blast_tac 1); |
|
55 |
qed "Collect_ball_eq"; |
|
56 |
||
57 |
Goal "{x. EX y. P x y} = (UN y. {x. P x y})"; |
|
58 |
by (Blast_tac 1); |
|
59 |
qed "Collect_ex_eq"; |
|
60 |
||
61 |
Goal "{x. EX y: A. P x y} = (UN y: A. {x. P x y})"; |
|
62 |
by (Blast_tac 1); |
|
63 |
qed "Collect_bex_eq"; |
|
64 |
||
4748 | 65 |
|
1548 | 66 |
section "insert"; |
923 | 67 |
|
1531 | 68 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a {}*) |
5069 | 69 |
Goal "insert a A = {a} Un A"; |
2891 | 70 |
by (Blast_tac 1); |
1531 | 71 |
qed "insert_is_Un"; |
72 |
||
5069 | 73 |
Goal "insert a A ~= {}"; |
4089 | 74 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
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qed"insert_not_empty"; |
1531 | 76 |
Addsimps[insert_not_empty]; |
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77 |
|
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78 |
bind_thm("empty_not_insert",insert_not_empty RS not_sym); |
1531 | 79 |
Addsimps[empty_not_insert]; |
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80 |
|
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81 |
Goal "a:A ==> insert a A = A"; |
2891 | 82 |
by (Blast_tac 1); |
923 | 83 |
qed "insert_absorb"; |
6832 | 84 |
(* Addsimps [insert_absorb] causes recursive calls |
85 |
when there are nested inserts, with QUADRATIC running time |
|
4605 | 86 |
*) |
923 | 87 |
|
5069 | 88 |
Goal "insert x (insert x A) = insert x A"; |
2891 | 89 |
by (Blast_tac 1); |
1531 | 90 |
qed "insert_absorb2"; |
91 |
Addsimps [insert_absorb2]; |
|
92 |
||
5069 | 93 |
Goal "insert x (insert y A) = insert y (insert x A)"; |
2891 | 94 |
by (Blast_tac 1); |
1879 | 95 |
qed "insert_commute"; |
96 |
||
5069 | 97 |
Goal "(insert x A <= B) = (x:B & A <= B)"; |
2891 | 98 |
by (Blast_tac 1); |
923 | 99 |
qed "insert_subset"; |
1531 | 100 |
Addsimps[insert_subset]; |
101 |
||
102 |
(* use new B rather than (A-{a}) to avoid infinite unfolding *) |
|
9206 | 103 |
Goal "a:A ==> EX B. A = insert a B & a ~: B"; |
1553 | 104 |
by (res_inst_tac [("x","A-{a}")] exI 1); |
2891 | 105 |
by (Blast_tac 1); |
1531 | 106 |
qed "mk_disjoint_insert"; |
923 | 107 |
|
4882 | 108 |
bind_thm ("insert_Collect", prove_goal thy |
5590 | 109 |
"insert a (Collect P) = {u. u ~= a --> P u}" (K [Auto_tac])); |
4882 | 110 |
|
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111 |
Goal "u: A ==> (UN x:A. insert a (B x)) = insert a (UN x:A. B x)"; |
2891 | 112 |
by (Blast_tac 1); |
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113 |
qed "UN_insert_distrib"; |
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114 |
|
1660 | 115 |
section "``"; |
923 | 116 |
|
5069 | 117 |
Goal "f``{} = {}"; |
2891 | 118 |
by (Blast_tac 1); |
923 | 119 |
qed "image_empty"; |
1531 | 120 |
Addsimps[image_empty]; |
923 | 121 |
|
5069 | 122 |
Goal "f``insert a B = insert (f a) (f``B)"; |
2891 | 123 |
by (Blast_tac 1); |
923 | 124 |
qed "image_insert"; |
1531 | 125 |
Addsimps[image_insert]; |
923 | 126 |
|
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127 |
Goal "x:A ==> (%x. c) `` A = {c}"; |
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128 |
by (Blast_tac 1); |
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129 |
qed "image_constant"; |
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|
130 |
|
5069 | 131 |
Goal "f``(g``A) = (%x. f (g x)) `` A"; |
3457 | 132 |
by (Blast_tac 1); |
4059 | 133 |
qed "image_image"; |
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|
134 |
|
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|
135 |
Goal "x:A ==> insert (f x) (f``A) = f``A"; |
2891 | 136 |
by (Blast_tac 1); |
1884 | 137 |
qed "insert_image"; |
138 |
Addsimps [insert_image]; |
|
139 |
||
5069 | 140 |
Goal "(f``A = {}) = (A = {})"; |
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|
141 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
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|
142 |
qed "image_is_empty"; |
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|
143 |
AddIffs [image_is_empty]; |
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144 |
|
5281 | 145 |
Goal "f `` {x. P x} = {f x | x. P x}"; |
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146 |
by (Blast_tac 1); |
5281 | 147 |
qed "image_Collect"; |
148 |
Addsimps [image_Collect]; |
|
149 |
||
5590 | 150 |
Goalw [image_def] "(%x. if P x then f x else g x) `` S \ |
151 |
\ = (f `` (S Int {x. P x})) Un (g `` (S Int {x. ~(P x)}))"; |
|
4686 | 152 |
by (Simp_tac 1); |
2891 | 153 |
by (Blast_tac 1); |
1748 | 154 |
qed "if_image_distrib"; |
155 |
Addsimps[if_image_distrib]; |
|
156 |
||
5590 | 157 |
val prems = Goal "[|M = N; !!x. x:N ==> f x = g x|] ==> f``M = g``N"; |
4136 | 158 |
by (simp_tac (simpset() addsimps image_def::prems) 1); |
159 |
qed "image_cong"; |
|
160 |
||
1748 | 161 |
|
7958 | 162 |
section "range"; |
163 |
||
9206 | 164 |
Goal "{u. EX x. u = f x} = range f"; |
7958 | 165 |
by Auto_tac; |
166 |
qed "full_SetCompr_eq"; |
|
167 |
||
8161 | 168 |
Goal "range (%x. f (g x)) = f``range g"; |
169 |
by (stac image_image 1); |
|
170 |
by (Simp_tac 1); |
|
171 |
qed "range_composition"; |
|
172 |
Addsimps[range_composition]; |
|
7958 | 173 |
|
1548 | 174 |
section "Int"; |
923 | 175 |
|
5069 | 176 |
Goal "A Int A = A"; |
2891 | 177 |
by (Blast_tac 1); |
