author | paulson |
Wed, 07 Jun 2000 12:06:36 +0200 | |
changeset 9041 | 3730ae0f513a |
parent 8913 | 0bc13d5e60b8 |
child 9075 | e8521ed7f35b |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: HOL/set |
923 | 2 |
ID: $Id$ |
1465 | 3 |
Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
923 | 4 |
Copyright 1991 University of Cambridge |
5 |
||
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
1937
diff
changeset
|
6 |
Set theory for higher-order logic. A set is simply a predicate. |
923 | 7 |
*) |
8 |
||
1548 | 9 |
section "Relating predicates and sets"; |
10 |
||
3469
61d927bd57ec
Now Collect_mem_eq is a default simprule (how could it have ever been omitted?
paulson
parents:
3420
diff
changeset
|
11 |
Addsimps [Collect_mem_eq]; |
61d927bd57ec
Now Collect_mem_eq is a default simprule (how could it have ever been omitted?
paulson
parents:
3420
diff
changeset
|
12 |
AddIffs [mem_Collect_eq]; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
13 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
14 |
Goal "P(a) ==> a : {x. P(x)}"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
15 |
by (Asm_simp_tac 1); |
923 | 16 |
qed "CollectI"; |
17 |
||
5316 | 18 |
Goal "a : {x. P(x)} ==> P(a)"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
19 |
by (Asm_full_simp_tac 1); |
923 | 20 |
qed "CollectD"; |
21 |
||
7658 | 22 |
bind_thm ("CollectE", make_elim CollectD); |
23 |
||
5316 | 24 |
val [prem] = Goal "[| !!x. (x:A) = (x:B) |] ==> A = B"; |
923 | 25 |
by (rtac (prem RS ext RS arg_cong RS box_equals) 1); |
26 |
by (rtac Collect_mem_eq 1); |
|
27 |
by (rtac Collect_mem_eq 1); |
|
28 |
qed "set_ext"; |
|
29 |
||
5316 | 30 |
val [prem] = Goal "[| !!x. P(x)=Q(x) |] ==> {x. P(x)} = {x. Q(x)}"; |
923 | 31 |
by (rtac (prem RS ext RS arg_cong) 1); |
32 |
qed "Collect_cong"; |
|
33 |
||
34 |
val CollectE = make_elim CollectD; |
|
35 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
36 |
AddSIs [CollectI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
37 |
AddSEs [CollectE]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
38 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
39 |
|
1548 | 40 |
section "Bounded quantifiers"; |
923 | 41 |
|
5316 | 42 |
val prems = Goalw [Ball_def] |
9041 | 43 |
"[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)"; |
923 | 44 |
by (REPEAT (ares_tac (prems @ [allI,impI]) 1)); |
45 |
qed "ballI"; |
|
46 |
||
8839 | 47 |
bind_thms ("strip", [impI, allI, ballI]); |
48 |
||
9041 | 49 |
Goalw [Ball_def] "[| ALL x:A. P(x); x:A |] ==> P(x)"; |
5316 | 50 |
by (Blast_tac 1); |
923 | 51 |
qed "bspec"; |
52 |
||
5316 | 53 |
val major::prems = Goalw [Ball_def] |
9041 | 54 |
"[| ALL x:A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q"; |
923 | 55 |
by (rtac (major RS spec RS impCE) 1); |
56 |
by (REPEAT (eresolve_tac prems 1)); |
|
57 |
qed "ballE"; |
|
58 |
||
9041 | 59 |
(*Takes assumptions ALL x:A.P(x) and a:A; creates assumption P(a)*) |
923 | 60 |
fun ball_tac i = etac ballE i THEN contr_tac (i+1); |
61 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
62 |
AddSIs [ballI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
63 |
AddEs [ballE]; |
7441 | 64 |
AddXDs [bspec]; |
5521 | 65 |
(* gives better instantiation for bound: *) |
66 |
claset_ref() := claset() addWrapper ("bspec", fn tac2 => |
|
67 |
(dtac bspec THEN' atac) APPEND' tac2); |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
68 |
|
6006 | 69 |
(*Normally the best argument order: P(x) constrains the choice of x:A*) |
9041 | 70 |
Goalw [Bex_def] "[| P(x); x:A |] ==> EX x:A. P(x)"; |
5316 | 71 |
by (Blast_tac 1); |
923 | 72 |
qed "bexI"; |
73 |
||
6006 | 74 |
(*The best argument order when there is only one x:A*) |
9041 | 75 |
Goalw [Bex_def] "[| x:A; P(x) |] ==> EX x:A. P(x)"; |
6006 | 76 |
by (Blast_tac 1); |
77 |
qed "rev_bexI"; |
|
78 |
||
7031 | 79 |
val prems = Goal |
9041 | 80 |
"[| ALL x:A. ~P(x) ==> P(a); a:A |] ==> EX x:A. P(x)"; |
7007 | 81 |
by (rtac classical 1); |
82 |
by (REPEAT (ares_tac (prems@[bexI,ballI,notI,notE]) 1)) ; |
|
83 |
qed "bexCI"; |
|
923 | 84 |
|
5316 | 85 |
val major::prems = Goalw [Bex_def] |
9041 | 86 |
"[| EX x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q"; |
923 | 87 |
by (rtac (major RS exE) 1); |
88 |
by (REPEAT (eresolve_tac (prems @ [asm_rl,conjE]) 1)); |
|
89 |
qed "bexE"; |
|
90 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
91 |
AddIs [bexI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
92 |
AddSEs [bexE]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
93 |
|
3420 | 94 |
(*Trival rewrite rule*) |
9041 | 95 |
Goal "(ALL x:A. P) = ((EX x. x:A) --> P)"; |
4089 | 96 |
by (simp_tac (simpset() addsimps [Ball_def]) 1); |
3420 | 97 |
qed "ball_triv"; |
1816 | 98 |
|
1882 | 99 |
(*Dual form for existentials*) |
9041 | 100 |
Goal "(EX x:A. P) = ((EX x. x:A) & P)"; |
4089 | 101 |
