src/HOL/Lifting_Set.thy
 author kuncar Thu Apr 10 17:48:15 2014 +0200 (2014-04-10) changeset 56520 3373f5d1e074 parent 56519 c1048f5bbb45 child 56524 f4ba736040fa permissions -rw-r--r--
abstract Domainp in relator_domain rules => more natural statement of the rule
1 (*  Title:      HOL/Lifting_Set.thy
2     Author:     Brian Huffman and Ondrej Kuncar
3 *)
5 header {* Setup for Lifting/Transfer for the set type *}
7 theory Lifting_Set
8 imports Lifting
9 begin
11 subsection {* Relator and predicator properties *}
13 definition rel_set :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
14   where "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
16 lemma rel_setI:
17   assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"
18   assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y"
19   shows "rel_set R A B"
20   using assms unfolding rel_set_def by simp
22 lemma rel_setD1: "\<lbrakk> rel_set R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y"
23   and rel_setD2: "\<lbrakk> rel_set R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y"
24 by(simp_all add: rel_set_def)
26 lemma rel_set_conversep [simp]: "rel_set A\<inverse>\<inverse> = (rel_set A)\<inverse>\<inverse>"
27   unfolding rel_set_def by auto
29 lemma rel_set_eq [relator_eq]: "rel_set (op =) = (op =)"
30   unfolding rel_set_def fun_eq_iff by auto
32 lemma rel_set_mono[relator_mono]:
33   assumes "A \<le> B"
34   shows "rel_set A \<le> rel_set B"
35 using assms unfolding rel_set_def by blast
37 lemma rel_set_OO[relator_distr]: "rel_set R OO rel_set S = rel_set (R OO S)"
38   apply (rule sym)
39   apply (intro ext, rename_tac X Z)
40   apply (rule iffI)
41   apply (rule_tac b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}" in relcomppI)
42   apply (simp add: rel_set_def, fast)
43   apply (simp add: rel_set_def, fast)
44   apply (simp add: rel_set_def, fast)
45   done
47 lemma Domainp_set[relator_domain]:
48   "Domainp (rel_set T) = (\<lambda>A. Ball A (Domainp T))"
49 unfolding rel_set_def Domainp_iff[abs_def]
50 apply (intro ext)
51 apply (rule iffI)
52 apply blast
53 apply (rename_tac A, rule_tac x="{y. \<exists>x\<in>A. T x y}" in exI, fast)
54 done
56 lemma left_total_rel_set[transfer_rule]:
57   "left_total A \<Longrightarrow> left_total (rel_set A)"
58   unfolding left_total_def rel_set_def
59   apply safe
60   apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast)
61 done
63 lemma left_unique_rel_set[transfer_rule]:
64   "left_unique A \<Longrightarrow> left_unique (rel_set A)"
65   unfolding left_unique_def rel_set_def
66   by fast
68 lemma right_total_rel_set [transfer_rule]:
69   "right_total A \<Longrightarrow> right_total (rel_set A)"
70 using left_total_rel_set[of "A\<inverse>\<inverse>"] by simp
72 lemma right_unique_rel_set [transfer_rule]:
73   "right_unique A \<Longrightarrow> right_unique (rel_set A)"
74   unfolding right_unique_def rel_set_def by fast
76 lemma bi_total_rel_set [transfer_rule]:
77   "bi_total A \<Longrightarrow> bi_total (rel_set A)"
78 by(simp add: bi_total_conv_left_right left_total_rel_set right_total_rel_set)
80 lemma bi_unique_rel_set [transfer_rule]:
81   "bi_unique A \<Longrightarrow> bi_unique (rel_set A)"
82   unfolding bi_unique_def rel_set_def by fast
84 lemma set_relator_eq_onp [relator_eq_onp]:
85   "rel_set (eq_onp P) = eq_onp (\<lambda>A. Ball A P)"
86   unfolding fun_eq_iff rel_set_def eq_onp_def Ball_def by fast
88 subsection {* Quotient theorem for the Lifting package *}
90 lemma Quotient_set[quot_map]:
91   assumes "Quotient R Abs Rep T"
92   shows "Quotient (rel_set R) (image Abs) (image Rep) (rel_set T)"
93   using assms unfolding Quotient_alt_def4
94   apply (simp add: rel_set_OO[symmetric])
95   apply (simp add: rel_set_def, fast)
96   done
98 subsection {* Transfer rules for the Transfer package *}
100 subsubsection {* Unconditional transfer rules *}
102 context
103 begin
104 interpretation lifting_syntax .
