author | wenzelm |
Fri, 06 Jun 1997 21:49:47 +0200 | |
changeset 3430 | d21b920363ab |
parent 3427 | e7cef2081106 |
child 3439 | 54785105178c |
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 |
||
9 |
open Finite; |
|
10 |
||
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
11 |
section "finite"; |
1531 | 12 |
|
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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13 |
(* |
923 | 14 |
goalw Finite.thy Fin.defs "!!A B. A<=B ==> Fin(A) <= Fin(B)"; |
1465 | 15 |
by (rtac lfp_mono 1); |
923 | 16 |
by (REPEAT (ares_tac basic_monos 1)); |
17 |
qed "Fin_mono"; |
|
18 |
||
19 |
goalw Finite.thy Fin.defs "Fin(A) <= Pow(A)"; |
|
2922 | 20 |
by (blast_tac (!claset addSIs [lfp_lowerbound]) 1); |
923 | 21 |
qed "Fin_subset_Pow"; |
22 |
||
23 |
(* A : Fin(B) ==> A <= B *) |
|
24 |
val FinD = Fin_subset_Pow RS subsetD RS PowD; |
|
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25 |
*) |
923 | 26 |
|
27 |
(*Discharging ~ x:y entails extra work*) |
|
28 |
val major::prems = goal Finite.thy |
|
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
29 |
"[| finite F; P({}); \ |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
30 |
\ !!F x. [| finite F; x ~: F; P(F) |] ==> P(insert x F) \ |
923 | 31 |
\ |] ==> P(F)"; |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
32 |
by (rtac (major RS Finites.induct) 1); |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
33 |
by (excluded_middle_tac "a:A" 2); |
923 | 34 |
by (etac (insert_absorb RS ssubst) 3 THEN assume_tac 3); (*backtracking!*) |
35 |
by (REPEAT (ares_tac prems 1)); |
|
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
36 |
qed "finite_induct"; |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
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|
37 |
|
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
diff
changeset
|
38 |
val major::prems = goal Finite.thy |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
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|
39 |
"[| finite F; \ |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
40 |
\ P({}); \ |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
41 |
\ !!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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parents:
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|
42 |
\ |] ==> F <= A --> P(F)"; |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
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changeset
|
43 |
by (rtac (major RS finite_induct) 1); |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
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|
44 |
by(ALLGOALS (blast_tac (!claset addIs prems))); |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
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|
45 |
val lemma = result(); |
923 | 46 |
|
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parents:
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|
47 |
val prems = goal Finite.thy |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
48 |
"[| finite F; F <= A; \ |
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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
|
49 |
\ P({}); \ |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
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|
50 |
\ !!F a. [| finite F; a:A; a ~: F; P(F) |] ==> P(insert a F) \ |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
51 |
\ |] ==> P(F)"; |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
52 |
by(blast_tac (HOL_cs addIs ((lemma RS mp)::prems)) 1); |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
53 |
qed "finite_subset_induct"; |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
54 |
|
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
55 |
Addsimps Finites.intrs; |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