by (simp_tac (simpset() addsimps [Bex_def]) 1); |
3420 | 102 |
qed "bex_triv"; |
1882 | 103 |
|
3420 | 104 |
Addsimps [ball_triv, bex_triv]; |
923 | 105 |
|
106 |
(** Congruence rules **) |
|
107 |
||
6291 | 108 |
val prems = Goalw [Ball_def] |
923 | 109 |
"[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ |
9041 | 110 |
\ (ALL x:A. P(x)) = (ALL x:B. Q(x))"; |
6291 | 111 |
by (asm_simp_tac (simpset() addsimps prems) 1); |
923 | 112 |
qed "ball_cong"; |
113 |
||
6291 | 114 |
val prems = Goalw [Bex_def] |
923 | 115 |
"[| A=B; !!x. x:B ==> P(x) = Q(x) |] ==> \ |
9041 | 116 |
\ (EX x:A. P(x)) = (EX x:B. Q(x))"; |
6291 | 117 |
by (asm_simp_tac (simpset() addcongs [conj_cong] addsimps prems) 1); |
923 | 118 |
qed "bex_cong"; |
119 |
||
6291 | 120 |
Addcongs [ball_cong,bex_cong]; |
121 |
||
1548 | 122 |
section "Subsets"; |
923 | 123 |
|
5316 | 124 |
val prems = Goalw [subset_def] "(!!x. x:A ==> x:B) ==> A <= B"; |
923 | 125 |
by (REPEAT (ares_tac (prems @ [ballI]) 1)); |
126 |
qed "subsetI"; |
|
127 |
||
5649 | 128 |
(*Map the type ('a set => anything) to just 'a. |
129 |
For overloading constants whose first argument has type "'a set" *) |
|
130 |
fun overload_1st_set s = Blast.overloaded (s, HOLogic.dest_setT o domain_type); |
|
131 |
||
4059 | 132 |
(*While (:) is not, its type must be kept |
133 |
for overloading of = to work.*) |
|
4240
8ba60a4cd380
Renamed "overload" to "overloaded" for sml/nj compatibility
paulson
parents:
4231
diff
changeset
|
134 |
Blast.overloaded ("op :", domain_type); |
5649 | 135 |
|
136 |
overload_1st_set "Ball"; (*need UNION, INTER also?*) |
|
137 |
overload_1st_set "Bex"; |
|
4059 | 138 |
|
4469 | 139 |
(*Image: retain the type of the set being expressed*) |
8005 | 140 |
Blast.overloaded ("image", domain_type); |
2881 | 141 |
|
923 | 142 |
(*Rule in Modus Ponens style*) |
5316 | 143 |
Goalw [subset_def] "[| A <= B; c:A |] ==> c:B"; |
144 |
by (Blast_tac 1); |
|
923 | 145 |
qed "subsetD"; |
7658 | 146 |
AddXIs [subsetD]; |
923 | 147 |
|
148 |
(*The same, with reversed premises for use with etac -- cf rev_mp*) |
|
7007 | 149 |
Goal "[| c:A; A <= B |] ==> c:B"; |
150 |
by (REPEAT (ares_tac [subsetD] 1)) ; |
|
151 |
qed "rev_subsetD"; |
|
7658 | 152 |
AddXIs [rev_subsetD]; |
923 | 153 |
|
1920 | 154 |
(*Converts A<=B to x:A ==> x:B*) |
155 |
fun impOfSubs th = th RSN (2, rev_subsetD); |
|
156 |
||
7007 | 157 |
Goal "[| A <= B; c ~: B |] ==> c ~: A"; |
158 |
by (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ; |
|
159 |
qed "contra_subsetD"; |
|
1841 | 160 |
|
7007 | 161 |
Goal "[| c ~: B; A <= B |] ==> c ~: A"; |
162 |
by (REPEAT (eresolve_tac [asm_rl, contrapos, subsetD] 1)) ; |
|
163 |
qed "rev_contra_subsetD"; |
|
1841 | 164 |
|
923 | 165 |
(*Classical elimination rule*) |
5316 | 166 |
val major::prems = Goalw [subset_def] |
923 | 167 |
"[| A <= B; c~:A ==> P; c:B ==> P |] ==> P"; |
168 |
by (rtac (major RS ballE) 1); |
|
169 |
by (REPEAT (eresolve_tac prems 1)); |
|
170 |
qed "subsetCE"; |
|
171 |
||
172 |
(*Takes assumptions A<=B; c:A and creates the assumption c:B *) |
|
173 |
fun set_mp_tac i = etac subsetCE i THEN mp_tac i; |
|
174 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
175 |
AddSIs [subsetI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
176 |
AddEs [subsetD, subsetCE]; |
923 | 177 |
|
7007 | 178 |
Goal "A <= (A::'a set)"; |
179 |
by (Fast_tac 1); |
|
180 |
qed "subset_refl"; (*Blast_tac would try order_refl and fail*) |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
181 |
|
5316 | 182 |
Goal "[| A<=B; B<=C |] ==> A<=(C::'a set)"; |
2891 | 183 |
by (Blast_tac 1); |
923 | 184 |
qed "subset_trans"; |
185 |
||
186 |
||
1548 | 187 |
section "Equality"; |
923 | 188 |
|
189 |
(*Anti-symmetry of the subset relation*) |
|
5316 | 190 |
Goal "[| A <= B; B <= A |] ==> A = (B::'a set)"; |
5318 | 191 |
by (rtac set_ext 1); |
5316 | 192 |
by (blast_tac (claset() addIs [subsetD]) 1); |
923 | 193 |
qed "subset_antisym"; |
194 |
val equalityI = subset_antisym; |
|
195 |
||
1762 | 196 |
AddSIs [equalityI]; |
197 |
||
923 | 198 |
(* Equality rules from ZF set theory -- are they appropriate here? *) |
5316 | 199 |
Goal "A = B ==> A<=(B::'a set)"; |
200 |
by (etac ssubst 1); |
|
923 | 201 |
by (rtac subset_refl 1); |
202 |
qed "equalityD1"; |
|
203 |
||
5316 | 204 |
Goal "A = B ==> B<=(A::'a set)"; |
205 |
by (etac ssubst 1); |
|
923 | 206 |
by (rtac subset_refl 1); |
207 |
qed "equalityD2"; |
|
208 |
||
5316 | 209 |
val prems = Goal |
923 | 210 |
"[| A = B; [| A<=B; B<=(A::'a set) |] ==> P |] ==> P"; |
211 |
by (resolve_tac prems 1); |
|
212 |
by (REPEAT (resolve_tac (prems RL [equalityD1,equalityD2]) 1)); |
|
213 |
qed "equalityE"; |
|
214 |
||
9041 | 215 |
(*This could be tried. Most things build fine with it. However, some proofs |
216 |
become very slow or even fail. |
|
217 |
AddEs [equalityE]; |
|
218 |
*) |
|
219 |
