author | paulson |
Mon, 22 May 2000 12:27:11 +0200 | |
changeset 8911 | c35b74bad518 |
parent 8889 | 2ec6371fde54 |
child 8963 | 0d4abacae6aa |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: HOL/Finite.thy |
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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 |
||
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9 |
section "finite"; |
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|
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(*Discharging ~ x:y entails extra work*) |
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val major::prems = Goal |
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nipkow
parents:
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"[| finite F; P({}); \ |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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\ !!F x. [| finite F; x ~: F; P(F) |] ==> P(insert x F) \ |
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\ |] ==> 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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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 |
|
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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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changeset
|
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)"; |
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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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parents:
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31 |
|
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nipkow
parents:
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|
32 |
Addsimps Finites.intrs; |
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|
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AddSIs Finites.intrs; |
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|
35 |
(*The union of two finite sets is finite*) |
|
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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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parents:
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qed "finite_UnI"; |
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|
41 |
(*Every subset of a finite set is finite*) |
|
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Goal "finite B ==> ALL A. A<=B --> finite A"; |
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by (etac finite_induct 1); |
44 |
by (Simp_tac 1); |
|
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by (safe_tac (claset() addSDs [subset_insert_iff RS iffD1])); |
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by (eres_inst_tac [("t","A")] (insert_Diff RS subst) 2); |
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by (ALLGOALS Asm_simp_tac); |
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val lemma = result(); |
49 |
||
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Goal "[| A<=B; finite B |] ==> finite A"; |
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by (dtac lemma 1); |
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by (Blast_tac 1); |
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parents:
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|
53 |
qed "finite_subset"; |
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|
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Goal "finite(F Un G) = (finite F & finite G)"; |
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by (blast_tac (claset() |
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addIs [inst "B" "?X Un ?Y" finite_subset, finite_UnI]) 1); |
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58 |
qed "finite_Un"; |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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AddIffs[finite_Un]; |
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|
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61 |
(*The converse obviously fails*) |
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Goal "finite F | finite G ==> finite(F Int G)"; |
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by (blast_tac (claset() addIs [finite_subset]) 1); |
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qed "finite_Int"; |
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|
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Addsimps [finite_Int]; |
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AddIs [finite_Int]; |
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|
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Goal "finite(insert a A) = finite A"; |
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by (stac insert_is_Un 1); |
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by (simp_tac (HOL_ss addsimps [finite_Un]) 1); |
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by (Blast_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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73 |
qed "finite_insert"; |
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Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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Addsimps[finite_insert]; |
1531 | 75 |
|
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(*The image of a finite set is finite *) |
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77 |
Goal "finite F ==> finite(h``F)"; |
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parents:
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|
78 |
by (etac finite_induct 1); |
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by (Simp_tac 1); |
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parents:
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|
80 |
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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|
81 |
qed "finite_imageI"; |
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|
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Goal "finite (range g) ==> finite (range (%x. f (g x)))"; |
84 |
by (Simp_tac 1); |
|
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by (etac finite_imageI 1); |
|
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qed "finite_range_imageI"; |
|
87 |
||
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val major::prems = Goal |
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|
89 |
"[| finite c; finite b; \ |
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\ P(b); \ |
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|
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\ !!x y. [| finite y; x:y; P(y) |] ==> P(y-{x}) \ |
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\ |] ==> c<=b --> P(b-c)"; |
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parents:
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|
93 |
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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(simpset() addsimps prems@[Diff_subset RS finite_subset]))); |
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val lemma = result(); |
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|
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val prems = Goal |
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|
