author | paulson |
Mon, 26 May 1997 12:40:51 +0200 | |
changeset 3344 | b3e39a2987c1 |
parent 3340 | a886795c9dce |
child 3352 | 04502e5431fb |
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 |
|
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|
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Finite sets and their cardinality |
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*) |
8 |
||
9 |
open Finite; |
|
10 |
||
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section "The finite powerset operator -- Fin"; |
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|
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goalw Finite.thy Fin.defs "!!A B. A<=B ==> Fin(A) <= Fin(B)"; |
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by (rtac lfp_mono 1); |
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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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by (blast_tac (!claset addSIs [lfp_lowerbound]) 1); |
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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({}); \ |
|
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\ !!F x. [| x:A; F:Fin(A); x~:F; P(F) |] ==> P(insert x F) \ |
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\ |] ==> 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 |
||
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Addsimps Fin.intrs; |
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|
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]); |
|
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by (rtac (fin RS Fin_induct) 1); |
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by (simp_tac (!simpset addsimps [subset_Un_eq]) 1); |
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best_tac, deepen_tac and safe_tac now also use default claset.
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changeset
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52 |
by (safe_tac (!claset addSDs [subset_insert_iff RS iffD1])); |
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by (eres_inst_tac [("t","C")] (insert_Diff RS subst) 2); |
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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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by (blast_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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by (blast_tac (!claset addSIs Fin.intrs addDs [FinD]) 1); |
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qed "insert_Fin"; |
68 |
Addsimps[insert_Fin]; |
|
69 |
||
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(*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] |
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delsimps [insert_Fin]) 1); |
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qed "Fin_imageI"; |
79 |
||
80 |
val major::prems = goal Finite.thy |
|
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"[| c: Fin(A); b: Fin(A); \ |
82 |
\ P(b); \ |
|
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\ !!(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); |
|
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by (stac Diff_insert 2); |
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by (ALLGOALS (asm_simp_tac |
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(!simpset addsimps (prems@[Diff_subset RS Fin_subset])))); |
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val lemma = result(); |
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|
91 |
val prems = goal Finite.thy |
|
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"[| b: Fin(A); \ |
93 |
\ P(b); \ |
|
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\ !!x y. [| x:A; y: Fin(A); x:y; P(y) |] ==> P(y-{x}) \ |
95 |
\ |] ==> P({})"; |
|
96 |
by (rtac (Diff_cancel RS subst) 1); |
|
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by (rtac (lemma RS mp) 1); |
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by (REPEAT (ares_tac (subset_refl::prems) 1)); |
99 |
qed "Fin_empty_induct"; |
|
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|
101 |
||
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section "The predicate 'finite'"; |
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|
