author | wenzelm |
Tue, 20 May 1997 19:29:50 +0200 | |
changeset 3257 | 4e3724e0659f |
parent 3222 | 726a9b069947 |
child 3348 | 3f9a806f061e |
permissions | -rw-r--r-- |
1465 | 1 |
(* 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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Equalities involving union, intersection, inclusion, etc. |
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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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1531 | 15 |
goal Set.thy "{x.False} = {}"; |
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by (Blast_tac 1); |
1531 | 17 |
qed "Collect_False_empty"; |
18 |
Addsimps [Collect_False_empty]; |
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19 |
||
20 |
goal Set.thy "(A <= {}) = (A = {})"; |
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by (Blast_tac 1); |
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qed "subset_empty"; |
23 |
Addsimps [subset_empty]; |
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24 |
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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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section "insert"; |
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(*NOT SUITABLE FOR REWRITING since {a} == insert a {}*) |
33 |
goal Set.thy "insert a A = {a} Un A"; |
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2891 | 34 |
by (Blast_tac 1); |
1531 | 35 |
qed "insert_is_Un"; |
36 |
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goal Set.thy "insert a A ~= {}"; |
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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); |
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Addsimps[empty_not_insert]; |
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goal Set.thy "!!a. 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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||
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goal Set.thy "insert x (insert x A) = insert x A"; |
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by (Blast_tac 1); |
1531 | 51 |
qed "insert_absorb2"; |
52 |
Addsimps [insert_absorb2]; |
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goal Set.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 |
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goal Set.thy "(insert x A <= B) = (x:B & A <= B)"; |
2891 | 59 |
by (Blast_tac 1); |
923 | 60 |
qed "insert_subset"; |
1531 | 61 |
Addsimps[insert_subset]; |
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goal Set.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 *) |
68 |
goal Set.thy "!!a. a:A ==> ? B. A = insert a B & a ~: B"; |
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1553 | 69 |
by (res_inst_tac [("x","A-{a}")] exI 1); |
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by (Blast_tac 1); |
1531 | 71 |
qed "mk_disjoint_insert"; |
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goal Set.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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goal Set.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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84 |
goal Set.thy "f``{} = {}"; |
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2891 | 85 |
by (Blast_tac 1); |
923 | 86 |
qed "image_empty"; |
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Addsimps[image_empty]; |
923 | 88 |
|
89 |
goal Set.thy "f``insert a B = insert (f a) (f``B)"; |
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2891 | 90 |
by (Blast_tac 1); |
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qed "image_insert"; |
1531 | 92 |
Addsimps[image_insert]; |
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goal Set.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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goal Set.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 |
|
