author | oheimb |
Tue, 04 Nov 1997 20:52:20 +0100 | |
changeset 4136 | ba267836dd7a |
parent 4089 | 96fba19bcbe2 |
child 4159 | 4aff9b7e5597 |
permissions | -rw-r--r-- |
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(* Title: HOL/equalities |
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ID: $Id$ |
1465 | 3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1994 University of Cambridge |
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||
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Equalities involving union, intersection, inclusion, etc. |
|
7 |
*) |
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||
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writeln"File HOL/equalities"; |
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AddSIs [equalityI]; |
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section "{}"; |
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||
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goal thy "{x. False} = {}"; |
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by (Blast_tac 1); |
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qed "Collect_False_empty"; |
18 |
Addsimps [Collect_False_empty]; |
|
19 |
||
4059 | 20 |
goal thy "(A <= {}) = (A = {})"; |
2891 | 21 |
by (Blast_tac 1); |
1531 | 22 |
qed "subset_empty"; |
23 |
Addsimps [subset_empty]; |
|
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||
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goalw thy [psubset_def] "~ (A < {})"; |
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by (Blast_tac 1); |
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qed "not_psubset_empty"; |
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AddIffs [not_psubset_empty]; |
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|
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section "insert"; |
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|
1531 | 32 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a {}*) |
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goal thy "insert a A = {a} Un A"; |
2891 | 34 |
by (Blast_tac 1); |
1531 | 35 |
qed "insert_is_Un"; |
36 |
||
4059 | 37 |
goal thy "insert a A ~= {}"; |
4089 | 38 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
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qed"insert_not_empty"; |
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Addsimps[insert_not_empty]; |
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bind_thm("empty_not_insert",insert_not_empty RS not_sym); |
1531 | 43 |
Addsimps[empty_not_insert]; |
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44 |
|
4059 | 45 |
goal thy "!!a. a:A ==> insert a A = A"; |
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by (Blast_tac 1); |
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qed "insert_absorb"; |
48 |
||
4059 | 49 |
goal thy "insert x (insert x A) = insert x A"; |
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by (Blast_tac 1); |
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qed "insert_absorb2"; |
52 |
Addsimps [insert_absorb2]; |
|
53 |
||
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goal thy "insert x (insert y A) = insert y (insert x A)"; |
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by (Blast_tac 1); |
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qed "insert_commute"; |
57 |
||
4059 | 58 |
goal thy "(insert x A <= B) = (x:B & A <= B)"; |
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by (Blast_tac 1); |
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qed "insert_subset"; |
1531 | 61 |
Addsimps[insert_subset]; |
62 |
||
4059 | 63 |
goal thy "!!a. insert a A ~= insert a B ==> A ~= B"; |
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by (Blast_tac 1); |
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qed "insert_lim"; |
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(* use new B rather than (A-{a}) to avoid infinite unfolding *) |
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goal thy "!!a. a:A ==> ? B. A = insert a B & a ~: B"; |
1553 | 69 |
by (res_inst_tac [("x","A-{a}")] exI 1); |
2891 | 70 |
by (Blast_tac 1); |
1531 | 71 |
qed "mk_disjoint_insert"; |
