src/HOL/Quotient_Examples/Quotient_FSet.thy
 author haftmann Sun Oct 08 22:28:22 2017 +0200 (23 months ago) changeset 66816 212a3334e7da parent 66453 cc19f7ca2ed6 child 67399 eab6ce8368fa permissions -rw-r--r--
more fundamental definition of div and mod on int
1 (*  Title:      HOL/Quotient_Examples/Quotient_FSet.thy
2     Author:     Cezary Kaliszyk, TU Munich
3     Author:     Christian Urban, TU Munich
5 Type of finite sets.
6 *)
8 (********************************************************************
9   WARNING: There is a formalization of 'a fset as a subtype of sets in
10   HOL/Library/FSet.thy using Lifting/Transfer. The user should use
11   that file rather than this file unless there are some very specific
12   reasons.
13 *********************************************************************)
15 theory Quotient_FSet
16 imports "HOL-Library.Multiset" "HOL-Library.Quotient_List"
17 begin
19 text \<open>
20   The type of finite sets is created by a quotient construction
21   over lists. The definition of the equivalence:
22 \<close>
24 definition
25   list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)
26 where
27   [simp]: "xs \<approx> ys \<longleftrightarrow> set xs = set ys"
29 lemma list_eq_reflp:
30   "reflp list_eq"
31   by (auto intro: reflpI)
33 lemma list_eq_symp:
34   "symp list_eq"
35   by (auto intro: sympI)
37 lemma list_eq_transp:
38   "transp list_eq"
39   by (auto intro: transpI)
41 lemma list_eq_equivp:
42   "equivp list_eq"
43   by (auto intro: equivpI list_eq_reflp list_eq_symp list_eq_transp)
45 text \<open>The \<open>fset\<close> type\<close>
47 quotient_type
48   'a fset = "'a list" / "list_eq"
49   by (rule list_eq_equivp)
51 text \<open>
52   Definitions for sublist, cardinality,
53   intersection, difference and respectful fold over
54   lists.
55 \<close>
57 declare List.member_def [simp]
59 definition
60   sub_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
61 where
62   [simp]: "sub_list xs ys \<longleftrightarrow> set xs \<subseteq> set ys"
64 definition
65   card_list :: "'a list \<Rightarrow> nat"
66 where
67   [simp]: "card_list xs = card (set xs)"
69 definition
70   inter_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
71 where
72   [simp]: "inter_list xs ys = [x \<leftarrow> xs. x \<in> set xs \<and> x \<in> set ys]"
74 definition
75   diff_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
76 where
77   [simp]: "diff_list xs ys = [x \<leftarrow> xs. x \<notin> set ys]"
79 definition
80   rsp_fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool"
81 where
82   "rsp_fold f \<longleftrightarrow> (\<forall>u v. f u \<circ> f v = f v \<circ> f u)"
84 lemma rsp_foldI:
85   "(\<And>u v. f u \<circ> f v = f v \<circ> f u) \<Longrightarrow> rsp_fold f"
88 lemma rsp_foldE:
89   assumes "rsp_fold f"
90   obtains "f u \<circ> f v = f v \<circ> f u"
91   using assms by (simp add: rsp_fold_def)
93 definition
94   fold_once :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
95 where
96   "fold_once f xs = (if rsp_fold f then fold f (remdups xs) else id)"
98 lemma fold_once_default [simp]:
99   "\<not> rsp_fold f \<Longrightarrow> fold_once f xs = id"
102 lemma fold_once_fold_remdups:
103   "rsp_fold f \<Longrightarrow> fold_once f xs = fold f (remdups xs)"
107 section \<open>Quotient composition lemmas\<close>
109 lemma list_all2_refl':
110   assumes q: "equivp R"
111   shows "(list_all2 R) r r"
112   by (rule list_all2_refl) (metis equivp_def q)
114 lemma compose_list_refl:
115   assumes q: "equivp R"
116   shows "(list_all2 R OOO op \<approx>) r r"
117 proof
118   have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
119   show "list_all2 R r r" by (rule list_all2_refl'[OF q])
120   with * show "(op \<approx> OO list_all2 R) r r" ..