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goal Set.thy "f``(range g) = range (%x. f (g x))"; |
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by(Blast_tac 1); |
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104 |
qed "image_range"; |
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105 |
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1660 | 106 |
qed_goal "ball_image" Set.thy "(!y:F``S. P y) = (!x:S. P (F x))" |
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(fn _ => [Blast_tac 1]); |
1660 | 108 |
|
1884 | 109 |
goal Set.thy "!!x. x:A ==> insert (f x) (f``A) = f``A"; |
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by (Blast_tac 1); |
1884 | 111 |
qed "insert_image"; |
112 |
Addsimps [insert_image]; |
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113 |
||
1748 | 114 |
goalw Set.thy [image_def] |
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"(%x. if P x then f x else g x) `` S \ |
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\ = (f `` ({x.x:S & P x})) Un (g `` ({x.x:S & ~(P x)}))"; |
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by (split_tac [expand_if] 1); |
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by (Blast_tac 1); |
1748 | 119 |
qed "if_image_distrib"; |
120 |
Addsimps[if_image_distrib]; |
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121 |
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122 |
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section "range"; |
124 |
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125 |
qed_goal "ball_range" Set.thy "(!y:range f. P y) = (!x. P (f x))" |
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2891 | 126 |
(fn _ => [Blast_tac 1]); |
1660 | 127 |
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128 |
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section "Int"; |
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|
131 |
goal Set.thy "A Int A = A"; |
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by (Blast_tac 1); |
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qed "Int_absorb"; |
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Addsimps[Int_absorb]; |
923 | 135 |
|
136 |
goal Set.thy "A Int B = B Int A"; |
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by (Blast_tac 1); |
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qed "Int_commute"; |
139 |
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140 |
goal Set.thy "(A Int B) Int C = A Int (B Int C)"; |
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by (Blast_tac 1); |
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qed "Int_assoc"; |
143 |
||
144 |
goal Set.thy "{} Int B = {}"; |
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by (Blast_tac 1); |
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qed "Int_empty_left"; |
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Addsimps[Int_empty_left]; |
923 | 148 |
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149 |
goal Set.thy "A Int {} = {}"; |
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by (Blast_tac 1); |
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qed "Int_empty_right"; |
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Addsimps[Int_empty_right]; |
153 |
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154 |
goal Set.thy "UNIV Int B = B"; |
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2891 | 155 |
by (Blast_tac 1); |
1531 | 156 |
qed "Int_UNIV_left"; |
157 |
Addsimps[Int_UNIV_left]; |
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158 |
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159 |
goal Set.thy "A Int UNIV = A"; |
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by (Blast_tac 1); |
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qed "Int_UNIV_right"; |
162 |
Addsimps[Int_UNIV_right]; |
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923 | 163 |
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164 |
goal Set.thy "A Int (B Un C) = (A Int B) Un (A Int C)"; |