923 | 178 |
qed "Int_absorb"; |
1531 | 179 |
Addsimps[Int_absorb]; |
923 | 180 |
|
5590 | 181 |
Goal "A Int (A Int B) = A Int B"; |
4609 | 182 |
by (Blast_tac 1); |
183 |
qed "Int_left_absorb"; |
|
184 |
||
5590 | 185 |
Goal "A Int B = B Int A"; |
2891 | 186 |
by (Blast_tac 1); |
923 | 187 |
qed "Int_commute"; |
188 |
||
5069 | 189 |
Goal "A Int (B Int C) = B Int (A Int C)"; |
4609 | 190 |
by (Blast_tac 1); |
191 |
qed "Int_left_commute"; |
|
192 |
||
5590 | 193 |
Goal "(A Int B) Int C = A Int (B Int C)"; |
2891 | 194 |
by (Blast_tac 1); |
923 | 195 |
qed "Int_assoc"; |
196 |
||
4609 | 197 |
(*Intersection is an AC-operator*) |
7648 | 198 |
bind_thms ("Int_ac", [Int_assoc, Int_left_absorb, Int_commute, Int_left_commute]); |
4609 | 199 |
|
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200 |
Goal "B<=A ==> A Int B = B"; |
4662 | 201 |
by (Blast_tac 1); |
202 |
qed "Int_absorb1"; |
|
203 |
||
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204 |
Goal "A<=B ==> A Int B = A"; |
4662 | 205 |
by (Blast_tac 1); |
206 |
qed "Int_absorb2"; |
|
207 |
||
5069 | 208 |
Goal "{} Int B = {}"; |
2891 | 209 |
by (Blast_tac 1); |
923 | 210 |
qed "Int_empty_left"; |
1531 | 211 |
Addsimps[Int_empty_left]; |
923 | 212 |
|
5069 | 213 |
Goal "A Int {} = {}"; |
2891 | 214 |
by (Blast_tac 1); |
923 | 215 |
qed "Int_empty_right"; |
1531 | 216 |
Addsimps[Int_empty_right]; |
217 |
||
5490 | 218 |
Goal "(A Int B = {}) = (A <= -B)"; |
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219 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
3356 | 220 |
qed "disjoint_eq_subset_Compl"; |
221 |
||
7877 | 222 |
Goal "(A Int B = {}) = (ALL x:A. ALL y:B. x ~= y)"; |
223 |
by (Blast_tac 1); |
|
224 |
qed "disjoint_iff_not_equal"; |
|
225 |
||
5069 | 226 |
Goal "UNIV Int B = B"; |
2891 | 227 |
by (Blast_tac 1); |
1531 | 228 |
qed "Int_UNIV_left"; |
229 |
Addsimps[Int_UNIV_left]; |
|
230 |
||
5069 | 231 |
Goal "A Int UNIV = A"; |
2891 | 232 |
by (Blast_tac 1); |
1531 | 233 |
qed "Int_UNIV_right"; |
234 |
Addsimps[Int_UNIV_right]; |
|
923 | 235 |
|
5069 | 236 |
Goal "A Int B = Inter{A,B}"; |
4634 | 237 |
by (Blast_tac 1); |
238 |
qed "Int_eq_Inter"; |
|
239 |
||
5590 | 240 |
Goal "A Int (B Un C) = (A Int B) Un (A Int C)"; |
2891 | 241 |
by (Blast_tac 1); |
923 | 242 |
qed "Int_Un_distrib"; |
243 |
||
5590 | 244 |
Goal "(B Un C) Int A = (B Int A) Un (C Int A)"; |
2891 | 245 |
by (Blast_tac 1); |
1618 | 246 |
qed "Int_Un_distrib2"; |
247 |
||
5069 | 248 |
Goal "(A Int B = UNIV) = (A = UNIV & B = UNIV)"; |
4089 | 249 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
1531 | 250 |
qed "Int_UNIV"; |
251 |
Addsimps[Int_UNIV]; |
|
252 |
||
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253 |
Goal "(C <= A Int B) = (C <= A & C <= B)"; |
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|
254 |
by (Blast_tac 1); |
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Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
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|
255 |
qed "Int_subset_iff"; |
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|
256 |
|
7713 | 257 |
Goal "(x : A Int {x. P x}) = (x : A & P x)"; |
258 |
by (Blast_tac 1); |
|
259 |
qed "Int_Collect"; |
|
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|
260 |
|
1548 | 261 |
section "Un"; |
923 | 262 |
|
5069 | 263 |
Goal "A Un A = A"; |
2891 | 264 |
by (Blast_tac 1); |
923 | 265 |
qed "Un_absorb"; |
1531 | 266 |
Addsimps[Un_absorb]; |
923 | 267 |
|
5069 | 268 |
Goal " A Un (A Un B) = A Un B"; |
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|
269 |
by (Blast_tac 1); |
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diff
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|
270 |
qed "Un_left_absorb"; |
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|
271 |
|
5590 | 272 |
Goal "A Un B = B Un A"; |
2891 | 273 |
by (Blast_tac 1); |
923 | 274 |
qed "Un_commute"; |
275 |
||
5069 | 276 |
Goal "A Un (B Un C) = B Un (A Un C)"; |
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|
277 |
by (Blast_tac 1); |
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parents:
2922
diff
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|
278 |
qed "Un_left_commute"; |
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|
279 |
|
5590 | 280 |
Goal "(A Un B) Un C = A Un (B Un C)"; |
2891 | 281 |
by (Blast_tac 1); |
923 | 282 |
qed "Un_assoc"; |
283 |
||
4609 | 284 |
(*Union is an AC-operator*) |
7648 | 285 |
bind_thms ("Un_ac", [Un_assoc, Un_left_absorb, Un_commute, Un_left_commute]); |
4609 | 286 |
|
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|
287 |
Goal "A<=B ==> A Un B = B"; |
4662 | 288 |
by (Blast_tac 1); |
289 |
qed "Un_absorb1"; |
|
290 |
||
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|
291 |
Goal "B<=A ==> A Un B = A"; |
4662 | 292 |
by (Blast_tac 1); |
293 |
qed "Un_absorb2"; |
|
294 |
||
5069 | 295 |
Goal "{} Un B = B"; |
2891 | 296 |
by (Blast_tac 1); |
923 | 297 |
qed "Un_empty_left"; |
1531 | 298 |
Addsimps[Un_empty_left]; |
923 | 299 |
|
5069 | 300 |
Goal "A Un {} = A"; |
2891 | 301 |
by (Blast_tac 1); |
923 | 302 |
qed "Un_empty_right"; |
1531 | 303 |
Addsimps[Un_empty_right]; |
304 |
||
5069 | 305 |
Goal "UNIV Un B = UNIV"; |
2891 | 306 |
by (Blast_tac 1); |
1531 | 307 |
qed "Un_UNIV_left"; |
308 |
