author | wenzelm |
Thu, 01 Jul 1999 21:19:45 +0200 | |
changeset 6874 | 747f656e04ec |
parent 6832 | 0c92ccb3c4ba |
child 7127 | 48e235179ffb |
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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||
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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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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); |
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qed "insert_absorb"; |
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(* 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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Goal "insert a A ~= insert a B ==> A ~= B"; |
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by (Blast_tac 1); |
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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 |
|
5069 | 104 |
Goal "(%x. x) `` Y = Y"; |
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105 |
by (Blast_tac 1); |
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106 |
qed "image_id"; |
5967 | 107 |
Addsimps [image_id]; |
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|
108 |
|
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new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
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109 |
Goal "x:A ==> (%x. c) `` A = {c}"; |
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110 |
by (Blast_tac 1); |
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111 |
qed "image_constant"; |
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112 |
|
5069 | 113 |
Goal "f``(g``A) = (%x. f (g x)) `` A"; |
3457 | 114 |
by (Blast_tac 1); |
4059 | 115 |
qed "image_image"; |
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116 |
|
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117 |
Goal "x:A ==> insert (f x) (f``A) = f``A"; |
2891 | 118 |
by (Blast_tac 1); |
1884 | 119 |
qed "insert_image"; |
120 |
Addsimps [insert_image]; |
|
121 |
||
5069 | 122 |
Goal "(f``A = {}) = (A = {})"; |
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123 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
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124 |
qed "image_is_empty"; |
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125 |
AddIffs [image_is_empty]; |
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126 |
|
5281 | 127 |
Goal "f `` {x. P x} = {f x | x. P x}"; |
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128 |
by (Blast_tac 1); |
5281 | 129 |
qed "image_Collect"; |
130 |
Addsimps [image_Collect]; |
|
131 |
||
5590 | 132 |
Goalw [image_def] "(%x. if P x then f x else g x) `` S \ |
133 |
\ = (f `` (S Int {x. P x})) Un (g `` (S Int {x. ~(P x)}))"; |
|
4686 | 134 |
by (Simp_tac 1); |
2891 | 135 |
by (Blast_tac 1); |
1748 | 136 |
qed "if_image_distrib"; |
137 |
Addsimps[if_image_distrib]; |
|
138 |
||
5590 | 139 |
val prems = Goal "[|M = N; !!x. x:N ==> f x = g x|] ==> f``M = g``N"; |
4136 | 140 |
by (simp_tac (simpset() addsimps image_def::prems) 1); |
141 |
qed "image_cong"; |
|
142 |
||
1748 | 143 |
|
1548 | 144 |
section "Int"; |
923 | 145 |
|
5069 | 146 |