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by (Blast_tac 1); |
923 | 166 |
qed "Int_Un_distrib"; |
167 |
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goal Set.thy "(B Un C) Int A = (B Int A) Un (C Int A)"; |
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by (Blast_tac 1); |
1618 | 170 |
qed "Int_Un_distrib2"; |
171 |
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goal Set.thy "(A<=B) = (A Int B = A)"; |
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by (blast_tac (!claset addSEs [equalityE]) 1); |
923 | 174 |
qed "subset_Int_eq"; |
175 |
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goal Set.thy "(A Int B = UNIV) = (A = UNIV & B = UNIV)"; |
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by (blast_tac (!claset addEs [equalityCE]) 1); |
1531 | 178 |
qed "Int_UNIV"; |
179 |
Addsimps[Int_UNIV]; |
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180 |
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section "Un"; |
923 | 182 |
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183 |
goal Set.thy "A Un A = A"; |
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by (Blast_tac 1); |
923 | 185 |
qed "Un_absorb"; |
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Addsimps[Un_absorb]; |
923 | 187 |
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goal Set.thy " A Un (A Un B) = A Un B"; |
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by (Blast_tac 1); |
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qed "Un_left_absorb"; |
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191 |
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goal Set.thy "A Un B = B Un A"; |
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by (Blast_tac 1); |
923 | 194 |
qed "Un_commute"; |
195 |
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goal Set.thy " A Un (B Un C) = B Un (A Un C)"; |
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by (Blast_tac 1); |
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qed "Un_left_commute"; |
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199 |
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goal Set.thy "(A Un B) Un C = A Un (B Un C)"; |
2891 | 201 |
by (Blast_tac 1); |
923 | 202 |
qed "Un_assoc"; |
203 |
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204 |
goal Set.thy "{} Un B = B"; |
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2891 | 205 |
by (Blast_tac 1); |
923 | 206 |
qed "Un_empty_left"; |
1531 | 207 |
Addsimps[Un_empty_left]; |
923 | 208 |
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209 |
goal Set.thy "A Un {} = A"; |
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2891 | 210 |
by (Blast_tac 1); |
923 | 211 |
qed "Un_empty_right"; |
1531 | 212 |
Addsimps[Un_empty_right]; |
213 |
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214 |
goal Set.thy "UNIV Un B = UNIV"; |
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2891 | 215 |
by (Blast_tac 1); |
1531 | 216 |
qed "Un_UNIV_left"; |
217 |
Addsimps[Un_UNIV_left]; |
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218 |
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219 |
goal Set.thy "A Un UNIV = UNIV"; |
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by (Blast_tac 1); |
1531 | 221 |
qed "Un_UNIV_right"; |
222 |
Addsimps[Un_UNIV_right]; |
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923 | 223 |
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224 |
goal Set.thy "(insert a B) Un C = insert a (B Un C)"; |
2891 | 225 |
by (Blast_tac 1); |
923 | 226 |
qed "Un_insert_left"; |
227 |
||
1917 | 228 |
goal Set.thy "A Un (insert a B) = insert a (A Un B)"; |
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by (Blast_tac 1); |
1917 | 230 |
qed "Un_insert_right"; |
231 |
||
923 | 232 |
goal Set.thy "(A Int B) Un C = (A Un C) Int (B Un C)"; |
2891 | 233 |
by (Blast_tac 1); |
923 | 234 |
qed "Un_Int_distrib"; |
235 |
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236 |
goal Set.thy |
|
237 |
