author | nipkow |
Tue, 06 Oct 1998 14:39:53 +0200 | |
changeset 5616 | 497eeeace3fc |
parent 5537 | c2bd39a2c0ee |
child 5626 | f67c34721486 |
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 |
923 | 7 |
*) |
8 |
||
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9 |
section "finite"; |
1531 | 10 |
|
923 | 11 |
(*Discharging ~ x:y entails extra work*) |
5316 | 12 |
val major::prems = Goal |
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"[| finite F; P({}); \ |
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14 |
\ !!F x. [| finite F; x ~: F; P(F) |] ==> P(insert x F) \ |
923 | 15 |
\ |] ==> P(F)"; |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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16 |
by (rtac (major RS Finites.induct) 1); |
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17 |
by (excluded_middle_tac "a:A" 2); |
923 | 18 |
by (etac (insert_absorb RS ssubst) 3 THEN assume_tac 3); (*backtracking!*) |
19 |
by (REPEAT (ares_tac prems 1)); |
|
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20 |
qed "finite_induct"; |
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21 |
|
5316 | 22 |
val major::subs::prems = Goal |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
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|
23 |
"[| finite F; F <= A; \ |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
24 |
\ P({}); \ |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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25 |
\ !!F a. [| finite F; a:A; a ~: F; P(F) |] ==> P(insert a F) \ |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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26 |
\ |] ==> P(F)"; |
4386 | 27 |
by (rtac (subs RS rev_mp) 1); |
28 |
by (rtac (major RS finite_induct) 1); |
|
29 |
by (ALLGOALS (blast_tac (claset() addIs prems))); |
|
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30 |
qed "finite_subset_induct"; |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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31 |
|
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
32 |
Addsimps Finites.intrs; |
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|
33 |
AddSIs Finites.intrs; |
923 | 34 |
|
35 |
(*The union of two finite sets is finite*) |
|
5316 | 36 |
Goal "[| finite F; finite G |] ==> finite(F Un G)"; |
37 |
by (etac finite_induct 1); |
|
38 |
by (ALLGOALS Asm_simp_tac); |
|
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39 |
qed "finite_UnI"; |
923 | 40 |
|
41 |
(*Every subset of a finite set is finite*) |
|
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42 |
Goal "finite B ==> ALL A. A<=B --> finite A"; |
4304 | 43 |
by (etac finite_induct 1); |
44 |
by (Simp_tac 1); |
|
4089 | 45 |
by (safe_tac (claset() addSDs [subset_insert_iff RS iffD1])); |
4304 | 46 |
by (eres_inst_tac [("t","A")] (insert_Diff RS subst) 2); |
1264 | 47 |
by (ALLGOALS Asm_simp_tac); |
4304 | 48 |
val lemma = result(); |
49 |
||
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50 |
Goal "[| A<=B; finite B |] ==> finite A"; |
4423 | 51 |
by (dtac lemma 1); |
4304 | 52 |
by (Blast_tac 1); |
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|
53 |
qed "finite_subset"; |
923 | 54 |
|
5069 | 55 |
Goal "finite(F Un G) = (finite F & finite G)"; |
4304 | 56 |
by (blast_tac (claset() |
5413 | 57 |
addIs [read_instantiate [("B", "?X Un ?Y")] finite_subset, |
4304 | 58 |
finite_UnI]) 1); |
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59 |
qed "finite_Un"; |
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|
60 |
AddIffs[finite_Un]; |
1531 | 61 |
|
5413 | 62 |
Goal "finite F ==> finite(F Int G)"; |
63 |
by (blast_tac (claset() addIs [finite_subset]) 1); |
|
64 |
qed "finite_Int1"; |
|
65 |
||
66 |
Goal "finite G ==> finite(F Int G)"; |
|
67 |
by (blast_tac (claset() addIs [finite_subset]) 1); |
|
68 |
qed "finite_Int2"; |
|
69 |
||
70 |
Addsimps[finite_Int1, finite_Int2]; |
|
71 |
AddIs[finite_Int1, finite_Int2]; |
|
72 |
