author | paulson |
Fri, 29 Nov 1996 18:03:21 +0100 | |
changeset 2284 | 80ebd1a213fd |
parent 2031 | 03a843f0f447 |
child 2922 | 580647a879cf |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: HOL/Finite.thy |
923 | 2 |
ID: $Id$ |
1531 | 3 |
Author: Lawrence C Paulson & Tobias Nipkow |
4 |
Copyright 1995 University of Cambridge & TU Muenchen |
|
923 | 5 |
|
1531 | 6 |
Finite sets and their cardinality |
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*) |
8 |
||
9 |
open Finite; |
|
10 |
||
1548 | 11 |
section "The finite powerset operator -- Fin"; |
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|
923 | 13 |
goalw Finite.thy Fin.defs "!!A B. A<=B ==> Fin(A) <= Fin(B)"; |
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by (rtac lfp_mono 1); |
923 | 15 |
by (REPEAT (ares_tac basic_monos 1)); |
16 |
qed "Fin_mono"; |
|
17 |
||
18 |
goalw Finite.thy Fin.defs "Fin(A) <= Pow(A)"; |
|
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19 |
by (fast_tac (!claset addSIs [lfp_lowerbound]) 1); |
923 | 20 |
qed "Fin_subset_Pow"; |
21 |
||
22 |
(* A : Fin(B) ==> A <= B *) |
|
23 |
val FinD = Fin_subset_Pow RS subsetD RS PowD; |
|
24 |
||
25 |
(*Discharging ~ x:y entails extra work*) |
|
26 |
val major::prems = goal Finite.thy |
|
27 |
"[| F:Fin(A); P({}); \ |
|
1465 | 28 |
\ !!F x. [| x:A; F:Fin(A); x~:F; P(F) |] ==> P(insert x F) \ |
923 | 29 |
\ |] ==> P(F)"; |
30 |
by (rtac (major RS Fin.induct) 1); |
|
31 |
by (excluded_middle_tac "a:b" 2); |
|
32 |
by (etac (insert_absorb RS ssubst) 3 THEN assume_tac 3); (*backtracking!*) |
|
33 |
by (REPEAT (ares_tac prems 1)); |
|
34 |
qed "Fin_induct"; |
|
35 |
||
1264 | 36 |
Addsimps Fin.intrs; |
923 | 37 |
|
38 |
(*The union of two finite sets is finite*) |
|
39 |
val major::prems = goal Finite.thy |
|
40 |
"[| F: Fin(A); G: Fin(A) |] ==> F Un G : Fin(A)"; |
|
41 |
by (rtac (major RS Fin_induct) 1); |
|
1264 | 42 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps (prems @ [Un_insert_left])))); |
923 | 43 |
qed "Fin_UnI"; |
44 |
||
45 |
(*Every subset of a finite set is finite*) |
|
46 |
val [subs,fin] = goal Finite.thy "[| A<=B; B: Fin(M) |] ==> A: Fin(M)"; |
|
47 |
by (EVERY1 [subgoal_tac "ALL C. C<=B --> C: Fin(M)", |
|
1465 | 48 |
rtac mp, etac spec, |
49 |
rtac subs]); |
|
923 | 50 |
by (rtac (fin RS Fin_induct) 1); |
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by (simp_tac (!simpset addsimps [subset_Un_eq]) 1); |
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|
52 |
by (safe_tac (!claset addSDs [subset_insert_iff RS iffD1])); |
923 | 53 |
by (eres_inst_tac [("t","C")] (insert_Diff RS subst) 2); |
1264 | 54 |
by (ALLGOALS Asm_simp_tac); |
923 | 55 |
qed "Fin_subset"; |
56 |
||
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goal Finite.thy "(F Un G : Fin(A)) = (F: Fin(A) & G: Fin(A))"; |
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|
58 |
by (fast_tac (!claset addIs [Fin_UnI] addDs |
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[Un_upper1 RS Fin_subset, Un_upper2 RS Fin_subset]) 1); |
60 |
qed "subset_Fin"; |
|
61 |
Addsimps[subset_Fin]; |
|
62 |
||
63 |
goal Finite.thy "(insert a A : Fin M) = (a:M & A : Fin M)"; |
|