100 |
"[| finite 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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|
101 |
\ P(A); \ |
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
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parents:
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|
102 |
\ !!a A. [| finite A; a:A; P(A) |] ==> P(A-{a}) \ |
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\ |] ==> P({})"; |
104 |
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)); |
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107 |
qed "finite_empty_induct"; |
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|
109 |
||
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(* finite B ==> finite (B - Ba) *) |
111 |
bind_thm ("finite_Diff", Diff_subset RS finite_subset); |
|
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Addsimps [finite_Diff]; |
113 |
||
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Goal "finite(A - insert a B) = finite(A-B)"; |
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by (stac Diff_insert 1); |
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by (case_tac "a : A-B" 1); |
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by (rtac (finite_insert RS sym RS trans) 1); |
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by (stac insert_Diff 1); |
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by (ALLGOALS Asm_full_simp_tac); |
120 |
qed "finite_Diff_insert"; |
|
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AddIffs [finite_Diff_insert]; |
|
122 |
||
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|
123 |
(*lemma merely for classical reasoner in the proof below: force_tac can't |
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|
124 |
prove it.*) |
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Goal "finite(A-{}) = finite A"; |
126 |
by (Simp_tac 1); |
|
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val lemma = result(); |
|
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|
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(*Lemma for proving finite_imageD*) |
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130 |
Goal "finite B ==> !A. f``A = B --> inj_on f A --> finite A"; |
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by (etac finite_induct 1); |
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132 |
by (ALLGOALS Asm_simp_tac); |
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by (Clarify_tac 1); |
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134 |
by (subgoal_tac "EX y:A. f y = x & F = f``(A-{y})" 1); |
3708 | 135 |
by (Clarify_tac 1); |
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by (full_simp_tac (simpset() addsimps [inj_on_def]) 1); |
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137 |
by (blast_tac (claset() addSDs [lemma RS iffD1]) 1); |
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by (thin_tac "ALL A. ?PP(A)" 1); |
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139 |
by (forward_tac [[equalityD2, insertI1] MRS subsetD] 1); |
3708 | 140 |
by (Clarify_tac 1); |
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by (res_inst_tac [("x","xa")] bexI 1); |
4059 | 142 |
by (ALLGOALS |
4830 | 143 |
(asm_full_simp_tac (simpset() addsimps [inj_on_image_set_diff]))); |
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val lemma = result(); |
145 |
||
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146 |
Goal "[| finite(f``A); inj_on f A |] ==> finite A"; |
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by (dtac lemma 1); |
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by (Blast_tac 1); |
149 |
qed "finite_imageD"; |
|
150 |
||
4014 | 151 |
(** The finite UNION of finite sets **) |
152 |
||
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Goal "finite A ==> (!a:A. finite(B a)) --> finite(UN a:A. B a)"; |
154 |
by (etac finite_induct 1); |
|
4153 | 155 |
by (ALLGOALS Asm_simp_tac); |
4014 | 156 |
bind_thm("finite_UnionI", ballI RSN (2, result() RS mp)); |
157 |
Addsimps [finite_UnionI]; |
|
158 |
||
159 |
(** Sigma of finite sets **) |
|
160 |
||
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Goalw [Sigma_def] |
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"[| finite A; !a:A. finite(B a) |] ==> finite(SIGMA a:A. B a)"; |
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by (blast_tac (claset() addSIs [finite_UnionI]) 1); |
4014 | 164 |
bind_thm("finite_SigmaI", ballI RSN (2,result())); |
165 |
Addsimps [finite_SigmaI]; |
|
3368 | 166 |
|
8262 | 167 |
Goal "[| finite (UNIV::'a set); finite (UNIV::'b set)|] ==> finite (UNIV::('a * 'b) set)"; |
168 |
by (subgoal_tac "(UNIV::('a * 'b) set) = Sigma UNIV (%x. UNIV)" 1); |
|
169 |
by (etac ssubst 1); |
|
170 |
by (etac finite_SigmaI 1); |
|
171 |
by Auto_tac; |
|
172 |
qed "finite_Prod_UNIV"; |
|
173 |
||
174 |
Goal "finite (UNIV :: ('a::finite * 'b::finite) set)"; |
|
8320 | 175 |
by (rtac (finite_Prod_UNIV) 1); |
176 |
by (rtac finite 1); |
|
177 |
by (rtac finite 1); |
|
8262 | 178 |
qed "finite_Prod"; |
179 |
||
3368 | 180 |
(** The powerset of a finite set **) |
181 |
||
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182 |
Goal "finite(Pow A) ==> finite A"; |
3368 | 183 |
by (subgoal_tac "finite ((%x.{x})``A)" 1); |
3457 | 184 |
by (rtac finite_subset 2); |
185 |
by (assume_tac 3); |
|
3368 | 186 |
by (ALLGOALS |
4830 | 187 |
(fast_tac (claset() addSDs [rewrite_rule [inj_on_def] finite_imageD]))); |
3368 | 188 |
val lemma = result(); |
189 |
||
5069 | 190 |
Goal "finite(Pow A) = finite A"; |
3457 | 191 |
by (rtac iffI 1); |
192 |
by (etac lemma 1); |
|
3368 | 193 |
(*Opposite inclusion: finite A ==> finite (Pow A) *) |
3340 | 194 |
by (etac finite_induct 1); |
195 |
by (ALLGOALS |
|
196 |
(asm_simp_tac |
|
4089 | 197 |
(simpset() addsimps [finite_UnI, finite_imageI, Pow_insert]))); |
3368 | 198 |
qed "finite_Pow_iff"; |
199 |
AddIffs [finite_Pow_iff]; |
|
3340 | 200 |
|
5069 | 201 |
Goal "finite(r^-1) = finite r"; |
3457 | 202 |
by (subgoal_tac "r^-1 = (%(x,y).(y,x))``r" 1); |
203 |
by (Asm_simp_tac 1); |
|
204 |
by (rtac iffI 1); |
|
4830 | 205 |
by (etac (rewrite_rule [inj_on_def] finite_imageD) 1); |
206 |
by (simp_tac (simpset() addsplits [split_split]) 1); |
|
3457 | 207 |
by (etac finite_imageI 1); |
4746 | 208 |
by (simp_tac (simpset() addsimps [converse_def,image_def]) 1); |
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209 |
by Auto_tac; |
5516 | 210 |
by (rtac bexI 1); |
211 |
by (assume_tac 2); |
|
4763 | 212 |
by (Simp_tac 1); |
4746 | 213 |
qed "finite_converse"; |
214 |
AddIffs [finite_converse]; |
|
1531 | 215 |
|
8889 | 216 |
(* A bounded set of natural numbers is finite *) |
217 |
Goal "!N. (!i:N. i<(n::nat)) --> finite N"; |
|
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by (induct_tac "n" 1); |
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219 |
by (Simp_tac 1); |
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by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1); |
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221 |
by (rtac allI 1); |
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222 |
by (case_tac "n : N" 1); |
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223 |
by (ftac insert_Diff 1); |
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224 |
by (etac subst 1); |