||
73 |
||
5069 | 74 |
Goal "finite(insert a A) = finite A"; |
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by (stac insert_is_Un 1); |
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76 |
by (simp_tac (HOL_ss addsimps [finite_Un]) 1); |
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77 |
by (Blast_tac 1); |
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parents:
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78 |
qed "finite_insert"; |
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79 |
Addsimps[finite_insert]; |
1531 | 80 |
|
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81 |
(*The image of a finite set is finite *) |
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82 |
Goal "finite F ==> finite(h``F)"; |
3413
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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changeset
|
83 |
by (etac finite_induct 1); |
1264 | 84 |
by (Simp_tac 1); |
3413
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parents:
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|
85 |
by (Asm_simp_tac 1); |
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parents:
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|
86 |
qed "finite_imageI"; |
923 | 87 |
|
5316 | 88 |
val major::prems = Goal |
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89 |
"[| finite c; finite b; \ |
1465 | 90 |
\ P(b); \ |
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|
91 |
\ !!x y. [| finite y; x:y; P(y) |] ==> P(y-{x}) \ |
923 | 92 |
\ |] ==> c<=b --> P(b-c)"; |
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|
93 |
by (rtac (major RS finite_induct) 1); |
2031 | 94 |
by (stac Diff_insert 2); |
923 | 95 |
by (ALLGOALS (asm_simp_tac |
5537 | 96 |
(simpset() addsimps prems@[Diff_subset RS finite_subset]))); |
1531 | 97 |
val lemma = result(); |
923 | 98 |
|
5316 | 99 |
val prems = Goal |
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|
100 |
"[| finite A; \ |
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|
101 |
\ P(A); \ |
c1f63cc3a768
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parents:
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|
102 |
\ !!a A. [| finite A; a:A; P(A) |] ==> P(A-{a}) \ |
923 | 103 |
\ |] ==> P({})"; |
104 |
by (rtac (Diff_cancel RS subst) 1); |
|
1531 | 105 |
by (rtac (lemma RS mp) 1); |
923 | 106 |
by (REPEAT (ares_tac (subset_refl::prems) 1)); |
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107 |
qed "finite_empty_induct"; |
1531 | 108 |
|
109 |
||
1618 | 110 |
(* finite B ==> finite (B - Ba) *) |
111 |
bind_thm ("finite_Diff", Diff_subset RS finite_subset); |
|
1531 | 112 |
Addsimps [finite_Diff]; |
113 |
||
5069 | 114 |
Goal "finite(A-{a}) = finite(A)"; |
3368 | 115 |
by (case_tac "a:A" 1); |
3457 | 116 |
by (rtac (finite_insert RS sym RS trans) 1); |
3368 | 117 |
by (stac insert_Diff 1); |
118 |
by (ALLGOALS Asm_simp_tac); |
|
119 |
qed "finite_Diff_singleton"; |
|
120 |
AddIffs [finite_Diff_singleton]; |
|
121 |
||
4059 | 122 |
(*Lemma for proving finite_imageD*) |
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123 |
Goal "finite B ==> !A. f``A = B --> inj_on f A --> finite A"; |
1553 | 124 |
by (etac finite_induct 1); |
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125 |
by (ALLGOALS Asm_simp_tac); |
3708 | 126 |
by (Clarify_tac 1); |
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|
127 |
by (subgoal_tac "EX y:A. f y = x & F = f``(A-{y})" 1); |
3708 | 128 |
by (Clarify_tac 1); |
4830 | 129 |
by (full_simp_tac (simpset() addsimps [inj_on_def]) 1); |
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130 |
by (Blast_tac 1); |
3368 | 131 |
by (thin_tac "ALL A. ?PP(A)" 1); |
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132 |
by (forward_tac [[equalityD2, insertI1] MRS subsetD] 1); |
3708 | 133 |
by (Clarify_tac 1); |
3368 | 134 |
by (res_inst_tac [("x","xa")] bexI 1); |
4059 | 135 |
by (ALLGOALS |
4830 | 136 |
(asm_full_simp_tac (simpset() addsimps [inj_on_image_set_diff]))); |
