author | haftmann |
Tue, 13 Oct 2015 09:21:15 +0200 | |
changeset 61424 | c3658c18b7bc |
parent 60247 | 6a5015b096a2 |
child 62324 | ae44f16dcea5 |
permissions | -rw-r--r-- |
60247 | 1 |
(* Title: HOL/Cardinals/Bounded_Set.thy |
59747 | 2 |
Author: Dmitriy Traytel, TU Muenchen |
3 |
Copyright 2015 |
|
4 |
||
5 |
Bounded powerset type. |
|
6 |
*) |
|
7 |
||
8 |
section \<open>Sets Strictly Bounded by an Infinite Cardinal\<close> |
|
9 |
||
10 |
theory Bounded_Set |
|
11 |
imports Cardinals |
|
12 |
begin |
|
13 |
||
14 |
typedef ('a, 'k) bset ("_ set[_]" [22, 21] 21) = |
|
15 |
"{A :: 'a set. |A| <o natLeq +c |UNIV :: 'k set|}" |
|
16 |
morphisms set_bset Abs_bset |
|
17 |
by (rule exI[of _ "{}"]) (auto simp: card_of_empty4 csum_def) |
|
18 |
||
19 |
setup_lifting type_definition_bset |
|
20 |
||
21 |
lift_definition map_bset :: |
|
22 |
"('a \<Rightarrow> 'b) \<Rightarrow> 'a set['k] \<Rightarrow> 'b set['k]" is image |
|
23 |
using card_of_image ordLeq_ordLess_trans by blast |
|
24 |
||
25 |
lift_definition rel_bset :: |
|
26 |
"('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set['k] \<Rightarrow> 'b set['k] \<Rightarrow> bool" is rel_set |
|
27 |
. |
|
28 |
||
29 |
lift_definition bempty :: "'a set['k]" is "{}" |
|
30 |
by (auto simp: card_of_empty4 csum_def) |
|
31 |
||
32 |
lift_definition binsert :: "'a \<Rightarrow> 'a set['k] \<Rightarrow> 'a set['k]" is "insert" |
|
33 |
using infinite_card_of_insert ordIso_ordLess_trans Field_card_of Field_natLeq UNIV_Plus_UNIV |
|
34 |
csum_def finite_Plus_UNIV_iff finite_insert finite_ordLess_infinite2 infinite_UNIV_nat by metis |
|
35 |
||
36 |
definition bsingleton where |
|
37 |
"bsingleton x = binsert x bempty" |
|
38 |
||
39 |
lemma set_bset_to_set_bset: "|A| <o natLeq +c |UNIV :: 'k set| \<Longrightarrow> |
|
40 |
set_bset (the_inv set_bset A :: 'a set['k]) = A" |
|
41 |
apply (rule f_the_inv_into_f[unfolded inj_on_def]) |
|
42 |
apply (simp add: set_bset_inject range_eqI Abs_bset_inverse[symmetric]) |
|
43 |
apply (rule range_eqI Abs_bset_inverse[symmetric] CollectI)+ |
|
44 |
. |
|
45 |
||
46 |
lemma rel_bset_aux_infinite: |
|
47 |
fixes a :: "'a set['k]" and b :: "'b set['k]" |
|
48 |
shows "(\<forall>t \<in> set_bset a. \<exists>u \<in> set_bset b. R t u) \<and> (\<forall>u \<in> set_bset b. \<exists>t \<in> set_bset a. R t u) \<longleftrightarrow> |
|
49 |
((BNF_Def.Grp {a. set_bset a \<subseteq> {(a, b). R a b}} (map_bset fst))\<inverse>\<inverse> OO |
|
50 |
BNF_Def.Grp {a. set_bset a \<subseteq> {(a, b). R a b}} (map_bset snd)) a b" (is "?L \<longleftrightarrow> ?R") |
|
51 |
proof |
|
52 |
assume ?L |
|
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
60247
diff
changeset
|
53 |