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|
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goal thy |
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"!!A. A~={} ==> (UN x:A. insert a (B x)) = insert a (UN x:A. B x)"; |
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by (Blast_tac 1); |
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qed "UN_insert_distrib"; |
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|
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goal thy "(UN x. insert a (B x)) = insert a (UN x. B x)"; |
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by (Blast_tac 1); |
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qed "UN1_insert_distrib"; |
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section "``"; |
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|
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goal thy "f``{} = {}"; |
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by (Blast_tac 1); |
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qed "image_empty"; |
1531 | 87 |
Addsimps[image_empty]; |
923 | 88 |
|
4059 | 89 |
goal thy "f``insert a B = insert (f a) (f``B)"; |
2891 | 90 |
by (Blast_tac 1); |
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qed "image_insert"; |
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Addsimps[image_insert]; |
923 | 93 |
|
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goal thy "(f `` (UNION A B)) = (UN x:A.(f `` (B x)))"; |
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by (Blast_tac 1); |
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qed "image_UNION"; |
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4059 | 98 |
goal thy "(%x. x) `` Y = Y"; |
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by (Blast_tac 1); |
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qed "image_id"; |
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101 |
|
4059 | 102 |
goal thy "f``(g``A) = (%x. f (g x)) `` A"; |
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by (Blast_tac 1); |
4059 | 104 |
qed "image_image"; |
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|
4059 | 106 |
goal thy "!!x. x:A ==> insert (f x) (f``A) = f``A"; |
2891 | 107 |
by (Blast_tac 1); |
1884 | 108 |
qed "insert_image"; |
109 |
Addsimps [insert_image]; |
|
110 |
||
4059 | 111 |
goal thy "(f``A = {}) = (A = {})"; |
4089 | 112 |
by (blast_tac (claset() addSEs [equalityE]) 1); |
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113 |
qed "image_is_empty"; |
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114 |
AddIffs [image_is_empty]; |
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115 |
|
4059 | 116 |
goalw thy [image_def] |
1763 | 117 |
"(%x. if P x then f x else g x) `` S \ |
3842 | 118 |
\ = (f `` ({x. x:S & P x})) Un (g `` ({x. x:S & ~(P x)}))"; |
2031 | 119 |
by (split_tac [expand_if] 1); |
2891 | 120 |
by (Blast_tac 1); |
1748 | 121 |
qed "if_image_distrib"; |
122 |
Addsimps[if_image_distrib]; |
|
123 |
||
4136 | 124 |
val prems= goal thy "[|M = N; !!x. x:N ==> f x = g x|] ==> f``M = g``N"; |
125 |
by (rtac set_ext 1); |
|
126 |
by (simp_tac (simpset() addsimps image_def::prems) 1); |
|
127 |
qed "image_cong"; |
|
128 |
||
1748 | 129 |
|
1548 | 130 |
section "Int"; |
923 | 131 |
|
4059 | 132 |
goal thy "A Int A = A"; |
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by (Blast_tac 1); |
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qed "Int_absorb"; |
1531 | 135 |
Addsimps[Int_absorb]; |
923 | 136 |
|
4059 | 137 |
goal thy "A Int B = B Int A"; |
2891 | 138 |
by (Blast_tac 1); |
923 | 139 |
qed "Int_commute"; |
140 |
||
4059 | 141 |
goal thy "(A Int B) Int C = A Int (B Int C)"; |
2891 | 142 |
by (Blast_tac 1); |
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qed "Int_assoc"; |
144 |
||
4059 | 145 |
goal thy "{} Int B = {}"; |
2891 | 146 |
by (Blast_tac 1); |
923 | 147 |
qed "Int_empty_left"; |
1531 | 148 |
Addsimps[Int_empty_left]; |
923 | 149 |
|
4059 | 150 |
goal thy "A Int {} = {}"; |
2891 | 151 |
by (Blast_tac 1); |
923 | 152 |
qed "Int_empty_right"; |
1531 | 153 |
Addsimps[Int_empty_right]; |
154 |
||
4059 | 155 |
goal thy "(A Int B = {}) = (A <= Compl B)"; |
4089 | 156 |
by (blast_tac (claset() addSEs [equalityE]) 1); |