||
5316 | 220 |
val major::prems = Goal |
923 | 221 |
"[| A = B; [| c:A; c:B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P"; |
222 |
by (rtac (major RS equalityE) 1); |
|
223 |
by (REPEAT (contr_tac 1 ORELSE eresolve_tac ([asm_rl,subsetCE]@prems) 1)); |
|
224 |
qed "equalityCE"; |
|
225 |
||
226 |
(*Lemma for creating induction formulae -- for "pattern matching" on p |
|
227 |
To make the induction hypotheses usable, apply "spec" or "bspec" to |
|
228 |
put universal quantifiers over the free variables in p. *) |
|
5316 | 229 |
val prems = Goal |
923 | 230 |
"[| p:A; !!z. z:A ==> p=z --> R |] ==> R"; |
231 |
by (rtac mp 1); |
|
232 |
by (REPEAT (resolve_tac (refl::prems) 1)); |
|
233 |
qed "setup_induction"; |
|
234 |
||
8053 | 235 |
Goal "A = B ==> (x : A) = (x : B)"; |
236 |
by (Asm_simp_tac 1); |
|
237 |
qed "eqset_imp_iff"; |
|
238 |
||
923 | 239 |
|
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
240 |
section "The universal set -- UNIV"; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
241 |
|
7031 | 242 |
Goalw [UNIV_def] "x : UNIV"; |
243 |
by (rtac CollectI 1); |
|
244 |
by (rtac TrueI 1); |
|
245 |
qed "UNIV_I"; |
|
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
246 |
|
4434 | 247 |
Addsimps [UNIV_I]; |
248 |
AddIs [UNIV_I]; (*unsafe makes it less likely to cause problems*) |
|
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
249 |
|
7031 | 250 |
Goal "A <= UNIV"; |
251 |
by (rtac subsetI 1); |
|
252 |
by (rtac UNIV_I 1); |
|
253 |
qed "subset_UNIV"; |
|
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
254 |
|
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
255 |
(** Eta-contracting these two rules (to remove P) causes them to be ignored |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
256 |
because of their interaction with congruence rules. **) |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
257 |
|
5069 | 258 |
Goalw [Ball_def] "Ball UNIV P = All P"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
259 |
by (Simp_tac 1); |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
260 |
qed "ball_UNIV"; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
261 |
|
5069 | 262 |
Goalw [Bex_def] "Bex UNIV P = Ex P"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
263 |
by (Simp_tac 1); |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
264 |
qed "bex_UNIV"; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
265 |
Addsimps [ball_UNIV, bex_UNIV]; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
266 |
|
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
267 |
|
2858 | 268 |
section "The empty set -- {}"; |
269 |
||
7007 | 270 |
Goalw [empty_def] "(c : {}) = False"; |
271 |
by (Blast_tac 1) ; |
|
272 |
qed "empty_iff"; |
|
2858 | 273 |
|
274 |
Addsimps [empty_iff]; |
|
275 |
||
7007 | 276 |
Goal "a:{} ==> P"; |
277 |
by (Full_simp_tac 1); |
|
278 |
qed "emptyE"; |
|
2858 | 279 |
|
280 |
AddSEs [emptyE]; |
|
281 |
||
7007 | 282 |
Goal "{} <= A"; |
283 |
by (Blast_tac 1) ; |
|
284 |
qed "empty_subsetI"; |
|
2858 | 285 |
|
5256 | 286 |
(*One effect is to delete the ASSUMPTION {} <= A*) |
287 |
AddIffs [empty_subsetI]; |
|
288 |
||
7031 | 289 |
val [prem]= Goal "[| !!y. y:A ==> False |] ==> A={}"; |
7007 | 290 |
by (blast_tac (claset() addIs [prem RS FalseE]) 1) ; |
291 |
qed "equals0I"; |
|
2858 | 292 |
|
5256 | 293 |
(*Use for reasoning about disjointness: A Int B = {} *) |
7007 | 294 |
Goal "A={} ==> a ~: A"; |
295 |
by (Blast_tac 1) ; |
|
296 |
qed "equals0D"; |
|
2858 | 297 |
|
9041 | 298 |
(* [| A = {}; a : A |] ==> R *) |
5450
fe9d103464a4
Changed equals0E back to equals0D and gave it the correct destruct form
paulson
parents:
5336
diff
changeset
|
299 |
AddDs [equals0D, sym RS equals0D]; |
5256 | 300 |
|
5069 | 301 |
Goalw [Ball_def] "Ball {} P = True"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
302 |
by (Simp_tac 1); |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
303 |
qed "ball_empty"; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
304 |
|
5069 | 305 |
Goalw [Bex_def] "Bex {} P = False"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
306 |
by (Simp_tac 1); |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
307 |
qed "bex_empty"; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
308 |
Addsimps [ball_empty, bex_empty]; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
309 |
|
5069 | 310 |
Goal "UNIV ~= {}"; |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
311 |
by (blast_tac (claset() addEs [equalityE]) 1); |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
312 |
qed "UNIV_not_empty"; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
313 |
AddIffs [UNIV_not_empty]; |
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
314 |
|
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
315 |
|
2858 | 316 |
|
317 |
section "The Powerset operator -- Pow"; |
