src/HOL/Lifting_Set.thy
 author hoelzl Fri Feb 19 13:40:50 2016 +0100 (2016-02-19) changeset 62378 85ed00c1fe7c parent 62343 24106dc44def child 63040 eb4ddd18d635 permissions -rw-r--r--
generalize more theorems to support enat and ennreal
1 (*  Title:      HOL/Lifting_Set.thy
2     Author:     Brian Huffman and Ondrej Kuncar
3 *)
5 section \<open>Setup for Lifting/Transfer for the set type\<close>
7 theory Lifting_Set
8 imports Lifting
9 begin
11 subsection \<open>Relator and predicator properties\<close>
13 lemma rel_setD1: "\<lbrakk> rel_set R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y"
14   and rel_setD2: "\<lbrakk> rel_set R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y"
15   by (simp_all add: rel_set_def)
17 lemma rel_set_conversep [simp]: "rel_set A\<inverse>\<inverse> = (rel_set A)\<inverse>\<inverse>"
18   unfolding rel_set_def by auto
20 lemma rel_set_eq [relator_eq]: "rel_set (op =) = (op =)"
21   unfolding rel_set_def fun_eq_iff by auto
23 lemma rel_set_mono[relator_mono]:
24   assumes "A \<le> B"
25   shows "rel_set A \<le> rel_set B"
26   using assms unfolding rel_set_def by blast
28 lemma rel_set_OO[relator_distr]: "rel_set R OO rel_set S = rel_set (R OO S)"
29   apply (rule sym)
30   apply (intro ext)
31   subgoal for X Z
32     apply (rule iffI)
33     apply (rule relcomppI [where b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}"])
34     apply (simp add: rel_set_def, fast)+
35     done
36   done
38 lemma Domainp_set[relator_domain]:
39   "Domainp (rel_set T) = (\<lambda>A. Ball A (Domainp T))"
40   unfolding rel_set_def Domainp_iff[abs_def]
41   apply (intro ext)
42   apply (rule iffI)
43   apply blast
44   subgoal for A by (rule exI [where x="{y. \<exists>x\<in>A. T x y}"]) fast
45   done
47 lemma left_total_rel_set[transfer_rule]:
48   "left_total A \<Longrightarrow> left_total (rel_set A)"
49   unfolding left_total_def rel_set_def
50   apply safe
51   subgoal for X by (rule exI [where x="{y. \<exists>x\<in>X. A x y}"]) fast
52   done
54 lemma left_unique_rel_set[transfer_rule]:
55   "left_unique A \<Longrightarrow> left_unique (rel_set A)"
56   unfolding left_unique_def rel_set_def
57   by fast
59 lemma right_total_rel_set [transfer_rule]:
60   "right_total A \<Longrightarrow> right_total (rel_set A)"
61   using left_total_rel_set[of "A\<inverse>\<inverse>"] by simp
63 lemma right_unique_rel_set [transfer_rule]:
64   "right_unique A \<Longrightarrow> right_unique (rel_set A)"
65   unfolding right_unique_def rel_set_def by fast
67 lemma bi_total_rel_set [transfer_rule]:
68   "bi_total A \<Longrightarrow> bi_total (rel_set A)"
69   by(simp add: bi_total_alt_def left_total_rel_set right_total_rel_set)
71 lemma bi_unique_rel_set [transfer_rule]:
72   "bi_unique A \<Longrightarrow> bi_unique (rel_set A)"
73   unfolding bi_unique_def rel_set_def by fast
75 lemma set_relator_eq_onp [relator_eq_onp]:
76   "rel_set (eq_onp P) = eq_onp (\<lambda>A. Ball A P)"
77   unfolding fun_eq_iff rel_set_def eq_onp_def Ball_def by fast
79 lemma bi_unique_rel_set_lemma:
80   assumes "bi_unique R" and "rel_set R X Y"
81   obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
82 proof
83   def f \<equiv> "\<lambda>x. THE y. R x y"
84   { fix x assume "x \<in> X"
85     with \<open>rel_set R X Y\<close> \<open>bi_unique R\<close> have "R x (f x)"
86       by (simp add: bi_unique_def rel_set_def f_def) (metis theI)
87     with assms \<open>x \<in> X\<close>
88     have  "R x (f x)" "\<forall>x'\<in>X. R x' (f x) \<longrightarrow> x = x'" "\<forall>y\<in>Y. R x y \<longrightarrow> y = f x" "f x \<in> Y"
89       by (fastforce simp add: bi_unique_def rel_set_def)+ }
90   note * = this
91   moreover
92   { fix y assume "y \<in> Y"
93     with \<open>rel_set R X Y\<close> *(3) \<open>y \<in> Y\<close> have "\<exists>x\<in>X. y = f x"
94       by (fastforce simp: rel_set_def) }
95   ultimately show "\<forall>x\<in>X. R x (f x)" "Y = image f X" "inj_on f X"
96     by (auto simp: inj_on_def image_iff)
97 qed
99 subsection \<open>Quotient theorem for the Lifting package\<close>
101 lemma Quotient_set[quot_map]:
102   assumes "Quotient R Abs Rep T"
103   shows "Quotient (rel_set R) (image Abs) (image Rep) (rel_set T)"
104   using assms unfolding Quotient_alt_def4
105   apply (simp add: rel_set_OO[symmetric])
106   apply (simp add: rel_set_def)
107   apply fast
108   done
111 subsection \<open>Transfer rules for the Transfer package\<close>
113 subsubsection \<open>Unconditional transfer rules\<close>
115 context
116 begin
118 interpretation lifting_syntax .