"(A Int B) Un (B Int C) Un (C Int A) = (A Un B) Int (B Un C) Int (C Un A)"; |
|
2891 | 238 |
by (Blast_tac 1); |
923 | 239 |
qed "Un_Int_crazy"; |
240 |
||
241 |
goal Set.thy "(A<=B) = (A Un B = B)"; |
|
2922 | 242 |
by (blast_tac (!claset addSEs [equalityE]) 1); |
923 | 243 |
qed "subset_Un_eq"; |
244 |
||
245 |
goal Set.thy "(A <= insert b C) = (A <= C | b:A & A-{b} <= C)"; |
|
2891 | 246 |
by (Blast_tac 1); |
923 | 247 |
qed "subset_insert_iff"; |
248 |
||
249 |
goal Set.thy "(A Un B = {}) = (A = {} & B = {})"; |
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2922 | 250 |
by (blast_tac (!claset addEs [equalityCE]) 1); |
923 | 251 |
qed "Un_empty"; |
1531 | 252 |
Addsimps[Un_empty]; |
923 | 253 |
|
1548 | 254 |
section "Compl"; |
923 | 255 |
|
256 |
goal Set.thy "A Int Compl(A) = {}"; |
|
2891 | 257 |
by (Blast_tac 1); |
923 | 258 |
qed "Compl_disjoint"; |
1531 | 259 |
Addsimps[Compl_disjoint]; |
923 | 260 |
|
1531 | 261 |
goal Set.thy "A Un Compl(A) = UNIV"; |
2891 | 262 |
by (Blast_tac 1); |
923 | 263 |
qed "Compl_partition"; |
264 |
||
265 |
goal Set.thy "Compl(Compl(A)) = A"; |
|
2891 | 266 |
by (Blast_tac 1); |
923 | 267 |
qed "double_complement"; |
1531 | 268 |
Addsimps[double_complement]; |
923 | 269 |
|
270 |
goal Set.thy "Compl(A Un B) = Compl(A) Int Compl(B)"; |
|
2891 | 271 |
by (Blast_tac 1); |
923 | 272 |
qed "Compl_Un"; |
273 |
||
274 |
goal Set.thy "Compl(A Int B) = Compl(A) Un Compl(B)"; |
|
2891 | 275 |
by (Blast_tac 1); |
923 | 276 |
qed "Compl_Int"; |
277 |
||
278 |
goal Set.thy "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))"; |
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2891 | 279 |
by (Blast_tac 1); |
923 | 280 |
qed "Compl_UN"; |
281 |
||
282 |
goal Set.thy "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))"; |
|
2891 | 283 |
by (Blast_tac 1); |
923 | 284 |
qed "Compl_INT"; |
285 |
||
286 |
(*Halmos, Naive Set Theory, page 16.*) |
|
287 |
||
288 |
goal Set.thy "((A Int B) Un C = A Int (B Un C)) = (C<=A)"; |
|
2922 | 289 |
by (blast_tac (!claset addSEs [equalityE]) 1); |
923 | 290 |
qed "Un_Int_assoc_eq"; |
291 |
||
292 |
||
1548 | 293 |
section "Union"; |
923 | 294 |
|
295 |
goal Set.thy "Union({}) = {}"; |
|
2891 | 296 |
by (Blast_tac 1); |
923 | 297 |
qed "Union_empty"; |
1531 | 298 |
Addsimps[Union_empty]; |
299 |
||
300 |
goal Set.thy "Union(UNIV) = UNIV"; |
|
2891 | 301 |
by (Blast_tac 1); |
1531 | 302 |
qed "Union_UNIV"; |
303 |
Addsimps[Union_UNIV]; |
|
923 | 304 |
|
305 |
goal Set.thy "Union(insert a B) = a Un Union(B)"; |
|
2891 | 306 |
by (Blast_tac 1); |
923 | 307 |
qed "Union_insert"; |
1531 | 308 |
Addsimps[Union_insert]; |
923 | 309 |
|
310 |
goal Set.thy "Union(A Un B) = Union(A) Un Union(B)"; |
|
2891 | 311 |
by (Blast_tac 1); |
923 | 312 |
qed "Union_Un_distrib"; |
1531 | 313 |
Addsimps[Union_Un_distrib]; |
923 | 314 |
|
315 |
goal Set.thy "Union(A Int B) <= Union(A) Int Union(B)"; |
|
2891 | 316 |
by (Blast_tac 1); |
923 | 317 |
qed "Union_Int_subset"; |
318 |
||
319 |
val prems = goal Set.thy |
|
320 |
"(Union(C) Int A = {}) = (! B:C. B Int A = {})"; |
|
2922 | 321 |
by (blast_tac (!claset addSEs [equalityE]) 1); |
923 | 322 |
qed "Union_disjoint"; |
323 |
||
1548 | 324 |
section "Inter"; |
325 |
||
1531 | 326 |
goal Set.thy "Inter({}) = UNIV"; |
2891 | 327 |
by (Blast_tac 1); |
1531 | 328 |
qed "Inter_empty"; |
329 |
Addsimps[Inter_empty]; |
|
330 |
||
331 |
goal Set.thy "Inter(UNIV) = {}"; |
|
2891 | 332 |
by (Blast_tac 1); |
1531 | 333 |
qed "Inter_UNIV"; |
334 |
Addsimps[Inter_UNIV]; |
|
335 |
||
336 |
goal Set.thy "Inter(insert a B) = a Int Inter(B)"; |
|
2891 | 337 |
by (Blast_tac 1); |
1531 | 338 |
qed "Inter_insert"; |
339 |
Addsimps[Inter_insert]; |
|
340 |
||
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|
341 |
goal Set.thy "Inter(A) Un Inter(B) <= Inter(A Int B)"; |
2891 | 342 |
by (Blast_tac 1); |
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|
343 |
qed "Inter_Un_subset"; |
1531 | 344 |
|
923 | 345 |
goal Set.thy "Inter(A Un B) = Inter(A) Int Inter(B)"; |