|
318 |
||
7007 | 319 |
Goalw [Pow_def] "(A : Pow(B)) = (A <= B)"; |
320 |
by (Asm_simp_tac 1); |
|
321 |
qed "Pow_iff"; |
|
2858 | 322 |
|
323 |
AddIffs [Pow_iff]; |
|
324 |
||
7031 | 325 |
Goalw [Pow_def] "A <= B ==> A : Pow(B)"; |
7007 | 326 |
by (etac CollectI 1); |
327 |
qed "PowI"; |
|
2858 | 328 |
|
7031 | 329 |
Goalw [Pow_def] "A : Pow(B) ==> A<=B"; |
7007 | 330 |
by (etac CollectD 1); |
331 |
qed "PowD"; |
|
332 |
||
2858 | 333 |
|
334 |
val Pow_bottom = empty_subsetI RS PowI; (* {}: Pow(B) *) |
|
335 |
val Pow_top = subset_refl RS PowI; (* A : Pow(A) *) |
|
336 |
||
337 |
||
5931 | 338 |
section "Set complement"; |
923 | 339 |
|
7031 | 340 |
Goalw [Compl_def] "(c : -A) = (c~:A)"; |
341 |
by (Blast_tac 1); |
|
342 |
qed "Compl_iff"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
343 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
344 |
Addsimps [Compl_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
345 |
|
5490 | 346 |
val prems = Goalw [Compl_def] "[| c:A ==> False |] ==> c : -A"; |
923 | 347 |
by (REPEAT (ares_tac (prems @ [CollectI,notI]) 1)); |
348 |
qed "ComplI"; |
|
349 |
||
350 |
(*This form, with negated conclusion, works well with the Classical prover. |
|
351 |
Negated assumptions behave like formulae on the right side of the notional |
|
352 |
turnstile...*) |
|
5490 | 353 |
Goalw [Compl_def] "c : -A ==> c~:A"; |
5316 | 354 |
by (etac CollectD 1); |
923 | 355 |
qed "ComplD"; |
356 |
||
357 |
val ComplE = make_elim ComplD; |
|
358 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
359 |
AddSIs [ComplI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
360 |
AddSEs [ComplE]; |
1640 | 361 |
|
923 | 362 |
|
1548 | 363 |
section "Binary union -- Un"; |
923 | 364 |
|
7031 | 365 |
Goalw [Un_def] "(c : A Un B) = (c:A | c:B)"; |
366 |
by (Blast_tac 1); |
|
367 |
qed "Un_iff"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
368 |
Addsimps [Un_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
369 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
370 |
Goal "c:A ==> c : A Un B"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
371 |
by (Asm_simp_tac 1); |
923 | 372 |
qed "UnI1"; |
373 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
374 |
Goal "c:B ==> c : A Un B"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
375 |
by (Asm_simp_tac 1); |
923 | 376 |
qed "UnI2"; |
377 |
||
378 |
(*Classical introduction rule: no commitment to A vs B*) |
|
7007 | 379 |
|
7031 | 380 |
val prems = Goal "(c~:B ==> c:A) ==> c : A Un B"; |
7007 | 381 |
by (Simp_tac 1); |
382 |
by (REPEAT (ares_tac (prems@[disjCI]) 1)) ; |
|
383 |
qed "UnCI"; |
|
923 | 384 |
|
5316 | 385 |
val major::prems = Goalw [Un_def] |
923 | 386 |
"[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P"; |
387 |
by (rtac (major RS CollectD RS disjE) 1); |
|
388 |
by (REPEAT (eresolve_tac prems 1)); |
|
389 |
qed "UnE"; |
|
390 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
391 |
AddSIs [UnCI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
392 |
AddSEs [UnE]; |
1640 | 393 |
|
923 | 394 |
|
1548 | 395 |
section "Binary intersection -- Int"; |
923 | 396 |
|
7031 | 397 |
Goalw [Int_def] "(c : A Int B) = (c:A & c:B)"; |
398 |
by (Blast_tac 1); |
|
399 |
qed "Int_iff"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
400 |
Addsimps [Int_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
401 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
402 |
Goal "[| c:A; c:B |] ==> c : A Int B"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
403 |
by (Asm_simp_tac 1); |
923 | 404 |
qed "IntI"; |
405 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
406 |
Goal "c : A Int B ==> c:A"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
407 |
by (Asm_full_simp_tac 1); |
923 | 408 |
qed "IntD1"; |
409 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
410 |
Goal "c : A Int B ==> c:B"; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
411 |
by (Asm_full_simp_tac 1); |
923 | 412 |
qed "IntD2"; |
413 |
||
5316 | 414 |
val [major,minor] = Goal |
923 | 415 |
"[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P"; |
416 |
by (rtac minor 1); |
|
417 |
by (rtac (major RS IntD1) 1); |
|
418 |
by (rtac (major RS IntD2) 1); |
|
419 |
qed "IntE"; |
|
420 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
421 |
AddSIs [IntI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
422 |
AddSEs [IntE]; |
923 | 423 |
|
1548 | 424 |
section "Set difference"; |
923 | 425 |
|
7031 | 426 |
Goalw [set_diff_def] "(c : A-B) = (c:A & c~:B)"; |
427 |
by (Blast_tac 1); |
|
428 |
qed "Diff_iff"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
429 |
Addsimps [Diff_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
430 |
|
7007 | 431 |
Goal "[| c : A; c ~: B |] ==> c : A - B"; |
432 |
by (Asm_simp_tac 1) ; |
|
433 |
qed "DiffI"; |
|
923 | 434 |
|