Goal "A Int A = A"; |
2891 | 147 |
by (Blast_tac 1); |
923 | 148 |
qed "Int_absorb"; |
1531 | 149 |
Addsimps[Int_absorb]; |
923 | 150 |
|
5590 | 151 |
Goal "A Int (A Int B) = A Int B"; |
4609 | 152 |
by (Blast_tac 1); |
153 |
qed "Int_left_absorb"; |
|
154 |
||
5590 | 155 |
Goal "A Int B = B Int A"; |
2891 | 156 |
by (Blast_tac 1); |
923 | 157 |
qed "Int_commute"; |
158 |
||
5069 | 159 |
Goal "A Int (B Int C) = B Int (A Int C)"; |
4609 | 160 |
by (Blast_tac 1); |
161 |
qed "Int_left_commute"; |
|
162 |
||
5590 | 163 |
Goal "(A Int B) Int C = A Int (B Int C)"; |
2891 | 164 |
by (Blast_tac 1); |
923 | 165 |
qed "Int_assoc"; |
166 |
||
4609 | 167 |
(*Intersection is an AC-operator*) |
168 |
val Int_ac = [Int_assoc, Int_left_absorb, Int_commute, Int_left_commute]; |
|
169 |
||
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170 |
Goal "B<=A ==> A Int B = B"; |
4662 | 171 |
by (Blast_tac 1); |
172 |
qed "Int_absorb1"; |
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173 |
||
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174 |
Goal "A<=B ==> A Int B = A"; |
4662 | 175 |
by (Blast_tac 1); |
176 |
qed "Int_absorb2"; |
|
177 |
||
5069 | 178 |
Goal "{} Int B = {}"; |
2891 | 179 |
by (Blast_tac 1); |
923 | 180 |
qed "Int_empty_left"; |
1531 | 181 |
Addsimps[Int_empty_left]; |
923 | 182 |
|
5069 | 183 |
Goal "A Int {} = {}"; |
2891 | 184 |
by (Blast_tac 1); |
923 | 185 |
qed "Int_empty_right"; |
1531 | 186 |
Addsimps[Int_empty_right]; |
187 |
||
5490 | 188 |
Goal "(A Int B = {}) = (A <= -B)"; |
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189 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
3356 | 190 |
qed "disjoint_eq_subset_Compl"; |
191 |
||
5069 | 192 |
Goal "UNIV Int B = B"; |
2891 | 193 |
by (Blast_tac 1); |
1531 | 194 |
qed "Int_UNIV_left"; |
195 |
Addsimps[Int_UNIV_left]; |
|
196 |
||
5069 | 197 |
Goal "A Int UNIV = A"; |
2891 | 198 |
by (Blast_tac 1); |
1531 | 199 |
qed "Int_UNIV_right"; |
200 |
Addsimps[Int_UNIV_right]; |
|
923 | 201 |
|
5069 | 202 |
Goal "A Int B = Inter{A,B}"; |
4634 | 203 |
by (Blast_tac 1); |
204 |
qed "Int_eq_Inter"; |
|
205 |
||
5590 | 206 |
Goal "A Int (B Un C) = (A Int B) Un (A Int C)"; |
2891 | 207 |
by (Blast_tac 1); |
923 | 208 |
qed "Int_Un_distrib"; |
209 |
||
5590 | 210 |
Goal "(B Un C) Int A = (B Int A) Un (C Int A)"; |
2891 | 211 |
by (Blast_tac 1); |
1618 | 212 |
qed "Int_Un_distrib2"; |
213 |
||
5069 | 214 |
Goal "(A Int B = UNIV) = (A = UNIV & B = UNIV)"; |
4089 | 215 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
1531 | 216 |
qed "Int_UNIV"; |
217 |
Addsimps[Int_UNIV]; |
|
218 |
||
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219 |
Goal "(C <= A Int B) = (C <= A & C <= B)"; |
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220 |
by (Blast_tac 1); |
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221 |
qed "Int_subset_iff"; |
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222 |
|
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223 |
|
1548 | 224 |
section "Un"; |
923 | 225 |
|
5069 | 226 |
Goal "A Un A = A"; |
2891 | 227 |
by (Blast_tac 1); |
923 | 228 |
qed "Un_absorb"; |
1531 | 229 |
Addsimps[Un_absorb]; |
923 | 230 |
|
5069 | 231 |
Goal " A Un (A Un B) = A Un B"; |