2891 | 346 |
by (Blast_tac 1); |
923 | 347 |
qed "Inter_Un_distrib"; |
348 |
||
1548 | 349 |
section "UN and INT"; |
923 | 350 |
|
351 |
(*Basic identities*) |
|
352 |
||
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|
353 |
goal Set.thy "(UN x:{}. B x) = {}"; |
2891 | 354 |
by (Blast_tac 1); |
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|
355 |
qed "UN_empty"; |
1531 | 356 |
Addsimps[UN_empty]; |
357 |
||
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358 |
goal Set.thy "(UN x:A. {}) = {}"; |
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|
359 |
by(Blast_tac 1); |
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|
360 |
qed "UN_empty2"; |
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|
361 |
Addsimps[UN_empty2]; |
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|
362 |
|
1531 | 363 |
goal Set.thy "(UN x:UNIV. B x) = (UN x. B x)"; |
2891 | 364 |
by (Blast_tac 1); |
1531 | 365 |
qed "UN_UNIV"; |
366 |
Addsimps[UN_UNIV]; |
|
367 |
||
368 |
goal Set.thy "(INT x:{}. B x) = UNIV"; |
|
2891 | 369 |
by (Blast_tac 1); |
1531 | 370 |
qed "INT_empty"; |
371 |
Addsimps[INT_empty]; |
|
372 |
||
373 |
goal Set.thy "(INT x:UNIV. B x) = (INT x. B x)"; |
|
2891 | 374 |
by (Blast_tac 1); |
1531 | 375 |
qed "INT_UNIV"; |
376 |
Addsimps[INT_UNIV]; |
|
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|
377 |
|
7678408f9751
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|
378 |
goal Set.thy "(UN x:insert a A. B x) = B a Un UNION A B"; |
2891 | 379 |
by (Blast_tac 1); |
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|
380 |
qed "UN_insert"; |
1531 | 381 |
Addsimps[UN_insert]; |
382 |
||
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|
383 |
goal Set.thy "(UN i: A Un B. M i) = ((UN i: A. M i) Un (UN i:B. M i))"; |
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|
384 |
by (Blast_tac 1); |
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|
385 |
qed "UN_Un"; |
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|
386 |
|
1531 | 387 |
goal Set.thy "(INT x:insert a A. B x) = B a Int INTER A B"; |
2891 | 388 |
by (Blast_tac 1); |
1531 | 389 |
qed "INT_insert"; |
390 |
Addsimps[INT_insert]; |
|
1179
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|
391 |
|
2021 | 392 |
goal Set.thy |
393 |
"!!A. A~={} ==> (INT x:A. insert a (B x)) = insert a (INT x:A. B x)"; |
|
2891 | 394 |
by (Blast_tac 1); |
2021 | 395 |
qed "INT_insert_distrib"; |
396 |
||
397 |
goal Set.thy "(INT x. insert a (B x)) = insert a (INT x. B x)"; |
|
2891 | 398 |
by (Blast_tac 1); |
2021 | 399 |
qed "INT1_insert_distrib"; |
400 |
||
923 | 401 |
goal Set.thy "Union(range(f)) = (UN x.f(x))"; |
2891 | 402 |
by (Blast_tac 1); |
923 | 403 |
qed "Union_range_eq"; |
404 |
||
405 |
goal Set.thy "Inter(range(f)) = (INT x.f(x))"; |
|
2891 | 406 |
by (Blast_tac 1); |
923 | 407 |
qed "Inter_range_eq"; |
408 |
||
409 |
goal Set.thy "Union(B``A) = (UN x:A. B(x))"; |
|
2891 | 410 |
by (Blast_tac 1); |
923 | 411 |
qed "Union_image_eq"; |
412 |
||
413 |
goal Set.thy "Inter(B``A) = (INT x:A. B(x))"; |
|
2891 | 414 |
by (Blast_tac 1); |
923 | 415 |
qed "Inter_image_eq"; |
416 |
||
417 |
goal Set.thy "!!A. a: A ==> (UN y:A. c) = c"; |
|
2891 | 418 |
by (Blast_tac 1); |
923 | 419 |
qed "UN_constant"; |
420 |
||
421 |
goal Set.thy "!!A. a: A ==> (INT y:A. c) = c"; |
|
2891 | 422 |
by (Blast_tac 1); |
923 | 423 |
qed "INT_constant"; |
424 |
||
425 |
goal Set.thy "(UN x.B) = B"; |
|
2891 | 426 |
by (Blast_tac 1); |
923 | 427 |
qed "UN1_constant"; |
1531 | 428 |
Addsimps[UN1_constant]; |
923 | 429 |
|
430 |
goal Set.thy "(INT x.B) = B"; |
|
2891 | 431 |
by (Blast_tac 1); |
923 | 432 |
qed "INT1_constant"; |
1531 | 433 |
Addsimps[INT1_constant]; |
923 | 434 |
|
435 |
goal Set.thy "(UN x:A. B(x)) = Union({Y. ? x:A. Y=B(x)})"; |
|
2891 | 436 |
by (Blast_tac 1); |
923 | 437 |
qed "UN_eq"; |
438 |
||
439 |
(*Look: it has an EXISTENTIAL quantifier*) |
|
440 |
goal Set.thy "(INT x:A. B(x)) = Inter({Y. ? x:A. Y=B(x)})"; |
|