106 lemma empty_transfer [transfer_rule]: "(rel_set A) {} {}"
107   unfolding rel_set_def by simp
109 lemma insert_transfer [transfer_rule]:
110   "(A ===> rel_set A ===> rel_set A) insert insert"
111   unfolding rel_fun_def rel_set_def by auto
113 lemma union_transfer [transfer_rule]:
114   "(rel_set A ===> rel_set A ===> rel_set A) union union"
115   unfolding rel_fun_def rel_set_def by auto
117 lemma Union_transfer [transfer_rule]:
118   "(rel_set (rel_set A) ===> rel_set A) Union Union"
119   unfolding rel_fun_def rel_set_def by simp fast
121 lemma image_transfer [transfer_rule]:
122   "((A ===> B) ===> rel_set A ===> rel_set B) image image"
123   unfolding rel_fun_def rel_set_def by simp fast
125 lemma UNION_transfer [transfer_rule]:
126   "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) UNION UNION"
127   unfolding Union_image_eq [symmetric, abs_def] by transfer_prover
129 lemma Ball_transfer [transfer_rule]:
130   "(rel_set A ===> (A ===> op =) ===> op =) Ball Ball"
131   unfolding rel_set_def rel_fun_def by fast
133 lemma Bex_transfer [transfer_rule]:
134   "(rel_set A ===> (A ===> op =) ===> op =) Bex Bex"
135   unfolding rel_set_def rel_fun_def by fast
137 lemma Pow_transfer [transfer_rule]:
138   "(rel_set A ===> rel_set (rel_set A)) Pow Pow"
139   apply (rule rel_funI, rename_tac X Y, rule rel_setI)
140   apply (rename_tac X', rule_tac x="{y\<in>Y. \<exists>x\<in>X'. A x y}" in rev_bexI, clarsimp)
141   apply (simp add: rel_set_def, fast)
142   apply (rename_tac Y', rule_tac x="{x\<in>X. \<exists>y\<in>Y'. A x y}" in rev_bexI, clarsimp)
143   apply (simp add: rel_set_def, fast)
144   done
146 lemma rel_set_transfer [transfer_rule]:
147   "((A ===> B ===> op =) ===> rel_set A ===> rel_set B ===> op =) rel_set rel_set"
148   unfolding rel_fun_def rel_set_def by fast
150 lemma bind_transfer [transfer_rule]:
151   "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) Set.bind Set.bind"
152   unfolding bind_UNION [abs_def] by transfer_prover
154 lemma INF_parametric [transfer_rule]:
155   "(rel_set A ===> (A ===> HOL.eq) ===> HOL.eq) INFIMUM INFIMUM"
156   unfolding INF_def [abs_def] by transfer_prover
158 lemma SUP_parametric [transfer_rule]:
159   "(rel_set R ===> (R ===> HOL.eq) ===> HOL.eq) SUPREMUM SUPREMUM"
160   unfolding SUP_def [abs_def] by transfer_prover
163 subsubsection {* Rules requiring bi-unique, bi-total or right-total relations *}
165 lemma member_transfer [transfer_rule]:
166   assumes "bi_unique A"
167   shows "(A ===> rel_set A ===> op =) (op \<in>) (op \<in>)"
168   using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
170 lemma right_total_Collect_transfer[transfer_rule]:
171   assumes "right_total A"
172   shows "((A ===> op =) ===> rel_set A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
173   using assms unfolding right_total_def rel_set_def rel_fun_def Domainp_iff by fast
175 lemma Collect_transfer [transfer_rule]:
176   assumes "bi_total A"
177   shows "((A ===> op =) ===> rel_set A) Collect Collect"
178   using assms unfolding rel_fun_def rel_set_def bi_total_def by fast
180 lemma inter_transfer [transfer_rule]:
181   assumes "bi_unique A"
182   shows "(rel_set A ===> rel_set A ===> rel_set A) inter inter"
183   using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
185 lemma Diff_transfer [transfer_rule]:
186   assumes "bi_unique A"
187   shows "(rel_set A ===> rel_set A ===> rel_set A) (op -) (op -)"
188   using assms unfolding rel_fun_def rel_set_def bi_unique_def
189   unfolding Ball_def Bex_def Diff_eq
190   by (safe, simp, metis, simp, metis)
192 lemma subset_transfer [transfer_rule]:
193   assumes [transfer_rule]: "bi_unique A"
194   shows "(rel_set A ===> rel_set A ===> op =) (op \<subseteq>) (op \<subseteq>)"
195   unfolding subset_eq [abs_def] by transfer_prover
197 lemma right_total_UNIV_transfer[transfer_rule]:
198   assumes "right_total A"
199   shows "(rel_set A) (Collect (Domainp A)) UNIV"
200   using assms unfolding right_total_def rel_set_def Domainp_iff by blast
202 lemma UNIV_transfer [transfer_rule]:
203   assumes "bi_total A"
204   shows "(rel_set A) UNIV UNIV"
205   using assms unfolding rel_set_def bi_total_def by simp
207 lemma right_total_Compl_transfer [transfer_rule]:
208   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
209   shows "(rel_set A ===> rel_set A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
210   unfolding Compl_eq [abs_def]