3356 | 157 |
qed "disjoint_eq_subset_Compl"; |
158 |
||
4059 | 159 |
goal thy "UNIV Int B = B"; |
2891 | 160 |
by (Blast_tac 1); |
1531 | 161 |
qed "Int_UNIV_left"; |
162 |
Addsimps[Int_UNIV_left]; |
|
163 |
||
4059 | 164 |
goal thy "A Int UNIV = A"; |
2891 | 165 |
by (Blast_tac 1); |
1531 | 166 |
qed "Int_UNIV_right"; |
167 |
Addsimps[Int_UNIV_right]; |
|
923 | 168 |
|
4059 | 169 |
goal thy "A Int (B Un C) = (A Int B) Un (A Int C)"; |
2891 | 170 |
by (Blast_tac 1); |
923 | 171 |
qed "Int_Un_distrib"; |
172 |
||
4059 | 173 |
goal thy "(B Un C) Int A = (B Int A) Un (C Int A)"; |
2891 | 174 |
by (Blast_tac 1); |
1618 | 175 |
qed "Int_Un_distrib2"; |
176 |
||
4059 | 177 |
goal thy "(A<=B) = (A Int B = A)"; |
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by (blast_tac (claset() addSEs [equalityE]) 1); |
923 | 179 |
qed "subset_Int_eq"; |
180 |
||
4059 | 181 |
goal thy "(A Int B = UNIV) = (A = UNIV & B = UNIV)"; |
4089 | 182 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
1531 | 183 |
qed "Int_UNIV"; |
184 |
Addsimps[Int_UNIV]; |
|
185 |
||
1548 | 186 |
section "Un"; |
923 | 187 |
|
4059 | 188 |
goal thy "A Un A = A"; |
2891 | 189 |
by (Blast_tac 1); |
923 | 190 |
qed "Un_absorb"; |
1531 | 191 |
Addsimps[Un_absorb]; |
923 | 192 |
|
4059 | 193 |
goal thy " A Un (A Un B) = A Un B"; |
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by (Blast_tac 1); |
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195 |
qed "Un_left_absorb"; |
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196 |
|
4059 | 197 |
goal thy "A Un B = B Un A"; |
2891 | 198 |
by (Blast_tac 1); |
923 | 199 |
qed "Un_commute"; |
200 |
||
4059 | 201 |
goal thy " A Un (B Un C) = B Un (A Un C)"; |
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by (Blast_tac 1); |
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203 |
qed "Un_left_commute"; |
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204 |
|
4059 | 205 |
goal thy "(A Un B) Un C = A Un (B Un C)"; |
2891 | 206 |
by (Blast_tac 1); |
923 | 207 |
qed "Un_assoc"; |
208 |
||
4059 | 209 |
goal thy "{} Un B = B"; |
2891 | 210 |
by (Blast_tac 1); |
923 | 211 |
qed "Un_empty_left"; |
1531 | 212 |
Addsimps[Un_empty_left]; |
923 | 213 |
|
4059 | 214 |
goal thy "A Un {} = A"; |
2891 | 215 |
by (Blast_tac 1); |
923 | 216 |
qed "Un_empty_right"; |
1531 | 217 |
Addsimps[Un_empty_right]; |
218 |
||
4059 | 219 |
goal thy "UNIV Un B = UNIV"; |
2891 | 220 |
by (Blast_tac 1); |
1531 | 221 |
qed "Un_UNIV_left"; |
222 |
Addsimps[Un_UNIV_left]; |
|
223 |
||
4059 | 224 |
goal thy "A Un UNIV = UNIV"; |
2891 | 225 |
by (Blast_tac 1); |
1531 | 226 |
qed "Un_UNIV_right"; |
227 |
Addsimps[Un_UNIV_right]; |
|
923 | 228 |
|
4059 | 229 |
goal thy "(insert a B) Un C = insert a (B Un C)"; |
2891 | 230 |
by (Blast_tac 1); |
923 | 231 |
qed "Un_insert_left"; |
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232 |
Addsimps[Un_insert_left]; |
923 | 233 |
|
4059 | 234 |
goal thy "A Un (insert a B) = insert a (A Un B)"; |
2891 | 235 |
by (Blast_tac 1); |
1917 | 236 |
qed "Un_insert_right"; |
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237 |
Addsimps[Un_insert_right]; |
1917 | 238 |
|
4059 | 239 |
goal thy "(insert a B) Int C = (if a:C then insert a (B Int C) \ |
3356 | 240 |
\ else B Int C)"; |
4089 | 241 |
by (simp_tac (simpset() addsplits [expand_if]) 1); |
3356 | 242 |
by (Blast_tac 1); |
243 |
qed "Int_insert_left"; |
|
244 |
||
4059 | 245 |
goal thy "A Int (insert a B) = (if a:A then insert a (A Int B) \ |
3356 | 246 |
\ else A Int B)"; |
4089 | 247 |
by (simp_tac (simpset() addsplits [expand_if]) 1); |
3356 | 248 |
by (Blast_tac 1); |
249 |
qed "Int_insert_right"; |
|
250 |
||
4059 | 251 |
goal thy "(A Int B) Un C = (A Un C) Int (B Un C)"; |
2891 | 252 |
by (Blast_tac 1); |
923 | 253 |
qed "Un_Int_distrib"; |
254 |
||
4059 | 255 |
goal thy |
923 | 256 |
"(A Int B) Un (B Int C) Un (C Int A) = (A Un B) Int (B Un C) Int (C Un A)"; |
2891 | 257 |
by (Blast_tac 1); |
923 | 258 |
qed "Un_Int_crazy"; |
259 |
||
4059 | 260 |
goal thy "(A<=B) = (A Un B = B)"; |
4089 | 261 |
by (blast_tac (claset() addSEs [equalityE]) 1); |
923 | 262 |
qed "subset_Un_eq"; |
263 |
||
4059 | 264 |
goal thy "(A <= insert b C) = (A <= C | b:A & A-{b} <= C)"; |