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)"; |
|
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by (Asm_simp_tac 1); |
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qed "finite_insertI"; |
121 |
||
122 |
(*The union of two finite sets is finite*) |
|
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goalw Finite.thy [finite_def] |
|
124 |
"!!F. [| finite F; finite G |] ==> finite(F Un G)"; |
|
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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); |
|
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qed "finite_subset"; |
132 |
||
133 |
goalw Finite.thy [finite_def] "finite(F Un G) = (finite F & finite G)"; |
|
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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)"; |
|
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by (Simp_tac 1); |
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qed "insert_finite"; |
141 |
Addsimps[insert_finite]; |
|
142 |
||
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(* finite B ==> finite (B - Ba) *) |
144 |
bind_thm ("finite_Diff", Diff_subset RS finite_subset); |
|
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Addsimps [finite_Diff]; |
146 |
||
147 |
(*The image of a finite set is finite*) |
|
148 |
goal Finite.thy "!!F. finite F ==> finite(h``F)"; |
|
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by (etac finite_induct 1); |
150 |
by (ALLGOALS Asm_simp_tac); |
|
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qed "finite_imageI"; |
152 |
||
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(*The powerset of a finite set is finite*) |
154 |
goal Finite.thy "!!A. finite A ==> finite(Pow A)"; |
|
155 |
by (etac finite_induct 1); |
|
156 |
by (ALLGOALS |
|
157 |
(asm_simp_tac |
|
158 |
(!simpset addsimps [finite_UnI, finite_imageI, Pow_insert]))); |
|
159 |
qed "finite_PowI"; |
|
160 |
AddSIs [finite_PowI]; |
|
161 |
||
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val major::prems = goalw Finite.thy [finite_def] |
163 |
"[| finite A; \ |
|
164 |
\ P(A); \ |
|
165 |
\ !!a A. [| finite A; a:A; P(A) |] ==> P(A-{a}) \ |
|
166 |
\ |] ==> P({})"; |
|
167 |
by (rtac (major RS Fin_empty_induct) 1); |
|
168 |
by (REPEAT (ares_tac (subset_refl::prems) 1)); |
|
169 |
qed "finite_empty_induct"; |
|
170 |
||
171 |
||
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section "Finite cardinality -- 'card'"; |
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|
174 |
goal Set.thy "{f i |i. P i | i=n} = insert (f n) {f i|i. P i}"; |
|
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by (Blast_tac 1); |
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val Collect_conv_insert = result(); |
177 |
||
178 |
goalw Finite.thy [card_def] "card {} = 0"; |
|
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by (rtac Least_equality 1); |
180 |
by (ALLGOALS Asm_full_simp_tac); |
|
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qed "card_empty"; |
182 |
Addsimps [card_empty]; |
|
183 |
||
184 |
val [major] = goal Finite.thy |
|
185 |
"finite A ==> ? (n::nat) f. A = {f i |i. i<n}"; |
|
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by (rtac (major RS finite_induct) 1); |
187 |
by (res_inst_tac [("x","0")] exI 1); |
|
188 |
by (Simp_tac 1); |
|
189 |
by (etac exE 1); |
|
190 |
by (etac exE 1); |
|
191 |
by (hyp_subst_tac 1); |
|
192 |
by (res_inst_tac [("x","Suc n")] exI 1); |
|
193 |
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] |
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addcongs [rev_conj_cong]) 1); |
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qed "finite_has_card"; |
197 |
||
198 |
goal Finite.thy |
|
199 |
"!!A.[| x ~: A; insert x A = {f i|i.i<n} |] ==> \ |
|
200 |
\ ? m::nat. m<n & (? g. A = {g i|i.i<m})"; |
|
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by (res_inst_tac [("n","n")] natE 1); |
202 |
by (hyp_subst_tac 1); |
|
203 |
by (Asm_full_simp_tac 1); |
|
204 |
by (rename_tac "m" 1); |
|
205 |
by (hyp_subst_tac 1); |
|
206 |
by (case_tac "? a. a:A" 1); |
|
207 |
by (res_inst_tac [("x","0")] exI 2); |
|
208 |
by (Simp_tac 2); |
|
2922 | 209 |
by (Blast_tac 2); |
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by (etac exE 1); |