56 |
AddSIs Finites.intrs; |
923 | 57 |
|
58 |
(*The union of two finite sets is finite*) |
|
59 |
val major::prems = goal Finite.thy |
|
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|
60 |
"[| finite F; finite G |] ==> finite(F Un G)"; |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
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changeset
|
61 |
by (rtac (major RS finite_induct) 1); |
1264 | 62 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps (prems @ [Un_insert_left])))); |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
63 |
qed "finite_UnI"; |
923 | 64 |
|
65 |
(*Every subset of a finite set is finite*) |
|
3413
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66 |
val [subs,fin] = goal Finite.thy "[| A<=B; finite B |] ==> finite A"; |
c1f63cc3a768
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|
67 |
by (EVERY1 [subgoal_tac "ALL C. C<=B --> finite C", |
1465 | 68 |
rtac mp, etac spec, |
69 |
rtac subs]); |
|
3413
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
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changeset
|
70 |
by (rtac (fin RS finite_induct) 1); |
1264 | 71 |
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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parents:
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|
72 |
by (safe_tac (!claset addSDs [subset_insert_iff RS iffD1])); |
923 | 73 |
by (eres_inst_tac [("t","C")] (insert_Diff RS subst) 2); |
1264 | 74 |
by (ALLGOALS Asm_simp_tac); |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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75 |
qed "finite_subset"; |
923 | 76 |
|
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77 |
goal Finite.thy "finite(F Un G) = (finite F & finite G)"; |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
78 |
by (blast_tac (!claset addIs [finite_UnI] addDs |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
79 |
[Un_upper1 RS finite_subset, Un_upper2 RS finite_subset]) 1); |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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80 |
qed "finite_Un"; |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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81 |
AddIffs[finite_Un]; |
1531 | 82 |
|
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
83 |
goal Finite.thy "finite(insert a A) = finite A"; |
1553 | 84 |
by (stac insert_is_Un 1); |
3413
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|
85 |
by (simp_tac (HOL_ss addsimps [finite_Un]) 1); |
3427
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Removed a few redundant additions of simprules or classical rules
paulson
parents:
3416
diff
changeset
|
86 |
by (Blast_tac 1); |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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87 |
qed "finite_insert"; |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
88 |
Addsimps[finite_insert]; |
1531 | 89 |
|
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
90 |
(*The image of a finite set is finite *) |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
91 |
goal Finite.thy "!!F. finite F ==> finite(h``F)"; |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
92 |
by (etac finite_induct 1); |
1264 | 93 |
by (Simp_tac 1); |
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
|
94 |
by (Asm_simp_tac 1); |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
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|
95 |
qed "finite_imageI"; |
923 | 96 |
|
97 |
val major::prems = goal Finite.thy |
|
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|
98 |
"[| finite c; finite b; \ |
1465 | 99 |
\ P(b); \ |
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100 |
\ !!x y. [| finite y; x:y; P(y) |] ==> P(y-{x}) \ |
923 | 101 |
\ |] ==> c<=b --> P(b-c)"; |
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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