3368 | 137 |
val lemma = result(); |
138 |
||
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139 |
Goal "[| finite(f``A); inj_on f A |] ==> finite A"; |
3457 | 140 |
by (dtac lemma 1); |
3368 | 141 |
by (Blast_tac 1); |
142 |
qed "finite_imageD"; |
|
143 |
||
4014 | 144 |
(** The finite UNION of finite sets **) |
145 |
||
5316 | 146 |
Goal "finite A ==> (!a:A. finite(B a)) --> finite(UN a:A. B a)"; |
147 |
by (etac finite_induct 1); |
|
4153 | 148 |
by (ALLGOALS Asm_simp_tac); |
4014 | 149 |
bind_thm("finite_UnionI", ballI RSN (2, result() RS mp)); |
150 |
Addsimps [finite_UnionI]; |
|
151 |
||
152 |
(** Sigma of finite sets **) |
|
153 |
||
5069 | 154 |
Goalw [Sigma_def] |
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155 |
"[| finite A; !a:A. finite(B a) |] ==> finite(SIGMA a:A. B a)"; |
4153 | 156 |
by (blast_tac (claset() addSIs [finite_UnionI]) 1); |
4014 | 157 |
bind_thm("finite_SigmaI", ballI RSN (2,result())); |
158 |
Addsimps [finite_SigmaI]; |
|
3368 | 159 |
|
160 |
(** The powerset of a finite set **) |
|
161 |
||
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|
162 |
Goal "finite(Pow A) ==> finite A"; |
3368 | 163 |
by (subgoal_tac "finite ((%x.{x})``A)" 1); |
3457 | 164 |
by (rtac finite_subset 2); |
165 |
by (assume_tac 3); |
|
3368 | 166 |
by (ALLGOALS |
4830 | 167 |
(fast_tac (claset() addSDs [rewrite_rule [inj_on_def] finite_imageD]))); |
3368 | 168 |
val lemma = result(); |
169 |
||
5069 | 170 |
Goal "finite(Pow A) = finite A"; |
3457 | 171 |
by (rtac iffI 1); |
172 |
by (etac lemma 1); |
|
3368 | 173 |
(*Opposite inclusion: finite A ==> finite (Pow A) *) |
3340 | 174 |
by (etac finite_induct 1); |
175 |
by (ALLGOALS |
|
176 |
(asm_simp_tac |
|
4089 | 177 |
(simpset() addsimps [finite_UnI, finite_imageI, Pow_insert]))); |
3368 | 178 |
qed "finite_Pow_iff"; |
179 |
AddIffs [finite_Pow_iff]; |
|
3340 | 180 |
|
5069 | 181 |
Goal "finite(r^-1) = finite r"; |
3457 | 182 |
by (subgoal_tac "r^-1 = (%(x,y).(y,x))``r" 1); |
183 |
by (Asm_simp_tac 1); |
|
184 |
by (rtac iffI 1); |
|
4830 | 185 |
by (etac (rewrite_rule [inj_on_def] finite_imageD) 1); |
186 |
by (simp_tac (simpset() addsplits [split_split]) 1); |
|
3457 | 187 |
by (etac finite_imageI 1); |
4746 | 188 |
by (simp_tac (simpset() addsimps [converse_def,image_def]) 1); |
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|
189 |
by Auto_tac; |
5516 | 190 |
by (rtac bexI 1); |
191 |
by (assume_tac 2); |
|
4763 | 192 |
by (Simp_tac 1); |
4746 | 193 |
qed "finite_converse"; |
194 |
AddIffs [finite_converse]; |
|
1531 | 195 |
|
1548 | 196 |
section "Finite cardinality -- 'card'"; |
1531 | 197 |
|
5316 | 198 |
Goal "{f i |i. (P i | i=n)} = insert (f n) {f i|i. P i}"; |
2922 | 199 |
by (Blast_tac 1); |
1531 | 200 |
val Collect_conv_insert = result(); |
201 |
||
5069 | 202 |
Goalw [card_def] "card {} = 0"; |
1553 | 203 |
by (rtac Least_equality 1); |
204 |
by (ALLGOALS Asm_full_simp_tac); |
|
1531 | 205 |
qed "card_empty"; |
206 |
Addsimps [card_empty]; |
|
207 |
||
5316 | 208 |
Goal "finite A ==> ? (n::nat) f. A = {f i |i. i<n}"; |
209 |
by (etac finite_induct 1); |
|
1553 | 210 |
by (res_inst_tac [("x","0")] exI 1); |
211 |
by (Simp_tac 1); |
|
212 |
by (etac exE 1); |
|
213 |
by (etac exE 1); |
|
214 |
by (hyp_subst_tac 1); |
|
215 |
by (res_inst_tac [("x","Suc n")] exI 1); |
|
216 |
by (res_inst_tac [("x","%i. if i<n then f i else x")] exI 1); |
|
4089 | 217 |
by (asm_simp_tac (simpset() addsimps [Collect_conv_insert, less_Suc_eq] |
1548 | 218 |
addcongs [rev_conj_cong]) 1); |
1531 | 219 |
qed "finite_has_card"; |
220 |
||
5278 | 221 |
Goal "[| x ~: A; insert x A = {f i|i. i<n} |] \ |
222 |
\ ==> ? m::nat. m<n & (? g. A = {g i|i. i<m})"; |
|
5183 | 223 |
by (exhaust_tac "n" 1); |
1553 | 224 |
by (hyp_subst_tac 1); |
225 |
by (Asm_full_simp_tac 1); |
|
226 |
by (rename_tac "m" 1); |
|
227 |
by (hyp_subst_tac 1); |
|
228 |
by (case_tac "? a. a:A" 1); |
|
229 |
by (res_inst_tac [("x","0")] exI 2); |
|
230 |