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by (simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
1553 | 212 |
by (rtac exI 1); |
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by (rtac (refl RS disjI2 RS conjI) 1); |
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by (etac equalityE 1); |
215 |
by (asm_full_simp_tac |
|
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(!simpset addsimps [subset_insert,Collect_conv_insert, less_Suc_eq]) 1); |
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by (safe_tac (!claset)); |
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by (Asm_full_simp_tac 1); |
219 |
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
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220 |
by (SELECT_GOAL(safe_tac (!claset))1); |
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by (subgoal_tac "x ~= f m" 1); |
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by (Blast_tac 2); |
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by (subgoal_tac "? k. f k = x & k<m" 1); |
2922 | 224 |
by (Blast_tac 2); |
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); |
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by (res_inst_tac [("x","k")] exI 1); |
227 |
by (Asm_simp_tac 1); |
|
228 |
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1); |
|
2922 | 229 |
by (Blast_tac 1); |
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bd sym 1; |
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by (rotate_tac ~1 1); |
232 |
by (Asm_full_simp_tac 1); |
|
233 |
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
|
234 |
by (SELECT_GOAL(safe_tac (!claset))1); |
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by (subgoal_tac "x ~= f m" 1); |
2922 | 236 |
by (Blast_tac 2); |
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by (subgoal_tac "? k. f k = x & k<m" 1); |
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by (Blast_tac 2); |
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
239 |
by (SELECT_GOAL(safe_tac (!claset))1); |
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by (res_inst_tac [("x","k")] exI 1); |
241 |
by (Asm_simp_tac 1); |
|
242 |
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1); |
|
2922 | 243 |
by (Blast_tac 1); |
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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
|
245 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 246 |
by (subgoal_tac "x ~= f i" 1); |
2922 | 247 |
by (Blast_tac 2); |
1553 | 248 |
by (case_tac "x = f m" 1); |
249 |
by (res_inst_tac [("x","i")] exI 1); |
|
250 |
by (Asm_simp_tac 1); |
|
251 |
by (subgoal_tac "? k. f k = x & k<m" 1); |
|
2922 | 252 |
by (Blast_tac 2); |
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
253 |
by (SELECT_GOAL(safe_tac (!claset))1); |
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by (res_inst_tac [("x","k")] exI 1); |
255 |
by (Asm_simp_tac 1); |
|
256 |
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1); |
|
2922 | 257 |
by (Blast_tac 1); |
1531 | 258 |
val lemma = result(); |
259 |
||
260 |
goal Finite.thy "!!A. [| finite A; x ~: A |] ==> \ |
|
261 |
\ (LEAST n. ? f. insert x A = {f i|i.i<n}) = Suc(LEAST n. ? f. A={f i|i.i<n})"; |
|
1553 | 262 |
by (rtac Least_equality 1); |
1531 | 263 |
bd finite_has_card 1; |
264 |
be exE 1; |
|
1553 | 265 |
by (dres_inst_tac [("P","%n.? f. A={f i|i.i<n}")] LeastI 1); |
1531 | 266 |
be exE 1; |
1553 | 267 |
by (res_inst_tac |
1531 | 268 |
[("x","%i. if i<(LEAST n. ? f. A={f i |i. i < n}) then f i else x")] exI 1); |
1553 | 269 |
by (simp_tac |
1660 | 270 |
(!simpset addsimps [Collect_conv_insert, less_Suc_eq] |
2031 | 271 |
addcongs [rev_conj_cong]) 1); |
1531 | 272 |
be subst 1; |
273 |
br refl 1; |
|
1553 | 274 |
by (rtac notI 1); |
275 |
by (etac exE 1); |
|
276 |
by (dtac lemma 1); |
|
1531 | 277 |
ba 1; |
1553 | 278 |
by (etac exE 1); |
279 |
by (etac conjE 1); |
|
280 |
by (dres_inst_tac [("P","%x. ? g. A = {g i |i. i < x}")] Least_le 1); |
|
281 |
by (dtac le_less_trans 1 THEN atac 1); |
|
1660 | 282 |
by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
1553 | 283 |
by (etac disjE 1); |
284 |
by (etac less_asym 1 THEN atac 1); |
|
285 |
by (hyp_subst_tac 1); |
|
286 |
by (Asm_full_simp_tac 1); |
|
1531 | 287 |
val lemma = result(); |
288 |
||
289 |