def R' \<equiv> "the_inv set_bset (Collect (case_prod R) \<inter> (set_bset a \<times> set_bset b)) :: ('a \<times> 'b) set['k]" |
59747 | 54 |
(is "the_inv set_bset ?L'") |
55 |
have "|?L'| <o natLeq +c |UNIV :: 'k set|" |
|
56 |
unfolding csum_def Field_natLeq |
|
57 |
by (intro ordLeq_ordLess_trans[OF card_of_mono1[OF Int_lower2]] |
|
58 |
card_of_Times_ordLess_infinite) |
|
59 |
(simp, (transfer, simp add: csum_def Field_natLeq)+) |
|
60 |
hence *: "set_bset R' = ?L'" unfolding R'_def by (intro set_bset_to_set_bset) |
|
61 |
show ?R unfolding Grp_def relcompp.simps conversep.simps |
|
62 |
proof (intro CollectI case_prodI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl) |
|
63 |
from * show "a = map_bset fst R'" using conjunct1[OF `?L`] |
|
64 |
by (transfer, auto simp add: image_def Int_def split: prod.splits) |
|
65 |
from * show "b = map_bset snd R'" using conjunct2[OF `?L`] |
|
66 |
by (transfer, auto simp add: image_def Int_def split: prod.splits) |
|
67 |
qed (auto simp add: *) |
|
68 |
next |
|
69 |
assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps |
|
70 |
by transfer force |
|
71 |
qed |
|
72 |
||
73 |
bnf "'a set['k]" |
|
74 |
map: map_bset |
|
75 |
sets: set_bset |
|
76 |
bd: "natLeq +c |UNIV :: 'k set|" |
|
77 |
wits: bempty |
|
78 |
rel: rel_bset |
|
79 |
proof - |
|
80 |
show "map_bset id = id" by (rule ext, transfer) simp |
|
81 |
next |
|
82 |
fix f g |
|
83 |
show "map_bset (f o g) = map_bset f o map_bset g" by (rule ext, transfer) auto |
|
84 |
next |
|
85 |
fix X f g |
|
86 |
assume "\<And>z. z \<in> set_bset X \<Longrightarrow> f z = g z" |
|
87 |
then show "map_bset f X = map_bset g X" by transfer force |
|
88 |
next |
|
89 |
fix f |
|
90 |
show "set_bset \<circ> map_bset f = op ` f \<circ> set_bset" by (rule ext, transfer) auto |
|
91 |
next |
|
92 |
fix X :: "'a set['k]" |
|
93 |
show "|set_bset X| \<le>o natLeq +c |UNIV :: 'k set|" |
|
94 |
by transfer (blast dest: ordLess_imp_ordLeq) |
|
95 |
next |
|
96 |
fix R S |
|
97 |
show "rel_bset R OO rel_bset S \<le> rel_bset (R OO S)" |
|
98 |
by (rule predicate2I, transfer) (auto simp: rel_set_OO[symmetric]) |
|
99 |
next |
|
100 |
fix R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" |
|
101 |
show "rel_bset R = ((BNF_Def.Grp {x. set_bset x \<subseteq> {(x, y). R x y}} (map_bset fst))\<inverse>\<inverse> OO |
|
102 |
BNF_Def.Grp {x. set_bset x \<subseteq> {(x, y). R x y}} (map_bset snd) :: |
|
103 |
'a set['k] \<Rightarrow> 'b set['k] \<Rightarrow> bool)" |
|
104 |
by (simp add: rel_bset_def map_fun_def o_def rel_set_def rel_bset_aux_infinite) |
|
105 |
next |
|
106 |
fix x |
|
107 |
assume "x \<in> set_bset bempty" |
|
108 |
then show False by transfer simp |
|
109 |
qed (simp_all add: card_order_csum natLeq_card_order cinfinite_csum natLeq_cinfinite) |
|
110 |
||