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225 |
by (rtac impI 1); |
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226 |
by (rtac (finite_insert RS iffD2) 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
227 |
by (EVERY1[etac allE, etac mp]); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
228 |
by (Blast_tac 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
229 |
by (Blast_tac 1); |
8889 | 230 |
qed_spec_mp "bounded_nat_set_is_finite"; |
231 |
||
232 |
||
1548 | 233 |
section "Finite cardinality -- 'card'"; |
1531 | 234 |
|
5626 | 235 |
(* Ugly proofs for the traditional definition |
236 |
||
5316 | 237 |
Goal "{f i |i. (P i | i=n)} = insert (f n) {f i|i. P i}"; |
2922 | 238 |
by (Blast_tac 1); |
1531 | 239 |
val Collect_conv_insert = result(); |
240 |
||
5069 | 241 |
Goalw [card_def] "card {} = 0"; |
1553 | 242 |
by (rtac Least_equality 1); |
243 |
by (ALLGOALS Asm_full_simp_tac); |
|
1531 | 244 |
qed "card_empty"; |
245 |
Addsimps [card_empty]; |
|
246 |
||
5316 | 247 |
Goal "finite A ==> ? (n::nat) f. A = {f i |i. i<n}"; |
248 |
by (etac finite_induct 1); |
|
1553 | 249 |
by (res_inst_tac [("x","0")] exI 1); |
250 |
by (Simp_tac 1); |
|
251 |
by (etac exE 1); |
|
252 |
by (etac exE 1); |
|
253 |
by (hyp_subst_tac 1); |
|
254 |
by (res_inst_tac [("x","Suc n")] exI 1); |
|
255 |
by (res_inst_tac [("x","%i. if i<n then f i else x")] exI 1); |
|
4089 | 256 |
by (asm_simp_tac (simpset() addsimps [Collect_conv_insert, less_Suc_eq] |
1548 | 257 |
addcongs [rev_conj_cong]) 1); |
1531 | 258 |
qed "finite_has_card"; |
259 |
||
5278 | 260 |
Goal "[| x ~: A; insert x A = {f i|i. i<n} |] \ |
261 |
\ ==> ? m::nat. m<n & (? g. A = {g i|i. i<m})"; |
|
8442
96023903c2df
case_tac now subsumes both boolean and datatype cases;
wenzelm
parents:
8423
diff
changeset
|
262 |
by (case_tac "n" 1); |
1553 | 263 |
by (hyp_subst_tac 1); |
264 |
by (Asm_full_simp_tac 1); |
|
265 |
by (rename_tac "m" 1); |
|
266 |
by (hyp_subst_tac 1); |
|
267 |
by (case_tac "? a. a:A" 1); |
|
268 |
by (res_inst_tac [("x","0")] exI 2); |
|
269 |
by (Simp_tac 2); |
|
2922 | 270 |
by (Blast_tac 2); |
1553 | 271 |
by (etac exE 1); |
4089 | 272 |
by (simp_tac (simpset() addsimps [less_Suc_eq]) 1); |
1553 | 273 |
by (rtac exI 1); |
1782 | 274 |
by (rtac (refl RS disjI2 RS conjI) 1); |
1553 | 275 |
by (etac equalityE 1); |
276 |
by (asm_full_simp_tac |
|
4089 | 277 |
(simpset() addsimps [subset_insert,Collect_conv_insert, less_Suc_eq]) 1); |
4153 | 278 |
by Safe_tac; |
1553 | 279 |
by (Asm_full_simp_tac 1); |
280 |
by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1); |
|
4153 | 281 |
by (SELECT_GOAL Safe_tac 1); |
1553 | 282 |
by (subgoal_tac "x ~= f m" 1); |
2922 | 283 |
by (Blast_tac 2); |
1553 | 284 |
by (subgoal_tac "? k. f k = x & k<m" 1); |
2922 | 285 |
by (Blast_tac 2); |
4153 | 286 |
by (SELECT_GOAL Safe_tac 1); |
1553 | 287 |
by (res_inst_tac [("x","k")] exI 1); |
288 |
by (Asm_simp_tac 1); |
|
4686 | 289 |
by (Simp_tac 1); |
2922 | 290 |
by (Blast_tac 1); |
3457 | 291 |
by (dtac sym 1); |
1553 | 292 |
by (rotate_tac ~1 1); |
293 |
by (Asm_full_simp_tac 1); |
|
294 |
by (res_inst_tac [("x","%i. if f i = f m then a else f i")] exI 1); |
|
4153 | 295 |
by (SELECT_GOAL Safe_tac 1); |
1553 | 296 |
by (subgoal_tac "x ~= f m" 1); |
2922 | 297 |
by (Blast_tac 2); |
1553 | 298 |
by (subgoal_tac "? k. f k = x & k<m" 1); |
2922 | 299 |
by (Blast_tac 2); |
4153 | 300 |
by (SELECT_GOAL Safe_tac 1); |
1553 | 301 |
by (res_inst_tac [("x","k")] exI 1); |
302 |
by (Asm_simp_tac 1); |
|
4686 | 303 |
by (Simp_tac 1); |
2922 | 304 |
by (Blast_tac 1); |
1553 | 305 |
by (res_inst_tac [("x","%j. if f j = f i then f m else f j")] exI 1); |
4153 | 306 |
by (SELECT_GOAL Safe_tac 1); |
1553 | 307 |
by (subgoal_tac "x ~= f i" 1); |
2922 | 308 |
by (Blast_tac 2); |
1553 | 309 |
by (case_tac "x = f m" 1); |
310 |
by (res_inst_tac [("x","i")] exI 1); |
|
311 |
by (Asm_simp_tac 1); |
|
312 |
by (subgoal_tac "? k. f k = x & k<m" 1); |
|
2922 | 313 |
by (Blast_tac 2); |
4153 | 314 |
by (SELECT_GOAL Safe_tac 1); |
1553 | 315 |
by (res_inst_tac [("x","k")] exI 1); |
316 |
by (Asm_simp_tac 1); |
|
4686 | 317 |
by (Simp_tac 1); |
2922 | 318 |
by (Blast_tac 1); |
1531 | 319 |
val lemma = result(); |
320 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
321 |
Goal "[| finite A; x ~: A |] ==> \ |
3842 | 322 |
\ (LEAST n. ? f. insert x A = {f i|i. i<n}) = Suc(LEAST n. ? f. A={f i|i. i<n})"; |
1553 | 323 |
by (rtac Least_equality 1); |
3457 | 324 |
by (dtac finite_has_card 1); |
325 |
by (etac exE 1); |
|
3842 | 326 |
by (dres_inst_tac [("P","%n.? f. A={f i|i. i<n}")] LeastI 1); |
3457 | 327 |
by (etac exE 1); |
1553 | 328 |
by (res_inst_tac |
1531 | 329 |
[("x","%i. if i<(LEAST n. ? f. A={f i |i. i < n}) then f i else x")] exI 1); |
1553 | 330 |
by (simp_tac |
4089 | 331 |
(simpset() addsimps [Collect_conv_insert, less_Suc_eq] |
2031 | 332 |
addcongs [rev_conj_cong]) 1); |
3457 | 333 |
by (etac subst 1); |
334 |
by (rtac refl 1); |
|
1553 | 335 |
by (rtac notI 1); |
336 |
by (etac exE 1); |
|
337 |
by (dtac lemma 1); |
|
3457 | 338 |
by (assume_tac 1); |
1553 | 339 |
by (etac exE 1); |
340 |
by (etac conjE 1); |
|
341 |
by (dres_inst_tac [("P","%x. ? g. A = {g i |i. i < x}")] Least_le 1); |
|
342 |
by (dtac le_less_trans 1 THEN atac 1); |
|
4089 | 343 |
by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1); |
1553 | 344 |
by (etac disjE 1); |
345 |
by (etac less_asym 1 THEN atac 1); |
|
346 |
by (hyp_subst_tac 1); |
|
347 |
by (Asm_full_simp_tac 1); |
|
1531 | 348 |
val lemma = result(); |
349 |
||
5416
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
350 |
Goalw [card_def] "[| finite A; x ~: A |] ==> card(insert x A) = Suc(card A)"; |
1553 | 351 |
by (etac lemma 1); |
352 |
by (assume_tac 1); |
|
1531 | 353 |
qed "card_insert_disjoint"; |
3352 | 354 |
Addsimps [card_insert_disjoint]; |
5626 | 355 |
*) |
356 |
||
6141 | 357 |
val cardR_emptyE = cardR.mk_cases "({},n) : cardR"; |
5626 | 358 |
AddSEs [cardR_emptyE]; |
6141 | 359 |
val cardR_insertE = cardR.mk_cases "(insert a A,n) : cardR"; |
5626 | 360 |
AddSIs cardR.intrs; |
361 |
||
362 |
Goal "[| (A,n) : cardR |] ==> a : A --> (? m. n = Suc m)"; |
|
6162 | 363 |
by (etac cardR.induct 1); |
364 |
by (Blast_tac 1); |
|
365 |
by (Blast_tac 1); |
|
5626 | 366 |
qed "cardR_SucD"; |
367 |
||
368 |
Goal "(A,m): cardR ==> (!n a. m = Suc n --> a:A --> (A-{a},n) : cardR)"; |
|
6162 | 369 |
by (etac cardR.induct 1); |
8911
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
370 |
by Auto_tac; |
6162 | 371 |
by (asm_simp_tac (simpset() addsimps [insert_Diff_if]) 1); |
8911
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
372 |
by Auto_tac; |
7499 | 373 |
by (ftac cardR_SucD 1); |
6162 | 374 |
by (Blast_tac 1); |
5626 | 375 |
val lemma = result(); |
376 |
||
377 |
Goal "[| (insert a A, Suc m) : cardR; a ~: A |] ==> (A,m) : cardR"; |
|
6162 | 378 |
by (dtac lemma 1); |
379 |
by (Asm_full_simp_tac 1); |
|
5626 | 380 |
val lemma = result(); |
381 |
||
382 |
Goal "(A,m): cardR ==> (!n. (A,n) : cardR --> n=m)"; |
|
6162 | 383 |
by (etac cardR.induct 1); |
384 |
by (safe_tac (claset() addSEs [cardR_insertE])); |
|
385 |
by (rename_tac "B b m" 1); |
|
386 |
by (case_tac "a = b" 1); |
|
387 |
by (subgoal_tac "A = B" 1); |
|
388 |
by (blast_tac (claset() addEs [equalityE]) 2); |
|
389 |
by (Blast_tac 1); |