120 lemma empty_transfer [transfer_rule]: "(rel_set A) {} {}"
121   unfolding rel_set_def by simp
123 lemma insert_transfer [transfer_rule]:
124   "(A ===> rel_set A ===> rel_set A) insert insert"
125   unfolding rel_fun_def rel_set_def by auto
127 lemma union_transfer [transfer_rule]:
128   "(rel_set A ===> rel_set A ===> rel_set A) union union"
129   unfolding rel_fun_def rel_set_def by auto
131 lemma Union_transfer [transfer_rule]:
132   "(rel_set (rel_set A) ===> rel_set A) Union Union"
133   unfolding rel_fun_def rel_set_def by simp fast
135 lemma image_transfer [transfer_rule]:
136   "((A ===> B) ===> rel_set A ===> rel_set B) image image"
137   unfolding rel_fun_def rel_set_def by simp fast
139 lemma UNION_transfer [transfer_rule]:
140   "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) UNION UNION"
141   by transfer_prover
143 lemma Ball_transfer [transfer_rule]:
144   "(rel_set A ===> (A ===> op =) ===> op =) Ball Ball"
145   unfolding rel_set_def rel_fun_def by fast
147 lemma Bex_transfer [transfer_rule]:
148   "(rel_set A ===> (A ===> op =) ===> op =) Bex Bex"
149   unfolding rel_set_def rel_fun_def by fast
151 lemma Pow_transfer [transfer_rule]:
152   "(rel_set A ===> rel_set (rel_set A)) Pow Pow"
153   apply (rule rel_funI)
154   apply (rule rel_setI)
155   subgoal for X Y X'
156     apply (rule rev_bexI [where x="{y\<in>Y. \<exists>x\<in>X'. A x y}"])
157     apply clarsimp
158     apply (simp add: rel_set_def)
159     apply fast
160     done
161   subgoal for X Y Y'
162     apply (rule rev_bexI [where x="{x\<in>X. \<exists>y\<in>Y'. A x y}"])
163     apply clarsimp
164     apply (simp add: rel_set_def)
165     apply fast
166     done
167   done
169 lemma rel_set_transfer [transfer_rule]:
170   "((A ===> B ===> op =) ===> rel_set A ===> rel_set B ===> op =) rel_set rel_set"
171   unfolding rel_fun_def rel_set_def by fast
173 lemma bind_transfer [transfer_rule]:
174   "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) Set.bind Set.bind"
175   unfolding bind_UNION [abs_def] by transfer_prover
177 lemma INF_parametric [transfer_rule]:
178   "(rel_set A ===> (A ===> HOL.eq) ===> HOL.eq) INFIMUM INFIMUM"
179   by transfer_prover
181 lemma SUP_parametric [transfer_rule]:
182   "(rel_set R ===> (R ===> HOL.eq) ===> HOL.eq) SUPREMUM SUPREMUM"
183   by transfer_prover
186 subsubsection \<open>Rules requiring bi-unique, bi-total or right-total relations\<close>
188 lemma member_transfer [transfer_rule]:
189   assumes "bi_unique A"
190   shows "(A ===> rel_set A ===> op =) (op \<in>) (op \<in>)"
191   using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
193 lemma right_total_Collect_transfer[transfer_rule]:
194   assumes "right_total A"
195   shows "((A ===> op =) ===> rel_set A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
196   using assms unfolding right_total_def rel_set_def rel_fun_def Domainp_iff by fast
198 lemma Collect_transfer [transfer_rule]:
199   assumes "bi_total A"
200   shows "((A ===> op =) ===> rel_set A) Collect Collect"
201   using assms unfolding rel_fun_def rel_set_def bi_total_def by fast
203 lemma inter_transfer [transfer_rule]:
204   assumes "bi_unique A"
205   shows "(rel_set A ===> rel_set A ===> rel_set A) inter inter"
206   using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
208 lemma Diff_transfer [transfer_rule]:
209   assumes "bi_unique A"
210   shows "(rel_set A ===> rel_set A ===> rel_set A) (op -) (op -)"
211   using assms unfolding rel_fun_def rel_set_def bi_unique_def
212   unfolding Ball_def Bex_def Diff_eq
213   by (safe, simp, metis, simp, metis)
