author | paulson |
Thu, 25 Sep 1997 12:13:18 +0200 | |
changeset 3708 | 56facaebf3e3 |
parent 3517 | 2547f33fa33a |
child 3724 | f33e301a89f5 |
permissions | -rw-r--r-- |
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(* Title: HOL/Finite.thy |
923 | 2 |
ID: $Id$ |
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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; |
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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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13 |
(* |
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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)); |
17 |
qed "Fin_mono"; |
|
18 |
||
19 |
goalw Finite.thy Fin.defs "Fin(A) <= Pow(A)"; |
|
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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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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); |
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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'.
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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'.
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parents:
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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({}); \ |
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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'.
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parents:
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|
43 |
by (rtac (major RS finite_induct) 1); |
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by (ALLGOALS (blast_tac (!claset addIs prems))); |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
45 |
val lemma = result(); |
923 | 46 |
|
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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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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|
49 |
\ 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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|
50 |
\ !!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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|
51 |
\ |] ==> P(F)"; |
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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"; |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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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; |
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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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"[| finite F; finite G |] ==> finite(F Un G)"; |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
61 |
by (rtac (major RS finite_induct) 1); |
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Removed redundant addsimps of Un_insert_left, which is now a default simprule
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|
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by (ALLGOALS (asm_simp_tac (!simpset addsimps prems))); |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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qed "finite_UnI"; |
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|
65 |
(*Every subset of a finite set is finite*) |
|
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val [subs,fin] = goal Finite.thy "[| A<=B; finite B |] ==> finite A"; |
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by (EVERY1 [subgoal_tac "ALL C. C<=B --> finite C", |
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rtac mp, etac spec, |
69 |
rtac subs]); |
|
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
70 |
by (rtac (fin RS finite_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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|
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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); |
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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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parents:
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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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|
79 |
[Un_upper1 RS finite_subset, Un_upper2 RS finite_subset]) 1); |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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|
80 |
qed "finite_Un"; |
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AddIffs[finite_Un]; |
1531 | 82 |
|
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
83 |
goal Finite.thy "finite(insert a A) = finite A"; |
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by (stac insert_is_Un 1); |
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|
85 |
by (simp_tac (HOL_ss addsimps [finite_Un]) 1); |
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Removed a few redundant additions of simprules or classical rules
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|
86 |
by (Blast_tac 1); |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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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 *) |
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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); |
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by (Simp_tac 1); |
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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'.
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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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|
102 |
by (rtac (major RS finite_induct) 1); |
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by (stac Diff_insert 2); |
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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); |
|
3457 | 125 |
by (rtac (finite_insert RS sym RS trans) 1); |
3368 | 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"; |
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by (etac finite_induct 1); |
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133 |
by (ALLGOALS Asm_simp_tac); |
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by (Clarify_tac 1); |
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|
135 |
by (subgoal_tac "EX y:A. f y = x & F = f``(A-{y})" 1); |
3708 | 136 |
by (Clarify_tac 1); |
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by (rewtac 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); |
3708 | 141 |
by (Clarify_tac 1); |
3368 | 142 |
by (res_inst_tac [("x","xa")] bexI 1); |
143 |
by (ALLGOALS Asm_simp_tac); |
|
3708 | 144 |
by (blast_tac (!claset addEs [equalityE]) 1); |
3368 | 145 |
val lemma = result(); |
146 |
||
147 |
goal Finite.thy "!!A. [| finite(f``A); inj_onto f A |] ==> finite A"; |
|
3457 | 148 |
by (dtac lemma 1); |
3368 | 149 |
by (Blast_tac 1); |
150 |
qed "finite_imageD"; |
|
151 |
||
152 |
||
153 |
(** The powerset of a finite set **) |
|
154 |
||
155 |
goal Finite.thy "!!A. finite(Pow A) ==> finite A"; |
|
156 |
by (subgoal_tac "finite ((%x.{x})``A)" 1); |
|
3457 | 157 |
by (rtac finite_subset 2); |
158 |
by (assume_tac 3); |
|
3368 | 159 |
by (ALLGOALS |
160 |
(fast_tac (!claset addSDs [rewrite_rule [inj_onto_def] finite_imageD]))); |
|
161 |
val lemma = result(); |
|
162 |
||
163 |
goal Finite.thy "finite(Pow A) = finite A"; |
|
3457 | 164 |
by (rtac iffI 1); |
165 |
by (etac lemma 1); |
|
3368 | 166 |
(*Opposite inclusion: finite A ==> finite (Pow A) *) |
3340 | 167 |
by (etac finite_induct 1); |
168 |
by (ALLGOALS |
|
169 |
(asm_simp_tac |
|
170 |
(!simpset addsimps [finite_UnI, finite_imageI, Pow_insert]))); |
|
3368 | 171 |
qed "finite_Pow_iff"; |
172 |
AddIffs [finite_Pow_iff]; |
|
3340 | 173 |
|
3439 | 174 |
goal Finite.thy "finite(r^-1) = finite r"; |
3457 | 175 |
by (subgoal_tac "r^-1 = (%(x,y).(y,x))``r" 1); |
176 |
by (Asm_simp_tac 1); |
|
177 |
by (rtac iffI 1); |
|
178 |
by (etac (rewrite_rule [inj_onto_def] finite_imageD) 1); |
|
179 |
by (simp_tac (!simpset setloop (split_tac[expand_split])) 1); |
|
180 |
by (etac finite_imageI 1); |
|
181 |
by (simp_tac (!simpset addsimps [inverse_def,image_def]) 1); |
|
182 |
by (Auto_tac()); |
|
183 |
by (rtac bexI 1); |
|
184 |
by (assume_tac 2); |
|
185 |
by (Simp_tac 1); |
|