2891 | 265 |
by (Blast_tac 1); |
923 | 266 |
qed "subset_insert_iff"; |
267 |
||
4059 | 268 |
goal thy "(A Un B = {}) = (A = {} & B = {})"; |
4089 | 269 |
by (blast_tac (claset() addEs [equalityCE]) 1); |
923 | 270 |
qed "Un_empty"; |
1531 | 271 |
Addsimps[Un_empty]; |
923 | 272 |
|
1548 | 273 |
section "Compl"; |
923 | 274 |
|
4059 | 275 |
goal thy "A Int Compl(A) = {}"; |
2891 | 276 |
by (Blast_tac 1); |
923 | 277 |
qed "Compl_disjoint"; |
1531 | 278 |
Addsimps[Compl_disjoint]; |
923 | 279 |
|
4059 | 280 |
goal thy "A Un Compl(A) = UNIV"; |
2891 | 281 |
by (Blast_tac 1); |
923 | 282 |
qed "Compl_partition"; |
283 |
||
4059 | 284 |
goal thy "Compl(Compl(A)) = A"; |
2891 | 285 |
by (Blast_tac 1); |
923 | 286 |
qed "double_complement"; |
1531 | 287 |
Addsimps[double_complement]; |
923 | 288 |
|
4059 | 289 |
goal thy "Compl(A Un B) = Compl(A) Int Compl(B)"; |
2891 | 290 |
by (Blast_tac 1); |
923 | 291 |
qed "Compl_Un"; |
292 |
||
4059 | 293 |
goal thy "Compl(A Int B) = Compl(A) Un Compl(B)"; |
2891 | 294 |
by (Blast_tac 1); |
923 | 295 |
qed "Compl_Int"; |
296 |
||
4059 | 297 |
goal thy "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))"; |
2891 | 298 |
by (Blast_tac 1); |
923 | 299 |
qed "Compl_UN"; |
300 |
||
4059 | 301 |
goal thy "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))"; |
2891 | 302 |
by (Blast_tac 1); |
923 | 303 |
qed "Compl_INT"; |
304 |
||
305 |
(*Halmos, Naive Set Theory, page 16.*) |
|
306 |
||
4059 | 307 |
goal thy "((A Int B) Un C = A Int (B Un C)) = (C<=A)"; |
4089 | 308 |
by (blast_tac (claset() addSEs [equalityE]) 1); |
923 | 309 |
qed "Un_Int_assoc_eq"; |
310 |
||
311 |
||
1548 | 312 |
section "Union"; |
923 | 313 |
|
4059 | 314 |
goal thy "Union({}) = {}"; |
2891 | 315 |
by (Blast_tac 1); |
923 | 316 |
qed "Union_empty"; |
1531 | 317 |
Addsimps[Union_empty]; |
318 |
||
4059 | 319 |
goal thy "Union(UNIV) = UNIV"; |
2891 | 320 |
by (Blast_tac 1); |
1531 | 321 |
qed "Union_UNIV"; |
322 |
Addsimps[Union_UNIV]; |
|
923 | 323 |
|
4059 | 324 |
goal thy "Union(insert a B) = a Un Union(B)"; |
2891 | 325 |
by (Blast_tac 1); |
923 | 326 |
qed "Union_insert"; |
1531 | 327 |
Addsimps[Union_insert]; |
923 | 328 |
|
4059 | 329 |
goal thy "Union(A Un B) = Union(A) Un Union(B)"; |
2891 | 330 |
by (Blast_tac 1); |
923 | 331 |
qed "Union_Un_distrib"; |
1531 | 332 |
Addsimps[Union_Un_distrib]; |
923 | 333 |
|
4059 | 334 |
goal thy "Union(A Int B) <= Union(A) Int Union(B)"; |
2891 | 335 |
by (Blast_tac 1); |
923 | 336 |
qed "Union_Int_subset"; |
337 |
||
4059 | 338 |
goal thy "(Union M = {}) = (! A : M. A = {})"; |
4089 | 339 |
by (blast_tac (claset() addEs [equalityE]) 1); |
4003 | 340 |
qed"Union_empty_conv"; |
341 |
AddIffs [Union_empty_conv]; |
|
342 |
||
4059 | 343 |
val prems = goal thy |
923 | 344 |
"(Union(C) Int A = {}) = (! B:C. B Int A = {})"; |
4089 | 345 |
by (blast_tac (claset() addSEs [equalityE]) 1); |
923 | 346 |
qed "Union_disjoint"; |
347 |
||
1548 | 348 |
section "Inter"; |
349 |
||
4059 | 350 |
goal thy "Inter({}) = UNIV"; |
2891 | 351 |
by (Blast_tac 1); |
1531 | 352 |
qed "Inter_empty"; |
353 |
Addsimps[Inter_empty]; |
|
354 |
||
4059 | 355 |
goal thy "Inter(UNIV) = {}"; |
2891 | 356 |
by (Blast_tac 1); |
1531 | 357 |
qed "Inter_UNIV"; |
358 |
Addsimps[Inter_UNIV]; |
|
359 |
||
4059 | 360 |
goal thy "Inter(insert a B) = a Int Inter(B)"; |
2891 | 361 |
by (Blast_tac 1); |
1531 | 362 |
qed "Inter_insert"; |
363 |
Addsimps[Inter_insert]; |
|
364 |
||
4059 | 365 |
goal thy "Inter(A) Un Inter(B) <= Inter(A Int B)"; |
2891 | 366 |
by (Blast_tac 1); |
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|
367 |
qed "Inter_Un_subset"; |
1531 | 368 |
|
4059 | 369 |
goal thy "Inter(A Un B) = Inter(A) Int Inter(B)"; |
2891 | 370 |
by (Blast_tac 1); |
923 | 371 |
qed "Inter_Un_distrib"; |
372 |
||
1548 | 373 |
section "UN and INT"; |
923 | 374 |
|
375 |
(*Basic identities*) |
|
376 |
||
4136 | 377 |
val not_empty = prove_goal Set.thy "A Û {} = (Ãx. xÎA)" (K [Blast_tac 1]); |
378 |
(*Addsimps[not_empty];*) |
|
379 |
||
4059 | 380 |
goal thy "(UN x:{}. B x) = {}"; |
2891 | 381 |
by (Blast_tac 1); |
1179
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Added insert_not_empty, UN_empty and UN_insert (to set_ss).