111 |
||
112 |
lemma map_bset_bempty[simp]: "map_bset f bempty = bempty" |
|
113 |
by transfer auto |
|
114 |
||
115 |
lemma map_bset_binsert[simp]: "map_bset f (binsert x X) = binsert (f x) (map_bset f X)" |
|
116 |
by transfer auto |
|
117 |
||
118 |
lemma map_bset_bsingleton: "map_bset f (bsingleton x) = bsingleton (f x)" |
|
119 |
unfolding bsingleton_def by simp |
|
120 |
||
121 |
lemma bempty_not_binsert: "bempty \<noteq> binsert x X" "binsert x X \<noteq> bempty" |
|
122 |
by (transfer, auto)+ |
|
123 |
||
124 |
lemma bempty_not_bsingleton[simp]: "bempty \<noteq> bsingleton x" "bsingleton x \<noteq> bempty" |
|
125 |
unfolding bsingleton_def by (simp_all add: bempty_not_binsert) |
|
126 |
||
127 |
lemma bsingleton_inj[simp]: "bsingleton x = bsingleton y \<longleftrightarrow> x = y" |
|
128 |
unfolding bsingleton_def by transfer auto |
|
129 |
||
130 |
lemma rel_bsingleton[simp]: |
|
131 |
"rel_bset R (bsingleton x1) (bsingleton x2) = R x1 x2" |
|
132 |
unfolding bsingleton_def |
|
133 |
by transfer (auto simp: rel_set_def) |
|
134 |
||
135 |
lemma rel_bset_bsingleton[simp]: |
|
136 |
"rel_bset R (bsingleton x1) = (\<lambda>X. X \<noteq> bempty \<and> (\<forall>x2\<in>set_bset X. R x1 x2))" |
|
137 |
"rel_bset R X (bsingleton x2) = (X \<noteq> bempty \<and> (\<forall>x1\<in>set_bset X. R x1 x2))" |
|
138 |
unfolding bsingleton_def fun_eq_iff |
|
139 |
by (transfer, force simp add: rel_set_def)+ |
|
140 |
||
141 |
lemma rel_bset_bempty[simp]: |
|
142 |
"rel_bset R bempty X = (X = bempty)" |
|
143 |
"rel_bset R Y bempty = (Y = bempty)" |
|
144 |
by (transfer, simp add: rel_set_def)+ |
|
145 |
||
146 |
definition bset_of_option where |
|
147 |
"bset_of_option = case_option bempty bsingleton" |
|
148 |
||
149 |
lift_definition bgraph :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<times> 'b) set['a set]" is |
|
150 |
"\<lambda>f. {(a, b). f a = Some b}" |
|
151 |
proof - |
|
152 |
fix f :: "'a \<Rightarrow> 'b option" |
|
153 |
have "|{(a, b). f a = Some b}| \<le>o |UNIV :: 'a set|" |
|
154 |
by (rule surj_imp_ordLeq[of _ "\<lambda>x. (x, the (f x))"]) auto |
|
155 |
also have "|UNIV :: 'a set| <o |UNIV :: 'a set set|" |
|
156 |
by simp |
|
157 |
also have "|UNIV :: 'a set set| \<le>o natLeq +c |UNIV :: 'a set set|" |
|
158 |
by (rule ordLeq_csum2) simp |
|
159 |
finally show "|{(a, b). f a = Some b}| <o natLeq +c |UNIV :: 'a set set|" . |
|
160 |
qed |
|
161 |
||
162 |
lemma rel_bset_False[simp]: "rel_bset (\<lambda>x y. False) x y = (x = bempty \<and> y = bempty)" |
|
163 |
by transfer (auto simp: rel_set_def) |
|
164 |
||
165 |
lemma rel_bset_of_option[simp]: |
|
166 |
"rel_bset R (bset_of_option x1) (bset_of_option x2) = rel_option R x1 x2" |
|
167 |
unfolding bset_of_option_def bsingleton_def[abs_def] |