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232 |
by (Blast_tac 1); |
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233 |
qed "Un_left_absorb"; |
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234 |
|
5590 | 235 |
Goal "A Un B = B Un A"; |
2891 | 236 |
by (Blast_tac 1); |
923 | 237 |
qed "Un_commute"; |
238 |
||
5069 | 239 |
Goal "A Un (B Un C) = B Un (A Un C)"; |
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|
240 |
by (Blast_tac 1); |
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|
241 |
qed "Un_left_commute"; |
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242 |
|
5590 | 243 |
Goal "(A Un B) Un C = A Un (B Un C)"; |
2891 | 244 |
by (Blast_tac 1); |
923 | 245 |
qed "Un_assoc"; |
246 |
||
4609 | 247 |
(*Union is an AC-operator*) |
248 |
val Un_ac = [Un_assoc, Un_left_absorb, Un_commute, Un_left_commute]; |
|
249 |
||
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250 |
Goal "A<=B ==> A Un B = B"; |
4662 | 251 |
by (Blast_tac 1); |
252 |
qed "Un_absorb1"; |
|
253 |
||
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|
254 |
Goal "B<=A ==> A Un B = A"; |
4662 | 255 |
by (Blast_tac 1); |
256 |
qed "Un_absorb2"; |
|
257 |
||
5069 | 258 |
Goal "{} Un B = B"; |
2891 | 259 |
by (Blast_tac 1); |
923 | 260 |
qed "Un_empty_left"; |
1531 | 261 |
Addsimps[Un_empty_left]; |
923 | 262 |
|
5069 | 263 |
Goal "A Un {} = A"; |
2891 | 264 |
by (Blast_tac 1); |
923 | 265 |
qed "Un_empty_right"; |
1531 | 266 |
Addsimps[Un_empty_right]; |
267 |
||
5069 | 268 |
Goal "UNIV Un B = UNIV"; |
2891 | 269 |
by (Blast_tac 1); |
1531 | 270 |
qed "Un_UNIV_left"; |
271 |
Addsimps[Un_UNIV_left]; |
|
272 |
||
5069 | 273 |
Goal "A Un UNIV = UNIV"; |
2891 | 274 |
by (Blast_tac 1); |
1531 | 275 |
qed "Un_UNIV_right"; |
276 |
Addsimps[Un_UNIV_right]; |
|
923 | 277 |
|
5069 | 278 |
Goal "A Un B = Union{A,B}"; |
4634 | 279 |
by (Blast_tac 1); |
280 |
qed "Un_eq_Union"; |
|
281 |
||
5069 | 282 |
Goal "(insert a B) Un C = insert a (B Un C)"; |
2891 | 283 |
by (Blast_tac 1); |
923 | 284 |
qed "Un_insert_left"; |
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|
285 |
Addsimps[Un_insert_left]; |
923 | 286 |
|
5069 | 287 |
Goal "A Un (insert a B) = insert a (A Un B)"; |
2891 | 288 |
by (Blast_tac 1); |
1917 | 289 |
qed "Un_insert_right"; |
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|
290 |
Addsimps[Un_insert_right]; |
1917 | 291 |
|
5069 | 292 |
Goal "(insert a B) Int C = (if a:C then insert a (B Int C) \ |
5590 | 293 |
\ else B Int C)"; |
4686 | 294 |
by (Simp_tac 1); |
3356 | 295 |
by (Blast_tac 1); |
296 |
qed "Int_insert_left"; |
|
297 |
||
5069 | 298 |
Goal "A Int (insert a B) = (if a:A then insert a (A Int B) \ |
5590 | 299 |
\ else A Int B)"; |
4686 | 300 |
by (Simp_tac 1); |
3356 | 301 |
by (Blast_tac 1); |
302 |
qed "Int_insert_right"; |
|
303 |
||
5590 | 304 |
Goal "A Un (B Int C) = (A Un B) Int (A Un C)"; |
2891 | 305 |
by (Blast_tac 1); |
923 | 306 |
qed "Un_Int_distrib"; |
307 |
||
5590 | 308 |
Goal "(B Int C) Un A = (B Un A) Int (C Un A)"; |
4609 | 309 |
by (Blast_tac 1); |
310 |
qed "Un_Int_distrib2"; |
|
311 |
||
5590 | 312 |
Goal "(A Int B) Un (B Int C) Un (C Int A) = \ |
313 |
\ (A Un B) Int (B Un C) Int (C Un A)"; |