|
390 |
by (subgoal_tac "? C. A = insert b C & B = insert a C" 1); |
|
391 |
by (res_inst_tac [("x","A Int B")] exI 2); |
|
392 |
by (blast_tac (claset() addEs [equalityE]) 2); |
|
393 |
by (forw_inst_tac [("A","B")] cardR_SucD 1); |
|
394 |
by (blast_tac (claset() addDs [lemma]) 1); |
|
5626 | 395 |
qed_spec_mp "cardR_determ"; |
396 |
||
397 |
Goal "(A,n) : cardR ==> finite(A)"; |
|
398 |
by (etac cardR.induct 1); |
|
399 |
by Auto_tac; |
|
400 |
qed "cardR_imp_finite"; |
|
401 |
||
402 |
Goal "finite(A) ==> EX n. (A, n) : cardR"; |
|
403 |
by (etac finite_induct 1); |
|
404 |
by Auto_tac; |
|
405 |
qed "finite_imp_cardR"; |
|
406 |
||
407 |
Goalw [card_def] "(A,n) : cardR ==> card A = n"; |
|
408 |
by (blast_tac (claset() addIs [cardR_determ]) 1); |
|
409 |
qed "card_equality"; |
|
410 |
||
411 |
Goalw [card_def] "card {} = 0"; |
|
412 |
by (Blast_tac 1); |
|
413 |
qed "card_empty"; |
|
414 |
Addsimps [card_empty]; |
|
415 |
||
416 |
Goal "x ~: A ==> \ |
|
417 |
\ ((insert x A, n) : cardR) = \ |
|
418 |
\ (EX m. (A, m) : cardR & n = Suc m)"; |
|
419 |
by Auto_tac; |
|
420 |
by (res_inst_tac [("A1", "A")] (finite_imp_cardR RS exE) 1); |
|
421 |
by (force_tac (claset() addDs [cardR_imp_finite], simpset()) 1); |
|
422 |
by (blast_tac (claset() addIs [cardR_determ]) 1); |
|
423 |
val lemma = result(); |
|
424 |
||
425 |
Goalw [card_def] |
|
426 |
"[| finite A; x ~: A |] ==> card (insert x A) = Suc(card A)"; |
|
427 |
by (asm_simp_tac (simpset() addsimps [lemma]) 1); |
|
428 |
by (rtac select_equality 1); |
|
429 |
by (auto_tac (claset() addIs [finite_imp_cardR], |
|
430 |
simpset() addcongs [conj_cong] |
|
431 |
addsimps [symmetric card_def, |
|
432 |
card_equality])); |
|
433 |
qed "card_insert_disjoint"; |
|
434 |
Addsimps [card_insert_disjoint]; |
|
435 |
||
436 |
(* Delete rules to do with cardR relation: obsolete *) |
|
437 |
Delrules [cardR_emptyE]; |
|
438 |
Delrules cardR.intrs; |
|
439 |
||
7958 | 440 |
Goal "finite A ==> (card A = 0) = (A = {})"; |
441 |
by Auto_tac; |
|
442 |
by (dres_inst_tac [("a","x")] mk_disjoint_insert 1); |
|
443 |
by (Clarify_tac 1); |
|
444 |
by (rotate_tac ~1 1); |
|
445 |
by Auto_tac; |
|
446 |
qed "card_0_eq"; |
|
447 |
Addsimps[card_0_eq]; |
|
448 |
||
5626 | 449 |
Goal "finite A ==> card(insert x A) = (if x:A then card A else Suc(card(A)))"; |
450 |
by (asm_simp_tac (simpset() addsimps [insert_absorb]) 1); |
|
451 |
qed "card_insert_if"; |
|
452 |
||
7821 | 453 |
Goal "[| finite A; x: A |] ==> Suc (card (A-{x})) = card A"; |
5626 | 454 |
by (res_inst_tac [("t", "A")] (insert_Diff RS subst) 1); |
455 |
by (assume_tac 1); |
|
456 |
by (Asm_simp_tac 1); |
|
457 |
qed "card_Suc_Diff1"; |
|
458 |
||
7821 | 459 |
Goal "[| finite A; x: A |] ==> card (A-{x}) = card A - 1"; |
460 |
by (asm_simp_tac (simpset() addsimps [card_Suc_Diff1 RS sym]) 1); |
|
461 |
qed "card_Diff_singleton"; |
|
462 |
||
5626 | 463 |
Goal "finite A ==> card(insert x A) = Suc(card(A-{x}))"; |
464 |
by (asm_simp_tac (simpset() addsimps [card_insert_if,card_Suc_Diff1]) 1); |
|
465 |
qed "card_insert"; |
|
3352 | 466 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
467 |
Goal "finite A ==> card A <= card (insert x A)"; |
5626 | 468 |
by (asm_simp_tac (simpset() addsimps [card_insert_if]) 1); |
4768
c342d63173e9
New theorems card_Diff_le and card_insert_le; tidied
paulson
parents:
4763
diff
changeset
|
469 |
qed "card_insert_le"; |
c342d63173e9
New theorems card_Diff_le and card_insert_le; tidied
paulson
parents:
4763
diff
changeset
|
470 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
471 |
Goal "finite A ==> !B. B <= A --> card(B) <= card(A)"; |
3352 | 472 |
by (etac finite_induct 1); |
473 |
by (Simp_tac 1); |
|
3708 | 474 |
by (Clarify_tac 1); |
3352 | 475 |
by (case_tac "x:B" 1); |
3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
diff
changeset
|
476 |
by (dres_inst_tac [("A","B")] mk_disjoint_insert 1); |
8741
61bc5ed22b62
removal of less_SucI, le_SucI from default simpset
paulson
parents:
8554
diff
changeset
|
477 |
by (asm_full_simp_tac (simpset() addsimps [le_SucI, subset_insert_iff]) 2); |
61bc5ed22b62
removal of less_SucI, le_SucI from default simpset
paulson
parents:
8554
diff
changeset
|
478 |
by (force_tac (claset(), |
61bc5ed22b62
removal of less_SucI, le_SucI from default simpset
paulson
parents:
8554
diff
changeset
|
479 |
simpset() addsimps [subset_insert_iff, finite_subset] |
61bc5ed22b62
removal of less_SucI, le_SucI from default simpset
paulson
parents:
8554
diff
changeset
|
480 |
delsimps [insert_subset]) 1); |
3352 | 481 |
qed_spec_mp "card_mono"; |
482 |
||
5416
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
483 |
|
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
484 |
Goal "[| finite A; finite B |] \ |
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
485 |
\ ==> card A + card B = card (A Un B) + card (A Int B)"; |
3352 | 486 |
by (etac finite_induct 1); |
5416
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
487 |
by (Simp_tac 1); |
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
488 |
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
|
489 |
qed "card_Un_Int"; |
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
490 |
|
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
491 |
Goal "[| finite A; finite B; A Int B = {} |] \ |
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
492 |
\ ==> card (A Un B) = card A + card B"; |
9f029e382b5d
New law card_Un_Int. Removed card_insert from simpset
paulson
parents:
5413
diff
changeset
|
493 |
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
|
494 |
qed "card_Un_disjoint"; |
3352 | 495 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
496 |
Goal "[| finite A; B<=A |] ==> card A - card B = card (A - B)"; |
3352 | 497 |
by (subgoal_tac "(A-B) Un B = A" 1); |
498 |
by (Blast_tac 2); |
|
3457 | 499 |
by (rtac (add_right_cancel RS iffD1) 1); |
500 |
by (rtac (card_Un_disjoint RS subst) 1); |
|
501 |
by (etac ssubst 4); |
|
3352 | 502 |
by (Blast_tac 3); |
503 |
by (ALLGOALS |
|
504 |
(asm_simp_tac |
|
4089 | 505 |
(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
|
506 |
add_diff_inverse, card_mono, finite_subset]))); |
3352 | 507 |
qed "card_Diff_subset"; |
1531 | 508 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
509 |
Goal "[| finite A; x: A |] ==> card(A-{x}) < card A"; |
2031 | 510 |
by (rtac Suc_less_SucD 1); |
5626 | 511 |
by (asm_simp_tac (simpset() addsimps [card_Suc_Diff1]) 1); |
512 |
qed "card_Diff1_less"; |
|
1618 | 513 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
514 |
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
|
515 |
by (case_tac "x: A" 1); |
5626 | 516 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [card_Diff1_less, less_imp_le]))); |
517 |
qed "card_Diff1_le"; |
|
1531 | 518 |
|
5148
74919e8f221c
More tidying and removal of "\!\!... from Goal commands
paulson
parents:
5143
diff
changeset
|
519 |
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
|
520 |
by (etac finite_induct 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
521 |
by (Simp_tac 1); |
3708 | 522 |
by (Clarify_tac 1); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
523 |
by (case_tac "x:A" 1); |
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
524 |
(*1*) |
3413
c1f63cc3a768
Finite.ML Finite.thy: Replaced `finite subset of' by mere `finite'.