by (Simp_tac 2); |
|
2922 | 231 |
by (Blast_tac 2); |
1553 | 232 |
by (etac exE 1); |
4089 | 233 |
by (simp_tac (simpset() addsimps [less_Suc_eq]) 1); |
1553 | 234 |
by (rtac exI 1); |
1782 | 235 |
by (rtac (refl RS disjI2 RS conjI) 1); |
1553 | 236 |
by (etac equalityE 1); |
237 |
by (asm_full_simp_tac |
|
4089 | 238 |
(simpset() addsimps [subset_insert,Collect_conv_insert, less_Suc_eq]) 1); |
4153 | 239 |
by Safe_tac; |
1553 | 240 |
by (Asm_full_simp_tac 1); |
241 |
by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1); |
|
4153 | 242 |
by (SELECT_GOAL Safe_tac 1); |
1553 | 243 |
by (subgoal_tac "x ~= f m" 1); |
2922 | 244 |
by (Blast_tac 2); |
1553 | 245 |
by (subgoal_tac "? k. f k = x & k<m" 1); |
2922 | 246 |
by (Blast_tac 2); |
4153 | 247 |
by (SELECT_GOAL Safe_tac 1); |
1553 | 248 |
by (res_inst_tac [("x","k")] exI 1); |
249 |
by (Asm_simp_tac 1); |
|
4686 | 250 |
by (Simp_tac 1); |
2922 | 251 |
by (Blast_tac 1); |
3457 | 252 |
by (dtac sym 1); |
1553 | 253 |
by (rotate_tac ~1 1); |
254 |
by (Asm_full_simp_tac 1); |
|
255 |
by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1); |
|
4153 | 256 |
by (SELECT_GOAL Safe_tac 1); |
1553 | 257 |
by (subgoal_tac "x ~= f m" 1); |
2922 | 258 |
by (Blast_tac 2); |
1553 | 259 |
by (subgoal_tac "? k. f k = x & k<m" 1); |
2922 | 260 |
by (Blast_tac 2); |
4153 | 261 |
by (SELECT_GOAL Safe_tac 1); |
1553 | 262 |
by (res_inst_tac [("x","k")] exI 1); |
263 |
by (Asm_simp_tac 1); |
|
4686 | 264 |
by (Simp_tac 1); |
2922 | 265 |
by (Blast_tac 1); |
1553 | 266 |
by (res_inst_tac [("x","%j. if f j = f i then f m else f j")] exI 1); |
4153 | 267 |
by (SELECT_GOAL Safe_tac 1); |
1553 | 268 |
by (subgoal_tac "x ~= f i" 1); |
2922 | 269 |
by (Blast_tac 2); |
1553 | 270 |
by (case_tac "x = f m" 1); |
271 |
by (res_inst_tac [("x","i")] exI 1); |
|
272 |
by (Asm_simp_tac 1); |
|
273 |
by (subgoal_tac "? k. f k = x & k<m" 1); |
|
2922 | 274 |
by (Blast_tac 2); |
4153 | 275 |
by (SELECT_GOAL Safe_tac 1); |
1553 | 276 |
by (res_inst_tac [("x","k")] exI 1); |
277 |
by (Asm_simp_tac 1); |
|
4686 | 278 |
by (Simp_tac 1); |
2922 | 279 |
by (Blast_tac 1); |
1531 | 280 |
val lemma = result(); |
281 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
282 |
Goal "[| finite A; x ~: A |] ==> \ |
3842 | 283 |
\ (LEAST n. ? f. insert x A = {f i|i. i<n}) = Suc(LEAST n. ? f. A={f i|i. i<n})"; |
1553 | 284 |
by (rtac Least_equality 1); |
3457 | 285 |
by (dtac finite_has_card 1); |
286 |
by (etac exE 1); |
|
3842 | 287 |
by (dres_inst_tac [("P","%n.? f. A={f i|i. i<n}")] LeastI 1); |
3457 | 288 |
by (etac exE 1); |
1553 | 289 |
by (res_inst_tac |
1531 | 290 |
[("x","%i. if i<(LEAST n. ? f. A={f i |i. i < n}) then f i else x")] exI 1); |
1553 | 291 |
by (simp_tac |
4089 | 292 |
(simpset() addsimps [Collect_conv_insert, less_Suc_eq] |
2031 | 293 |
addcongs [rev_conj_cong]) 1); |
3457 | 294 |
by (etac subst 1); |
295 |
by (rtac refl 1); |
|
1553 | 296 |
by (rtac notI 1); |
297 |
by (etac exE 1); |
|
298 |
by (dtac lemma 1); |
|
3457 | 299 |
by (assume_tac 1); |
1553 | 300 |
by (etac exE 1); |
301 |
by (etac conjE 1); |
|
302 |
by (dres_inst_tac [("P","%x. ? g. A = {g i |i. i < x}")] Least_le 1); |
|
303 |
by (dtac le_less_trans 1 THEN atac 1); |
|
4089 | 304 |
by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1); |
1553 | 305 |
by (etac disjE 1); |
306 |
by (etac less_asym 1 THEN atac 1); |
|
307 |
by (hyp_subst_tac 1); |
|
308 |
by (Asm_full_simp_tac 1); |
|
1531 | 309 |
val lemma = result(); |
310 |
||
5416
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
311 |
Goalw [card_def] "[| finite A; x ~: A |] ==> card(insert x A) = Suc(card A)"; |
1553 | 312 |
by (etac lemma 1); |
313 |
by (assume_tac 1); |
|
1531 | 314 |
qed "card_insert_disjoint"; |
3352 | 315 |
Addsimps [card_insert_disjoint]; |
316 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
317 |
Goal "finite A ==> card A <= card (insert x A)"; |
4768
c342d63173e9
New theorems card_Diff_le and card_insert_le; tidied
paulson
parents:
4763
diff
changeset
|
318 |
by (case_tac "x: A" 1); |
c342d63173e9
New theorems card_Diff_le and card_insert_le; tidied
paulson
parents:
4763
diff
changeset
|
319 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [insert_absorb]))); |
c342d63173e9
New theorems card_Diff_le and card_insert_le; tidied
paulson
parents:
4763
diff
changeset
|
320 |
qed "card_insert_le"; |
c342d63173e9
New theorems card_Diff_le and card_insert_le; tidied
paulson
parents:
4763
diff
changeset
|
321 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
322 |
Goal "finite A ==> !B. B <= A --> card(B) <= card(A)"; |
3352 | 323 |
by (etac finite_induct 1); |
324 |
by (Simp_tac 1); |
|
3708 | 325 |
by (Clarify_tac 1); |
3352 | 326 |
by (case_tac "x:B" 1); |
3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
diff
changeset
|
327 |
by (dres_inst_tac [("A","B")] mk_disjoint_insert 1); |
5476 | 328 |
by (asm_full_simp_tac (simpset() addsimps [subset_insert_iff]) 2); |
4775 | 329 |
by (fast_tac (claset() addss |
5477 | 330 |
(simpset() addsimps [subset_insert_iff, finite_subset] |
331 |
delsimps [insert_subset])) 1); |
|
3352 | 332 |
qed_spec_mp "card_mono"; |
333 |
||
5416
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
334 |
|
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
335 |
Goal "[| finite A; finite B |] \ |
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
336 |
\ ==> card A + card B = card (A Un B) + card (A Int B)"; |
3352 | 337 |
by (etac finite_induct 1); |
5416
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
338 |
by (Simp_tac 1); |
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
339 |
by (asm_simp_tac (simpset() addsimps [insert_absorb, Int_insert_left]) 1); |
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
340 |
qed "card_Un_Int"; |
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
341 |
|
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
342 |
Goal "[| finite A; finite B; A Int B = {} |] \ |
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
343 |
\ ==> card (A Un B) = card A + card B"; |
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
344 |
by (asm_simp_tac (simpset() addsimps [card_Un_Int]) 1); |
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
345 |
qed "card_Un_disjoint"; |
3352 | 346 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
347 |
Goal "[| finite A; B<=A |] ==> card A - card B = card (A - B)"; |
3352 | 348 |
by (subgoal_tac "(A-B) Un B = A" 1); |
349 |
by (Blast_tac 2); |
|
3457 | 350 |
by (rtac (add_right_cancel RS iffD1) 1); |
351 |
by (rtac (card_Un_disjoint RS subst) 1); |
|
352 |
by (etac ssubst 4); |
|
3352 | 353 |
by (Blast_tac 3); |
354 |
by (ALLGOALS |
|
355 |
(asm_simp_tac |
|
4089 | 356 |
(simpset() addsimps [add_commute, not_less_iff_le, |
5416
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
357 |
add_diff_inverse, card_mono, finite_subset]))); |
3352 | 358 |
qed "card_Diff_subset"; |
1531 | 359 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
360 |
Goal "[| finite A; x: A |] ==> Suc(card(A-{x})) = card A"; |
1618 | 361 |
by (res_inst_tac [("t", "A")] (insert_Diff RS subst) 1); |
362 |
by (assume_tac 1); |
|
3352 | 363 |
by (Asm_simp_tac 1); |
1618 | 364 |
qed "card_Suc_Diff"; |
365 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
366 |
Goal "[| finite A; x: A |] ==> card(A-{x}) < card A"; |
2031 | 367 |
by (rtac Suc_less_SucD 1); |
4089 | 368 |
by (asm_simp_tac (simpset() addsimps [card_Suc_Diff]) 1); |
1618 | 369 |
qed "card_Diff"; |
370 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
371 |
Goal "finite A ==> card(A-{x}) <= card A"; |
4768
c342d63173e9
New theorems card_Diff_le and card_insert_le; tidied
paulson
parents:
4763
diff
changeset
|
372 |
by (case_tac "x: A" 1); |
c342d63173e9
New theorems card_Diff_le and card_insert_le; tidied
paulson
parents:
4763
diff
changeset
|
373 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [card_Diff, less_imp_le]))); |
c342d63173e9
New theorems card_Diff_le and card_insert_le; tidied