|
2891 | 314 |
by (Blast_tac 1); |
923 | 315 |
qed "Un_Int_crazy"; |
316 |
||
5069 | 317 |
Goal "(A<=B) = (A Un B = B)"; |
4306
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Tidying and using equalityCE instead of the slower equalityE
paulson
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4231
diff
changeset
|
318 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
923 | 319 |
qed "subset_Un_eq"; |
320 |
||
5069 | 321 |
Goal "(A <= insert b C) = (A <= C | b:A & A-{b} <= C)"; |
2891 | 322 |
by (Blast_tac 1); |
923 | 323 |
qed "subset_insert_iff"; |
324 |
||
5069 | 325 |
Goal "(A Un B = {}) = (A = {} & B = {})"; |
4089 | 326 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
923 | 327 |
qed "Un_empty"; |
1531 | 328 |
Addsimps[Un_empty]; |
923 | 329 |
|
5319
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
330 |
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
|
331 |
by (Blast_tac 1); |
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
332 |
qed "Un_subset_iff"; |
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
333 |
|
5331 | 334 |
Goal "(A - B) Un (A Int B) = A"; |
335 |
by (Blast_tac 1); |
|
336 |
qed "Un_Diff_Int"; |
|
337 |
||
5319
7356d0c88b1b
Moved Un_subset_iff and Int_subset_iff from UNITY to equalities.ML
paulson
parents:
5316
diff
changeset
|
338 |
|
5931 | 339 |
section "Set complement"; |
923 | 340 |
|
5490 | 341 |
Goal "A Int -A = {}"; |
2891 | 342 |
by (Blast_tac 1); |
923 | 343 |
qed "Compl_disjoint"; |
1531 | 344 |
Addsimps[Compl_disjoint]; |
923 | 345 |
|
5490 | 346 |
Goal "A Un -A = UNIV"; |
2891 | 347 |
by (Blast_tac 1); |
923 | 348 |
qed "Compl_partition"; |
349 |
||
5490 | 350 |
Goal "- -A = (A:: 'a set)"; |
2891 | 351 |
by (Blast_tac 1); |
923 | 352 |
qed "double_complement"; |
1531 | 353 |
Addsimps[double_complement]; |
923 | 354 |
|
5490 | 355 |
Goal "-(A Un B) = -A Int -B"; |
2891 | 356 |
by (Blast_tac 1); |
923 | 357 |
qed "Compl_Un"; |
358 |
||
5490 | 359 |
Goal "-(A Int B) = -A Un -B"; |
2891 | 360 |
by (Blast_tac 1); |
923 | 361 |
qed "Compl_Int"; |
362 |
||
5490 | 363 |
Goal "-(UN x:A. B(x)) = (INT x:A. -B(x))"; |
2891 | 364 |
by (Blast_tac 1); |
923 | 365 |
qed "Compl_UN"; |
366 |
||
5490 | 367 |
Goal "-(INT x:A. B(x)) = (UN x:A. -B(x))"; |
2891 | 368 |
by (Blast_tac 1); |
923 | 369 |
qed "Compl_INT"; |
370 |
||
4615 | 371 |
Addsimps [Compl_Un, Compl_Int, Compl_UN, Compl_INT]; |
372 |
||
923 | 373 |
(*Halmos, Naive Set Theory, page 16.*) |
374 |
||
5069 | 375 |
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
|
376 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
923 | 377 |
qed "Un_Int_assoc_eq"; |
378 |
||
379 |
||
1548 | 380 |
section "Union"; |
923 | 381 |
|
5069 | 382 |
Goal "Union({}) = {}"; |
2891 | 383 |
by (Blast_tac 1); |
923 | 384 |
qed "Union_empty"; |
1531 | 385 |
Addsimps[Union_empty]; |
386 |
||
5069 | 387 |
Goal "Union(UNIV) = UNIV"; |
2891 | 388 |
by (Blast_tac 1); |
1531 | 389 |
qed "Union_UNIV"; |
390 |
Addsimps[Union_UNIV]; |
|
923 | 391 |
|
5069 | 392 |
Goal "Union(insert a B) = a Un Union(B)"; |
2891 | 393 |
by (Blast_tac 1); |
923 | 394 |
qed "Union_insert"; |
1531 | 395 |
Addsimps[Union_insert]; |
923 | 396 |
|
5069 | 397 |
Goal "Union(A Un B) = Union(A) Un Union(B)"; |