nipkow
parents:
3389
diff
changeset
|
525 |
by (dres_inst_tac [("A","A")]mk_disjoint_insert 1); |
4775 | 526 |
by (Clarify_tac 1); |
527 |
by (rotate_tac ~3 1); |
|
528 |
by (asm_full_simp_tac (simpset() addsimps [finite_subset]) 1); |
|
3708 | 529 |
by (Blast_tac 1); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
530 |
(*2*) |
3708 | 531 |
by (eres_inst_tac [("P","?a<?b")] notE 1); |
4775 | 532 |
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
|
533 |
by (case_tac "A=F" 1); |
8741
61bc5ed22b62
removal of less_SucI, le_SucI from default simpset
paulson
parents:
8554
diff
changeset
|
534 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_SucI]))); |
3222
726a9b069947
Distributed Psubset stuff to basic set theory files, incl Finite.
nipkow
parents:
2922
diff
changeset
|
535 |
qed_spec_mp "psubset_card" ; |
3368 | 536 |
|
7821 | 537 |
Goal "[| A <= B; card B <= card A; finite B |] ==> A = B"; |
5626 | 538 |
by (case_tac "A < B" 1); |
7497 | 539 |
by (datac psubset_card 1 1); |
5626 | 540 |
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [psubset_eq]))); |
541 |
qed "card_seteq"; |
|
542 |
||
543 |
Goal "[| finite B; A <= B; card A < card B |] ==> A < B"; |
|
544 |
by (etac psubsetI 1); |
|
545 |
by (Blast_tac 1); |
|
546 |
qed "card_psubset"; |
|
547 |
||
548 |
(*** Cardinality of image ***) |
|
549 |
||
550 |
Goal "finite A ==> card (f `` A) <= card A"; |
|
551 |
by (etac finite_induct 1); |
|
552 |
by (Simp_tac 1); |
|
8741
61bc5ed22b62
removal of less_SucI, le_SucI from default simpset
paulson
parents:
8554
diff
changeset
|
553 |
by (asm_simp_tac (simpset() addsimps [le_SucI,finite_imageI,card_insert_if]) 1); |
5626 | 554 |
qed "card_image_le"; |
555 |
||
556 |
Goal "finite(A) ==> inj_on f A --> card (f `` A) = card A"; |
|
557 |
by (etac finite_induct 1); |
|
558 |
by (ALLGOALS Asm_simp_tac); |
|
559 |
by Safe_tac; |
|
560 |
by (rewtac inj_on_def); |
|
561 |
by (Blast_tac 1); |
|
562 |
by (stac card_insert_disjoint 1); |
|
563 |
by (etac finite_imageI 1); |
|
564 |
by (Blast_tac 1); |
|
565 |
by (Blast_tac 1); |
|
566 |
qed_spec_mp "card_image"; |
|
567 |
||
568 |
Goal "[| finite A; f``A <= A; inj_on f A |] ==> f``A = A"; |
|
7497 | 569 |
by (etac card_seteq 1); |
570 |
by (dtac (card_image RS sym) 1); |
|
571 |
by Auto_tac; |
|
5626 | 572 |
qed "endo_inj_surj"; |
573 |
||
574 |
(*** Cardinality of the Powerset ***) |
|
575 |
||
576 |
Goal "finite A ==> card (Pow A) = 2 ^ card A"; |
|
577 |
by (etac finite_induct 1); |
|
578 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [Pow_insert]))); |
|
579 |
by (stac card_Un_disjoint 1); |
|
580 |
by (EVERY (map (blast_tac (claset() addIs [finite_imageI])) [3,2,1])); |
|
581 |
by (subgoal_tac "inj_on (insert x) (Pow F)" 1); |
|
582 |
by (asm_simp_tac (simpset() addsimps [card_image, Pow_insert]) 1); |
|
583 |
by (rewtac inj_on_def); |
|
584 |
by (blast_tac (claset() addSEs [equalityE]) 1); |
|
585 |
qed "card_Pow"; |
|
586 |
||
3368 | 587 |
|
3430 | 588 |
(*Relates to equivalence classes. Based on a theorem of F. Kammueller's. |
3368 | 589 |
The "finite C" premise is redundant*) |
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5069
diff
changeset
|
590 |
Goal "finite C ==> finite (Union C) --> \ |
3368 | 591 |
\ (! c : C. k dvd card c) --> \ |
592 |
\ (! c1: C. ! c2: C. c1 ~= c2 --> c1 Int c2 = {}) \ |
|
593 |
\ --> k dvd card(Union C)"; |
|
594 |
by (etac finite_induct 1); |
|
595 |
by (ALLGOALS Asm_simp_tac); |
|
3708 | 596 |
by (Clarify_tac 1); |
3368 | 597 |
by (stac card_Un_disjoint 1); |
598 |
by (ALLGOALS |
|
4089 | 599 |
(asm_full_simp_tac (simpset() |
3368 | 600 |
addsimps [dvd_add, disjoint_eq_subset_Compl]))); |
601 |
by (thin_tac "!c:F. ?PP(c)" 1); |
|
602 |
by (thin_tac "!c:F. ?PP(c) & ?QQ(c)" 1); |
|
3708 | 603 |
by (Clarify_tac 1); |
3368 | 604 |
by (ball_tac 1); |
605 |
by (Blast_tac 1); |
|
606 |
qed_spec_mp "dvd_partition"; |
|
607 |
||
5616 | 608 |
|
609 |
(*** foldSet ***) |
|
610 |
||
6141 | 611 |
val empty_foldSetE = foldSet.mk_cases "({}, x) : foldSet f e"; |
5616 | 612 |
|
613 |
AddSEs [empty_foldSetE]; |
|
614 |
AddIs foldSet.intrs; |
|
615 |
||
616 |
Goal "[| (A-{x},y) : foldSet f e; x: A |] ==> (A, f x y) : foldSet f e"; |
|
617 |
by (etac (insert_Diff RS subst) 1 THEN resolve_tac foldSet.intrs 1); |
|
618 |
by Auto_tac; |
|
5626 | 619 |
qed "Diff1_foldSet"; |
5616 | 620 |
|
621 |
Goal "(A, x) : foldSet f e ==> finite(A)"; |
|
622 |
by (eresolve_tac [foldSet.induct] 1); |
|
623 |
by Auto_tac; |
|
624 |
qed "foldSet_imp_finite"; |
|
625 |
||
626 |
Addsimps [foldSet_imp_finite]; |
|
627 |
||
628 |
||
629 |
Goal "finite(A) ==> EX x. (A, x) : foldSet f e"; |
|
630 |
by (etac finite_induct 1); |
|
631 |
by Auto_tac; |
|
632 |
qed "finite_imp_foldSet"; |
|
633 |
||
634 |
||
635 |
Open_locale "LC"; |
|
636 |
||
5782
7559f116cb10
locales now implicitly quantify over free variables
paulson
parents:
5626
diff
changeset
|
637 |
val f_lcomm = thm "lcomm"; |
5616 | 638 |
|
639 |
||
640 |
Goal "ALL A x. card(A) < n --> (A, x) : foldSet f e --> \ |
|
641 |
\ (ALL y. (A, y) : foldSet f e --> y=x)"; |
|
642 |
by (induct_tac "n" 1); |
|
643 |
by (auto_tac (claset(), simpset() addsimps [less_Suc_eq])); |
|
644 |
by (etac foldSet.elim 1); |
|
645 |
by (Blast_tac 1); |
|
646 |
by (etac foldSet.elim 1); |
|
647 |
by (Blast_tac 1); |
|
648 |
by (Clarify_tac 1); |
|
649 |
(*force simplification of "card A < card (insert ...)"*) |
|
650 |
by (etac rev_mp 1); |
|
651 |
by (asm_simp_tac (simpset() addsimps [less_Suc_eq_le]) 1); |
|
652 |
by (rtac impI 1); |
|
653 |
(** LEVEL 10 **) |
|
654 |
by (rename_tac "Aa xa ya Ab xb yb" 1); |
|
655 |
by (case_tac "xa=xb" 1); |
|
656 |