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by (stac insert_is_Un 1); |
65 |
by (Simp_tac 1); |
|
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|
66 |
by (fast_tac (!claset addSIs Fin.intrs addDs [FinD]) 1); |
1531 | 67 |
qed "insert_Fin"; |
68 |
Addsimps[insert_Fin]; |
|
69 |
||
923 | 70 |
(*The image of a finite set is finite*) |
71 |
val major::_ = goal Finite.thy |
|
72 |
"F: Fin(A) ==> h``F : Fin(h``A)"; |
|
73 |
by (rtac (major RS Fin_induct) 1); |
|
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by (Simp_tac 1); |
75 |
by (asm_simp_tac |
|
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(!simpset addsimps [image_eqI RS Fin.insertI, image_insert] |
2031 | 77 |
delsimps [insert_Fin]) 1); |
923 | 78 |
qed "Fin_imageI"; |
79 |
||
80 |
val major::prems = goal Finite.thy |
|
1465 | 81 |
"[| c: Fin(A); b: Fin(A); \ |
82 |
\ P(b); \ |
|
923 | 83 |
\ !!(x::'a) y. [| x:A; y: Fin(A); x:y; P(y) |] ==> P(y-{x}) \ |
84 |
\ |] ==> c<=b --> P(b-c)"; |
|
85 |
by (rtac (major RS Fin_induct) 1); |
|
2031 | 86 |
by (stac Diff_insert 2); |
923 | 87 |
by (ALLGOALS (asm_simp_tac |
1264 | 88 |
(!simpset addsimps (prems@[Diff_subset RS Fin_subset])))); |
1531 | 89 |
val lemma = result(); |
923 | 90 |
|
91 |
val prems = goal Finite.thy |
|
1465 | 92 |
"[| b: Fin(A); \ |
93 |
\ P(b); \ |
|
923 | 94 |
\ !!x y. [| x:A; y: Fin(A); x:y; P(y) |] ==> P(y-{x}) \ |
95 |
\ |] ==> P({})"; |
|
96 |
by (rtac (Diff_cancel RS subst) 1); |
|
1531 | 97 |
by (rtac (lemma RS mp) 1); |
923 | 98 |
by (REPEAT (ares_tac (subset_refl::prems) 1)); |
99 |
qed "Fin_empty_induct"; |
|
1531 | 100 |
|
101 |
||
1548 | 102 |
section "The predicate 'finite'"; |
1531 | 103 |
|
104 |
val major::prems = goalw Finite.thy [finite_def] |
|
105 |
"[| finite F; P({}); \ |
|
106 |
\ !!F x. [| finite F; x~:F; P(F) |] ==> P(insert x F) \ |
|
107 |
\ |] ==> P(F)"; |
|
108 |
by (rtac (major RS Fin_induct) 1); |
|
109 |
by (REPEAT (ares_tac prems 1)); |
|
110 |
qed "finite_induct"; |
|
111 |
||
112 |
||
113 |
goalw Finite.thy [finite_def] "finite {}"; |
|
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by (Simp_tac 1); |
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qed "finite_emptyI"; |
116 |
Addsimps [finite_emptyI]; |
|
117 |
||
118 |
goalw Finite.thy [finite_def] "!!A. finite A ==> finite(insert a A)"; |
|
1553 | 119 |
by (Asm_simp_tac 1); |
1531 | 120 |
qed "finite_insertI"; |
121 |
||
122 |
(*The union of two finite sets is finite*) |
|
123 |
goalw Finite.thy [finite_def] |
|
124 |
"!!F. [| finite F; finite G |] ==> finite(F Un G)"; |
|
1553 | 125 |
by (Asm_simp_tac 1); |
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qed "finite_UnI"; |
127 |
||
128 |
goalw Finite.thy [finite_def] "!!A. [| A<=B; finite B |] ==> finite A"; |
|
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by (etac Fin_subset 1); |
130 |
by (assume_tac 1); |
|
1531 | 131 |
qed "finite_subset"; |
132 |
||
133 |
goalw Finite.thy [finite_def] "finite(F Un G) = (finite F & finite G)"; |
|
1553 | 134 |
by (Simp_tac 1); |
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qed "subset_finite"; |
136 |
Addsimps[subset_finite]; |
|
137 |
||
138 |
goalw Finite.thy [finite_def] "finite(insert a A) = finite(A)"; |