2891 | 398 |
by (Blast_tac 1); |
923 | 399 |
qed "Union_Un_distrib"; |
1531 | 400 |
Addsimps[Union_Un_distrib]; |
923 | 401 |
|
5069 | 402 |
Goal "Union(A Int B) <= Union(A) Int Union(B)"; |
2891 | 403 |
by (Blast_tac 1); |
923 | 404 |
qed "Union_Int_subset"; |
405 |
||
5069 | 406 |
Goal "(Union M = {}) = (! A : M. A = {})"; |
4306
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
407 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
408 |
qed "Union_empty_conv"; |
4003 | 409 |
AddIffs [Union_empty_conv]; |
410 |
||
5069 | 411 |
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
|
412 |
by (blast_tac (claset() addSEs [equalityCE]) 1); |
923 | 413 |
qed "Union_disjoint"; |
414 |
||
1548 | 415 |
section "Inter"; |
416 |
||
5069 | 417 |
Goal "Inter({}) = UNIV"; |
2891 | 418 |
by (Blast_tac 1); |
1531 | 419 |
qed "Inter_empty"; |
420 |
Addsimps[Inter_empty]; |
|
421 |
||
5069 | 422 |
Goal "Inter(UNIV) = {}"; |
2891 | 423 |
by (Blast_tac 1); |
1531 | 424 |
qed "Inter_UNIV"; |
425 |
Addsimps[Inter_UNIV]; |
|
426 |
||
5069 | 427 |
Goal "Inter(insert a B) = a Int Inter(B)"; |
2891 | 428 |
by (Blast_tac 1); |
1531 | 429 |
qed "Inter_insert"; |
430 |
Addsimps[Inter_insert]; |
|
431 |
||
5069 | 432 |
Goal "Inter(A) Un Inter(B) <= Inter(A Int B)"; |
2891 | 433 |
by (Blast_tac 1); |
1564
822575c737bd
Deleted faulty comment; proved new rule Inter_Un_subset
paulson
parents:
1553
diff
changeset
|
434 |
qed "Inter_Un_subset"; |
1531 | 435 |
|
5069 | 436 |
Goal "Inter(A Un B) = Inter(A) Int Inter(B)"; |
2891 | 437 |
by (Blast_tac 1); |
923 | 438 |
qed "Inter_Un_distrib"; |
439 |
||
1548 | 440 |
section "UN and INT"; |
923 | 441 |
|
442 |
(*Basic identities*) |
|
443 |
||
4200 | 444 |
val not_empty = prove_goal Set.thy "(A ~= {}) = (? x. x:A)" (K [Blast_tac 1]); |
4136 | 445 |
|
5069 | 446 |
Goal "(UN x:{}. B x) = {}"; |
2891 | 447 |
by (Blast_tac 1); |
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
changeset
|
448 |
qed "UN_empty"; |
1531 | 449 |
Addsimps[UN_empty]; |
450 |
||
5069 | 451 |
Goal "(UN x:A. {}) = {}"; |
3457 | 452 |
by (Blast_tac 1); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
453 |
qed "UN_empty2"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
454 |
Addsimps[UN_empty2]; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
455 |
|
5069 | 456 |
Goal "(UN x:A. {x}) = A"; |
4645 | 457 |
by (Blast_tac 1); |
458 |
qed "UN_singleton"; |
|
459 |
Addsimps [UN_singleton]; |
|
460 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
461 |
Goal "k:I ==> A k Un (UN i:I. A i) = (UN i:I. A i)"; |
4634 | 462 |
by (Blast_tac 1); |
463 |
qed "UN_absorb"; |
|
464 |
||
5069 | 465 |
Goal "(INT x:{}. B x) = UNIV"; |
2891 | 466 |
by (Blast_tac 1); |
1531 | 467 |
qed "INT_empty"; |
468 |
Addsimps[INT_empty]; |
|
469 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
470 |
Goal "k:I ==> A k Int (INT i:I. A i) = (INT i:I. A i)"; |
4634 | 471 |
by (Blast_tac 1); |
472 |
qed "INT_absorb"; |
|
473 |
||
5069 | 474 |
Goal "(UN x:insert a A. B x) = B a Un UNION A B"; |
2891 | 475 |
by (Blast_tac 1); |
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
changeset
|
476 |
qed "UN_insert"; |
1531 | 477 |
Addsimps[UN_insert]; |
478 |
||
5999 | 479 |