|
102 |
by (rtac (major RS finite_induct) 1); |
2031 | 103 |
by (stac Diff_insert 2); |
923 | 104 |
by (ALLGOALS (asm_simp_tac |
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|
105 |
(!simpset addsimps (prems@[Diff_subset RS finite_subset])))); |
1531 | 106 |
val lemma = result(); |
923 | 107 |
|
108 |
val prems = goal Finite.thy |
|
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|
109 |
"[| finite A; \ |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
110 |
\ P(A); \ |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
111 |
\ !!a A. [| finite A; a:A; P(A) |] ==> P(A-{a}) \ |
923 | 112 |
\ |] ==> P({})"; |
113 |
by (rtac (Diff_cancel RS subst) 1); |
|
1531 | 114 |
by (rtac (lemma RS mp) 1); |
923 | 115 |
by (REPEAT (ares_tac (subset_refl::prems) 1)); |
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|
116 |
qed "finite_empty_induct"; |
1531 | 117 |
|
118 |
||
1618 | 119 |
(* finite B ==> finite (B - Ba) *) |
120 |
bind_thm ("finite_Diff", Diff_subset RS finite_subset); |
|
1531 | 121 |
Addsimps [finite_Diff]; |
122 |
||
3368 | 123 |
goal Finite.thy "finite(A-{a}) = finite(A)"; |
124 |
by (case_tac "a:A" 1); |
|
125 |
br (finite_insert RS sym RS trans) 1; |
|
126 |
by (stac insert_Diff 1); |
|
127 |
by (ALLGOALS Asm_simp_tac); |
|
128 |
qed "finite_Diff_singleton"; |
|
129 |
AddIffs [finite_Diff_singleton]; |
|
130 |
||
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|
131 |
goal Finite.thy "!!A. finite B ==> !A. f``A = B --> inj_onto f A --> finite A"; |
1553 | 132 |
by (etac finite_induct 1); |
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133 |
by (ALLGOALS Asm_simp_tac); |
3368 | 134 |
by (Step_tac 1); |
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|
135 |
by (subgoal_tac "EX y:A. f y = x & F = f``(A-{y})" 1); |
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|
136 |
by (Step_tac 1); |
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|
137 |
bw inj_onto_def; |
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|
138 |
by (Blast_tac 1); |
3368 | 139 |
by (thin_tac "ALL A. ?PP(A)" 1); |
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|
140 |
by (forward_tac [[equalityD2, insertI1] MRS subsetD] 1); |
3368 | 141 |
by (Step_tac 1); |
142 |
by (res_inst_tac [("x","xa")] bexI 1); |
|
143 |
by (ALLGOALS Asm_simp_tac); |
|
144 |
be equalityE 1; |
|
145 |
br equalityI 1; |
|
146 |
by (Blast_tac 2); |
|
147 |
by (Best_tac 1); |
|
148 |
val lemma = result(); |
|
149 |
||
150 |
goal Finite.thy "!!A. [| finite(f``A); inj_onto f A |] ==> finite A"; |
|
151 |
bd lemma 1; |
|
152 |
by (Blast_tac 1); |
|
153 |
qed "finite_imageD"; |
|
154 |
||
155 |
||
156 |
(** The powerset of a finite set **) |
|
157 |
||
158 |
goal Finite.thy "!!A. finite(Pow A) ==> finite A"; |
|
159 |
by (subgoal_tac "finite ((%x.{x})``A)" 1); |
|
160 |
br finite_subset 2; |
|
161 |
ba 3; |
|
162 |
by (ALLGOALS |
|
163 |
(fast_tac (!claset addSDs [rewrite_rule [inj_onto_def] finite_imageD]))); |
|
164 |
val lemma = result(); |
|
165 |
||
166 |
goal Finite.thy "finite(Pow A) = finite A"; |
|
167 |
br iffI 1; |
|
168 |
be lemma 1; |
|
169 |
(*Opposite inclusion: finite A ==> finite (Pow A) *) |
|
3340 | 170 |
by (etac finite_induct 1); |
171 |
by (ALLGOALS |
|
172 |
(asm_simp_tac |
|
173 |
(!simpset addsimps [finite_UnI, finite_imageI, Pow_insert]))); |
|
3368 | 174 |
qed "finite_Pow_iff"; |
175 |
AddIffs [finite_Pow_iff]; |
|
3340 | 176 |
|
3416 | 177 |
goal Finite.thy "finite(converse r) = finite r"; |
178 |
by(subgoal_tac "converse r = (%(x,y).(y,x))``r" 1); |
|
179 |
by(Asm_simp_tac 1); |
|
180 |
br iffI 1; |
|
181 |
be (rewrite_rule [inj_onto_def] finite_imageD) 1; |
|
182 |
by(simp_tac (!simpset setloop (split_tac[expand_split])) 1); |
|
183 |
be finite_imageI 1; |
|
184 |
by(simp_tac (!simpset addsimps [converse_def,image_def]) 1); |
|