|
1553 | 139 |
by (Simp_tac 1); |
1531 | 140 |
qed "insert_finite"; |
141 |
Addsimps[insert_finite]; |
|
142 |
||
1618 | 143 |
(* finite B ==> finite (B - Ba) *) |
144 |
bind_thm ("finite_Diff", Diff_subset RS finite_subset); |
|
1531 | 145 |
Addsimps [finite_Diff]; |
146 |
||
147 |
(*The image of a finite set is finite*) |
|
148 |
goal Finite.thy "!!F. finite F ==> finite(h``F)"; |
|
1553 | 149 |
by (etac finite_induct 1); |
150 |
by (ALLGOALS Asm_simp_tac); |
|
1531 | 151 |
qed "finite_imageI"; |
152 |
||
153 |
val major::prems = goalw Finite.thy [finite_def] |
|
154 |
"[| finite A; \ |
|
155 |
\ P(A); \ |
|
156 |
\ !!a A. [| finite A; a:A; P(A) |] ==> P(A-{a}) \ |
|
157 |
\ |] ==> P({})"; |
|
158 |
by (rtac (major RS Fin_empty_induct) 1); |
|
159 |
by (REPEAT (ares_tac (subset_refl::prems) 1)); |
|
160 |
qed "finite_empty_induct"; |
|
161 |
||
162 |
||
1548 | 163 |
section "Finite cardinality -- 'card'"; |
1531 | 164 |
|
165 |
goal Set.thy "{f i |i. P i | i=n} = insert (f n) {f i|i. P i}"; |
|
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|
166 |
by (Fast_tac 1); |
1531 | 167 |
val Collect_conv_insert = result(); |
168 |
||
169 |
goalw Finite.thy [card_def] "card {} = 0"; |
|
1553 | 170 |
by (rtac Least_equality 1); |
171 |
by (ALLGOALS Asm_full_simp_tac); |
|
1531 | 172 |
qed "card_empty"; |
173 |
Addsimps [card_empty]; |
|
174 |
||
175 |
val [major] = goal Finite.thy |
|
176 |
"finite A ==> ? (n::nat) f. A = {f i |i. i<n}"; |
|
1553 | 177 |
by (rtac (major RS finite_induct) 1); |
178 |
by (res_inst_tac [("x","0")] exI 1); |
|
179 |
by (Simp_tac 1); |
|
180 |
by (etac exE 1); |
|
181 |
by (etac exE 1); |
|
182 |
by (hyp_subst_tac 1); |
|
183 |
by (res_inst_tac [("x","Suc n")] exI 1); |
|
184 |
by (res_inst_tac [("x","%i. if i<n then f i else x")] exI 1); |
|
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by (asm_simp_tac (!simpset addsimps [Collect_conv_insert, less_Suc_eq] |
1548 | 186 |
addcongs [rev_conj_cong]) 1); |
1531 | 187 |
qed "finite_has_card"; |
188 |
||
189 |
goal Finite.thy |
|
190 |
"!!A.[| x ~: A; insert x A = {f i|i.i<n} |] ==> \ |
|
191 |
\ ? m::nat. m<n & (? g. A = {g i|i.i<m})"; |
|
1553 | 192 |
by (res_inst_tac [("n","n")] natE 1); |
193 |
by (hyp_subst_tac 1); |
|
194 |
by (Asm_full_simp_tac 1); |
|
195 |
by (rename_tac "m" 1); |
|
196 |
by (hyp_subst_tac 1); |
|
197 |
by (case_tac "? a. a:A" 1); |
|
198 |
by (res_inst_tac [("x","0")] exI 2); |
|
199 |
by (Simp_tac 2); |
|
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diff
changeset
|
200 |
by (Fast_tac 2); |
1553 | 201 |
by (etac exE 1); |
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by (simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
1553 | 203 |
by (rtac exI 1); |
1782 | 204 |
by (rtac (refl RS disjI2 RS conjI) 1); |
1553 | 205 |
by (etac equalityE 1); |
206 |
by (asm_full_simp_tac |
|
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(!simpset addsimps [subset_insert,Collect_conv_insert, less_Suc_eq]) 1); |
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changeset
|
208 |
by (SELECT_GOAL(safe_tac (!claset))1); |
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by (Asm_full_simp_tac 1); |
210 |