by (subgoal_tac "Aa = Ab" 1); |
|
657 |
by (blast_tac (claset() addEs [equalityE]) 2); |
|
658 |
by (Blast_tac 1); |
|
659 |
(*case xa ~= xb*) |
|
660 |
by (subgoal_tac "Aa-{xb} = Ab-{xa} & xb : Aa & xa : Ab" 1); |
|
661 |
by (blast_tac (claset() addEs [equalityE]) 2); |
|
662 |
by (Clarify_tac 1); |
|
663 |
by (subgoal_tac "Aa = insert xb Ab - {xa}" 1); |
|
664 |
by (blast_tac (claset() addEs [equalityE]) 2); |
|
665 |
(** LEVEL 20 **) |
|
666 |
by (subgoal_tac "card Aa <= card Ab" 1); |
|
667 |
by (rtac (Suc_le_mono RS subst) 2); |
|
5626 | 668 |
by (asm_simp_tac (simpset() addsimps [card_Suc_Diff1]) 2); |
5616 | 669 |
by (res_inst_tac [("A1", "Aa-{xb}"), ("f1","f")] |
670 |
(finite_imp_foldSet RS exE) 1); |
|
671 |
by (blast_tac (claset() addIs [foldSet_imp_finite, finite_Diff]) 1); |
|
7499 | 672 |
by (ftac Diff1_foldSet 1 THEN assume_tac 1); |
5616 | 673 |
by (subgoal_tac "ya = f xb x" 1); |
674 |
by (Blast_tac 2); |
|
675 |
by (subgoal_tac "(Ab - {xa}, x) : foldSet f e" 1); |
|
676 |
by (Asm_full_simp_tac 2); |
|
677 |
by (subgoal_tac "yb = f xa x" 1); |
|
5626 | 678 |
by (blast_tac (claset() addDs [Diff1_foldSet]) 2); |
5616 | 679 |
by (asm_simp_tac (simpset() addsimps [f_lcomm]) 1); |
680 |
val lemma = result(); |
|
681 |
||
682 |
||
683 |
Goal "[| (A, x) : foldSet f e; (A, y) : foldSet f e |] ==> y=x"; |
|
684 |
by (blast_tac (claset() addIs [normalize_thm [RSspec, RSmp] lemma]) 1); |
|
685 |
qed "foldSet_determ"; |
|
686 |
||
687 |
Goalw [fold_def] "(A,y) : foldSet f e ==> fold f e A = y"; |
|
688 |
by (blast_tac (claset() addIs [foldSet_determ]) 1); |
|
689 |
qed "fold_equality"; |
|
690 |
||
691 |
Goalw [fold_def] "fold f e {} = e"; |
|
692 |
by (Blast_tac 1); |
|
693 |
qed "fold_empty"; |
|
694 |
Addsimps [fold_empty]; |
|
695 |
||
5626 | 696 |
|
5616 | 697 |
Goal "x ~: A ==> \ |
698 |
\ ((insert x A, v) : foldSet f e) = \ |
|
699 |
\ (EX y. (A, y) : foldSet f e & v = f x y)"; |
|
700 |
by Auto_tac; |
|
701 |
by (res_inst_tac [("A1", "A"), ("f1","f")] (finite_imp_foldSet RS exE) 1); |
|
702 |
by (force_tac (claset() addDs [foldSet_imp_finite], simpset()) 1); |
|
703 |
by (blast_tac (claset() addIs [foldSet_determ]) 1); |
|
704 |
val lemma = result(); |
|
705 |
||
706 |
Goalw [fold_def] |
|
707 |
"[| finite A; x ~: A |] ==> fold f e (insert x A) = f x (fold f e A)"; |
|
708 |
by (asm_simp_tac (simpset() addsimps [lemma]) 1); |
|
709 |
by (rtac select_equality 1); |
|
710 |
by (auto_tac (claset() addIs [finite_imp_foldSet], |
|
711 |
simpset() addcongs [conj_cong] |
|
712 |
addsimps [symmetric fold_def, |
|
713 |
fold_equality])); |
|
714 |
qed "fold_insert"; |
|
715 |
||
8911
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
716 |
Goal "finite A ==> ALL e. f x (fold f e A) = fold f (f x e) A"; |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
717 |
by (etac finite_induct 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
718 |
by (Simp_tac 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
719 |
by (asm_simp_tac (simpset() addsimps [f_lcomm, fold_insert]) 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
720 |
qed_spec_mp "fold_commute"; |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
721 |
|
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
722 |
Goal "[| finite A; finite B |] \ |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
723 |
\ ==> fold f (fold f e B) A = fold f (fold f e (A Int B)) (A Un B)"; |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
724 |
by (etac finite_induct 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
725 |
by (Simp_tac 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
726 |
by (asm_simp_tac (simpset() addsimps [fold_insert, fold_commute, |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
727 |
Int_insert_left, insert_absorb]) 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
728 |
qed "fold_nest_Un_Int"; |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
729 |
|
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
730 |
Goal "[| finite A; finite B; A Int B = {} |] \ |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
731 |
\ ==> fold f e (A Un B) = fold f (fold f e B) A"; |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
732 |
by (asm_simp_tac (simpset() addsimps [fold_nest_Un_Int]) 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
733 |
qed "fold_nest_Un_disjoint"; |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
734 |
|
5626 | 735 |
(* Delete rules to do with foldSet relation: obsolete *) |
736 |
Delsimps [foldSet_imp_finite]; |
|
737 |
Delrules [empty_foldSetE]; |
|
738 |
Delrules foldSet.intrs; |
|
739 |
||
6024 | 740 |
Close_locale "LC"; |
5616 | 741 |
|
742 |
Open_locale "ACe"; |
|
743 |
||
8911
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
744 |
(*We enter a more restrictive framework, with f :: ['a,'a] => 'a |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
745 |
instead of ['b,'a] => 'a |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
746 |
At present, none of these results are used!*) |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
747 |
|
5782
7559f116cb10
locales now implicitly quantify over free variables
paulson
parents:
5626
diff
changeset
|
748 |
val f_ident = thm "ident"; |
7559f116cb10
locales now implicitly quantify over free variables
paulson
parents:
5626
diff
changeset
|
749 |
val f_commute = thm "commute"; |
7559f116cb10
locales now implicitly quantify over free variables
paulson
parents:
5626
diff
changeset
|
750 |
val f_assoc = thm "assoc"; |
5616 | 751 |
|
752 |
||
753 |
Goal "f x (f y z) = f y (f x z)"; |
|
754 |
by (rtac (f_commute RS trans) 1); |
|
755 |
by (rtac (f_assoc RS trans) 1); |
|
756 |