2891 | 441 |
by (Blast_tac 1); |
923 | 442 |
qed "INT_eq"; |
443 |
||
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|
444 |
goalw Set.thy [o_def] "UNION A (g o f) = UNION (f``A) g"; |
726a9b069947
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|
445 |
by (Blast_tac 1); |
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|
446 |
qed "UNION_o"; |
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|
447 |
|
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|
448 |
|
923 | 449 |
(*Distributive laws...*) |
450 |
||
451 |
goal Set.thy "A Int Union(B) = (UN C:B. A Int C)"; |
|
2891 | 452 |
by (Blast_tac 1); |
923 | 453 |
qed "Int_Union"; |
454 |
||
2912 | 455 |
(* Devlin, Setdamentals of Contemporary Set Theory, page 12, exercise 5: |
923 | 456 |
Union of a family of unions **) |
457 |
goal Set.thy "(UN x:C. A(x) Un B(x)) = Union(A``C) Un Union(B``C)"; |
|
2891 | 458 |
by (Blast_tac 1); |
923 | 459 |
qed "Un_Union_image"; |
460 |
||
461 |
(*Equivalent version*) |
|
462 |
goal Set.thy "(UN i:I. A(i) Un B(i)) = (UN i:I. A(i)) Un (UN i:I. B(i))"; |
|
2891 | 463 |
by (Blast_tac 1); |
923 | 464 |
qed "UN_Un_distrib"; |
465 |
||
466 |
goal Set.thy "A Un Inter(B) = (INT C:B. A Un C)"; |
|
2891 | 467 |
by (Blast_tac 1); |
923 | 468 |
qed "Un_Inter"; |
469 |
||
470 |
goal Set.thy "(INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C)"; |
|
2891 | 471 |
by (Blast_tac 1); |
923 | 472 |
qed "Int_Inter_image"; |
473 |
||
474 |
(*Equivalent version*) |
|
475 |
goal Set.thy "(INT i:I. A(i) Int B(i)) = (INT i:I. A(i)) Int (INT i:I. B(i))"; |
|
2891 | 476 |
by (Blast_tac 1); |
923 | 477 |
qed "INT_Int_distrib"; |
478 |
||
479 |
(*Halmos, Naive Set Theory, page 35.*) |
|
480 |
goal Set.thy "B Int (UN i:I. A(i)) = (UN i:I. B Int A(i))"; |
|
2891 | 481 |
by (Blast_tac 1); |
923 | 482 |
qed "Int_UN_distrib"; |
483 |
||
484 |
goal Set.thy "B Un (INT i:I. A(i)) = (INT i:I. B Un A(i))"; |
|
2891 | 485 |
by (Blast_tac 1); |
923 | 486 |
qed "Un_INT_distrib"; |
487 |
||
488 |
goal Set.thy |
|
489 |
"(UN i:I. A(i)) Int (UN j:J. B(j)) = (UN i:I. UN j:J. A(i) Int B(j))"; |
|
2891 | 490 |
by (Blast_tac 1); |
923 | 491 |
qed "Int_UN_distrib2"; |
492 |
||
493 |
goal Set.thy |
|
494 |
"(INT i:I. A(i)) Un (INT j:J. B(j)) = (INT i:I. INT j:J. A(i) Un B(j))"; |
|
2891 | 495 |
by (Blast_tac 1); |
923 | 496 |
qed "Un_INT_distrib2"; |
497 |
||
2512 | 498 |
|
499 |
section"Bounded quantifiers"; |
|
500 |
||
2519 | 501 |
(** These are not added to the default simpset because (a) they duplicate the |
502 |
body and (b) there are no similar rules for Int. **) |
|
2512 | 503 |
|
2519 | 504 |
goal Set.thy "(ALL x:A Un B.P x) = ((ALL x:A.P x) & (ALL x:B.P x))"; |
2891 | 505 |
by (Blast_tac 1); |
2519 | 506 |
qed "ball_Un"; |
507 |
||
508 |
goal Set.thy "(EX x:A Un B.P x) = ((EX x:A.P x) | (EX x:B.P x))"; |
|
2891 | 509 |
by (Blast_tac 1); |
2519 | 510 |
qed "bex_Un"; |
2512 | 511 |
|
512 |
||
1548 | 513 |
section "-"; |
923 | 514 |
|
515 |
goal Set.thy "A-A = {}"; |
|
2891 | 516 |
by (Blast_tac 1); |
923 | 517 |
qed "Diff_cancel"; |
1531 | 518 |
Addsimps[Diff_cancel]; |
923 | 519 |
|
520 |
goal Set.thy "{}-A = {}"; |
|
2891 | 521 |
by (Blast_tac 1); |
923 | 522 |
qed "empty_Diff"; |
1531 | 523 |
Addsimps[empty_Diff]; |
923 | 524 |
|
525 |
goal Set.thy "A-{} = A"; |
|
2891 | 526 |
by (Blast_tac 1); |
923 | 527 |
qed "Diff_empty"; |
1531 | 528 |
Addsimps[Diff_empty]; |
529 |
||
530 |
goal Set.thy "A-UNIV = {}"; |
|
2891 | 531 |
by (Blast_tac 1); |
1531 | 532 |
qed "Diff_UNIV"; |
533 |
Addsimps[Diff_UNIV]; |
|
534 |
||
535 |
goal Set.thy "!!x. x~:A ==> A - insert x B = A-B"; |
|
2891 | 536 |
by (Blast_tac 1); |
1531 | 537 |
qed "Diff_insert0"; |
538 |
Addsimps [Diff_insert0]; |
|
923 | 539 |
|
540 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a 0*) |
|
541 |
goal Set.thy "A - insert a B = A - B - {a}"; |
|
2891 | 542 |
by (Blast_tac 1); |
923 | 543 |
qed "Diff_insert"; |
544 |
||
545 |
(*NOT SUITABLE FOR REWRITING since {a} == insert a 0*) |
|
546 |