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|
382 |
qed "UN_empty"; |
1531 | 383 |
Addsimps[UN_empty]; |
384 |
||
4059 | 385 |
goal thy "(UN x:A. {}) = {}"; |
3457 | 386 |
by (Blast_tac 1); |
3222
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|
387 |
qed "UN_empty2"; |
726a9b069947
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|
388 |
Addsimps[UN_empty2]; |
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diff
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|
389 |
|
4059 | 390 |
goal thy "(UN x:UNIV. B x) = (UN x. B x)"; |
2891 | 391 |
by (Blast_tac 1); |
1531 | 392 |
qed "UN_UNIV"; |
393 |
Addsimps[UN_UNIV]; |
|
394 |
||
4059 | 395 |
goal thy "(INT x:{}. B x) = UNIV"; |
2891 | 396 |
by (Blast_tac 1); |
1531 | 397 |
qed "INT_empty"; |
398 |
Addsimps[INT_empty]; |
|
399 |
||
4059 | 400 |
goal thy "(INT x:UNIV. B x) = (INT x. B x)"; |
2891 | 401 |
by (Blast_tac 1); |
1531 | 402 |
qed "INT_UNIV"; |
403 |
Addsimps[INT_UNIV]; |
|
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
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923
diff
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|
404 |
|
4059 | 405 |
goal thy "(UN x:insert a A. B x) = B a Un UNION A B"; |
2891 | 406 |
by (Blast_tac 1); |
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
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|
407 |
qed "UN_insert"; |
1531 | 408 |
Addsimps[UN_insert]; |
409 |
||
4059 | 410 |
goal thy "(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.
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2922
diff
changeset
|
411 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
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2922
diff
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|
412 |
qed "UN_Un"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
413 |
|
4059 | 414 |
goal thy "(INT x:insert a A. B x) = B a Int INTER A B"; |
2891 | 415 |
by (Blast_tac 1); |
1531 | 416 |
qed "INT_insert"; |
417 |
Addsimps[INT_insert]; |
|
1179
7678408f9751
Added insert_not_empty, UN_empty and UN_insert (to set_ss).
nipkow
parents:
923
diff
changeset
|
418 |
|
4059 | 419 |
goal thy |
2021 | 420 |
"!!A. A~={} ==> (INT x:A. insert a (B x)) = insert a (INT x:A. B x)"; |
2891 | 421 |
by (Blast_tac 1); |
2021 | 422 |
qed "INT_insert_distrib"; |
423 |
||
4059 | 424 |
goal thy "(INT x. insert a (B x)) = insert a (INT x. B x)"; |
2891 | 425 |
by (Blast_tac 1); |
2021 | 426 |
qed "INT1_insert_distrib"; |
427 |
||
4059 | 428 |
goal thy "Union(B``A) = (UN x:A. B(x))"; |
2891 | 429 |
by (Blast_tac 1); |
923 | 430 |
qed "Union_image_eq"; |
431 |
||
4059 | 432 |
goal thy "Inter(B``A) = (INT x:A. B(x))"; |
2891 | 433 |
by (Blast_tac 1); |
923 | 434 |
qed "Inter_image_eq"; |
435 |
||
4059 | 436 |
goal thy "!!A. a: A ==> (UN y:A. c) = c"; |
2891 | 437 |
by (Blast_tac 1); |
923 | 438 |
qed "UN_constant"; |
439 |
||
4059 | 440 |
goal thy "!!A. a: A ==> (INT y:A. c) = c"; |
2891 | 441 |
by (Blast_tac 1); |
923 | 442 |
qed "INT_constant"; |
443 |
||
4059 | 444 |
goal thy "(UN x. B) = B"; |
2891 | 445 |
by (Blast_tac 1); |
923 | 446 |
qed "UN1_constant"; |
1531 | 447 |
Addsimps[UN1_constant]; |
923 | 448 |
|
4059 | 449 |
goal thy "(INT x. B) = B"; |
2891 | 450 |
by (Blast_tac 1); |
923 | 451 |
qed "INT1_constant"; |
1531 | 452 |
Addsimps[INT1_constant]; |
923 | 453 |
|
4059 | 454 |
goal thy "(UN x:A. B(x)) = Union({Y. ? x:A. Y=B(x)})"; |
2891 | 455 |
by (Blast_tac 1); |
923 | 456 |
qed "UN_eq"; |
457 |
||
458 |
(*Look: it has an EXISTENTIAL quantifier*) |
|
4059 | 459 |
goal thy "(INT x:A. B(x)) = Inter({Y. ? x:A. Y=B(x)})"; |
2891 | 460 |
by (Blast_tac 1); |
923 | 461 |
qed "INT_eq"; |
462 |
||
4059 | 463 |
goalw thy [o_def] "UNION A (g o f) = UNION (f``A) g"; |
3222
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nipkow
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diff
changeset
|
464 |
by (Blast_tac 1); |
726a9b069947
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2922
diff
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|
465 |
qed "UNION_o"; |
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changeset