Addsimps[Un_UNIV_left]; |
|
309 |
||
5069 | 310 |
Goal "A Un UNIV = UNIV"; |
2891 | 311 |
by (Blast_tac 1); |
1531 | 312 |
qed "Un_UNIV_right"; |
313 |
Addsimps[Un_UNIV_right]; |
|
923 | 314 |
|
5069 | 315 |
Goal "A Un B = Union{A,B}"; |
4634 | 316 |
by (Blast_tac 1); |
317 |
qed "Un_eq_Union"; |
|
318 |
||
5069 | 319 |
Goal "(insert a B) Un C = insert a (B Un C)"; |
2891 | 320 |
by (Blast_tac 1); |
923 | 321 |
qed "Un_insert_left"; |
3384
5ef99c94e1fb
Now Un_insert_left, Un_insert_right are default rewrite rules
paulson
parents:
3356
diff
changeset
|
322 |
Addsimps[Un_insert_left]; |
923 | 323 |
|
5069 | 324 |
Goal "A Un (insert a B) = insert a (A Un B)"; |
2891 | 325 |
by (Blast_tac 1); |
1917 | 326 |
qed "Un_insert_right"; |
3384
5ef99c94e1fb
Now Un_insert_left, Un_insert_right are default rewrite rules
paulson
parents:
3356
diff
changeset
|
327 |
Addsimps[Un_insert_right]; |
1917 | 328 |
|
5069 | 329 |
Goal "(insert a B) Int C = (if a:C then insert a (B Int C) \ |
5590 | 330 |
\ else B Int C)"; |
4686 | 331 |
by (Simp_tac 1); |
3356 | 332 |
by (Blast_tac 1); |
333 |
qed "Int_insert_left"; |
|
334 |
||
5069 | 335 |
Goal "A Int (insert a B) = (if a:A then insert a (A Int B) \ |
5590 | 336 |
\ else A Int B)"; |
4686 | 337 |
by (Simp_tac 1); |
3356 | 338 |
by (Blast_tac 1); |
339 |
qed "Int_insert_right"; |
|
340 |
||
5590 | 341 |
Goal "A Un (B Int C) = (A Un B) Int (A Un C)"; |
2891 | 342 |
by (Blast_tac 1); |
923 | 343 |
qed "Un_Int_distrib"; |
344 |
||
5590 | 345 |
Goal "(B Int C) Un A = (B Un A) Int (C Un A)"; |
4609 | 346 |
by (Blast_tac 1); |
347 |
qed "Un_Int_distrib2"; |
|
348 |
||
5590 | 349 |
Goal "(A Int B) Un (B Int C) Un (C Int A) = \ |
350 |
\ (A Un B) Int (B Un C) Int (C Un A)"; |
|
2891 | 351 |
by (Blast_tac 1); |
923 | 352 |
qed "Un_Int_crazy"; |
353 |
||
5069 | 354 |
Goal "(A<=B) = (A Un B = B)"; |
4306
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
355 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
923 | 356 |
qed "subset_Un_eq"; |
357 |
||
5069 | 358 |
Goal "(A <= insert b C) = (A <= C | b:A & A-{b} <= C)"; |
2891 | 359 |
by (Blast_tac 1); |
923 | 360 |
qed "subset_insert_iff"; |
361 |
||
5069 | 362 |
Goal "(A Un B = {}) = (A = {} & B = {})"; |
4089 | 363 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
923 | 364 |
qed "Un_empty"; |
9098
3a0912a127ec
new theorem UN_empty; it and Un_empty inserted by AddIffs
paulson
parents:
9077
diff
changeset
|
365 |
AddIffs[Un_empty]; |
923 | 366 |
|
5319
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
367 |
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
|
368 |
by (Blast_tac 1); |
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
369 |
qed "Un_subset_iff"; |
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
370 |
|
5331 | 371 |
Goal "(A - B) Un (A Int B) = A"; |
372 |
by (Blast_tac 1); |
|
373 |
qed "Un_Diff_Int"; |
|
374 |
||
5319
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
375 |
|
5931 | 376 |
section "Set complement"; |
923 | 377 |
|
8333 | 378 |
Goal "A Int -A = {}"; |
2891 | 379 |
by (Blast_tac 1); |
923 | 380 |
qed "Compl_disjoint"; |
8333 | 381 |
|
382 |
Goal "-A Int A = {}"; |
|
383 |
by (Blast_tac 1); |
|
384 |
qed "Compl_disjoint2"; |
|
385 |
Addsimps[Compl_disjoint, Compl_disjoint2]; |
|
923 | 386 |
|
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
387 |
Goal "A Un (-A) = UNIV"; |
2891 | 388 |
by (Blast_tac 1); |
923 | 389 |
qed "Compl_partition"; |
390 |
||
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
391 |
Goal "- (-A) = (A:: 'a set)"; |
2891 | 392 |
by (Blast_tac 1); |
923 | 393 |
qed "double_complement"; |
1531 | 394 |
Addsimps[double_complement]; |
923 | 395 |
|
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
396 |
Goal "-(A Un B) = (-A) Int (-B)"; |
2891 | 397 |
by (Blast_tac 1); |
923 | 398 |
qed "Compl_Un"; |
399 |
||
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
400 |
Goal "-(A Int B) = (-A) Un (-B)"; |
2891 | 401 |
by (Blast_tac 1); |
923 | 402 |
qed "Compl_Int"; |
403 |
||
5490 | 404 |
Goal "-(UN x:A. B(x)) = (INT x:A. -B(x))"; |
2891 | 405 |
by (Blast_tac 1); |
923 | 406 |
qed "Compl_UN"; |
407 |
||
5490 | 408 |
Goal "-(INT x:A. B(x)) = (UN x:A. -B(x))"; |
2891 | 409 |
by (Blast_tac 1); |
923 | 410 |
qed "Compl_INT"; |
411 |
||
4615 | 412 |
Addsimps [Compl_Un, Compl_Int, Compl_UN, Compl_INT]; |
413 |
||
8121 | 414 |
Goal "(A <= -A) = (A = {})"; |
415 |
by (Blast_tac 1); |
|
416 |
qed "subset_Compl_self_eq"; |
|
417 |
||
923 | 418 |
(*Halmos, Naive Set Theory, page 16.*) |
5069 | 419 |
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
|
420 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
923 | 421 |
qed "Un_Int_assoc_eq"; |
422 |
||
8333 | 423 |
Goal "-UNIV = {}"; |
424 |
by (Blast_tac 1); |
|
425 |
qed "Compl_UNIV_eq"; |
|
426 |
||
427 |
Goal "-{} = UNIV"; |
|
428 |
by (Blast_tac 1); |
|
429 |
qed "Compl_empty_eq"; |
|
430 |
||
431 |
Addsimps [Compl_UNIV_eq, Compl_empty_eq]; |
|
432 |
||
923 | 433 |
|
1548 | 434 |
section "Union"; |
923 | 435 |
|
5069 | 436 |
Goal "Union({}) = {}"; |
2891 | 437 |
by (Blast_tac 1); |
923 | 438 |
qed "Union_empty"; |
1531 | 439 |
Addsimps[Union_empty]; |
440 |
||
5069 | 441 |
Goal "Union(UNIV) = UNIV"; |
2891 | 442 |
by (Blast_tac 1); |
1531 | 443 |
qed "Union_UNIV"; |
444 |
Addsimps[Union_UNIV]; |
|