by (rtac (f_commute RS arg_cong) 1); |
|
757 |
qed "f_left_commute"; |
|
758 |
||
759 |
val f_ac = [f_assoc, f_commute, f_left_commute]; |
|
760 |
||
761 |
Goal "f e x = x"; |
|
762 |
by (stac f_commute 1); |
|
763 |
by (rtac f_ident 1); |
|
764 |
qed "f_left_ident"; |
|
765 |
||
766 |
val f_idents = [f_left_ident, f_ident]; |
|
767 |
||
768 |
Goal "[| finite A; finite B |] \ |
|
769 |
\ ==> f (fold f e A) (fold f e B) = \ |
|
770 |
\ f (fold f e (A Un B)) (fold f e (A Int B))"; |
|
771 |
by (etac finite_induct 1); |
|
772 |
by (simp_tac (simpset() addsimps f_idents) 1); |
|
773 |
by (asm_simp_tac (simpset() addsimps f_ac @ f_idents @ |
|
774 |
[export fold_insert,insert_absorb, Int_insert_left]) 1); |
|
775 |
qed "fold_Un_Int"; |
|
776 |
||
777 |
Goal "[| finite A; finite B; A Int B = {} |] \ |
|
778 |
\ ==> fold f e (A Un B) = f (fold f e A) (fold f e B)"; |
|
779 |
by (asm_simp_tac (simpset() addsimps fold_Un_Int::f_idents) 1); |
|
780 |
qed "fold_Un_disjoint"; |
|
781 |
||
782 |
Goal |
|
783 |
"[| finite A; finite B |] ==> A Int B = {} --> \ |
|
784 |
\ fold (f o g) e (A Un B) = f (fold (f o g) e A) (fold (f o g) e B)"; |
|
785 |
by (etac finite_induct 1); |
|
786 |
by (simp_tac (simpset() addsimps f_idents) 1); |
|
787 |
by (asm_full_simp_tac (simpset() addsimps f_ac @ f_idents @ |
|
788 |
[export fold_insert,insert_absorb, Int_insert_left]) 1); |
|
789 |
qed "fold_Un_disjoint2"; |
|
790 |
||
6024 | 791 |
Close_locale "ACe"; |
5616 | 792 |
|
793 |
||
794 |
(*** setsum ***) |
|
795 |
||
796 |
Goalw [setsum_def] "setsum f {} = 0"; |
|
6162 | 797 |
by (Simp_tac 1); |
5616 | 798 |
qed "setsum_empty"; |
799 |
Addsimps [setsum_empty]; |
|
800 |
||
801 |
Goalw [setsum_def] |
|
802 |
"[| finite F; a ~: F |] ==> setsum f (insert a F) = f(a) + setsum f F"; |
|
6162 | 803 |
by (asm_simp_tac (simpset() addsimps [export fold_insert]) 1); |
5616 | 804 |
qed "setsum_insert"; |
805 |
Addsimps [setsum_insert]; |
|
806 |
||
8911
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
807 |
(*Could allow many "card" proofs to be simplified*) |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
808 |
Goal "finite A ==> card A = setsum (%x. 1) A"; |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
809 |
by (etac finite_induct 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
810 |
by Auto_tac; |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
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parents:
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diff
changeset
|
811 |
qed "card_eq_setsum"; |
5616 | 812 |
|
8911
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
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diff
changeset
|
813 |
Goal "[| finite A; finite B |] \ |
c35b74bad518
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parents:
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changeset
|
814 |
\ ==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"; |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
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parents:
8889
diff
changeset
|
815 |
by (etac finite_induct 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
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parents:
8889
diff
changeset
|
816 |
by (Simp_tac 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
817 |
by (asm_full_simp_tac (simpset() addsimps [Int_insert_left, insert_absorb, |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
818 |
add_commute]) 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
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diff
changeset
|
819 |
qed "setsum_Un_Int"; |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
8889
diff
changeset
|
820 |
|
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
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changeset
|
821 |
Goal "[| finite A; finite B |] \ |
c35b74bad518
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parents:
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changeset
|
822 |
\ ==> setsum f (A Un B) = setsum f A + setsum f B - setsum f (A Int B)"; |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
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parents:
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|
823 |
by (stac (setsum_Un_Int RS sym) 1); |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
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parents:
8889
diff
changeset
|
824 |
by Auto_tac; |
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
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parents:
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changeset
|
825 |
qed "setsum_Un"; |
c35b74bad518
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parents:
8889
diff
changeset
|
826 |
|
c35b74bad518
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parents:
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changeset
|
827 |
Goal "finite F \ |
c35b74bad518
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diff
changeset
|
828 |
\ ==> setsum f (F-{a}) = (if a:F then setsum f F - f a else setsum f F)"; |
6162 | 829 |
by (etac finite_induct 1); |
830 |
by (auto_tac (claset(), simpset() addsimps [insert_Diff_if])); |
|
831 |
by (dres_inst_tac [("a","a")] mk_disjoint_insert 1); |
|
8911
c35b74bad518
fold_commute, fold_nest_Un_Int, setsum_Un and other new results
paulson
parents:
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changeset
|
832 |
by Auto_tac; |
5616 | 833 |
qed_spec_mp "setsum_diff1"; |
7834 | 834 |
|
8911
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parents:
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changeset
|
835 |