185 |
by(Auto_tac()); |
|
186 |
br bexI 1; |
|
187 |
ba 2; |
|
188 |
by(Simp_tac 1); |
|
189 |
by(split_all_tac 1); |
|
190 |
by(Asm_full_simp_tac 1); |
|
191 |
qed "finite_converse"; |
|
192 |
AddIffs [finite_converse]; |
|
1531 | 193 |
|
1548 | 194 |
section "Finite cardinality -- 'card'"; |
1531 | 195 |
|
196 |
goal Set.thy "{f i |i. P i | i=n} = insert (f n) {f i|i. P i}"; |
|
2922 | 197 |
by (Blast_tac 1); |
1531 | 198 |
val Collect_conv_insert = result(); |
199 |
||
200 |
goalw Finite.thy [card_def] "card {} = 0"; |
|
1553 | 201 |
by (rtac Least_equality 1); |
202 |
by (ALLGOALS Asm_full_simp_tac); |
|
1531 | 203 |
qed "card_empty"; |
204 |
Addsimps [card_empty]; |
|
205 |
||
206 |
val [major] = goal Finite.thy |
|
207 |
"finite A ==> ? (n::nat) f. A = {f i |i. i<n}"; |
|
1553 | 208 |
by (rtac (major RS finite_induct) 1); |
209 |
by (res_inst_tac [("x","0")] exI 1); |
|
210 |
by (Simp_tac 1); |
|
211 |
by (etac exE 1); |
|
212 |
by (etac exE 1); |
|
213 |
by (hyp_subst_tac 1); |
|
214 |
by (res_inst_tac [("x","Suc n")] exI 1); |
|
215 |
by (res_inst_tac [("x","%i. if i<n then f i else x")] exI 1); |
|
1660 | 216 |
by (asm_simp_tac (!simpset addsimps [Collect_conv_insert, less_Suc_eq] |
1548 | 217 |
addcongs [rev_conj_cong]) 1); |
1531 | 218 |
qed "finite_has_card"; |
219 |
||
220 |
goal Finite.thy |
|
221 |
"!!A.[| x ~: A; insert x A = {f i|i.i<n} |] ==> \ |
|
222 |
\ ? m::nat. m<n & (? g. A = {g i|i.i<m})"; |
|
1553 | 223 |
by (res_inst_tac [("n","n")] natE 1); |
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); |
1660 | 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 |
|
1660 | 238 |
(!simpset addsimps [subset_insert,Collect_conv_insert, less_Suc_eq]) 1); |
2922 | 239 |
by (safe_tac (!claset)); |
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); |
|
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
242 |
by (SELECT_GOAL(safe_tac (!claset))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); |
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
247 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 248 |
by (res_inst_tac [("x","k")] exI 1); |
249 |
by (Asm_simp_tac 1); |
|
250 |
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1); |
|
2922 | 251 |
by (Blast_tac 1); |
1531 | 252 |
bd 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); |
|
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
256 |
by (SELECT_GOAL(safe_tac (!claset))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); |
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
261 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 262 |
by (res_inst_tac [("x","k")] exI 1); |
263 |
by (Asm_simp_tac 1); |
|
264 |
by (simp_tac (!simpset setloop (split_tac [expand_if])) 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); |
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
267 |
by (SELECT_GOAL(safe_tac (!claset))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); |
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
275 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 276 |
by (res_inst_tac [("x","k")] exI 1); |
277 |
by (Asm_simp_tac 1); |
|
278 |
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1); |
|
2922 | 279 |
by (Blast_tac 1); |
1531 | 280 |
val lemma = result(); |
281 |
||
282 |
goal Finite.thy "!!A. [| finite A; x ~: A |] ==> \ |
|
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); |
1531 | 285 |
bd finite_has_card 1; |
286 |
be exE 1; |
|
1553 | 287 |
by (dres_inst_tac [("P","%n.? f. A={f i|i.i<n}")] LeastI 1); |
1531 | 288 |
be 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 |
1660 | 292 |
(!simpset addsimps [Collect_conv_insert, less_Suc_eq] |
2031 | 293 |
addcongs [rev_conj_cong]) 1); |
1531 | 294 |
be subst 1; |
295 |
br refl 1; |
|
1553 | 296 |
by (rtac notI 1); |
297 |
by (etac exE 1); |
|
298 |
by (dtac lemma 1); |
|
1531 | 299 |