923 | 445 |
|
5069 | 446 |
Goal "Union(insert a B) = a Un Union(B)"; |
2891 | 447 |
by (Blast_tac 1); |
923 | 448 |
qed "Union_insert"; |
1531 | 449 |
Addsimps[Union_insert]; |
923 | 450 |
|
5069 | 451 |
Goal "Union(A Un B) = Union(A) Un Union(B)"; |
2891 | 452 |
by (Blast_tac 1); |
923 | 453 |
qed "Union_Un_distrib"; |
1531 | 454 |
Addsimps[Union_Un_distrib]; |
923 | 455 |
|
5069 | 456 |
Goal "Union(A Int B) <= Union(A) Int Union(B)"; |
2891 | 457 |
by (Blast_tac 1); |
923 | 458 |
qed "Union_Int_subset"; |
459 |
||
9098
3a0912a127ec
new theorem UN_empty; it and Un_empty inserted by AddIffs
paulson
parents:
9077
diff
changeset
|
460 |
Goal "(Union A = {}) = (ALL x:A. x={})"; |
3a0912a127ec
new theorem UN_empty; it and Un_empty inserted by AddIffs
paulson
parents:
9077
diff
changeset
|
461 |
by Auto_tac; |
4306
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
462 |
qed "Union_empty_conv"; |
4003 | 463 |
AddIffs [Union_empty_conv]; |
464 |
||
9206 | 465 |
Goal "(Union(C) Int A = {}) = (ALL B:C. B Int A = {})"; |
4306
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
466 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
923 | 467 |
qed "Union_disjoint"; |
468 |
||
1548 | 469 |
section "Inter"; |
470 |
||
5069 | 471 |
Goal "Inter({}) = UNIV"; |
2891 | 472 |
by (Blast_tac 1); |
1531 | 473 |
qed "Inter_empty"; |
474 |
Addsimps[Inter_empty]; |
|
475 |
||
5069 | 476 |
Goal "Inter(UNIV) = {}"; |
2891 | 477 |
by (Blast_tac 1); |
1531 | 478 |
qed "Inter_UNIV"; |
479 |
Addsimps[Inter_UNIV]; |
|
480 |
||
5069 | 481 |
Goal "Inter(insert a B) = a Int Inter(B)"; |
2891 | 482 |
by (Blast_tac 1); |
1531 | 483 |
qed "Inter_insert"; |
484 |
Addsimps[Inter_insert]; |
|
485 |
||
5069 | 486 |
Goal "Inter(A) Un Inter(B) <= Inter(A Int B)"; |
2891 | 487 |
by (Blast_tac 1); |
1564
822575c737bd
Deleted faulty comment; proved new rule Inter_Un_subset
paulson
parents:
1553
diff
changeset
|
488 |
qed "Inter_Un_subset"; |
1531 | 489 |
|
5069 | 490 |
Goal "Inter(A Un B) = Inter(A) Int Inter(B)"; |
2891 | 491 |
by (Blast_tac 1); |
923 | 492 |
qed "Inter_Un_distrib"; |
493 |
||
1548 | 494 |
section "UN and INT"; |
923 | 495 |
|
496 |
(*Basic identities*) |
|
497 |
||
5069 | 498 |
Goal "(UN x:{}. B x) = {}"; |
2891 | 499 |
by (Blast_tac 1); |
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
changeset
|
500 |
qed "UN_empty"; |
1531 | 501 |
Addsimps[UN_empty]; |
502 |
||
5069 | 503 |
Goal "(UN x:A. {}) = {}"; |
3457 | 504 |
by (Blast_tac 1); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
505 |
qed "UN_empty2"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
506 |
Addsimps[UN_empty2]; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
507 |
|
5069 | 508 |
Goal "(UN x:A. {x}) = A"; |
4645 | 509 |
by (Blast_tac 1); |
510 |
qed "UN_singleton"; |
|
511 |
Addsimps [UN_singleton]; |
|
512 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
513 |
Goal "k:I ==> A k Un (UN i:I. A i) = (UN i:I. A i)"; |
4634 | 514 |
by (Blast_tac 1); |
515 |
qed "UN_absorb"; |
|
516 |
||
5069 | 517 |
Goal "(INT x:{}. B x) = UNIV"; |
2891 | 518 |
by (Blast_tac 1); |
1531 | 519 |
qed "INT_empty"; |
520 |
Addsimps[INT_empty]; |
|
521 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
522 |
Goal "k:I ==> A k Int (INT i:I. A i) = (INT i:I. A i)"; |
4634 | 523 |
by (Blast_tac 1); |
524 |
qed "INT_absorb"; |
|
525 |
||
5069 | 526 |
Goal "(UN x:insert a A. B x) = B a Un UNION A B"; |
2891 | 527 |
by (Blast_tac 1); |
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
changeset
|
528 |
qed "UN_insert"; |
1531 | 529 |
Addsimps[UN_insert]; |
530 |
||
5999 | 531 |
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
|
532 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
533 |
qed "UN_Un"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
534 |
|
5069 | 535 |
Goal "(UN x : (UN y:A. B y). C x) = (UN y:A. UN x: B y. C x)"; |
4771 | 536 |
by (Blast_tac 1); |
537 |
qed "UN_UN_flatten"; |
|
538 |
||
6292
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
539 |
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
|
540 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
541 |
qed "UN_subset_iff"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
542 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
543 |
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
|
544 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
545 |
qed "INT_subset_iff"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
546 |
|
5069 | 547 |
Goal "(INT x:insert a A. B x) = B a Int INTER A B"; |
2891 | 548 |
by (Blast_tac 1); |
1531 | 549 |
qed "INT_insert"; |
550 |
Addsimps[INT_insert]; |
|
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
changeset
|
551 |
|
5999 | 552 |
Goal "(INT i: A Un B. M i) = (INT i: A. M i) Int (INT i:B. M i)"; |
553 |
by (Blast_tac 1); |
|
554 |
qed "INT_Un"; |
|
555 |
||
5941
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
556 |
Goal "u: A ==> (INT x:A. insert a (B x)) = insert a (INT x:A. B x)"; |
2891 | 557 |
by (Blast_tac 1); |
2021 | 558 |
qed "INT_insert_distrib"; |
559 |
||
5069 | 560 |
Goal "Union(B``A) = (UN x:A. B(x))"; |
2891 | 561 |
by (Blast_tac 1); |
923 | 562 |
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
|
563 |
Addsimps [Union_image_eq]; |
923 | 564 |
|
6283 | 565 |