215 lemma subset_transfer [transfer_rule]:
216   assumes [transfer_rule]: "bi_unique A"
217   shows "(rel_set A ===> rel_set A ===> op =) (op \<subseteq>) (op \<subseteq>)"
218   unfolding subset_eq [abs_def] by transfer_prover
220 declare right_total_UNIV_transfer[transfer_rule]
222 lemma UNIV_transfer [transfer_rule]:
223   assumes "bi_total A"
224   shows "(rel_set A) UNIV UNIV"
225   using assms unfolding rel_set_def bi_total_def by simp
227 lemma right_total_Compl_transfer [transfer_rule]:
228   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
229   shows "(rel_set A ===> rel_set A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
230   unfolding Compl_eq [abs_def]
231   by (subst Collect_conj_eq[symmetric]) transfer_prover
233 lemma Compl_transfer [transfer_rule]:
234   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
235   shows "(rel_set A ===> rel_set A) uminus uminus"
236   unfolding Compl_eq [abs_def] by transfer_prover
238 lemma right_total_Inter_transfer [transfer_rule]:
239   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
240   shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>S. \<Inter>S \<inter> Collect (Domainp A)) Inter"
241   unfolding Inter_eq[abs_def]
242   by (subst Collect_conj_eq[symmetric]) transfer_prover
244 lemma Inter_transfer [transfer_rule]:
245   assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
246   shows "(rel_set (rel_set A) ===> rel_set A) Inter Inter"
247   unfolding Inter_eq [abs_def] by transfer_prover
249 lemma filter_transfer [transfer_rule]:
250   assumes [transfer_rule]: "bi_unique A"
251   shows "((A ===> op=) ===> rel_set A ===> rel_set A) Set.filter Set.filter"
252   unfolding Set.filter_def[abs_def] rel_fun_def rel_set_def by blast
254 lemma finite_transfer [transfer_rule]:
255   "bi_unique A \<Longrightarrow> (rel_set A ===> op =) finite finite"
256   by (rule rel_funI, erule (1) bi_unique_rel_set_lemma)
257      (auto dest: finite_imageD)
259 lemma card_transfer [transfer_rule]:
260   "bi_unique A \<Longrightarrow> (rel_set A ===> op =) card card"
261   by (rule rel_funI, erule (1) bi_unique_rel_set_lemma)
262      (simp add: card_image)
264 lemma vimage_parametric [transfer_rule]:
265   assumes [transfer_rule]: "bi_total A" "bi_unique B"
266   shows "((A ===> B) ===> rel_set B ===> rel_set A) vimage vimage"
267   unfolding vimage_def[abs_def] by transfer_prover
269 lemma Image_parametric [transfer_rule]:
270   assumes "bi_unique A"
271   shows "(rel_set (rel_prod A B) ===> rel_set A ===> rel_set B) op `` op ``"
272   by (intro rel_funI rel_setI)
273     (force dest: rel_setD1 bi_uniqueDr[OF assms], force dest: rel_setD2 bi_uniqueDl[OF assms])
275 end
277 lemma (in comm_monoid_set) F_parametric [transfer_rule]:
278   fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
279   assumes "bi_unique A"
280   shows "rel_fun (rel_fun A (op =)) (rel_fun (rel_set A) (op =)) F F"
281 proof (rule rel_funI)+
282   fix f :: "'b \<Rightarrow> 'a" and g S T
283   assume "rel_fun A (op =) f g" "rel_set A S T"
284   with \<open>bi_unique A\<close> obtain i where "bij_betw i S T" "\<And>x. x \<in> S \<Longrightarrow> f x = g (i x)"
285     by (auto elim: bi_unique_rel_set_lemma simp: rel_fun_def bij_betw_def)
286   then show "F f S = F g T"
287     by (simp add: reindex_bij_betw)
288 qed
290 lemmas setsum_parametric = setsum.F_parametric
291 lemmas setprod_parametric = setprod.F_parametric
293 lemma rel_set_UNION:
294   assumes [transfer_rule]: "rel_set Q A B" "rel_fun Q (rel_set R) f g"
295   shows "rel_set R (UNION A f) (UNION B g)"
296   by transfer_prover
298 end