|
168 |
by transfer (auto simp: rel_set_def split: option.splits) |
|
169 |
||
170 |
lemma rel_bgraph[simp]: |
|
171 |
"rel_bset (rel_prod (op =) R) (bgraph f1) (bgraph f2) = rel_fun (op =) (rel_option R) f1 f2" |
|
172 |
apply transfer |
|
173 |
apply (auto simp: rel_fun_def rel_option_iff rel_set_def split: option.splits) |
|
174 |
using option.collapse apply fastforce+ |
|
175 |
done |
|
176 |
||
177 |
lemma set_bset_bsingleton[simp]: |
|
178 |
"set_bset (bsingleton x) = {x}" |
|
179 |
unfolding bsingleton_def by transfer auto |
|
180 |
||
181 |
lemma binsert_absorb[simp]: "binsert a (binsert a x) = binsert a x" |
|
182 |
by transfer simp |
|
183 |
||
184 |
lemma map_bset_eq_bempty_iff[simp]: "map_bset f X = bempty \<longleftrightarrow> X = bempty" |
|
185 |
by transfer auto |
|
186 |
||
187 |
lemma map_bset_eq_bsingleton_iff[simp]: |
|
188 |
"map_bset f X = bsingleton x \<longleftrightarrow> (set_bset X \<noteq> {} \<and> (\<forall>y \<in> set_bset X. f y = x))" |
|
189 |
unfolding bsingleton_def by transfer auto |
|
190 |
||
191 |
lift_definition bCollect :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set['a set]" is Collect |
|
192 |
apply (rule ordLeq_ordLess_trans[OF card_of_mono1[OF subset_UNIV]]) |
|
193 |
apply (rule ordLess_ordLeq_trans[OF card_of_set_type]) |
|
194 |
apply (rule ordLeq_csum2[OF card_of_Card_order]) |
|
195 |
done |
|
196 |
||
197 |
lift_definition bmember :: "'a \<Rightarrow> 'a set['k] \<Rightarrow> bool" is "op \<in>" . |
|
198 |
||
199 |
lemma bmember_bCollect[simp]: "bmember a (bCollect P) = P a" |
|
200 |
by transfer simp |
|
201 |
||
202 |
lemma bset_eq_iff: "A = B \<longleftrightarrow> (\<forall>a. bmember a A = bmember a B)" |
|
203 |
by transfer auto |
|
204 |
||
205 |
(* FIXME: Lifting does not work with dead variables, |
|
206 |
that is why we are hiding the below setup in a locale*) |
|
207 |
locale bset_lifting |
|
208 |
begin |
|
209 |
||
210 |
declare bset.rel_eq[relator_eq] |
|
211 |
declare bset.rel_mono[relator_mono] |
|
212 |
declare bset.rel_compp[symmetric, relator_distr] |
|
213 |
declare bset.rel_transfer[transfer_rule] |
|
214 |
||
215 |
lemma bset_quot_map[quot_map]: "Quotient R Abs Rep T \<Longrightarrow> |
|
216 |
Quotient (rel_bset R) (map_bset Abs) (map_bset Rep) (rel_bset T)" |
|
217 |
unfolding Quotient_alt_def5 bset.rel_Grp[of UNIV, simplified, symmetric] |
|
218 |
bset.rel_conversep[symmetric] bset.rel_compp[symmetric] |
|
219 |
by (auto elim: bset.rel_mono_strong) |
|
220 |
||
221 |
lemma set_relator_eq_onp [relator_eq_onp]: |
|
222 |
"rel_bset (eq_onp P) = eq_onp (\<lambda>A. Ball (set_bset A) P)" |
|
223 |
unfolding fun_eq_iff eq_onp_def by transfer (auto simp: rel_set_def) |
|
224 |
||
225 |
end |
|
226 |
||
227 |
end |