Goal "f `` Union S = (UN x:S. f `` x)"; |
566 |
by (Blast_tac 1); |
|
8176 | 567 |
qed "image_Union"; |
6283 | 568 |
|
5069 | 569 |
Goal "Inter(B``A) = (INT x:A. B(x))"; |
2891 | 570 |
by (Blast_tac 1); |
923 | 571 |
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
|
572 |
Addsimps [Inter_image_eq]; |
923 | 573 |
|
8313
c7a87e79e9b1
replaced UN_constant, INT_constant by unconditional versions that rewrite
paulson
parents:
8176
diff
changeset
|
574 |
Goal "(UN y:A. c) = (if A={} then {} else c)"; |
c7a87e79e9b1
replaced UN_constant, INT_constant by unconditional versions that rewrite
paulson
parents:
8176
diff
changeset
|
575 |
by Auto_tac; |
923 | 576 |
qed "UN_constant"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
577 |
Addsimps[UN_constant]; |
923 | 578 |
|
8313
c7a87e79e9b1
replaced UN_constant, INT_constant by unconditional versions that rewrite
paulson
parents:
8176
diff
changeset
|
579 |
Goal "(INT y:A. c) = (if A={} then UNIV else c)"; |
c7a87e79e9b1
replaced UN_constant, INT_constant by unconditional versions that rewrite
paulson
parents:
8176
diff
changeset
|
580 |
by Auto_tac; |
923 | 581 |
qed "INT_constant"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
582 |
Addsimps[INT_constant]; |
923 | 583 |
|
9206 | 584 |
Goal "(UN x:A. B(x)) = Union({Y. EX x:A. Y=B(x)})"; |
2891 | 585 |
by (Blast_tac 1); |
923 | 586 |
qed "UN_eq"; |
587 |
||
588 |
(*Look: it has an EXISTENTIAL quantifier*) |
|
9206 | 589 |
Goal "(INT x:A. B(x)) = Inter({Y. EX x:A. Y=B(x)})"; |
2891 | 590 |
by (Blast_tac 1); |
923 | 591 |
qed "INT_eq"; |
592 |
||
9098
3a0912a127ec
new theorem UN_empty; it and Un_empty inserted by AddIffs
paulson
parents:
9077
diff
changeset
|
593 |
Goal "(UNION A B = {}) = (ALL x:A. B x = {})"; |
3a0912a127ec
new theorem UN_empty; it and Un_empty inserted by AddIffs
paulson
parents:
9077
diff
changeset
|
594 |
by Auto_tac; |
3a0912a127ec
new theorem UN_empty; it and Un_empty inserted by AddIffs
paulson
parents:
9077
diff
changeset
|
595 |
qed "UN_empty"; |
3a0912a127ec
new theorem UN_empty; it and Un_empty inserted by AddIffs
paulson
parents:
9077
diff
changeset
|
596 |
AddIffs [UN_empty]; |
3a0912a127ec
new theorem UN_empty; it and Un_empty inserted by AddIffs
paulson
parents:
9077
diff
changeset
|
597 |
|
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
598 |
|
923 | 599 |
(*Distributive laws...*) |
600 |
||
5069 | 601 |
Goal "A Int Union(B) = (UN C:B. A Int C)"; |
2891 | 602 |
by (Blast_tac 1); |
923 | 603 |
qed "Int_Union"; |
604 |
||
5069 | 605 |
Goal "Union(B) Int A = (UN C:B. C Int A)"; |
4674 | 606 |
by (Blast_tac 1); |
607 |
qed "Int_Union2"; |
|
608 |
||
4306
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
609 |
(* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: |
923 | 610 |
Union of a family of unions **) |
5069 | 611 |
Goal "(UN x:C. A(x) Un B(x)) = Union(A``C) Un Union(B``C)"; |
2891 | 612 |
by (Blast_tac 1); |
923 | 613 |
qed "Un_Union_image"; |
614 |
||
615 |
(*Equivalent version*) |
|
5069 | 616 |
Goal "(UN i:I. A(i) Un B(i)) = (UN i:I. A(i)) Un (UN i:I. B(i))"; |
2891 | 617 |
by (Blast_tac 1); |
923 | 618 |
qed "UN_Un_distrib"; |
619 |
||
5069 | 620 |
Goal "A Un Inter(B) = (INT C:B. A Un C)"; |
2891 | 621 |
by (Blast_tac 1); |
923 | 622 |
qed "Un_Inter"; |
623 |
||
5069 | 624 |
Goal "(INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C)"; |
2891 | 625 |
by (Blast_tac 1); |
923 | 626 |
qed "Int_Inter_image"; |
627 |
||
628 |
(*Equivalent version*) |
|
5069 | 629 |
Goal "(INT i:I. A(i) Int B(i)) = (INT i:I. A(i)) Int (INT i:I. B(i))"; |
2891 | 630 |
by (Blast_tac 1); |
923 | 631 |
qed "INT_Int_distrib"; |
632 |
||
633 |
(*Halmos, Naive Set Theory, page 35.*) |
|
5069 | 634 |
Goal "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))"; |
2891 | 635 |
by (Blast_tac 1); |
923 | 636 |
qed "Int_UN_distrib"; |
637 |
||
5069 | 638 |
Goal "B Un (INT i:I. A(i)) = (INT i:I. B Un A(i))"; |
2891 | 639 |
by (Blast_tac 1); |
923 | 640 |
qed "Un_INT_distrib"; |
641 |
||
5278 | 642 |
Goal "(UN i:I. A(i)) Int (UN j:J. B(j)) = (UN i:I. UN j:J. A(i) Int B(j))"; |
2891 | 643 |
by (Blast_tac 1); |
923 | 644 |
qed "Int_UN_distrib2"; |
645 |
||
5278 | 646 |
Goal "(INT i:I. A(i)) Un (INT j:J. B(j)) = (INT i:I. INT j:J. A(i) Un B(j))"; |
2891 | 647 |
by (Blast_tac 1); |
923 | 648 |
qed "Un_INT_distrib2"; |
649 |
||
2512 | 650 |
|
651 |
section"Bounded quantifiers"; |
|
652 |
||
3860 | 653 |
(** The following are not added to the default simpset because |
654 |
(a) they duplicate the body and (b) there are no similar rules for Int. **) |
|
2512 | 655 |
|
5069 | 656 |
Goal "(ALL x:A Un B. P x) = ((ALL x:A. P x) & (ALL x:B. P x))"; |
2891 | 657 |
by (Blast_tac 1); |
2519 | 658 |
qed "ball_Un"; |
659 |
||
5069 | 660 |
Goal "(EX x:A Un B. P x) = ((EX x:A. P x) | (EX x:B. P x))"; |
2891 | 661 |
by (Blast_tac 1); |
2519 | 662 |
qed "bex_Un"; |
2512 | 663 |
|
5069 | 664 |
Goal "(ALL z: UNION A B. P z) = (ALL x:A. ALL z:B x. P z)"; |
4771 | 665 |
by (Blast_tac 1); |
666 |
qed "ball_UN"; |
|
667 |
||
5069 | 668 |
Goal "(EX z: UNION A B. P z) = (EX x:A. EX z:B x. P z)"; |
4771 | 669 |
by (Blast_tac 1); |
670 |
qed "bex_UN"; |
|
671 |
||
2512 | 672 |
|
1548 | 673 |
section "-"; |
923 | 674 |
|
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
675 |
Goal "A-B = A Int (-B)"; |
4609 | 676 |