186 |
by (split_all_tac 1); |
|
187 |
by (Asm_full_simp_tac 1); |
|
3439 | 188 |
qed "finite_inverse"; |
189 |
AddIffs [finite_inverse]; |
|
1531 | 190 |
|
1548 | 191 |
section "Finite cardinality -- 'card'"; |
1531 | 192 |
|
193 |
goal Set.thy "{f i |i. P i | i=n} = insert (f n) {f i|i. P i}"; |
|
2922 | 194 |
by (Blast_tac 1); |
1531 | 195 |
val Collect_conv_insert = result(); |
196 |
||
197 |
goalw Finite.thy [card_def] "card {} = 0"; |
|
1553 | 198 |
by (rtac Least_equality 1); |
199 |
by (ALLGOALS Asm_full_simp_tac); |
|
1531 | 200 |
qed "card_empty"; |
201 |
Addsimps [card_empty]; |
|
202 |
||
203 |
val [major] = goal Finite.thy |
|
204 |
"finite A ==> ? (n::nat) f. A = {f i |i. i<n}"; |
|
1553 | 205 |
by (rtac (major RS finite_induct) 1); |
206 |
by (res_inst_tac [("x","0")] exI 1); |
|
207 |
by (Simp_tac 1); |
|
208 |
by (etac exE 1); |
|
209 |
by (etac exE 1); |
|
210 |
by (hyp_subst_tac 1); |
|
211 |
by (res_inst_tac [("x","Suc n")] exI 1); |
|
212 |
by (res_inst_tac [("x","%i. if i<n then f i else x")] exI 1); |
|
1660 | 213 |
by (asm_simp_tac (!simpset addsimps [Collect_conv_insert, less_Suc_eq] |
1548 | 214 |
addcongs [rev_conj_cong]) 1); |
1531 | 215 |
qed "finite_has_card"; |
216 |
||
217 |
goal Finite.thy |
|
218 |
"!!A.[| x ~: A; insert x A = {f i|i.i<n} |] ==> \ |
|
219 |
\ ? m::nat. m<n & (? g. A = {g i|i.i<m})"; |
|
1553 | 220 |
by (res_inst_tac [("n","n")] natE 1); |
221 |
by (hyp_subst_tac 1); |
|
222 |
by (Asm_full_simp_tac 1); |
|
223 |
by (rename_tac "m" 1); |
|
224 |
by (hyp_subst_tac 1); |
|
225 |
by (case_tac "? a. a:A" 1); |
|
226 |
by (res_inst_tac [("x","0")] exI 2); |
|
227 |
by (Simp_tac 2); |
|
2922 | 228 |
by (Blast_tac 2); |
1553 | 229 |
by (etac exE 1); |
1660 | 230 |
by (simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
1553 | 231 |
by (rtac exI 1); |
1782 | 232 |
by (rtac (refl RS disjI2 RS conjI) 1); |
1553 | 233 |
by (etac equalityE 1); |
234 |
by (asm_full_simp_tac |
|
1660 | 235 |
(!simpset addsimps [subset_insert,Collect_conv_insert, less_Suc_eq]) 1); |
2922 | 236 |
by (safe_tac (!claset)); |
1553 | 237 |
by (Asm_full_simp_tac 1); |
238 |
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
|
239 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 240 |
by (subgoal_tac "x ~= f m" 1); |
2922 | 241 |
by (Blast_tac 2); |
1553 | 242 |
by (subgoal_tac "? k. f k = x & k<m" 1); |
2922 | 243 |
by (Blast_tac 2); |
1786
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); |
|
2922 | 248 |
by (Blast_tac 1); |
3457 | 249 |
by (dtac sym 1); |
1553 | 250 |
by (rotate_tac ~1 1); |
251 |
by (Asm_full_simp_tac 1); |
|
252 |
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
|
253 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 254 |
by (subgoal_tac "x ~= f m" 1); |
2922 | 255 |
by (Blast_tac 2); |
1553 | 256 |
by (subgoal_tac "? k. f k = x & k<m" 1); |
2922 | 257 |
by (Blast_tac 2); |
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
258 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 259 |
by (res_inst_tac [("x","k")] exI 1); |
260 |
by (Asm_simp_tac 1); |
|
261 |
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1); |
|
2922 | 262 |
by (Blast_tac 1); |
1553 | 263 |
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
|
264 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 265 |
by (subgoal_tac "x ~= f i" 1); |
2922 | 266 |
by (Blast_tac 2); |
1553 | 267 |
by (case_tac "x = f m" 1); |
268 |
by (res_inst_tac [("x","i")] exI 1); |
|
269 |
by (Asm_simp_tac 1); |
|
270 |
by (subgoal_tac "? k. f k = x & k<m" 1); |
|
2922 | 271 |
by (Blast_tac 2); |
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1782
diff
changeset
|
272 |
by (SELECT_GOAL(safe_tac (!claset))1); |
1553 | 273 |
by (res_inst_tac [("x","k")] exI 1); |
274 |
by (Asm_simp_tac 1); |
|
275 |
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1); |
|
2922 | 276 |
by (Blast_tac 1); |
1531 | 277 |
val lemma = result(); |
278 |
||
279 |
goal Finite.thy "!!A. [| finite A; x ~: A |] ==> \ |
|
280 |
\ (LEAST n. ? f. insert x A = {f i|i.i<n}) = Suc(LEAST n. ? f. A={f i|i.i<n})"; |
|
1553 | 281 |
by (rtac Least_equality 1); |
3457 | 282 |
by (dtac finite_has_card 1); |
283 |
by (etac exE 1); |
|
1553 | 284 |