7007 | 435 |
Goal "c : A - B ==> c : A"; |
436 |
by (Asm_full_simp_tac 1) ; |
|
437 |
qed "DiffD1"; |
|
923 | 438 |
|
7007 | 439 |
Goal "[| c : A - B; c : B |] ==> P"; |
440 |
by (Asm_full_simp_tac 1) ; |
|
441 |
qed "DiffD2"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
442 |
|
7031 | 443 |
val prems = Goal "[| c : A - B; [| c:A; c~:B |] ==> P |] ==> P"; |
7007 | 444 |
by (resolve_tac prems 1); |
445 |
by (REPEAT (ares_tac (prems RL [DiffD1, DiffD2 RS notI]) 1)) ; |
|
446 |
qed "DiffE"; |
|
923 | 447 |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
448 |
AddSIs [DiffI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
449 |
AddSEs [DiffE]; |
923 | 450 |
|
451 |
||
1548 | 452 |
section "Augmenting a set -- insert"; |
923 | 453 |
|
7031 | 454 |
Goalw [insert_def] "a : insert b A = (a=b | a:A)"; |
455 |
by (Blast_tac 1); |
|
456 |
qed "insert_iff"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
457 |
Addsimps [insert_iff]; |
923 | 458 |
|
7031 | 459 |
Goal "a : insert a B"; |
7007 | 460 |
by (Simp_tac 1); |
461 |
qed "insertI1"; |
|
923 | 462 |
|
7007 | 463 |
Goal "!!a. a : B ==> a : insert b B"; |
464 |
by (Asm_simp_tac 1); |
|
465 |
qed "insertI2"; |
|
466 |
||
467 |
val major::prems = Goalw [insert_def] |
|
468 |
"[| a : insert b A; a=b ==> P; a:A ==> P |] ==> P"; |
|
469 |
by (rtac (major RS UnE) 1); |
|
470 |
by (REPEAT (eresolve_tac (prems @ [CollectE]) 1)); |
|
471 |
qed "insertE"; |
|
923 | 472 |
|
473 |
(*Classical introduction rule*) |
|
7031 | 474 |
val prems = Goal "(a~:B ==> a=b) ==> a: insert b B"; |
7007 | 475 |
by (Simp_tac 1); |
476 |
by (REPEAT (ares_tac (prems@[disjCI]) 1)) ; |
|
477 |
qed "insertCI"; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
478 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
479 |
AddSIs [insertCI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
480 |
AddSEs [insertE]; |
923 | 481 |
|
9041 | 482 |
Goal "A <= insert x B ==> A <= B & x ~: A | (EX B'. A = insert x B' & B' <= B)"; |
7496
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
483 |
by (case_tac "x:A" 1); |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
484 |
by (Fast_tac 2); |
7499 | 485 |
by (rtac disjI2 1); |
7496
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
486 |
by (res_inst_tac [("x","A-{x}")] exI 1); |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
487 |
by (Fast_tac 1); |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
488 |
qed "subset_insertD"; |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
489 |
|
1548 | 490 |
section "Singletons, using insert"; |
923 | 491 |
|
7007 | 492 |
Goal "a : {a}"; |
493 |
by (rtac insertI1 1) ; |
|
494 |
qed "singletonI"; |
|
923 | 495 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
496 |
Goal "b : {a} ==> b=a"; |
2891 | 497 |
by (Blast_tac 1); |
923 | 498 |
qed "singletonD"; |
499 |
||
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
500 |
bind_thm ("singletonE", make_elim singletonD); |
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
501 |
|
7007 | 502 |
Goal "(b : {a}) = (b=a)"; |
503 |
by (Blast_tac 1); |
|
504 |
qed "singleton_iff"; |
|
923 | 505 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
506 |
Goal "{a}={b} ==> a=b"; |
4089 | 507 |
by (blast_tac (claset() addEs [equalityE]) 1); |
923 | 508 |
qed "singleton_inject"; |
509 |
||
2858 | 510 |
(*Redundant? But unlike insertCI, it proves the subgoal immediately!*) |
511 |
AddSIs [singletonI]; |
|
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
512 |
AddSDs [singleton_inject]; |
3718 | 513 |
AddSEs [singletonE]; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
514 |
|
7969 | 515 |
Goal "{b} = insert a A = (a = b & A <= {b})"; |
8326
0e329578b0ef
tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents:
8053
diff
changeset
|
516 |
by (blast_tac (claset() addSEs [equalityE]) 1); |
7496
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
517 |
qed "singleton_insert_inj_eq"; |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
518 |
|
8326
0e329578b0ef
tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents:
8053
diff
changeset
|
519 |
Goal "(insert a A = {b}) = (a = b & A <= {b})"; |
0e329578b0ef
tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents:
8053
diff
changeset
|
520 |
by (blast_tac (claset() addSEs [equalityE]) 1); |
7969 | 521 |
qed "singleton_insert_inj_eq'"; |
522 |
||
8326
0e329578b0ef
tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents:
8053
diff
changeset
|
523 |
AddIffs [singleton_insert_inj_eq, singleton_insert_inj_eq']; |
0e329578b0ef
tidied the proofs of singleton_insert_inj_eq, singleton_insert_inj_eq' and
paulson
parents:
8053
diff
changeset
|
524 |
|
7496
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
525 |
Goal "A <= {x} ==> A={} | A = {x}"; |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
526 |