paulson
parents:
4763
diff
changeset
|
374 |
qed "card_Diff_le"; |
c342d63173e9
New theorems card_Diff_le and card_insert_le; tidied
paulson
parents:
4763
diff
changeset
|
375 |
|
3389
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
376 |
|
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
377 |
(*** Cardinality of the Powerset ***) |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
378 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
379 |
Goal "finite A ==> card(insert x A) = Suc(card(A-{x}))"; |
1553 | 380 |
by (case_tac "x:A" 1); |
4768
c342d63173e9
New theorems card_Diff_le and card_insert_le; tidied
paulson
parents:
4763
diff
changeset
|
381 |
by (ALLGOALS |
c342d63173e9
New theorems card_Diff_le and card_insert_le; tidied
paulson
parents:
4763
diff
changeset
|
382 |
(asm_simp_tac (simpset() addsimps [card_Suc_Diff, insert_absorb]))); |
1531 | 383 |
qed "card_insert"; |
384 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
385 |
Goal "finite(A) ==> inj_on f A --> card (f `` A) = card A"; |
3340 | 386 |
by (etac finite_induct 1); |
387 |
by (ALLGOALS Asm_simp_tac); |
|
3724 | 388 |
by Safe_tac; |
4830 | 389 |
by (rewtac inj_on_def); |
3340 | 390 |
by (Blast_tac 1); |
391 |
by (stac card_insert_disjoint 1); |
|
392 |
by (etac finite_imageI 1); |
|
393 |
by (Blast_tac 1); |
|
394 |
by (Blast_tac 1); |
|
395 |
qed_spec_mp "card_image"; |
|
396 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
397 |
Goal "finite A ==> card (Pow A) = 2 ^ card A"; |
3389
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
398 |
by (etac finite_induct 1); |
4089 | 399 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [Pow_insert]))); |
3389
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
400 |
by (stac card_Un_disjoint 1); |
4089 | 401 |
by (EVERY (map (blast_tac (claset() addIs [finite_imageI])) [3,2,1])); |
4830 | 402 |
by (subgoal_tac "inj_on (insert x) (Pow F)" 1); |
4089 | 403 |
by (asm_simp_tac (simpset() addsimps [card_image, Pow_insert]) 1); |
4830 | 404 |
by (rewtac inj_on_def); |
4089 | 405 |
by (blast_tac (claset() addSEs [equalityE]) 1); |
3389
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
406 |
qed "card_Pow"; |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
407 |
Addsimps [card_Pow]; |
3340 | 408 |
|
3389
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
409 |
|
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
410 |
(*Proper subsets*) |
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
411 |
Goalw [psubset_def] "finite B ==> !A. A < B --> card(A) < card(B)"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
412 |
by (etac finite_induct 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
413 |
by (Simp_tac 1); |
3708 | 414 |
by (Clarify_tac 1); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
415 |
by (case_tac "x:A" 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
416 |
(*1*) |
3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
diff
changeset
|
417 |
by (dres_inst_tac [("A","A")]mk_disjoint_insert 1); |
4775 | 418 |
by (Clarify_tac 1); |
419 |
by (rotate_tac ~3 1); |
|
420 |
by (asm_full_simp_tac (simpset() addsimps [finite_subset]) 1); |
|
3708 | 421 |
by (Blast_tac 1); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
422 |
(*2*) |
3708 | 423 |
by (eres_inst_tac [("P","?a<?b")] notE 1); |
4775 | 424 |
by (asm_full_simp_tac (simpset() addsimps [subset_insert_iff]) 1); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
425 |
by (case_tac "A=F" 1); |
3708 | 426 |
by (ALLGOALS Asm_simp_tac); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
427 |
qed_spec_mp "psubset_card" ; |
3368 | 428 |
|
429 |
||
3430 | 430 |
(*Relates to equivalence classes. Based on a theorem of F. Kammueller's. |
3368 | 431 |
The "finite C" premise is redundant*) |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
432 |
Goal "finite C ==> finite (Union C) --> \ |
3368 | 433 |
\ (! c : C. k dvd card c) --> \ |
434 |
\ (! c1: C. ! c2: C. c1 ~= c2 --> c1 Int c2 = {}) \ |