|
466 |
|
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|
467 |
|
923 | 468 |
(*Distributive laws...*) |
469 |
||
4059 | 470 |
goal thy "A Int Union(B) = (UN C:B. A Int C)"; |
2891 | 471 |
by (Blast_tac 1); |
923 | 472 |
qed "Int_Union"; |
473 |
||
2912 | 474 |
(* Devlin, Setdamentals of Contemporary Set Theory, page 12, exercise 5: |
923 | 475 |
Union of a family of unions **) |
4059 | 476 |
goal thy "(UN x:C. A(x) Un B(x)) = Union(A``C) Un Union(B``C)"; |
2891 | 477 |
by (Blast_tac 1); |
923 | 478 |
qed "Un_Union_image"; |
479 |
||
480 |
(*Equivalent version*) |
|
4059 | 481 |
goal thy "(UN i:I. A(i) Un B(i)) = (UN i:I. A(i)) Un (UN i:I. B(i))"; |
2891 | 482 |
by (Blast_tac 1); |
923 | 483 |
qed "UN_Un_distrib"; |
484 |
||
4059 | 485 |
goal thy "A Un Inter(B) = (INT C:B. A Un C)"; |
2891 | 486 |
by (Blast_tac 1); |
923 | 487 |
qed "Un_Inter"; |
488 |
||
4059 | 489 |
goal thy "(INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C)"; |
2891 | 490 |
by (Blast_tac 1); |
923 | 491 |
qed "Int_Inter_image"; |
492 |
||
493 |
(*Equivalent version*) |
|
4059 | 494 |
goal thy "(INT i:I. A(i) Int B(i)) = (INT i:I. A(i)) Int (INT i:I. B(i))"; |
2891 | 495 |
by (Blast_tac 1); |
923 | 496 |
qed "INT_Int_distrib"; |
497 |
||
498 |
(*Halmos, Naive Set Theory, page 35.*) |
|
4059 | 499 |
goal thy "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))"; |
2891 | 500 |
by (Blast_tac 1); |
923 | 501 |
qed "Int_UN_distrib"; |
502 |
||
4059 | 503 |
goal thy "B Un (INT i:I. A(i)) = (INT i:I. B Un A(i))"; |
2891 | 504 |
by (Blast_tac 1); |
923 | 505 |
qed "Un_INT_distrib"; |
506 |
||
4059 | 507 |
goal thy |
923 | 508 |
"(UN i:I. A(i)) Int (UN j:J. B(j)) = (UN i:I. UN j:J. A(i) Int B(j))"; |
2891 | 509 |
by (Blast_tac 1); |
923 | 510 |
qed "Int_UN_distrib2"; |
511 |
||
4059 | 512 |
goal thy |
923 | 513 |
"(INT i:I. A(i)) Un (INT j:J. B(j)) = (INT i:I. INT j:J. A(i) Un B(j))"; |
2891 | 514 |
by (Blast_tac 1); |
923 | 515 |
qed "Un_INT_distrib2"; |
516 |
||
2512 | 517 |
|
518 |
section"Bounded quantifiers"; |
|
519 |
||
3860 | 520 |
(** The following are not added to the default simpset because |
521 |
(a) they duplicate the body and (b) there are no similar rules for Int. **) |
|
2512 | 522 |
|
4059 | 523 |
goal thy "(ALL x:A Un B. P x) = ((ALL x:A. P x) & (ALL x:B. P x))"; |
2891 | 524 |
by (Blast_tac 1); |
2519 | 525 |
qed "ball_Un"; |
526 |
||
4059 | 527 |
goal thy "(EX x:A Un B. P x) = ((EX x:A. P x) | (EX x:B. P x))"; |
2891 | 528 |
by (Blast_tac 1); |
2519 | 529 |
qed "bex_Un"; |
2512 | 530 |
|
531 |
||
1548 | 532 |
section "-"; |
923 | 533 |
|
4059 | 534 |
goal thy "A-A = {}"; |
2891 | 535 |
by (Blast_tac 1); |
923 | 536 |
qed "Diff_cancel"; |
1531 | 537 |
Addsimps[Diff_cancel]; |
923 | 538 |
|
4059 | 539 |
goal thy "{}-A = {}"; |
2891 | 540 |
by (Blast_tac 1); |
923 | 541 |
qed "empty_Diff"; |
1531 | 542 |
Addsimps[empty_Diff]; |
923 | 543 |
|
4059 | 544 |
goal thy "A-{} = A"; |
2891 | 545 |
by (Blast_tac 1); |
923 | 546 |
qed "Diff_empty"; |
1531 | 547 |
Addsimps[Diff_empty]; |
548 |
||
4059 | 549 |
goal thy "A-UNIV = {}"; |
2891 | 550 |
by (Blast_tac 1); |
1531 | 551 |
qed "Diff_UNIV"; |
552 |
Addsimps[Diff_UNIV]; |
|
553 |
||
4059 | 554 |
goal thy "!!x. x~:A ==> A - insert x B = A-B"; |
2891 | 555 |
by (Blast_tac 1); |
1531 | 556 |
qed "Diff_insert0"; |
557 |
Addsimps [Diff_insert0]; |
|
923 | 558 |
|
559 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a 0*) |
|
4059 | 560 |
goal thy "A - insert a B = A - B - {a}"; |
2891 | 561 |
by (Blast_tac 1); |
923 | 562 |
qed "Diff_insert"; |
563 |
||
564 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a 0*) |
|
4059 | 565 |
goal thy "A - insert a B = A - {a} - B"; |
2891 | 566 |
by (Blast_tac 1); |
923 | 567 |
qed "Diff_insert2"; |
568 |
||
4059 | 569 |
goal thy "insert x A - B = (if x:B then A-B else insert x (A-B))"; |
4089 | 570 |
by (simp_tac (simpset() addsplits [expand_if]) 1); |
2891 | 571 |
by (Blast_tac 1); |
1531 | 572 |
qed "insert_Diff_if"; |
573 |
||
4059 | 574 |
goal thy "!!x. x:B ==> insert x A - B = A-B"; |
2891 | 575 |
by (Blast_tac 1); |
1531 | 576 |