by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1); |
|
1786
8a31d85d27b8
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berghofe
parents:
1782
diff
changeset
|
211 |
by (SELECT_GOAL(safe_tac (!claset))1); |
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by (subgoal_tac "x ~= f m" 1); |
1760
6f41a494f3b1
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berghofe
parents:
1660
diff
changeset
|
213 |
by (Fast_tac 2); |
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by (subgoal_tac "? k. f k = x & k<m" 1); |
1786
8a31d85d27b8
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berghofe
parents:
1782
diff
changeset
|
215 |
by (best_tac (!claset) 2); |
8a31d85d27b8
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1782
diff
changeset
|
216 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 217 |
by (res_inst_tac [("x","k")] exI 1); |
218 |
by (Asm_simp_tac 1); |
|
219 |
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1); |
|
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
220 |
by (best_tac (!claset) 1); |
1531 | 221 |
bd sym 1; |
1553 | 222 |
by (rotate_tac ~1 1); |
223 |
by (Asm_full_simp_tac 1); |
|
224 |
by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1); |
|
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
225 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 226 |
by (subgoal_tac "x ~= f m" 1); |
1760
6f41a494f3b1
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1660
diff
changeset
|
227 |
by (Fast_tac 2); |
1553 | 228 |
by (subgoal_tac "? k. f k = x & k<m" 1); |
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
229 |
by (best_tac (!claset) 2); |
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
230 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 231 |
by (res_inst_tac [("x","k")] exI 1); |
232 |
by (Asm_simp_tac 1); |
|
233 |
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1); |
|
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
234 |
by (best_tac (!claset) 1); |
1553 | 235 |
by (res_inst_tac [("x","%j. if f j = f i then f m else f j")] exI 1); |
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
236 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 237 |
by (subgoal_tac "x ~= f i" 1); |
1760
6f41a494f3b1
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1660
diff
changeset
|
238 |
by (Fast_tac 2); |
1553 | 239 |
by (case_tac "x = f m" 1); |
240 |
by (res_inst_tac [("x","i")] exI 1); |
|
241 |
by (Asm_simp_tac 1); |
|
242 |
by (subgoal_tac "? k. f k = x & k<m" 1); |
|
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
243 |
by (best_tac (!claset) 2); |
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
244 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 245 |
by (res_inst_tac [("x","k")] exI 1); |
246 |
by (Asm_simp_tac 1); |
|
247 |
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1); |
|
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
248 |
by (best_tac (!claset) 1); |
1531 | 249 |
val lemma = result(); |
250 |
||
251 |
goal Finite.thy "!!A. [| finite A; x ~: A |] ==> \ |
|
252 |
\ (LEAST n. ? f. insert x A = {f i|i.i<n}) = Suc(LEAST n. ? f. A={f i|i.i<n})"; |
|
1553 | 253 |
by (rtac Least_equality 1); |
1531 | 254 |