|
435 |
\ --> k dvd card(Union C)"; |
|
436 |
by (etac finite_induct 1); |
|
437 |
by (ALLGOALS Asm_simp_tac); |
|
3708 | 438 |
by (Clarify_tac 1); |
3368 | 439 |
by (stac card_Un_disjoint 1); |
440 |
by (ALLGOALS |
|
4089 | 441 |
(asm_full_simp_tac (simpset() |
3368 | 442 |
addsimps [dvd_add, disjoint_eq_subset_Compl]))); |
443 |
by (thin_tac "!c:F. ?PP(c)" 1); |
|
444 |
by (thin_tac "!c:F. ?PP(c) & ?QQ(c)" 1); |
|
3708 | 445 |
by (Clarify_tac 1); |
3368 | 446 |
by (ball_tac 1); |
447 |
by (Blast_tac 1); |
|
448 |
qed_spec_mp "dvd_partition"; |
|
449 |
||
5616 | 450 |
|
451 |
(*** foldSet ***) |
|
452 |
||
453 |
val empty_foldSetE = foldSet.mk_cases [] "({}, x) : foldSet f e"; |
|
454 |
||
455 |
AddSEs [empty_foldSetE]; |
|
456 |
AddIs foldSet.intrs; |
|
457 |
||
458 |
Goal "[| (A-{x},y) : foldSet f e; x: A |] ==> (A, f x y) : foldSet f e"; |
|
459 |
by (etac (insert_Diff RS subst) 1 THEN resolve_tac foldSet.intrs 1); |
|
460 |
by Auto_tac; |
|
461 |
qed "Diff_foldSet"; |
|
462 |
||
463 |
Goal "(A, x) : foldSet f e ==> finite(A)"; |
|
464 |
by (eresolve_tac [foldSet.induct] 1); |
|
465 |
by Auto_tac; |
|
466 |
qed "foldSet_imp_finite"; |
|
467 |
||
468 |
Addsimps [foldSet_imp_finite]; |
|
469 |
||
470 |
||
471 |
Goal "finite(A) ==> EX x. (A, x) : foldSet f e"; |
|
472 |
by (etac finite_induct 1); |
|
473 |
by Auto_tac; |
|
474 |
qed "finite_imp_foldSet"; |
|
475 |
||
476 |
||
477 |
Open_locale "LC"; |
|
478 |
||
479 |
(*Strip meta-quantifiers: perhaps the locale should do this?*) |
|
480 |
val f_lcomm = forall_elim_vars 0 (thm "lcomm"); |
|
481 |
||
482 |
||
483 |
Goal "ALL A x. card(A) < n --> (A, x) : foldSet f e --> \ |
|
484 |
\ (ALL y. (A, y) : foldSet f e --> y=x)"; |
|
485 |
by (induct_tac "n" 1); |
|
486 |
by (auto_tac (claset(), simpset() addsimps [less_Suc_eq])); |
|
487 |
by (etac foldSet.elim 1); |
|
488 |
by (Blast_tac 1); |
|
489 |
by (etac foldSet.elim 1); |
|
490 |
by (Blast_tac 1); |
|
491 |
by (Clarify_tac 1); |
|
492 |
(*force simplification of "card A < card (insert ...)"*) |
|
493 |
by (etac rev_mp 1); |
|
494 |
by (asm_simp_tac (simpset() addsimps [less_Suc_eq_le]) 1); |
|
495 |
by (rtac impI 1); |
|
496 |
(** LEVEL 10 **) |
|
497 |
by (rename_tac "Aa xa ya Ab xb yb" 1); |
|
498 |
by (case_tac "xa=xb" 1); |
|
499 |
by (subgoal_tac "Aa = Ab" 1); |
|
500 |
by (blast_tac (claset() addEs [equalityE]) 2); |
|
501 |
by (Blast_tac 1); |
|
502 |
(*case xa ~= xb*) |
|
503 |
by (subgoal_tac "Aa-{xb} = Ab-{xa} & xb : Aa & xa : Ab" 1); |
|
504 |
by (blast_tac (claset() addEs [equalityE]) 2); |
|
505 |
by (Clarify_tac 1); |
|
506 |
by (subgoal_tac "Aa = insert xb Ab - {xa}" 1); |
|
507 |
by (blast_tac (claset() addEs [equalityE]) 2); |
|
508 |
(** LEVEL 20 **) |
|
509 |
by (subgoal_tac "card Aa <= card Ab" 1); |
|
510 |
by (rtac (Suc_le_mono RS subst) 2); |
|
511 |
by (asm_simp_tac (simpset() addsimps [card_Suc_Diff]) 2); |
|
512 |
by (res_inst_tac [("A1", "Aa-{xb}"), ("f1","f")] |
|
513 |
(finite_imp_foldSet RS exE) 1); |
|
514 |
by (blast_tac (claset() addIs [foldSet_imp_finite, finite_Diff]) 1); |
|
515 |
by (forward_tac [Diff_foldSet] 1 THEN assume_tac 1); |
|
516 |
by (subgoal_tac "ya = f xb x" 1); |
|
517 |
by (Blast_tac 2); |
|
518 |
by (subgoal_tac "(Ab - {xa}, x) : foldSet f e" 1); |
|
519 |
by (Asm_full_simp_tac 2); |
|
520 |
by (subgoal_tac "yb = f xa x" 1); |
|
521 |
by (blast_tac (claset() addDs [Diff_foldSet]) 2); |
|
522 |
by (asm_simp_tac (simpset() addsimps [f_lcomm]) 1); |
|
523 |
val lemma = result(); |
|
524 |
||
525 |
||
526 |
Goal "[| (A, x) : foldSet f e; (A, y) : foldSet f e |] ==> y=x"; |
|
527 |
by (blast_tac (claset() addIs [normalize_thm [RSspec, RSmp] lemma]) 1); |
|
528 |
qed "foldSet_determ"; |
|
529 |
||
530 |
Goalw [fold_def] "(A,y) : foldSet f e ==> fold f e A = y"; |
|
531 |
by (blast_tac (claset() addIs [foldSet_determ]) 1); |
|
532 |
qed "fold_equality"; |
|
533 |
||
534 |
Goalw [fold_def] "fold f e {} = e"; |
|
535 |
by (Blast_tac 1); |
|
536 |