by (dres_inst_tac [("P","%n.? f. A={f i|i.i<n}")] LeastI 1); |
3457 | 285 |
by (etac exE 1); |
1553 | 286 |
by (res_inst_tac |
1531 | 287 |
[("x","%i. if i<(LEAST n. ? f. A={f i |i. i < n}) then f i else x")] exI 1); |
1553 | 288 |
by (simp_tac |
1660 | 289 |
(!simpset addsimps [Collect_conv_insert, less_Suc_eq] |
2031 | 290 |
addcongs [rev_conj_cong]) 1); |
3457 | 291 |
by (etac subst 1); |
292 |
by (rtac refl 1); |
|
1553 | 293 |
by (rtac notI 1); |
294 |
by (etac exE 1); |
|
295 |
by (dtac lemma 1); |
|
3457 | 296 |
by (assume_tac 1); |
1553 | 297 |
by (etac exE 1); |
298 |
by (etac conjE 1); |
|
299 |
by (dres_inst_tac [("P","%x. ? g. A = {g i |i. i < x}")] Least_le 1); |
|
300 |
by (dtac le_less_trans 1 THEN atac 1); |
|
1660 | 301 |
by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
1553 | 302 |
by (etac disjE 1); |
303 |
by (etac less_asym 1 THEN atac 1); |
|
304 |
by (hyp_subst_tac 1); |
|
305 |
by (Asm_full_simp_tac 1); |
|
1531 | 306 |
val lemma = result(); |
307 |
||
308 |
goalw Finite.thy [card_def] |
|
309 |
"!!A. [| finite A; x ~: A |] ==> card(insert x A) = Suc(card A)"; |
|
1553 | 310 |
by (etac lemma 1); |
311 |
by (assume_tac 1); |
|
1531 | 312 |
qed "card_insert_disjoint"; |
3352 | 313 |
Addsimps [card_insert_disjoint]; |
314 |
||
315 |
goal Finite.thy "!!A. finite A ==> !B. B <= A --> card(B) <= card(A)"; |
|
316 |
by (etac finite_induct 1); |
|
317 |
by (Simp_tac 1); |
|
3708 | 318 |
by (Clarify_tac 1); |
3352 | 319 |
by (case_tac "x:B" 1); |
3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
diff
changeset
|
320 |
by (dres_inst_tac [("A","B")] mk_disjoint_insert 1); |
3352 | 321 |
by (SELECT_GOAL(safe_tac (!claset))1); |
322 |
by (rotate_tac ~1 1); |
|
323 |
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1); |
|
324 |
by (rotate_tac ~1 1); |
|
325 |
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1); |
|
326 |
qed_spec_mp "card_mono"; |
|
327 |
||
328 |
goal Finite.thy "!!A B. [| finite A; finite B |]\ |
|
329 |
\ ==> A Int B = {} --> card(A Un B) = card A + card B"; |
|
330 |
by (etac finite_induct 1); |
|
331 |
by (ALLGOALS |
|
3517
2547f33fa33a
Removed redundant addsimps of Un_insert_left, which is now a default simprule
paulson
parents:
3457
diff
changeset
|
332 |
(asm_simp_tac (!simpset addsimps [Int_insert_left] |
2547f33fa33a
Removed redundant addsimps of Un_insert_left, which is now a default simprule
paulson
parents:
3457
diff
changeset
|
333 |
setloop split_tac [expand_if]))); |
3352 | 334 |
qed_spec_mp "card_Un_disjoint"; |
335 |
||
336 |
goal Finite.thy "!!A. [| finite A; B<=A |] ==> card A - card B = card (A - B)"; |
|
337 |
by (subgoal_tac "(A-B) Un B = A" 1); |
|
338 |
by (Blast_tac 2); |
|
3457 | 339 |
by (rtac (add_right_cancel RS iffD1) 1); |
340 |
by (rtac (card_Un_disjoint RS subst) 1); |
|
341 |
by (etac ssubst 4); |
|
3352 | 342 |
by (Blast_tac 3); |
343 |
by (ALLGOALS |
|
344 |
(asm_simp_tac |
|
345 |
(!simpset addsimps [add_commute, not_less_iff_le, |
|
346 |
add_diff_inverse, card_mono, finite_subset]))); |
|
347 |
qed "card_Diff_subset"; |
|
1531 | 348 |
|
1618 | 349 |
goal Finite.thy "!!A. [| finite A; x: A |] ==> Suc(card(A-{x})) = card A"; |
350 |
by (res_inst_tac [("t", "A")] (insert_Diff RS subst) 1); |
|
351 |
by (assume_tac 1); |
|
3352 | 352 |
by (Asm_simp_tac 1); |
1618 | 353 |
qed "card_Suc_Diff"; |
354 |
||
355 |
goal Finite.thy "!!A. [| finite A; x: A |] ==> card(A-{x}) < card A"; |
|
2031 | 356 |
by (rtac Suc_less_SucD 1); |
1618 | 357 |
by (asm_simp_tac (!simpset addsimps [card_Suc_Diff]) 1); |
358 |
qed "card_Diff"; |
|
359 |
||
3389
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
360 |
|
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
361 |
(*** Cardinality of the Powerset ***) |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
362 |
|
1531 | 363 |
val [major] = goal Finite.thy |
364 |
"finite A ==> card(insert x A) = Suc(card(A-{x}))"; |
|
1553 | 365 |
by (case_tac "x:A" 1); |
366 |
by (asm_simp_tac (!simpset addsimps [insert_absorb]) 1); |
|
367 |
by (dtac mk_disjoint_insert 1); |
|
368 |
by (etac exE 1); |
|
369 |
by (Asm_simp_tac 1); |
|
370 |
by (rtac card_insert_disjoint 1); |
|
371 |
by (rtac (major RSN (2,finite_subset)) 1); |
|
2922 | 372 |
by (Blast_tac 1); |
373 |