by (Blast_tac 1); |
4662 | 677 |
qed "Diff_eq"; |
4609 | 678 |
|
7516 | 679 |
Goal "(A-B = {}) = (A<=B)"; |
680 |
by (Blast_tac 1); |
|
681 |
qed "Diff_eq_empty_iff"; |
|
682 |
Addsimps[Diff_eq_empty_iff]; |
|
683 |
||
5069 | 684 |
Goal "A-A = {}"; |
2891 | 685 |
by (Blast_tac 1); |
923 | 686 |
qed "Diff_cancel"; |
1531 | 687 |
Addsimps[Diff_cancel]; |
923 | 688 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
689 |
Goal "A Int B = {} ==> A-B = A"; |
4674 | 690 |
by (blast_tac (claset() addEs [equalityE]) 1); |
691 |
qed "Diff_triv"; |
|
692 |
||
5069 | 693 |
Goal "{}-A = {}"; |
2891 | 694 |
by (Blast_tac 1); |
923 | 695 |
qed "empty_Diff"; |
1531 | 696 |
Addsimps[empty_Diff]; |
923 | 697 |
|
5069 | 698 |
Goal "A-{} = A"; |
2891 | 699 |
by (Blast_tac 1); |
923 | 700 |
qed "Diff_empty"; |
1531 | 701 |
Addsimps[Diff_empty]; |
702 |
||
5069 | 703 |
Goal "A-UNIV = {}"; |
2891 | 704 |
by (Blast_tac 1); |
1531 | 705 |
qed "Diff_UNIV"; |
706 |
Addsimps[Diff_UNIV]; |
|
707 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
708 |
Goal "x~:A ==> A - insert x B = A-B"; |
2891 | 709 |
by (Blast_tac 1); |
1531 | 710 |
qed "Diff_insert0"; |
711 |
Addsimps [Diff_insert0]; |
|
923 | 712 |
|
713 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a 0*) |
|
5069 | 714 |
Goal "A - insert a B = A - B - {a}"; |
2891 | 715 |
by (Blast_tac 1); |
923 | 716 |
qed "Diff_insert"; |
717 |
||
718 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a 0*) |
|
5069 | 719 |
Goal "A - insert a B = A - {a} - B"; |
2891 | 720 |
by (Blast_tac 1); |
923 | 721 |
qed "Diff_insert2"; |
722 |
||
5069 | 723 |
Goal "insert x A - B = (if x:B then A-B else insert x (A-B))"; |
4686 | 724 |
by (Simp_tac 1); |
2891 | 725 |
by (Blast_tac 1); |
1531 | 726 |
qed "insert_Diff_if"; |
727 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
728 |
Goal "x:B ==> insert x A - B = A-B"; |
2891 | 729 |
by (Blast_tac 1); |
1531 | 730 |
qed "insert_Diff1"; |
731 |
Addsimps [insert_Diff1]; |
|
732 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
733 |
Goal "a:A ==> insert a (A-{a}) = A"; |
2922 | 734 |
by (Blast_tac 1); |
923 | 735 |
qed "insert_Diff"; |
736 |
||
7824
1a85ba81d019
new default simprule Collect_const and new them Diff_insert_absorb
paulson
parents:
7713
diff
changeset
|
737 |
Goal "x ~: A ==> (insert x A) - {x} = A"; |
1a85ba81d019
new default simprule Collect_const and new them Diff_insert_absorb
paulson
parents:
7713
diff
changeset
|
738 |
by Auto_tac; |
1a85ba81d019
new default simprule Collect_const and new them Diff_insert_absorb
paulson
parents:
7713
diff
changeset
|
739 |
qed "Diff_insert_absorb"; |
1a85ba81d019
new default simprule Collect_const and new them Diff_insert_absorb
paulson
parents:
7713
diff
changeset
|
740 |
|
5069 | 741 |
Goal "A Int (B-A) = {}"; |
2891 | 742 |
by (Blast_tac 1); |
923 | 743 |
qed "Diff_disjoint"; |
1531 | 744 |
Addsimps[Diff_disjoint]; |
923 | 745 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
746 |
Goal "A<=B ==> A Un (B-A) = B"; |
2891 | 747 |
by (Blast_tac 1); |
923 | 748 |
qed "Diff_partition"; |
749 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
750 |
Goal "[| A<=B; B<= C |] ==> (B - (C - A)) = (A :: 'a set)"; |
2891 | 751 |
by (Blast_tac 1); |
923 | 752 |
qed "double_diff"; |
753 |
||
5069 | 754 |
Goal "A Un (B-A) = A Un B"; |
4645 | 755 |
by (Blast_tac 1); |
756 |
qed "Un_Diff_cancel"; |
|
757 |
||
5069 | 758 |
Goal "(B-A) Un A = B Un A"; |
4645 | 759 |
by (Blast_tac 1); |
760 |
qed "Un_Diff_cancel2"; |
|
761 |
||
762 |
Addsimps [Un_Diff_cancel, Un_Diff_cancel2]; |
|
763 |
||
5069 | 764 |
Goal "A - (B Un C) = (A-B) Int (A-C)"; |
2891 | 765 |
by (Blast_tac 1); |
923 | 766 |
qed "Diff_Un"; |
767 |
||
5069 | 768 |
Goal "A - (B Int C) = (A-B) Un (A-C)"; |
2891 | 769 |
by (Blast_tac 1); |
923 | 770 |
qed "Diff_Int"; |
771 |
||
5069 | 772 |
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
|
773 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
774 |
qed "Un_Diff"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
775 |
|
5069 | 776 |
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
|
777 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
778 |
qed "Int_Diff"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
779 |
|
5069 | 780 |
Goal "C Int (A-B) = (C Int A) - (C Int B)"; |
4748 | 781 |
by (Blast_tac 1); |
782 |
qed "Diff_Int_distrib"; |
|
783 |
||
5069 | 784 |
Goal "(A-B) Int C = (A Int C) - (B Int C)"; |
4645 | 785 |
by (Blast_tac 1); |
4748 | 786 |
qed "Diff_Int_distrib2"; |
4645 | 787 |
|
7127
48e235179ffb
added parentheses to cope with a possible reduction of the precedence of unary
paulson
parents:
6832
diff
changeset
|
788 |
Goal "A - (- B) = A Int B"; |
5632 | 789 |
by Auto_tac; |
790 |
qed "Diff_Compl"; |
|
791 |
Addsimps [Diff_Compl]; |
|
792 |
||
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
793 |
|
5238 | 794 |
section "Quantification over type \"bool\""; |
795 |
||
796 |
Goal "(ALL b::bool. P b) = (P True & P False)"; |
|
797 |
by Auto_tac; |
|
798 |
by (case_tac "b" 1); |
|
799 |
by Auto_tac; |
|
800 |
qed "all_bool_eq"; |
|
801 |
||
5762 | 802 |
bind_thm ("bool_induct", conjI RS (all_bool_eq RS iffD2) RS spec); |
803 |
||
5238 | 804 |
Goal "(EX b::bool. P b) = (P True | P False)"; |
805 |
by Auto_tac; |
|
806 |
by (case_tac "b" 1); |