ba 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); |
|
1660 | 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 |
||
311 |
goalw Finite.thy [card_def] |
|
312 |
"!!A. [| finite A; x ~: A |] ==> card(insert x A) = Suc(card A)"; |
|
1553 | 313 |
by (etac lemma 1); |
314 |
by (assume_tac 1); |
|
1531 | 315 |
qed "card_insert_disjoint"; |
3352 | 316 |
Addsimps [card_insert_disjoint]; |
317 |
||
318 |
goal Finite.thy "!!A. finite A ==> !B. B <= A --> card(B) <= card(A)"; |
|
319 |
by (etac finite_induct 1); |
|
320 |
by (Simp_tac 1); |
|
321 |
by (strip_tac 1); |
|
322 |
by (case_tac "x:B" 1); |
|
3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
diff
changeset
|
323 |
by (dres_inst_tac [("A","B")] mk_disjoint_insert 1); |
3352 | 324 |
by (SELECT_GOAL(safe_tac (!claset))1); |
325 |
by (rotate_tac ~1 1); |
|
326 |
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1); |
|
327 |
by (rotate_tac ~1 1); |
|
328 |
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1); |
|
329 |
qed_spec_mp "card_mono"; |
|
330 |
||
331 |
goal Finite.thy "!!A B. [| finite A; finite B |]\ |
|
332 |
\ ==> A Int B = {} --> card(A Un B) = card A + card B"; |
|
333 |
by (etac finite_induct 1); |
|
334 |
by (ALLGOALS |
|
335 |
(asm_simp_tac (!simpset addsimps [Un_insert_left, Int_insert_left] |
|
336 |
setloop split_tac [expand_if]))); |
|
337 |
qed_spec_mp "card_Un_disjoint"; |
|
338 |
||
339 |
goal Finite.thy "!!A. [| finite A; B<=A |] ==> card A - card B = card (A - B)"; |
|
340 |
by (subgoal_tac "(A-B) Un B = A" 1); |
|
341 |
by (Blast_tac 2); |
|
342 |
br (add_right_cancel RS iffD1) 1; |
|
343 |
br (card_Un_disjoint RS subst) 1; |
|
344 |
be ssubst 4; |
|
345 |
by (Blast_tac 3); |
|
346 |
by (ALLGOALS |
|
347 |
(asm_simp_tac |
|
348 |
(!simpset addsimps [add_commute, not_less_iff_le, |
|
349 |
add_diff_inverse, card_mono, finite_subset]))); |
|
350 |
qed "card_Diff_subset"; |
|
1531 | 351 |
|
1618 | 352 |
goal Finite.thy "!!A. [| finite A; x: A |] ==> Suc(card(A-{x})) = card A"; |
353 |
by (res_inst_tac [("t", "A")] (insert_Diff RS subst) 1); |
|
354 |
by (assume_tac 1); |
|
3352 | 355 |
by (Asm_simp_tac 1); |
1618 | 356 |
qed "card_Suc_Diff"; |
357 |
||
358 |
goal Finite.thy "!!A. [| finite A; x: A |] ==> card(A-{x}) < card A"; |
|
2031 | 359 |
by (rtac Suc_less_SucD 1); |
1618 | 360 |
by (asm_simp_tac (!simpset addsimps [card_Suc_Diff]) 1); |
361 |
qed "card_Diff"; |
|
362 |
||
3389
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
363 |
|
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
364 |
(*** Cardinality of the Powerset ***) |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
365 |
|
1531 | 366 |
val [major] = goal Finite.thy |
367 |
"finite A ==> card(insert x A) = Suc(card(A-{x}))"; |
|
1553 | 368 |
by (case_tac "x:A" 1); |
369 |
by (asm_simp_tac (!simpset addsimps [insert_absorb]) 1); |
|
370 |
by (dtac mk_disjoint_insert 1); |
|
371 |
by (etac exE 1); |
|
372 |
by (Asm_simp_tac 1); |
|
373 |
by (rtac card_insert_disjoint 1); |
|
374 |
by (rtac (major RSN (2,finite_subset)) 1); |
|
2922 | 375 |
by (Blast_tac 1); |
376 |
by (Blast_tac 1); |
|
1553 | 377 |
by (asm_simp_tac (!simpset addsimps [major RS card_insert_disjoint]) 1); |
1531 | 378 |
qed "card_insert"; |
379 |
Addsimps [card_insert]; |
|
380 |
||
3340 | 381 |
goal Finite.thy "!!A. finite(A) ==> inj_onto f A --> card (f `` A) = card A"; |
382 |
by (etac finite_induct 1); |
|
383 |
by (ALLGOALS Asm_simp_tac); |
|
384 |
by (Step_tac 1); |
|
385 |
bw inj_onto_def; |
|
386 |
by (Blast_tac 1); |
|
387 |
by (stac card_insert_disjoint 1); |
|
388 |
by (etac finite_imageI 1); |
|
389 |
by (Blast_tac 1); |
|
390 |
by (Blast_tac 1); |
|
391 |
qed_spec_mp "card_image"; |
|
392 |
||
3389
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
393 |
goal thy "!!A. finite A ==> card (Pow A) = 2 ^ card A"; |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
394 |