qed "insert_Diff1"; |
577 |
Addsimps [insert_Diff1]; |
|
578 |
||
4059 | 579 |
goal thy "!!a. a:A ==> insert a (A-{a}) = A"; |
2922 | 580 |
by (Blast_tac 1); |
923 | 581 |
qed "insert_Diff"; |
582 |
||
4059 | 583 |
goal thy "A Int (B-A) = {}"; |
2891 | 584 |
by (Blast_tac 1); |
923 | 585 |
qed "Diff_disjoint"; |
1531 | 586 |
Addsimps[Diff_disjoint]; |
923 | 587 |
|
4059 | 588 |
goal thy "!!A. A<=B ==> A Un (B-A) = B"; |
2891 | 589 |
by (Blast_tac 1); |
923 | 590 |
qed "Diff_partition"; |
591 |
||
4059 | 592 |
goal thy "!!A. [| A<=B; B<= C |] ==> (B - (C - A)) = (A :: 'a set)"; |
2891 | 593 |
by (Blast_tac 1); |
923 | 594 |
qed "double_diff"; |
595 |
||
4059 | 596 |
goal thy "A - (B Un C) = (A-B) Int (A-C)"; |
2891 | 597 |
by (Blast_tac 1); |
923 | 598 |
qed "Diff_Un"; |
599 |
||
4059 | 600 |
goal thy "A - (B Int C) = (A-B) Un (A-C)"; |
2891 | 601 |
by (Blast_tac 1); |
923 | 602 |
qed "Diff_Int"; |
603 |
||
4059 | 604 |
goal thy "(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
|
605 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
606 |
qed "Un_Diff"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
607 |
|
4059 | 608 |
goal thy "(A Int B) - C = (A - C) Int (B - C)"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
609 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
610 |
qed "Int_Diff"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
611 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
612 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
613 |
section "Miscellany"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
614 |
|
4059 | 615 |
goal thy "(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
|
616 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
617 |
qed "set_eq_subset"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
618 |
|
4059 | 619 |
goal thy "A <= B = (! t. t:A --> t:B)"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
620 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
621 |
qed "subset_iff"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
622 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
623 |
goalw thy [psubset_def] "((A::'a set) <= B) = ((A < B) | (A=B))"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
624 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
625 |
qed "subset_iff_psubset_eq"; |
2021 | 626 |
|
4059 | 627 |
goal thy "(!x. x ~: A) = (A={})"; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
628 |
by(Blast_tac 1); |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
629 |
qed "all_not_in_conv"; |
3907 | 630 |
AddIffs [all_not_in_conv]; |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
631 |
|
4059 | 632 |
goalw thy [Pow_def] "Pow {} = {{}}"; |
3348
3f9a806f061e
Two useful facts about Powersets suggested by Florian Kammueller
paulson
parents:
3222
diff
changeset
|
633 |
by (Auto_tac()); |
3f9a806f061e
Two useful facts about Powersets suggested by Florian Kammueller
paulson
parents:
3222
diff
changeset
|
634 |
qed "Pow_empty"; |
3f9a806f061e
Two useful facts about Powersets suggested by Florian Kammueller
paulson
parents:
3222
diff
changeset
|
635 |
Addsimps [Pow_empty]; |
3f9a806f061e
Two useful facts about Powersets suggested by Florian Kammueller
paulson
parents:
3222
diff
changeset
|
636 |
|
4059 | 637 |
goal thy "Pow (insert a A) = Pow A Un (insert a `` Pow A)"; |
3724 | 638 |
by Safe_tac; |
3457 | 639 |
by (etac swap 1); |
3348
3f9a806f061e
Two useful facts about Powersets suggested by Florian Kammueller
paulson
parents:
3222
diff
changeset
|
640 |
by (res_inst_tac [("x", "x-{a}")] image_eqI 1); |
3f9a806f061e
Two useful facts about Powersets suggested by Florian Kammueller
paulson
parents:
3222
diff
changeset
|
641 |
by (ALLGOALS Blast_tac); |
3f9a806f061e
Two useful facts about Powersets suggested by Florian Kammueller
paulson
parents:
3222
diff
changeset
|
642 |
qed "Pow_insert"; |
3f9a806f061e
Two useful facts about Powersets suggested by Florian Kammueller
paulson
parents:
3222
diff
changeset
|
643 |
|
2021 | 644 |
|
645 |
(** Miniscoping: pushing in big Unions and Intersections **) |
|
646 |
local |
|
4059 | 647 |