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
|
480 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
481 |
qed "UN_Un"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
482 |
|
5069 | 483 |
Goal "(UN x : (UN y:A. B y). C x) = (UN y:A. UN x: B y. C x)"; |
4771 | 484 |
by (Blast_tac 1); |
485 |
qed "UN_UN_flatten"; |
|
486 |
||
6292
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
487 |
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
|
488 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
489 |
qed "UN_subset_iff"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
490 |
|
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
491 |
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
|
492 |
by (Blast_tac 1); |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
493 |
qed "INT_subset_iff"; |
e50e1142dd06
new results e.g. about Pow; new simprules Union_image_eq, Inter_image_eq
paulson
parents:
6283
diff
changeset
|
494 |
|
5069 | 495 |
Goal "(INT x:insert a A. B x) = B a Int INTER A B"; |
2891 | 496 |
by (Blast_tac 1); |
1531 | 497 |
qed "INT_insert"; |
498 |
Addsimps[INT_insert]; |
|
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
changeset
|
499 |
|
5999 | 500 |
Goal "(INT i: A Un B. M i) = (INT i: A. M i) Int (INT i:B. M i)"; |
501 |
by (Blast_tac 1); |
|
502 |
qed "INT_Un"; |
|
503 |
||
5941
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
504 |
Goal "u: A ==> (INT x:A. insert a (B x)) = insert a (INT x:A. B x)"; |
2891 | 505 |
by (Blast_tac 1); |
2021 | 506 |
qed "INT_insert_distrib"; |
507 |
||
5069 | 508 |
Goal "Union(B``A) = (UN x:A. B(x))"; |
2891 | 509 |
by (Blast_tac 1); |
923 | 510 |
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
|
511 |
Addsimps [Union_image_eq]; |
923 | 512 |
|
6283 | 513 |
Goal "f `` Union S = (UN x:S. f `` x)"; |
514 |
by (Blast_tac 1); |
|
515 |
qed "image_Union_eq"; |
|
516 |
||
5069 | 517 |
Goal "Inter(B``A) = (INT x:A. B(x))"; |
2891 | 518 |
by (Blast_tac 1); |
923 | 519 |
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
|
520 |
Addsimps [Inter_image_eq]; |
923 | 521 |
|
5941
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
522 |
Goal "u: A ==> (UN y:A. c) = c"; |
2891 | 523 |
by (Blast_tac 1); |
923 | 524 |
qed "UN_constant"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
525 |
Addsimps[UN_constant]; |
923 | 526 |
|
5941
1db9fad40a4f
better miniscoping rules: the premise C~={} is not good
paulson
parents:
5931
diff
changeset
|
527 |
Goal "u: A ==> (INT y:A. c) = c"; |
2891 | 528 |
by (Blast_tac 1); |
923 | 529 |
qed "INT_constant"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
530 |
Addsimps[INT_constant]; |
923 | 531 |
|
5069 | 532 |
Goal "(UN x:A. B(x)) = Union({Y. ? x:A. Y=B(x)})"; |
2891 | 533 |
by (Blast_tac 1); |
923 | 534 |
qed "UN_eq"; |
535 |
||
536 |
(*Look: it has an EXISTENTIAL quantifier*) |
|
5069 | 537 |
Goal "(INT x:A. B(x)) = Inter({Y. ? x:A. Y=B(x)})"; |
2891 | 538 |
by (Blast_tac 1); |
923 | 539 |
qed "INT_eq"; |
540 |
||
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
541 |
|
923 | 542 |
(*Distributive laws...*) |
543 |
||
5069 | 544 |
Goal "A Int Union(B) = (UN C:B. A Int C)"; |
2891 | 545 |
by (Blast_tac 1); |
923 | 546 |