goal Set.thy "A - insert a B = A - {a} - B"; |
|
2891 | 547 |
by (Blast_tac 1); |
923 | 548 |
qed "Diff_insert2"; |
549 |
||
1531 | 550 |
goal Set.thy "insert x A - B = (if x:B then A-B else insert x (A-B))"; |
1553 | 551 |
by (simp_tac (!simpset setloop split_tac[expand_if]) 1); |
2891 | 552 |
by (Blast_tac 1); |
1531 | 553 |
qed "insert_Diff_if"; |
554 |
||
555 |
goal Set.thy "!!x. x:B ==> insert x A - B = A-B"; |
|
2891 | 556 |
by (Blast_tac 1); |
1531 | 557 |
qed "insert_Diff1"; |
558 |
Addsimps [insert_Diff1]; |
|
559 |
||
2922 | 560 |
goal Set.thy "!!a. a:A ==> insert a (A-{a}) = A"; |
561 |
by (Blast_tac 1); |
|
923 | 562 |
qed "insert_Diff"; |
563 |
||
564 |
goal Set.thy "A Int (B-A) = {}"; |
|
2891 | 565 |
by (Blast_tac 1); |
923 | 566 |
qed "Diff_disjoint"; |
1531 | 567 |
Addsimps[Diff_disjoint]; |
923 | 568 |
|
569 |
goal Set.thy "!!A. A<=B ==> A Un (B-A) = B"; |
|
2891 | 570 |
by (Blast_tac 1); |
923 | 571 |
qed "Diff_partition"; |
572 |
||
573 |
goal Set.thy "!!A. [| A<=B; B<= C |] ==> (B - (C - A)) = (A :: 'a set)"; |
|
2891 | 574 |
by (Blast_tac 1); |
923 | 575 |
qed "double_diff"; |
576 |
||
577 |
goal Set.thy "A - (B Un C) = (A-B) Int (A-C)"; |
|
2891 | 578 |
by (Blast_tac 1); |
923 | 579 |
qed "Diff_Un"; |
580 |
||
581 |
goal Set.thy "A - (B Int C) = (A-B) Un (A-C)"; |
|
2891 | 582 |
by (Blast_tac 1); |
923 | 583 |
qed "Diff_Int"; |
584 |
||
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|
585 |
goal Set.thy "(A Un B) - C = (A - C) Un (B - C)"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
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|
586 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
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diff
changeset
|
587 |
qed "Un_Diff"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
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|
588 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
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parents:
2922
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changeset
|
589 |
goal Set.thy "(A Int B) - C = (A - C) Int (B - C)"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
590 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
591 |
qed "Int_Diff"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
592 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
593 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
594 |
section "Miscellany"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
595 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
596 |
goal Set.thy "(A = B) = ((A <= (B::'a set)) & (B<=A))"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
597 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
598 |
qed "set_eq_subset"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
599 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
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diff
changeset
|
600 |
goal Set.thy "A <= B = (! t.t:A --> t:B)"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
601 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
602 |
qed "subset_iff"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
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2922
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changeset
|
603 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
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diff
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|
604 |
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
|
605 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
606 |
qed "subset_iff_psubset_eq"; |
2021 | 607 |
|
608 |
||
609 |
(** Miniscoping: pushing in big Unions and Intersections **) |
|
610 |
local |
|
2891 | 611 |
fun prover s = prove_goal Set.thy s (fn _ => [Blast_tac 1]) |
2021 | 612 |
in |
613 |
val UN1_simps = map prover |
|
2031 | 614 |
["(UN x. insert a (B x)) = insert a (UN x. B x)", |
615 |
"(UN x. A x Int B) = ((UN x.A x) Int B)", |
|
616 |
"(UN x. A Int B x) = (A Int (UN x.B x))", |