fun prover s = prove_goal thy s (fn _ => [Blast_tac 1]) |
2021 | 648 |
in |
649 |
val UN1_simps = map prover |
|
2031 | 650 |
["(UN x. insert a (B x)) = insert a (UN x. B x)", |
3842 | 651 |
"(UN x. A x Int B) = ((UN x. A x) Int B)", |
652 |
"(UN x. A Int B x) = (A Int (UN x. B x))", |
|
653 |
"(UN x. A x Un B) = ((UN x. A x) Un B)", |
|
654 |
"(UN x. A Un B x) = (A Un (UN x. B x))", |
|
655 |
"(UN x. A x - B) = ((UN x. A x) - B)", |
|
656 |
"(UN x. A - B x) = (A - (INT x. B x))"]; |
|
2021 | 657 |
|
658 |
val INT1_simps = map prover |
|
2031 | 659 |
["(INT x. insert a (B x)) = insert a (INT x. B x)", |
3842 | 660 |
"(INT x. A x Int B) = ((INT x. A x) Int B)", |
661 |
"(INT x. A Int B x) = (A Int (INT x. B x))", |
|
662 |
"(INT x. A x Un B) = ((INT x. A x) Un B)", |
|
663 |
"(INT x. A Un B x) = (A Un (INT x. B x))", |
|
664 |
"(INT x. A x - B) = ((INT x. A x) - B)", |
|
665 |
"(INT x. A - B x) = (A - (UN x. B x))"]; |
|
2021 | 666 |
|
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
667 |
val UN_simps = map prover |
3842 | 668 |
["(UN x:C. A x Int B) = ((UN x:C. A x) Int B)", |
669 |
"(UN x:C. A Int B x) = (A Int (UN x:C. B x))", |
|
670 |
"(UN x:C. A x - B) = ((UN x:C. A x) - B)", |
|
671 |
"(UN x:C. A - B x) = (A - (INT x:C. B x))"]; |
|
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
672 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
673 |
val INT_simps = map prover |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
674 |
["(INT x:C. insert a (B x)) = insert a (INT x:C. B x)", |
3842 | 675 |
"(INT x:C. A x Un B) = ((INT x:C. A x) Un B)", |
676 |
"(INT x:C. A Un B x) = (A Un (INT x:C. B x))"]; |
|
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
677 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
678 |
(*The missing laws for bounded Unions and Intersections are conditional |
2021 | 679 |
on the index set's being non-empty. Thus they are probably NOT worth |
680 |
adding as default rewrites.*) |
|
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
681 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
682 |
val ball_simps = map prover |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
683 |
["(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
|
684 |
"(ALL x:A. P | Q x) = (P | (ALL x:A. Q x))", |
3422 | 685 |
"(ALL x:A. P --> Q x) = (P --> (ALL x:A. Q x))", |
686 |
"(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
|
687 |
"(ALL x:{}. P x) = True", |
4136 | 688 |
"(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
|
689 |
"(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
|
690 |
"(ALL x:Union(A). P x) = (ALL y:A. ALL x:y. P x)", |
3860 | 691 |
"(ALL x:Collect Q. P x) = (ALL x. Q x --> P x)", |
692 |
"(ALL x:f``A. P x) = (ALL x:A. P(f x))", |
|
693 |
"(~(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
|
694 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
695 |
val ball_conj_distrib = |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
696 |
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
|
697 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
698 |
val bex_simps = map prover |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
699 |
["(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
|
700 |
"(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
|
701 |
"(EX x:{}. P x) = False", |
4136 | 702 |
"(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
|
703 |
"(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
|
704 |
"(EX x:Union(A). P x) = (EX y:A. EX x:y. P x)", |
3860 | 705 |
"(EX x:Collect Q. P x) = (EX x. Q x & P x)", |
706 |
"(EX x:f``A. P x) = (EX x:A. P(f x))", |
|
707 |
"(~(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
|
708 |
|
3426 | 709 |
val bex_disj_distrib = |
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
710 |
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
|
711 |
|
2021 | 712 |
end; |
713 |
||
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
714 |
Addsimps (UN1_simps @ INT1_simps @ UN_simps @ INT_simps @ |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
715 |
ball_simps @ bex_simps); |