211   by (subst Collect_conj_eq[symmetric]) transfer_prover
213 lemma Compl_transfer [transfer_rule]:
214   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
215   shows "(rel_set A ===> rel_set A) uminus uminus"
216   unfolding Compl_eq [abs_def] by transfer_prover
218 lemma right_total_Inter_transfer [transfer_rule]:
219   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
220   shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter"
221   unfolding Inter_eq[abs_def]
222   by (subst Collect_conj_eq[symmetric]) transfer_prover
224 lemma Inter_transfer [transfer_rule]:
225   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
226   shows "(rel_set (rel_set A) ===> rel_set A) Inter Inter"
227   unfolding Inter_eq [abs_def] by transfer_prover
229 lemma filter_transfer [transfer_rule]:
230   assumes [transfer_rule]: "bi_unique A"
231   shows "((A ===> op=) ===> rel_set A ===> rel_set A) Set.filter Set.filter"
232   unfolding Set.filter_def[abs_def] rel_fun_def rel_set_def by blast
234 lemma bi_unique_rel_set_lemma:
235   assumes "bi_unique R" and "rel_set R X Y"
236   obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
237 proof
238   let ?f = "\<lambda>x. THE y. R x y"
239   from assms show f: "\<forall>x\<in>X. R x (?f x)"
240     apply (clarsimp simp add: rel_set_def)
241     apply (drule (1) bspec, clarify)
242     apply (rule theI2, assumption)
243     apply (simp add: bi_unique_def)
244     apply assumption
245     done
246   from assms show "Y = image ?f X"
247     apply safe
248     apply (clarsimp simp add: rel_set_def)
249     apply (drule (1) bspec, clarify)
250     apply (rule image_eqI)
251     apply (rule the_equality [symmetric], assumption)
252     apply (simp add: bi_unique_def)
253     apply assumption
254     apply (clarsimp simp add: rel_set_def)
255     apply (frule (1) bspec, clarify)
256     apply (rule theI2, assumption)
257     apply (clarsimp simp add: bi_unique_def)
258     apply (simp add: bi_unique_def, metis)
259     done
260   show "inj_on ?f X"
261     apply (rule inj_onI)
262     apply (drule f [rule_format])
263     apply (drule f [rule_format])
264     apply (simp add: assms(1) [unfolded bi_unique_def])
265     done
266 qed
268 lemma finite_transfer [transfer_rule]:
269   "bi_unique A \<Longrightarrow> (rel_set A ===> op =) finite finite"
270   by (rule rel_funI, erule (1) bi_unique_rel_set_lemma,
271     auto dest: finite_imageD)
273 lemma card_transfer [transfer_rule]:
274   "bi_unique A \<Longrightarrow> (rel_set A ===> op =) card card"
275   by (rule rel_funI, erule (1) bi_unique_rel_set_lemma, simp add: card_image)
277 lemma vimage_parametric [transfer_rule]:
278   assumes [transfer_rule]: "bi_total A" "bi_unique B"
279   shows "((A ===> B) ===> rel_set B ===> rel_set A) vimage vimage"
280 unfolding vimage_def[abs_def] by transfer_prover
282 lemma setsum_parametric [transfer_rule]:
283   assumes "bi_unique A"
284   shows "((A ===> op =) ===> rel_set A ===> op =) setsum setsum"
285 proof(rule rel_funI)+
286   fix f :: "'a \<Rightarrow> 'c" and g S T
287   assume fg: "(A ===> op =) f g"
288     and ST: "rel_set A S T"
289   show "setsum f S = setsum g T"
290   proof(rule setsum_reindex_cong)
291     let ?f = "\<lambda>t. THE s. A s t"
292     show "S = ?f ` T"
293       by(blast dest: rel_setD1[OF ST] rel_setD2[OF ST] bi_uniqueDl[OF assms]
294            intro: rev_image_eqI the_equality[symmetric] subst[rotated, where P="\<lambda>x. x \<in> S"])
296     show "inj_on ?f T"
297     proof(rule inj_onI)
298       fix t1 t2
299       assume "t1 \<in> T" "t2 \<in> T" "?f t1 = ?f t2"
300       from ST `t1 \<in> T` obtain s1 where "A s1 t1" "s1 \<in> S" by(auto dest: rel_setD2)
301       hence "?f t1 = s1" by(auto dest: bi_uniqueDl[OF assms])
302       moreover
303       from ST `t2 \<in> T` obtain s2 where "A s2 t2" "s2 \<in> S" by(auto dest: rel_setD2)
304       hence "?f t2 = s2" by(auto dest: bi_uniqueDl[OF assms])
305       ultimately have "s1 = s2" using `?f t1 = ?f t2` by simp
306       with `A s1 t1` `A s2 t2` show "t1 = t2" by(auto dest: bi_uniqueDr[OF assms])
307     qed
309     fix t
310     assume "t \<in> T"
311     with ST obtain s where "A s t" "s \<in> S" by(auto dest: rel_setD2)
312     hence "?f t = s" by(auto dest: bi_uniqueDl[OF assms])
313     moreover from fg `A s t` have "f s = g t" by(rule rel_funD)
314     ultimately show "g t = f (?f t)" by simp
315   qed
316 qed
318 end
320 end