Goal "finite A ==> setsum (%x. f x + g x) A = setsum f A + setsum g A"; |
c35b74bad518
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paulson
parents:
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diff
changeset
|
836 |
by (etac finite_induct 1); |
c35b74bad518
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parents:
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diff
changeset
|
837 |
by Auto_tac; |
c35b74bad518
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parents:
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diff
changeset
|
838 |
qed "setsum_addf"; |
c35b74bad518
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parents:
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changeset
|
839 |
|
7834 | 840 |
|
841 |
(*** Basic theorem about "choose". By Florian Kammueller, tidied by LCP ***) |
|
842 |
||
843 |
Goal "finite S ==> (card S = 0) = (S = {})"; |
|
844 |
by (auto_tac (claset() addDs [card_Suc_Diff1], |
|
845 |
simpset())); |
|
846 |
qed "card_0_empty_iff"; |
|
847 |
||
848 |
Goal "finite A ==> card {B. B <= A & card B = 0} = 1"; |
|
849 |
by (asm_simp_tac (simpset() addcongs [conj_cong] |
|
850 |
addsimps [finite_subset RS card_0_empty_iff]) 1); |
|
851 |
by (simp_tac (simpset() addcongs [rev_conj_cong]) 1); |
|
852 |
qed "card_s_0_eq_empty"; |
|
853 |
||
854 |
Goal "[| finite M; x ~: M |] \ |
|
855 |
\ ==> {s. s <= insert x M & card(s) = Suc k} \ |
|
856 |
\ = {s. s <= M & card(s) = Suc k} Un \ |
|
857 |
\ {s. EX t. t <= M & card(t) = k & s = insert x t}"; |
|
858 |
by Safe_tac; |
|
859 |
by (auto_tac (claset() addIs [finite_subset RS card_insert_disjoint], |
|
860 |
simpset())); |
|
861 |
by (dres_inst_tac [("x","xa - {x}")] spec 1); |
|
862 |
by (subgoal_tac ("x ~: xa") 1); |
|
863 |
by Auto_tac; |
|
864 |
by (etac rev_mp 1 THEN stac card_Diff_singleton 1); |
|
7958 | 865 |
by (auto_tac (claset() addIs [finite_subset], simpset())); |
7834 | 866 |
qed "choose_deconstruct"; |
867 |
||
8140 | 868 |
Goal "[| finite(A); finite(B); f``A <= B; inj_on f A |] \ |
7834 | 869 |
\ ==> card A <= card B"; |
870 |
by (auto_tac (claset() addIs [card_mono], |
|
8140 | 871 |
simpset() addsimps [card_image RS sym])); |
7834 | 872 |
qed "card_inj_on_le"; |
873 |
||
874 |
Goal "[| finite A; finite B; \ |
|
8140 | 875 |
\ f``A <= B; inj_on f A; g``B <= A; inj_on g B |] \ |
7834 | 876 |
\ ==> card(A) = card(B)"; |
877 |
by (auto_tac (claset() addIs [le_anti_sym,card_inj_on_le], simpset())); |
|
878 |
qed "card_bij_eq"; |
|
879 |
||
880 |
Goal "[| finite M; x ~: M |] \ |
|
881 |
\ ==> card{s. EX t. t <= M & card(t) = k & s = insert x t} = \ |
|
882 |
\ card {s. s <= M & card(s) = k}"; |
|
8140 | 883 |
by (res_inst_tac [("f", "%s. s - {x}"), ("g","insert x")] card_bij_eq 1); |
7834 | 884 |
by (res_inst_tac [("B","Pow(insert x M)")] finite_subset 1); |
885 |
by (res_inst_tac [("B","Pow(M)")] finite_subset 3); |
|
8320 | 886 |
by (REPEAT(Fast_tac 1)); |
7834 | 887 |
(* arity *) |
8140 | 888 |
by (auto_tac (claset() addSEs [equalityE], simpset() addsimps [inj_on_def])); |
7834 | 889 |
by (stac Diff_insert0 1); |
890 |
by Auto_tac; |
|
891 |
qed "constr_bij"; |
|
892 |
||
893 |
(* Main theorem: combinatorial theorem about number of subsets of a set *) |
|
7842
6858c98385c3
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paulson
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7834
diff
changeset
|
894 |
Goal "(ALL A. finite A --> card {s. s <= A & card s = k} = (card A choose k))"; |
7834 | 895 |
by (induct_tac "k" 1); |
896 |
by (simp_tac (simpset() addsimps [card_s_0_eq_empty]) 1); |
|
897 |
(* first 0 case finished *) |
|
7842
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simplified and generalized n_sub_lemma and n_subsets
paulson
parents:
7834
diff
changeset
|
898 |
(* prepare finite set induction *) |
6858c98385c3
simplified and generalized n_sub_lemma and n_subsets
paulson
parents:
7834
diff
changeset
|
899 |
by (rtac allI 1 THEN rtac impI 1); |
7834 | 900 |
(* second induction *) |
901 |
by (etac finite_induct 1); |
|
7842
6858c98385c3
simplified and generalized n_sub_lemma and n_subsets
paulson
parents:
7834
diff
changeset
|
902 |
by (ALLGOALS |
6858c98385c3
simplified and generalized n_sub_lemma and n_subsets
paulson
parents:
7834
diff
changeset
|
903 |
(simp_tac (simpset() addcongs [conj_cong] addsimps [card_s_0_eq_empty]))); |
7834 | 904 |
by (stac choose_deconstruct 1); |
905 |
by (assume_tac 1); |
|
906 |
by (assume_tac 1); |
|
907 |
by (stac card_Un_disjoint 1); |
|
908 |
by (Force_tac 3); |
|
7842
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7834
diff
changeset
|
909 |
(** LEVEL 10 **) |
6858c98385c3
simplified and generalized n_sub_lemma and n_subsets
paulson
parents:
7834
diff
changeset
|
910 |
(* use bijection *) |
6858c98385c3
simplified and generalized n_sub_lemma and n_subsets
paulson
parents:
7834
diff
changeset
|
911 |
by (force_tac (claset(), simpset() addsimps [constr_bij]) 3); |
7834 | 912 |
(* finite goal *) |
913 |
by (res_inst_tac [("B","Pow F")] finite_subset 1); |
|
914 |
by (Blast_tac 1); |
|
915 |
by (etac (finite_Pow_iff RS iffD2) 1); |
|
916 |
(* finite goal *) |
|
917 |
by (res_inst_tac [("B","Pow (insert x F)")] finite_subset 1); |
|
918 |
by (Blast_tac 1); |
|
919 |
by (blast_tac (claset() addIs [finite_Pow_iff RS iffD2]) 1); |
|
920 |
qed "n_sub_lemma"; |
|
921 |
||
7842
6858c98385c3
simplified and generalized n_sub_lemma and n_subsets
paulson
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7834
diff
changeset
|
922 |
Goal "finite M ==> card {s. s <= M & card(s) = k} = ((card M) choose k)"; |
7834 | 923 |
by (asm_simp_tac (simpset() addsimps [n_sub_lemma]) 1); |
924 |
qed "n_subsets"; |