bd finite_has_card 1; |
255 |
be exE 1; |
|
1553 | 256 |
by (dres_inst_tac [("P","%n.? f. A={f i|i.i<n}")] LeastI 1); |
1531 | 257 |
be exE 1; |
1553 | 258 |
by (res_inst_tac |
1531 | 259 |
[("x","%i. if i<(LEAST n. ? f. A={f i |i. i < n}) then f i else x")] exI 1); |
1553 | 260 |
by (simp_tac |
1660 | 261 |
(!simpset addsimps [Collect_conv_insert, less_Suc_eq] |
2031 | 262 |
addcongs [rev_conj_cong]) 1); |
1531 | 263 |
be subst 1; |
264 |
br refl 1; |
|
1553 | 265 |
by (rtac notI 1); |
266 |
by (etac exE 1); |
|
267 |
by (dtac lemma 1); |
|
1531 | 268 |
ba 1; |
1553 | 269 |
by (etac exE 1); |
270 |
by (etac conjE 1); |
|
271 |
by (dres_inst_tac [("P","%x. ? g. A = {g i |i. i < x}")] Least_le 1); |
|
272 |
by (dtac le_less_trans 1 THEN atac 1); |
|
1660 | 273 |
by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
1553 | 274 |
by (etac disjE 1); |
275 |
by (etac less_asym 1 THEN atac 1); |
|
276 |
by (hyp_subst_tac 1); |
|
277 |
by (Asm_full_simp_tac 1); |
|
1531 | 278 |
val lemma = result(); |
279 |
||
280 |
goalw Finite.thy [card_def] |
|
281 |
"!!A. [| finite A; x ~: A |] ==> card(insert x A) = Suc(card A)"; |
|
1553 | 282 |
by (etac lemma 1); |
283 |
by (assume_tac 1); |
|
1531 | 284 |
qed "card_insert_disjoint"; |
285 |
||
1618 | 286 |
goal Finite.thy "!!A. [| finite A; x: A |] ==> Suc(card(A-{x})) = card A"; |
287 |
by (res_inst_tac [("t", "A")] (insert_Diff RS subst) 1); |
|
288 |
by (assume_tac 1); |
|
289 |
by (asm_simp_tac (!simpset addsimps [card_insert_disjoint]) 1); |
|
290 |
qed "card_Suc_Diff"; |
|
291 |
||
292 |
goal Finite.thy "!!A. [| finite A; x: A |] ==> card(A-{x}) < card A"; |
|
2031 | 293 |
by (rtac Suc_less_SucD 1); |
1618 | 294 |
by (asm_simp_tac (!simpset addsimps [card_Suc_Diff]) 1); |
295 |
qed "card_Diff"; |
|
296 |
||
1531 | 297 |
val [major] = goal Finite.thy |
298 |
"finite A ==> card(insert x A) = Suc(card(A-{x}))"; |
|
1553 | 299 |
by (case_tac "x:A" 1); |
300 |
by (asm_simp_tac (!simpset addsimps [insert_absorb]) 1); |
|
301 |
by (dtac mk_disjoint_insert 1); |
|
302 |
by (etac exE 1); |
|
303 |
by (Asm_simp_tac 1); |
|
304 |
by (rtac card_insert_disjoint 1); |
|
305 |
by (rtac (major RSN (2,finite_subset)) 1); |
|
1760
6f41a494f3b1
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1660
diff
changeset
|
306 |
by (Fast_tac 1); |
6f41a494f3b1
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1660
diff
changeset
|
307 |
by (Fast_tac 1); |
1553 | 308 |
by (asm_simp_tac (!simpset addsimps [major RS card_insert_disjoint]) 1); |
1531 | 309 |
qed "card_insert"; |
310 |
Addsimps [card_insert]; |
|
311 |
||
312 |
||
313 |
goal Finite.thy "!!A. finite A ==> !B. B <= A --> card(B) <= card(A)"; |
|
1553 | 314 |
by (etac finite_induct 1); |
315 |
by (Simp_tac 1); |
|
316 |
by (strip_tac 1); |
|
317 |
by (case_tac "x:B" 1); |
|
318 |
by (dtac mk_disjoint_insert 1); |
|
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
319 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 320 |
by (rotate_tac ~1 1); |
321 |
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1); |
|
322 |
by (rotate_tac ~1 1); |
|
323 |
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1); |
|
1531 | 324 |
qed_spec_mp "card_mono"; |