by (Fast_tac 1); |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
527 |
qed "subset_singletonD"; |
93ae11d887ff
added theorems subset_insertD, singleton_insert_inj_eq, subset_singletonD
oheimb
parents:
7441
diff
changeset
|
528 |
|
5069 | 529 |
Goal "{x. x=a} = {a}"; |
4423 | 530 |
by (Blast_tac 1); |
3582 | 531 |
qed "singleton_conv"; |
532 |
Addsimps [singleton_conv]; |
|
1531 | 533 |
|
5600 | 534 |
Goal "{x. a=x} = {a}"; |
6301 | 535 |
by (Blast_tac 1); |
5600 | 536 |
qed "singleton_conv2"; |
537 |
Addsimps [singleton_conv2]; |
|
538 |
||
1531 | 539 |
|
1548 | 540 |
section "Unions of families -- UNION x:A. B(x) is Union(B``A)"; |
923 | 541 |
|
5069 | 542 |
Goalw [UNION_def] "(b: (UN x:A. B(x))) = (EX x:A. b: B(x))"; |
2891 | 543 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
544 |
qed "UN_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
545 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
546 |
Addsimps [UN_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
547 |
|
923 | 548 |
(*The order of the premises presupposes that A is rigid; b may be flexible*) |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
549 |
Goal "[| a:A; b: B(a) |] ==> b: (UN x:A. B(x))"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4469
diff
changeset
|
550 |
by Auto_tac; |
923 | 551 |
qed "UN_I"; |
552 |
||
5316 | 553 |
val major::prems = Goalw [UNION_def] |
923 | 554 |
"[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"; |
555 |
by (rtac (major RS CollectD RS bexE) 1); |
|
556 |
by (REPEAT (ares_tac prems 1)); |
|
557 |
qed "UN_E"; |
|
558 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
559 |
AddIs [UN_I]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
560 |
AddSEs [UN_E]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
561 |
|
6291 | 562 |
val prems = Goalw [UNION_def] |
923 | 563 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
564 |
\ (UN x:A. C(x)) = (UN x:B. D(x))"; |
|
6291 | 565 |
by (asm_simp_tac (simpset() addsimps prems) 1); |
923 | 566 |
qed "UN_cong"; |
567 |
||
568 |
||
1548 | 569 |
section "Intersections of families -- INTER x:A. B(x) is Inter(B``A)"; |
923 | 570 |
|
5069 | 571 |
Goalw [INTER_def] "(b: (INT x:A. B(x))) = (ALL x:A. b: B(x))"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4469
diff
changeset
|
572 |
by Auto_tac; |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
573 |
qed "INT_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
574 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
575 |
Addsimps [INT_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
576 |
|
5316 | 577 |
val prems = Goalw [INTER_def] |
923 | 578 |
"(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"; |
579 |
by (REPEAT (ares_tac ([CollectI,ballI] @ prems) 1)); |
|
580 |
qed "INT_I"; |
|
581 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
582 |
Goal "[| b : (INT x:A. B(x)); a:A |] ==> b: B(a)"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4469
diff
changeset
|
583 |
by Auto_tac; |
923 | 584 |
qed "INT_D"; |
585 |
||
586 |
(*"Classical" elimination -- by the Excluded Middle on a:A *) |
|
5316 | 587 |
val major::prems = Goalw [INTER_def] |
923 | 588 |
"[| b : (INT x:A. B(x)); b: B(a) ==> R; a~:A ==> R |] ==> R"; |
589 |
by (rtac (major RS CollectD RS ballE) 1); |
|
590 |
by (REPEAT (eresolve_tac prems 1)); |
|
591 |
qed "INT_E"; |
|
592 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
593 |
AddSIs [INT_I]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
594 |
AddEs [INT_D, INT_E]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
595 |
|
6291 | 596 |
val prems = Goalw [INTER_def] |
923 | 597 |
"[| A=B; !!x. x:B ==> C(x) = D(x) |] ==> \ |
598 |
\ (INT x:A. C(x)) = (INT x:B. D(x))"; |
|
6291 | 599 |
by (asm_simp_tac (simpset() addsimps prems) 1); |
923 | 600 |
qed "INT_cong"; |
601 |
||
602 |
||
1548 | 603 |
section "Union"; |
923 | 604 |
|
5069 | 605 |
Goalw [Union_def] "(A : Union(C)) = (EX X:C. A:X)"; |
2891 | 606 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
607 |
qed "Union_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
608 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
609 |
Addsimps [Union_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
610 |
|
923 | 611 |
(*The order of the premises presupposes that C is rigid; A may be flexible*) |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
612 |
Goal "[| X:C; A:X |] ==> A : Union(C)"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4469
diff
changeset
|
613 |
by Auto_tac; |
923 | 614 |
qed "UnionI"; |
615 |
||
5316 | 616 |
val major::prems = Goalw [Union_def] |
923 | 617 |
"[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"; |
618 |
by (rtac (major RS UN_E) 1); |
|
619 |
by (REPEAT (ares_tac prems 1)); |
|
620 |
qed "UnionE"; |
|
621 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
622 |
AddIs [UnionI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