|
617 |
"(UN x. A x Un B) = ((UN x.A x) Un B)", |
|
618 |
"(UN x. A Un B x) = (A Un (UN x.B x))", |
|
619 |
"(UN x. A x - B) = ((UN x.A x) - B)", |
|
620 |
"(UN x. A - B x) = (A - (INT x.B x))"]; |
|
2021 | 621 |
|
622 |
val INT1_simps = map prover |
|
2031 | 623 |
["(INT x. insert a (B x)) = insert a (INT x. B x)", |
624 |
"(INT x. A x Int B) = ((INT x.A x) Int B)", |
|
625 |
"(INT x. A Int B x) = (A Int (INT x.B x))", |
|
626 |
"(INT x. A x Un B) = ((INT x.A x) Un B)", |
|
627 |
"(INT x. A Un B x) = (A Un (INT x.B x))", |
|
628 |
"(INT x. A x - B) = ((INT x.A x) - B)", |
|
629 |
"(INT x. A - B x) = (A - (UN x.B x))"]; |
|
2021 | 630 |
|
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
631 |
val UN_simps = map prover |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
632 |
["(UN x:C. A x Int B) = ((UN x:C.A x) Int B)", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
633 |
"(UN x:C. A Int B x) = (A Int (UN x:C.B x))", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
634 |
"(UN x:C. A x - B) = ((UN x:C.A x) - B)", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
635 |
"(UN x:C. A - B x) = (A - (INT x:C.B x))"]; |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
636 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
637 |
val INT_simps = map prover |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
638 |
["(INT x:C. insert a (B x)) = insert a (INT x:C. B x)", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
639 |
"(INT x:C. A x Un B) = ((INT x:C.A x) Un B)", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
640 |
"(INT x:C. A Un B x) = (A Un (INT x:C.B x))"]; |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
641 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
642 |
(*The missing laws for bounded Unions and Intersections are conditional |
2021 | 643 |
on the index set's being non-empty. Thus they are probably NOT worth |
644 |
adding as default rewrites.*) |
|
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
645 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
646 |
val ball_simps = map prover |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
647 |
["(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
|
648 |
"(ALL x:A. P | Q x) = (P | (ALL x:A. Q x))", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
649 |
"(ALL x:{}. P x) = True", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
650 |
"(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
|
651 |
"(ALL x:Union(A). P x) = (ALL y:A. ALL x:y. P x)", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
652 |
"(ALL x:Collect Q. P x) = (ALL x. Q x --> P x)"]; |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
653 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
654 |
val ball_conj_distrib = |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
655 |
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
|
656 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
657 |
val bex_simps = map prover |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
658 |
["(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
|
659 |
"(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
|
660 |
"(EX x:{}. P x) = False", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
661 |
"(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
|
662 |
"(EX x:Union(A). P x) = (EX y:A. EX x:y. P x)", |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
663 |
"(EX x:Collect Q. P x) = (EX x. Q x & P x)"]; |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
664 |
|
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
665 |
val bex_conj_distrib = |
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
666 |
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
|
667 |
|
2021 | 668 |
end; |
669 |
||
2513
d708d8cdc8e8
New miniscoping rules for the bounded quantifiers and UN/INT operators
paulson
parents:
2512
diff
changeset
|
670 |
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
|
671 |
ball_simps @ bex_simps); |