qed "fold_empty"; |
|
537 |
Addsimps [fold_empty]; |
|
538 |
||
539 |
Goal "x ~: A ==> \ |
|
540 |
\ ((insert x A, v) : foldSet f e) = \ |
|
541 |
\ (EX y. (A, y) : foldSet f e & v = f x y)"; |
|
542 |
by Auto_tac; |
|
543 |
by (res_inst_tac [("A1", "A"), ("f1","f")] (finite_imp_foldSet RS exE) 1); |
|
544 |
by (force_tac (claset() addDs [foldSet_imp_finite], simpset()) 1); |
|
545 |
by (blast_tac (claset() addIs [foldSet_determ]) 1); |
|
546 |
val lemma = result(); |
|
547 |
||
548 |
||
549 |
Goalw [fold_def] |
|
550 |
"[| finite A; x ~: A |] ==> fold f e (insert x A) = f x (fold f e A)"; |
|
551 |
by (asm_simp_tac (simpset() addsimps [lemma]) 1); |
|
552 |
by (rtac select_equality 1); |
|
553 |
by (auto_tac (claset() addIs [finite_imp_foldSet], |
|
554 |
simpset() addcongs [conj_cong] |
|
555 |
addsimps [symmetric fold_def, |
|
556 |
fold_equality])); |
|
557 |
qed "fold_insert"; |
|
558 |
||
559 |
Close_locale(); |
|
560 |
||
561 |
Open_locale "ACe"; |
|
562 |
||
563 |
(*Strip meta-quantifiers: perhaps the locale should do this?*) |
|
564 |
val f_ident = forall_elim_vars 0 (thm "ident"); |
|
565 |
val f_commute = forall_elim_vars 0 (thm "commute"); |
|
566 |
val f_assoc = forall_elim_vars 0 (thm "assoc"); |
|
567 |
||
568 |
||
569 |
Goal "f x (f y z) = f y (f x z)"; |
|
570 |
by (rtac (f_commute RS trans) 1); |
|
571 |
by (rtac (f_assoc RS trans) 1); |
|
572 |
by (rtac (f_commute RS arg_cong) 1); |
|
573 |
qed "f_left_commute"; |
|
574 |
||
575 |
val f_ac = [f_assoc, f_commute, f_left_commute]; |
|
576 |
||
577 |
Goal "f e x = x"; |
|
578 |
by (stac f_commute 1); |
|
579 |
by (rtac f_ident 1); |
|
580 |
qed "f_left_ident"; |
|
581 |
||
582 |
val f_idents = [f_left_ident, f_ident]; |
|
583 |
||
584 |
Goal "[| finite A; finite B |] \ |
|
585 |
\ ==> f (fold f e A) (fold f e B) = \ |
|
586 |
\ f (fold f e (A Un B)) (fold f e (A Int B))"; |
|
587 |
by (etac finite_induct 1); |
|
588 |
by (simp_tac (simpset() addsimps f_idents) 1); |
|
589 |
by (asm_simp_tac (simpset() addsimps f_ac @ f_idents @ |
|
590 |
[export fold_insert,insert_absorb, Int_insert_left]) 1); |
|
591 |
qed "fold_Un_Int"; |
|
592 |
||
593 |
Goal "[| finite A; finite B; A Int B = {} |] \ |
|
594 |
\ ==> fold f e (A Un B) = f (fold f e A) (fold f e B)"; |
|
595 |
by (asm_simp_tac (simpset() addsimps fold_Un_Int::f_idents) 1); |
|
596 |
qed "fold_Un_disjoint"; |
|
597 |
||
598 |
Goal |
|
599 |
"[| finite A; finite B |] ==> A Int B = {} --> \ |
|
600 |
\ fold (f o g) e (A Un B) = f (fold (f o g) e A) (fold (f o g) e B)"; |
|
601 |
by (etac finite_induct 1); |
|
602 |
by (simp_tac (simpset() addsimps f_idents) 1); |
|
603 |
by (asm_full_simp_tac (simpset() addsimps f_ac @ f_idents @ |
|
604 |
[export fold_insert,insert_absorb, Int_insert_left]) 1); |
|
605 |
qed "fold_Un_disjoint2"; |
|
606 |
||
607 |
Close_locale(); |
|
608 |
||
609 |
Delrules ([empty_foldSetE] @ foldSet.intrs); |
|
610 |
Delsimps [foldSet_imp_finite]; |
|
611 |
||
612 |
(*** setsum ***) |
|
613 |
||
614 |
Goalw [setsum_def] "setsum f {} = 0"; |
|
615 |
by(Simp_tac 1); |
|
616 |
qed "setsum_empty"; |
|
617 |
Addsimps [setsum_empty]; |
|
618 |
||
619 |
Goalw [setsum_def] |
|
620 |
"[| finite F; a ~: F |] ==> setsum f (insert a F) = f(a) + setsum f F"; |
|
621 |
by(asm_simp_tac (simpset() addsimps [export fold_insert]) 1); |
|
622 |
qed "setsum_insert"; |
|
623 |
Addsimps [setsum_insert]; |
|
624 |
||
625 |
Goalw [setsum_def] |
|
626 |
"[| finite A; finite B; A Int B = {} |] ==> \ |
|
627 |
\ setsum f (A Un B) = setsum f A + setsum f B"; |
|
628 |
by(asm_simp_tac (simpset() addsimps [export fold_Un_disjoint2]) 1); |
|
629 |
qed_spec_mp "setsum_disj_Un"; |
|
630 |
||
631 |
Goal "[| finite F |] ==> \ |
|
632 |
\ setsum f (F-{a}) = (if a:F then setsum f F - f a else setsum f F)"; |
|
633 |
be finite_induct 1; |
|
634 |
by(auto_tac (claset(), simpset() addsimps [insert_Diff_if])); |
|
635 |
by(dres_inst_tac [("a","a")] mk_disjoint_insert 1); |
|
636 |
by(Auto_tac); |
|
637 |
qed_spec_mp "setsum_diff1"; |