|
807 |
by Auto_tac; |
|
808 |
qed "ex_bool_eq"; |
|
809 |
||
810 |
Goal "A Un B = (UN b. if b then A else B)"; |
|
8026 | 811 |
by (auto_tac(claset(), simpset() addsimps [split_if_mem2])); |
5238 | 812 |
qed "Un_eq_UN"; |
813 |
||
814 |
Goal "(UN b::bool. A b) = (A True Un A False)"; |
|
815 |
by Auto_tac; |
|
816 |
by (case_tac "b" 1); |
|
817 |
by Auto_tac; |
|
818 |
qed "UN_bool_eq"; |
|
819 |
||
820 |
Goal "(INT b::bool. A b) = (A True Int A False)"; |
|
821 |
by Auto_tac; |
|
822 |
by (case_tac "b" 1); |
|
823 |
by Auto_tac; |
|
824 |
qed "INT_bool_eq"; |
|
825 |
||
826 |
||
6292
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
827 |
section "Pow"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
828 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
829 |
Goalw [Pow_def] "Pow {} = {{}}"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
830 |
by Auto_tac; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
831 |
qed "Pow_empty"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
832 |
Addsimps [Pow_empty]; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
833 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
834 |
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
|
835 |
by Safe_tac; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
836 |
by (etac swap 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
837 |
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
|
838 |
by (ALLGOALS Blast_tac); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
839 |
qed "Pow_insert"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
840 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
841 |
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
|
842 |
by Safe_tac; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
843 |
by (Blast_tac 2); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
844 |
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
|
845 |
by (ALLGOALS Blast_tac); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
846 |
qed "Pow_Compl"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
847 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
848 |
Goal "Pow UNIV = UNIV"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
849 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
850 |
qed "Pow_UNIV"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
851 |
Addsimps [Pow_UNIV]; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
852 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
853 |
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
|
854 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
855 |
qed "Un_Pow_subset"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
856 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
857 |
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
|
858 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
859 |
qed "UN_Pow_subset"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
860 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
861 |
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
|
862 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
863 |
qed "subset_Pow_Union"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
864 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
865 |
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
|
866 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
867 |
qed "Union_Pow_eq"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
868 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
869 |
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
|
870 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
871 |
qed "Pow_Int_eq"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
872 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
873 |
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
|
874 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
875 |
qed "Pow_INT_eq"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
876 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
877 |
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
|
878 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
879 |
|
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
880 |
section "Miscellany"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
881 |
|
5069 | 882 |
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
|
883 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
884 |
qed "set_eq_subset"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
885 |
|
9206 | 886 |
Goal "A <= B = (ALL t. t:A --> t:B)"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
887 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
888 |
qed "subset_iff"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
889 |
|
5069 | 890 |
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
|
891 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
892 |
qed "subset_iff_psubset_eq"; |
2021 | 893 |
|
9206 | 894 |
Goal "(ALL x. x ~: A) = (A={})"; |
4423 | 895 |
by (Blast_tac 1); |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
896 |
qed "all_not_in_conv"; |
3907 | 897 |
AddIffs [all_not_in_conv]; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
898 |
|
6007 | 899 |
|
5189
362e4d6213c5
Added theorem distinct_lemma (needed for datatypes).