623 |
AddSEs [UnionE]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
624 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
625 |
|
1548 | 626 |
section "Inter"; |
923 | 627 |
|
5069 | 628 |
Goalw [Inter_def] "(A : Inter(C)) = (ALL X:C. A:X)"; |
2891 | 629 |
by (Blast_tac 1); |
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
630 |
qed "Inter_iff"; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
631 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
632 |
Addsimps [Inter_iff]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
633 |
|
5316 | 634 |
val prems = Goalw [Inter_def] |
923 | 635 |
"[| !!X. X:C ==> A:X |] ==> A : Inter(C)"; |
636 |
by (REPEAT (ares_tac ([INT_I] @ prems) 1)); |
|
637 |
qed "InterI"; |
|
638 |
||
639 |
(*A "destruct" rule -- every X in C contains A as an element, but |
|
640 |
A:X can hold when X:C does not! This rule is analogous to "spec". *) |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
641 |
Goal "[| A : Inter(C); X:C |] ==> A:X"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4469
diff
changeset
|
642 |
by Auto_tac; |
923 | 643 |
qed "InterD"; |
644 |
||
645 |
(*"Classical" elimination rule -- does not require proving X:C *) |
|
5316 | 646 |
val major::prems = Goalw [Inter_def] |
2721 | 647 |
"[| A : Inter(C); X~:C ==> R; A:X ==> R |] ==> R"; |
923 | 648 |
by (rtac (major RS INT_E) 1); |
649 |
by (REPEAT (eresolve_tac prems 1)); |
|
650 |
qed "InterE"; |
|
651 |
||
2499
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
652 |
AddSIs [InterI]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
653 |
AddEs [InterD, InterE]; |
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
654 |
|
0bc87b063447
Tidying of proofs. New theorems are enterred immediately into the
paulson
parents:
2031
diff
changeset
|
655 |
|
2912 | 656 |
(*** Image of a set under a function ***) |
657 |
||
658 |
(*Frequently b does not have the syntactic form of f(x).*) |
|
5316 | 659 |
Goalw [image_def] "[| b=f(x); x:A |] ==> b : f``A"; |
660 |
by (Blast_tac 1); |
|
2912 | 661 |
qed "image_eqI"; |
3909 | 662 |
Addsimps [image_eqI]; |
2912 | 663 |
|
664 |
bind_thm ("imageI", refl RS image_eqI); |
|
665 |
||
8025 | 666 |
(*This version's more effective when we already have the required x*) |
667 |
Goalw [image_def] "[| x:A; b=f(x) |] ==> b : f``A"; |
|
668 |
by (Blast_tac 1); |
|
669 |
qed "rev_image_eqI"; |
|
670 |
||
2912 | 671 |
(*The eta-expansion gives variable-name preservation.*) |
5316 | 672 |
val major::prems = Goalw [image_def] |
3842 | 673 |
"[| b : (%x. f(x))``A; !!x.[| b=f(x); x:A |] ==> P |] ==> P"; |
2912 | 674 |
by (rtac (major RS CollectD RS bexE) 1); |
675 |
by (REPEAT (ares_tac prems 1)); |
|
676 |
qed "imageE"; |
|
677 |
||
678 |
AddIs [image_eqI]; |
|
679 |
AddSEs [imageE]; |
|
680 |
||
5069 | 681 |
Goal "f``(A Un B) = f``A Un f``B"; |
2935 | 682 |
by (Blast_tac 1); |
2912 | 683 |
qed "image_Un"; |
684 |
||
5069 | 685 |
Goal "(z : f``A) = (EX x:A. z = f x)"; |
3960 | 686 |
by (Blast_tac 1); |
687 |
qed "image_iff"; |
|
688 |
||
4523 | 689 |
(*This rewrite rule would confuse users if made default.*) |
5069 | 690 |
Goal "(f``A <= B) = (ALL x:A. f(x): B)"; |
4523 | 691 |
by (Blast_tac 1); |
692 |
qed "image_subset_iff"; |
|
693 |
||
694 |
(*Replaces the three steps subsetI, imageE, hyp_subst_tac, but breaks too |
|
695 |
many existing proofs.*) |
|
5316 | 696 |
val prems = Goal "(!!x. x:A ==> f(x) : B) ==> f``A <= B"; |
4510 | 697 |
by (blast_tac (claset() addIs prems) 1); |
698 |
qed "image_subsetI"; |
|
699 |
||
2912 | 700 |
|
701 |
(*** Range of a function -- just a translation for image! ***) |
|
702 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
703 |
Goal "b=f(x) ==> b : range(f)"; |
2912 | 704 |
by (EVERY1 [etac image_eqI, rtac UNIV_I]); |
705 |
bind_thm ("range_eqI", UNIV_I RSN (2,image_eqI)); |
|
706 |
||
707 |
bind_thm ("rangeI", UNIV_I RS imageI); |
|
708 |
||
5316 | 709 |
val [major,minor] = Goal |
3842 | 710 |
"[| b : range(%x. f(x)); !!x. b=f(x) ==> P |] ==> P"; |
2912 | 711 |
by (rtac (major RS imageE) 1); |
712 |
by (etac minor 1); |
|
713 |
qed "rangeE"; |
|
714 |
||
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
715 |
|
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
716 |
(*** Set reasoning tools ***) |
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
717 |
|
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
718 |
|
3912 | 719 |
(** Rewrite rules for boolean case-splitting: faster than |
4830 | 720 |
addsplits[split_if] |
3912 | 721 |
**) |
722 |
||
4830 | 723 |
bind_thm ("split_if_eq1", read_instantiate [("P", "%x. x = ?b")] split_if); |
724 |
bind_thm ("split_if_eq2", read_instantiate [("P", "%x. ?a = x")] split_if); |
|
3912 | 725 |
|
5237 | 726 |
(*Split ifs on either side of the membership relation. |
727 |