by (etac finite_induct 1); |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
395 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps [Pow_insert]))); |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
396 |
by (stac card_Un_disjoint 1); |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
397 |
by (EVERY (map (blast_tac (!claset addIs [finite_imageI])) [3,2,1])); |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
398 |
by (subgoal_tac "inj_onto (insert x) (Pow F)" 1); |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
399 |
by (asm_simp_tac (!simpset addsimps [card_image, Pow_insert]) 1); |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
400 |
bw inj_onto_def; |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
401 |
by (blast_tac (!claset addSEs [equalityE]) 1); |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
402 |
qed "card_Pow"; |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
403 |
Addsimps [card_Pow]; |
3340 | 404 |
|
3389
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
405 |
|
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
406 |
(*Proper subsets*) |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
407 |
goalw Finite.thy [psubset_def] |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
408 |
"!!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
|
409 |
by (etac finite_induct 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
410 |
by (Simp_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
411 |
by (Blast_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
412 |
by (strip_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
413 |
by (etac conjE 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
414 |
by (case_tac "x:A" 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
415 |
(*1*) |
3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
diff
changeset
|
416 |
by (dres_inst_tac [("A","A")]mk_disjoint_insert 1); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
417 |
by (etac exE 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
418 |
by (etac conjE 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
419 |
by (hyp_subst_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
420 |
by (rotate_tac ~1 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
421 |
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
|
422 |
by (dtac insert_lim 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
423 |
by (Asm_full_simp_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
424 |
(*2*) |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
425 |
by (rotate_tac ~1 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
426 |
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
|
427 |
by (case_tac "A=F" 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
428 |
by (Asm_simp_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
429 |
by (Asm_simp_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
430 |
qed_spec_mp "psubset_card" ; |
3368 | 431 |
|
432 |
||
3430 | 433 |
(*Relates to equivalence classes. Based on a theorem of F. Kammueller's. |
3368 | 434 |
The "finite C" premise is redundant*) |
435 |
goal thy "!!C. finite C ==> finite (Union C) --> \ |
|
436 |
\ (! c : C. k dvd card c) --> \ |
|
437 |
\ (! c1: C. ! c2: C. c1 ~= c2 --> c1 Int c2 = {}) \ |
|
438 |
\ --> k dvd card(Union C)"; |
|
439 |
by (etac finite_induct 1); |
|
440 |
by (ALLGOALS Asm_simp_tac); |
|
441 |
by (strip_tac 1); |
|
442 |
by (REPEAT (etac conjE 1)); |
|
443 |
by (stac card_Un_disjoint 1); |
|
444 |
by (ALLGOALS |
|
445 |
(asm_full_simp_tac (!simpset |
|
446 |
addsimps [dvd_add, disjoint_eq_subset_Compl]))); |
|
447 |
by (thin_tac "?PP-->?QQ" 1); |
|
448 |
by (thin_tac "!c:F. ?PP(c)" 1); |
|
449 |
by (thin_tac "!c:F. ?PP(c) & ?QQ(c)" 1); |
|
450 |
by (Step_tac 1); |
|
451 |
by (ball_tac 1); |
|
452 |
by (Blast_tac 1); |
|
453 |
qed_spec_mp "dvd_partition"; |
|
454 |