qed "Int_Union"; |
547 |
||
5069 | 548 |
Goal "Union(B) Int A = (UN C:B. C Int A)"; |
4674 | 549 |
by (Blast_tac 1); |
550 |
qed "Int_Union2"; |
|
551 |
||
4306
ddbe1a9722ab
Tidying and using equalityCE instead of the slower equalityE
paulson
parents:
4231
diff
changeset
|
552 |
(* Devlin, Fundamentals of Contemporary Set Theory, page 12, exercise 5: |
923 | 553 |
Union of a family of unions **) |
5069 | 554 |
Goal "(UN x:C. A(x) Un B(x)) = Union(A``C) Un Union(B``C)"; |
2891 | 555 |
by (Blast_tac 1); |
923 | 556 |
qed "Un_Union_image"; |
557 |
||
558 |
(*Equivalent version*) |
|
5069 | 559 |
Goal "(UN i:I. A(i) Un B(i)) = (UN i:I. A(i)) Un (UN i:I. B(i))"; |
2891 | 560 |
by (Blast_tac 1); |
923 | 561 |
qed "UN_Un_distrib"; |
562 |
||
5069 | 563 |
Goal "A Un Inter(B) = (INT C:B. A Un C)"; |
2891 | 564 |
by (Blast_tac 1); |
923 | 565 |
qed "Un_Inter"; |
566 |
||
5069 | 567 |
Goal "(INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C)"; |
2891 | 568 |
by (Blast_tac 1); |
923 | 569 |
qed "Int_Inter_image"; |
570 |
||
571 |
(*Equivalent version*) |
|
5069 | 572 |
Goal "(INT i:I. A(i) Int B(i)) = (INT i:I. A(i)) Int (INT i:I. B(i))"; |
2891 | 573 |
by (Blast_tac 1); |
923 | 574 |
qed "INT_Int_distrib"; |
575 |
||
576 |
(*Halmos, Naive Set Theory, page 35.*) |
|
5069 | 577 |
Goal "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))"; |
2891 | 578 |
by (Blast_tac 1); |
923 | 579 |
qed "Int_UN_distrib"; |
580 |
||
5069 | 581 |
Goal "B Un (INT i:I. A(i)) = (INT i:I. B Un A(i))"; |
2891 | 582 |
by (Blast_tac 1); |
923 | 583 |
qed "Un_INT_distrib"; |
584 |
||
5278 | 585 |
Goal "(UN i:I. A(i)) Int (UN j:J. B(j)) = (UN i:I. UN j:J. A(i) Int B(j))"; |
2891 | 586 |
by (Blast_tac 1); |
923 | 587 |
qed "Int_UN_distrib2"; |
588 |
||
5278 | 589 |
Goal "(INT i:I. A(i)) Un (INT j:J. B(j)) = (INT i:I. INT j:J. A(i) Un B(j))"; |
2891 | 590 |
by (Blast_tac 1); |
923 | 591 |
qed "Un_INT_distrib2"; |
592 |
||
2512 | 593 |
|
594 |
section"Bounded quantifiers"; |
|
595 |
||
3860 | 596 |
(** The following are not added to the default simpset because |
597 |
(a) they duplicate the body and (b) there are no similar rules for Int. **) |
|
2512 | 598 |
|
5069 | 599 |
Goal "(ALL x:A Un B. P x) = ((ALL x:A. P x) & (ALL x:B. P x))"; |
2891 | 600 |
by (Blast_tac 1); |
2519 | 601 |
qed "ball_Un"; |
602 |
||
5069 | 603 |
Goal "(EX x:A Un B. P x) = ((EX x:A. P x) | (EX x:B. P x))"; |
2891 | 604 |
by (Blast_tac 1); |
2519 | 605 |
qed "bex_Un"; |
2512 | 606 |
|
5069 | 607 |
Goal "(ALL z: UNION A B. P z) = (ALL x:A. ALL z:B x. P z)"; |
4771 | 608 |
by (Blast_tac 1); |
609 |
qed "ball_UN"; |
|
610 |
||
5069 | 611 |
Goal "(EX z: UNION A B. P z) = (EX x:A. EX z:B x. P z)"; |
4771 | 612 |
by (Blast_tac 1); |
613 |
qed "bex_UN"; |
|
614 |
||
2512 | 615 |
|
1548 | 616 |
section "-"; |
923 | 617 |
|
5490 | 618 |
Goal "A-B = A Int -B"; |
4609 | 619 |
by (Blast_tac 1); |
4662 | 620 |
qed "Diff_eq"; |
4609 | 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 |
|
5632 | 722 |
Goal "A - - B = A Int B"; |
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 |
||
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4136
diff
changeset
|
899 |
Addsimps (UN_simps @ INT_simps @ ball_simps @ bex_simps); |