Not for Addsimps -- can cause goals to blow up!*) |
|
4830 | 728 |
bind_thm ("split_if_mem1", |
6394 | 729 |
read_instantiate_sg (Theory.sign_of Set.thy) [("P", "%x. x : ?b")] split_if); |
4830 | 730 |
bind_thm ("split_if_mem2", |
6394 | 731 |
read_instantiate_sg (Theory.sign_of Set.thy) [("P", "%x. ?a : x")] split_if); |
3912 | 732 |
|
4830 | 733 |
val split_ifs = [if_bool_eq_conj, split_if_eq1, split_if_eq2, |
734 |
split_if_mem1, split_if_mem2]; |
|
3912 | 735 |
|
736 |
||
4089 | 737 |
(*Each of these has ALREADY been added to simpset() above.*) |
2024
909153d8318f
Rationalized the rewriting of membership for {} and insert
paulson
parents:
1985
diff
changeset
|
738 |
val mem_simps = [insert_iff, empty_iff, Un_iff, Int_iff, Compl_iff, Diff_iff, |
4159
4aff9b7e5597
UNIV now a constant; UNION1, INTER1 now translations and no longer have
paulson
parents:
4135
diff
changeset
|
739 |
mem_Collect_eq, UN_iff, Union_iff, INT_iff, Inter_iff]; |
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
740 |
|
9041 | 741 |
(*Would like to add these, but the existing code only searches for the |
742 |
outer-level constant, which in this case is just "op :"; we instead need |
|
743 |
to use term-nets to associate patterns with rules. Also, if a rule fails to |
|
744 |
apply, then the formula should be kept. |
|
745 |
[("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]), |
|
746 |
("op Int", [IntD1,IntD2]), |
|
747 |
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] |
|
748 |
*) |
|
749 |
val mksimps_pairs = |
|
750 |
[("Ball",[bspec])] @ mksimps_pairs; |
|
1776
d7e77cb8ce5c
moved mem_simps and the corresponding update of !simpset from Fun.ML to Set.ML,
oheimb
parents:
1762
diff
changeset
|
751 |
|
6291 | 752 |
simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
753 |
|
5256 | 754 |
Addsimps[subset_UNIV, subset_refl]; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
755 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
756 |
|
8001 | 757 |
(*** The 'proper subset' relation (<) ***) |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
758 |
|
5069 | 759 |
Goalw [psubset_def] "!!A::'a set. [| A <= B; A ~= B |] ==> A<B"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
760 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
761 |
qed "psubsetI"; |
8913 | 762 |
AddSIs [psubsetI]; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
763 |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
764 |
Goalw [psubset_def] "A < insert x B ==> (x ~: A) & A<=B | x:A & A-{x}<B"; |
4477
b3e5857d8d99
New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4469
diff
changeset
|
765 |
by Auto_tac; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2935
diff
changeset
|
766 |
qed "psubset_insertD"; |
4059 | 767 |
|
768 |
bind_thm ("psubset_eq", psubset_def RS meta_eq_to_obj_eq); |
|
6443 | 769 |
|
770 |
bind_thm ("psubset_imp_subset", psubset_eq RS iffD1 RS conjunct1); |
|
771 |
||
772 |
Goal"[| (A::'a set) < B; B <= C |] ==> A < C"; |
|
773 |
by (auto_tac (claset(), simpset() addsimps [psubset_eq])); |
|
774 |
qed "psubset_subset_trans"; |
|
775 |
||
776 |
Goal"[| (A::'a set) <= B; B < C|] ==> A < C"; |
|
777 |
by (auto_tac (claset(), simpset() addsimps [psubset_eq])); |
|
778 |
qed "subset_psubset_trans"; |
|
7717
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
779 |
|
8001 | 780 |
Goalw [psubset_def] "A < B ==> EX b. b : (B - A)"; |
781 |
by (Blast_tac 1); |
|
782 |
qed "psubset_imp_ex_mem"; |
|
783 |
||
7717
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
784 |
|
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
785 |
(* attributes *) |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
786 |
|
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
787 |
local |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
788 |
|
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
789 |
fun gen_rulify_prems x = |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
790 |
Attrib.no_args (Drule.rule_attribute (fn _ => (standard o |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
791 |
rule_by_tactic (REPEAT (ALLGOALS (resolve_tac [allI, ballI, impI])))))) x; |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
792 |
|
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
793 |
in |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
794 |
|
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
795 |
val rulify_prems_attrib_setup = |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
796 |
[Attrib.add_attributes |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
797 |
[("rulify_prems", (gen_rulify_prems, gen_rulify_prems), "put theorem into standard rule form")]]; |
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
798 |
|
e7ecfa617443
Added attribute rulify_prems (useful for modifying premises of introduction
berghofe
parents:
7658
diff
changeset
|
799 |
end; |