goalw Finite.thy [card_def] |
|
290 |
"!!A. [| finite A; x ~: A |] ==> card(insert x A) = Suc(card A)"; |
|
1553 | 291 |
by (etac lemma 1); |
292 |
by (assume_tac 1); |
|
1531 | 293 |
qed "card_insert_disjoint"; |
294 |
||
1618 | 295 |
goal Finite.thy "!!A. [| finite A; x: A |] ==> Suc(card(A-{x})) = card A"; |
296 |
by (res_inst_tac [("t", "A")] (insert_Diff RS subst) 1); |
|
297 |
by (assume_tac 1); |
|
298 |
by (asm_simp_tac (!simpset addsimps [card_insert_disjoint]) 1); |
|
299 |
qed "card_Suc_Diff"; |
|
300 |
||
301 |
goal Finite.thy "!!A. [| finite A; x: A |] ==> card(A-{x}) < card A"; |
|
2031 | 302 |
by (rtac Suc_less_SucD 1); |
1618 | 303 |
by (asm_simp_tac (!simpset addsimps [card_Suc_Diff]) 1); |
304 |
qed "card_Diff"; |
|
305 |
||
1531 | 306 |
val [major] = goal Finite.thy |
307 |
"finite A ==> card(insert x A) = Suc(card(A-{x}))"; |
|
1553 | 308 |
by (case_tac "x:A" 1); |
309 |
by (asm_simp_tac (!simpset addsimps [insert_absorb]) 1); |
|
310 |
by (dtac mk_disjoint_insert 1); |
|
311 |
by (etac exE 1); |
|
312 |
by (Asm_simp_tac 1); |
|
313 |
by (rtac card_insert_disjoint 1); |
|
314 |
by (rtac (major RSN (2,finite_subset)) 1); |
|
2922 | 315 |
by (Blast_tac 1); |
316 |
by (Blast_tac 1); |
|
1553 | 317 |
by (asm_simp_tac (!simpset addsimps [major RS card_insert_disjoint]) 1); |
1531 | 318 |
qed "card_insert"; |
319 |
Addsimps [card_insert]; |
|
320 |
||
321 |
||
3340 | 322 |
goal Finite.thy "!!A. finite(A) ==> inj_onto f A --> card (f `` A) = card A"; |
323 |
by (etac finite_induct 1); |
|
324 |
by (ALLGOALS Asm_simp_tac); |
|
325 |
by (Step_tac 1); |
|
326 |
bw inj_onto_def; |
|
327 |
by (Blast_tac 1); |
|
328 |
by (stac card_insert_disjoint 1); |
|
329 |
by (etac finite_imageI 1); |
|
330 |
by (Blast_tac 1); |
|
331 |
by (Blast_tac 1); |
|
332 |
qed_spec_mp "card_image"; |
|
333 |
||
334 |
||
1531 | 335 |
goal Finite.thy "!!A. finite A ==> !B. B <= A --> card(B) <= card(A)"; |
1553 | 336 |
by (etac finite_induct 1); |
337 |
by (Simp_tac 1); |
|
338 |
by (strip_tac 1); |
|
339 |
by (case_tac "x:B" 1); |
|
340 |
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
|
341 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 342 |
by (rotate_tac ~1 1); |
343 |
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1); |
|
344 |
by (rotate_tac ~1 1); |
|
345 |
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1); |
|
1531 | 346 |
qed_spec_mp "card_mono"; |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
347 |
|
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
348 |
goalw Finite.thy [psubset_def] |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
349 |
"!!B. finite B ==> !A. A < B --> card(A) < card(B)"; |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
350 |
by (etac finite_induct 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
351 |
by (Simp_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
352 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
353 |
by (strip_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
354 |
by (etac conjE 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
355 |
by (case_tac "x:A" 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
356 |
(*1*) |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
357 |
by (dtac mk_disjoint_insert 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
358 |
by (etac exE 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
359 |
by (etac conjE 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
360 |
by (hyp_subst_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
361 |
by (rotate_tac ~1 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
362 |
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
363 |
by (dtac insert_lim 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
364 |
by (Asm_full_simp_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
365 |
(*2*) |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
366 |
by (rotate_tac ~1 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
367 |
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
368 |
by (case_tac "A=F" 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
369 |
by (Asm_simp_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
370 |
by (Asm_simp_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
371 |
qed_spec_mp "psubset_card" ; |