berghofe
parents:
5148
diff
changeset
|
900 |
(** for datatypes **) |
362e4d6213c5
Added theorem distinct_lemma (needed for datatypes).
berghofe
parents:
5148
diff
changeset
|
901 |
Goal "f x ~= f y ==> x ~= y"; |
362e4d6213c5
Added theorem distinct_lemma (needed for datatypes).
berghofe
parents:
5148
diff
changeset
|
902 |
by (Fast_tac 1); |
362e4d6213c5
Added theorem distinct_lemma (needed for datatypes).
berghofe
parents:
5148
diff
changeset
|
903 |
qed "distinct_lemma"; |
362e4d6213c5
Added theorem distinct_lemma (needed for datatypes).
berghofe
parents:
5148
diff
changeset
|
904 |
|
2021 | 905 |
|
906 |
(** Miniscoping: pushing in big Unions and Intersections **) |
|
907 |
local |
|
4059 | 908 |
fun prover s = prove_goal thy s (fn _ => [Blast_tac 1]) |
2021 | 909 |
in |
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
910 |
val UN_simps = map prover |
5941
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
911 |
["!!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
|
912 |
"!!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
|
913 |
"!!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
|
914 |
"(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
|
915 |
"(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
|
916 |
"(UN x:C. A x - B) = ((UN x:C. A x) - B)", |
4231 | 917 |
"(UN x:C. A - B x) = (A - (INT x:C. B x))", |
7914 | 918 |
"(UN x: Union A. B x) = (UN y:A. UN x:y. B x)", |
919 |
"(UN z: UNION A B. C z) = (UN x:A. UN z: B(x). C z)", |
|
4231 | 920 |
"(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
|
921 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
922 |
val INT_simps = map prover |
5941
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
923 |
["!!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
|
924 |
"!!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
|
925 |
"!!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
|
926 |
"!!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
|
927 |
"(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
|
928 |
"(INT x:C. A x Un B) = ((INT x:C. A x) Un B)", |
4231 | 929 |
"(INT x:C. A Un B x) = (A Un (INT x:C. B x))", |
7914 | 930 |
"(INT x: Union A. B x) = (INT y:A. INT x:y. B x)", |
931 |
"(INT z: UNION A B. C z) = (INT x:A. INT z: B(x). C z)", |
|
4231 | 932 |
"(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
|
933 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
934 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
935 |
val ball_simps = map prover |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
936 |
["(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
|
937 |
"(ALL x:A. P | Q x) = (P | (ALL x:A. Q x))", |
3422 | 938 |
"(ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))", |
939 |
"(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
|
940 |
"(ALL x:{}. P x) = True", |
4136 | 941 |
"(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
|
942 |
"(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
|
943 |
"(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
|
944 |
"(ALL x: UNION A B. P x) = (ALL a:A. ALL x: B a. P x)", |
3860 | 945 |
"(ALL x:Collect Q. P x) = (ALL x. Q x --> P x)", |
946 |
"(ALL x:f``A. P x) = (ALL x:A. P(f x))", |
|
947 |
"(~(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
|
948 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
949 |
val ball_conj_distrib = |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
950 |
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
|
951 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
952 |
val bex_simps = map prover |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
953 |
["(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
|
954 |
"(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
|
955 |
"(EX x:{}. P x) = False", |
4136 | 956 |
"(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
|
957 |
"(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
|
958 |
"(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
|
959 |
"(EX x: UNION A B. P x) = (EX a:A. EX x: B a. P x)", |
3860 | 960 |
"(EX x:Collect Q. P x) = (EX x. Q x & P x)", |
961 |
"(EX x:f``A. P x) = (EX x:A. P(f x))", |
|
962 |
"(~(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
|
963 |
|
3426 | 964 |
val bex_disj_distrib = |
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
965 |
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
|
966 |
|
2021 | 967 |
end; |
968 |
||
7648 | 969 |
bind_thms ("UN_simps", UN_simps); |
970 |
bind_thms ("INT_simps", INT_simps); |
|
971 |
bind_thms ("ball_simps", ball_simps); |
|
972 |
bind_thms ("bex_simps", bex_simps); |
|
973 |
bind_thm ("ball_conj_distrib", ball_conj_distrib); |
|
974 |
bind_thm ("bex_disj_distrib", bex_disj_distrib); |
|
975 |
||
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
976 |
Addsimps (UN_simps @ INT_simps @ ball_simps @ bex_simps); |