by (Blast_tac 1); |
|
1553 | 374 |
by (asm_simp_tac (!simpset addsimps [major RS card_insert_disjoint]) 1); |
1531 | 375 |
qed "card_insert"; |
376 |
Addsimps [card_insert]; |
|
377 |
||
3340 | 378 |
goal Finite.thy "!!A. finite(A) ==> inj_onto f A --> card (f `` A) = card A"; |
379 |
by (etac finite_induct 1); |
|
380 |
by (ALLGOALS Asm_simp_tac); |
|
381 |
by (Step_tac 1); |
|
3457 | 382 |
by (rewtac inj_onto_def); |
3340 | 383 |
by (Blast_tac 1); |
384 |
by (stac card_insert_disjoint 1); |
|
385 |
by (etac finite_imageI 1); |
|
386 |
by (Blast_tac 1); |
|
387 |
by (Blast_tac 1); |
|
388 |
qed_spec_mp "card_image"; |
|
389 |
||
3389
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
390 |
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
|
391 |
by (etac finite_induct 1); |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
392 |
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
|
393 |
by (stac card_Un_disjoint 1); |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
394 |
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
|
395 |
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
|
396 |
by (asm_simp_tac (!simpset addsimps [card_image, Pow_insert]) 1); |
3457 | 397 |
by (rewtac inj_onto_def); |
3389
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
398 |
by (blast_tac (!claset addSEs [equalityE]) 1); |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
399 |
qed "card_Pow"; |
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
400 |
Addsimps [card_Pow]; |
3340 | 401 |
|
3389
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
402 |
|
3150eba724a1
New theorem about the cardinality of the powerset (uses exponentiation)
paulson
parents:
3382
diff
changeset
|
403 |
(*Proper subsets*) |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
404 |
goalw Finite.thy [psubset_def] |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
405 |
"!!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
|
406 |
by (etac finite_induct 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
407 |
by (Simp_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
408 |
by (Blast_tac 1); |
3708 | 409 |
by (Clarify_tac 1); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
410 |
by (case_tac "x:A" 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
411 |
(*1*) |
3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
diff
changeset
|
412 |
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
|
413 |
by (etac exE 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
414 |
by (etac conjE 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
415 |
by (hyp_subst_tac 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
416 |
by (rotate_tac ~1 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
417 |
by (asm_full_simp_tac (!simpset addsimps [subset_insert_iff,finite_subset]) 1); |
3708 | 418 |
by (Blast_tac 1); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
419 |
(*2*) |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
420 |
by (rotate_tac ~1 1); |
3708 | 421 |
by (eres_inst_tac [("P","?a<?b")] notE 1); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
422 |
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
|
423 |
by (case_tac "A=F" 1); |
3708 | 424 |
by (ALLGOALS Asm_simp_tac); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
425 |
qed_spec_mp "psubset_card" ; |
3368 | 426 |
|
427 |
||
3430 | 428 |
(*Relates to equivalence classes. Based on a theorem of F. Kammueller's. |
3368 | 429 |
The "finite C" premise is redundant*) |
430 |
goal thy "!!C. finite C ==> finite (Union C) --> \ |
|
431 |
\ (! c : C. k dvd card c) --> \ |
|
432 |
\ (! c1: C. ! c2: C. c1 ~= c2 --> c1 Int c2 = {}) \ |
|
433 |
\ --> k dvd card(Union C)"; |
|
434 |
by (etac finite_induct 1); |
|
435 |
by (ALLGOALS Asm_simp_tac); |
|
3708 | 436 |
by (Clarify_tac 1); |
3368 | 437 |
by (stac card_Un_disjoint 1); |
438 |
by (ALLGOALS |
|
439 |
(asm_full_simp_tac (!simpset |
|
440 |
addsimps [dvd_add, disjoint_eq_subset_Compl]))); |
|
441 |
by (thin_tac "!c:F. ?PP(c)" 1); |
|
442 |
by (thin_tac "!c:F. ?PP(c) & ?QQ(c)" 1); |
|
3708 | 443 |
by (Clarify_tac 1); |
3368 | 444 |
by (ball_tac 1); |
445 |
by (Blast_tac 1); |
|
446 |
qed_spec_mp "dvd_partition"; |
|
447 |