121 qed
123 lemma map_list_eq_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"
124   by (simp only: list_eq_def set_map)
126 lemma quotient_compose_list_g:
127   assumes q: "Quotient3 R Abs Rep"
128   and     e: "equivp R"
129   shows  "Quotient3 ((list_all2 R) OOO (op \<approx>))
130     (abs_fset \<circ> (map Abs)) ((map Rep) \<circ> rep_fset)"
131   unfolding Quotient3_def comp_def
132 proof (intro conjI allI)
133   fix a r s
134   show "abs_fset (map Abs (map Rep (rep_fset a))) = a"
135     by (simp add: abs_o_rep[OF q] Quotient3_abs_rep[OF Quotient3_fset] List.map.id)
136   have b: "list_all2 R (map Rep (rep_fset a)) (map Rep (rep_fset a))"
137     by (rule list_all2_refl'[OF e])
138   have c: "(op \<approx> OO list_all2 R) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
139     by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
140   show "(list_all2 R OOO op \<approx>) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
141     by (rule, rule list_all2_refl'[OF e]) (rule c)
142   show "(list_all2 R OOO op \<approx>) r s = ((list_all2 R OOO op \<approx>) r r \<and>
143         (list_all2 R OOO op \<approx>) s s \<and> abs_fset (map Abs r) = abs_fset (map Abs s))"
144   proof (intro iffI conjI)
145     show "(list_all2 R OOO op \<approx>) r r" by (rule compose_list_refl[OF e])
146     show "(list_all2 R OOO op \<approx>) s s" by (rule compose_list_refl[OF e])
147   next
148     assume a: "(list_all2 R OOO op \<approx>) r s"
149     then have b: "map Abs r \<approx> map Abs s"
150     proof (elim relcomppE)
151       fix b ba
152       assume c: "list_all2 R r b"
153       assume d: "b \<approx> ba"
154       assume e: "list_all2 R ba s"
155       have f: "map Abs r = map Abs b"
156         using Quotient3_rel[OF list_quotient3[OF q]] c by blast
157       have "map Abs ba = map Abs s"
158         using Quotient3_rel[OF list_quotient3[OF q]] e by blast
159       then have g: "map Abs s = map Abs ba" by simp
160       then show "map Abs r \<approx> map Abs s" using d f map_list_eq_cong by simp
161     qed
162     then show "abs_fset (map Abs r) = abs_fset (map Abs s)"
163       using Quotient3_rel[OF Quotient3_fset] by blast
164   next
165     assume a: "(list_all2 R OOO op \<approx>) r r \<and> (list_all2 R OOO op \<approx>) s s
166       \<and> abs_fset (map Abs r) = abs_fset (map Abs s)"
167     then have s: "(list_all2 R OOO op \<approx>) s s" by simp
168     have d: "map Abs r \<approx> map Abs s"
169       by (subst Quotient3_rel [OF Quotient3_fset, symmetric]) (simp add: a)
170     have b: "map Rep (map Abs r) \<approx> map Rep (map Abs s)"
171       by (rule map_list_eq_cong[OF d])
172     have y: "list_all2 R (map Rep (map Abs s)) s"
173       by (fact rep_abs_rsp_left[OF list_quotient3[OF q], OF list_all2_refl'[OF e, of s]])
174     have c: "(op \<approx> OO list_all2 R) (map Rep (map Abs r)) s"
175       by (rule relcomppI) (rule b, rule y)
176     have z: "list_all2 R r (map Rep (map Abs r))"
177       by (fact rep_abs_rsp[OF list_quotient3[OF q], OF list_all2_refl'[OF e, of r]])
178     then show "(list_all2 R OOO op \<approx>) r s"
179       using a c relcomppI by simp
180   qed
181 qed
183 lemma quotient_compose_list[quot_thm]:
184   shows  "Quotient3 ((list_all2 op \<approx>) OOO (op \<approx>))
185     (abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"
186   by (rule quotient_compose_list_g, rule Quotient3_fset, rule list_eq_equivp)
189 section \<open>Quotient definitions for fsets\<close>
192 subsection \<open>Finite sets are a bounded, distributive lattice with minus\<close>
194 instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
195 begin
197 quotient_definition
198   "bot :: 'a fset"
199   is "Nil :: 'a list" done
201 abbreviation
202   empty_fset  ("{||}")
203 where
204   "{||} \<equiv> bot :: 'a fset"
206 quotient_definition
207   "less_eq_fset :: ('a fset \<Rightarrow> 'a fset \<Rightarrow> bool)"
208   is "sub_list :: ('a list \<Rightarrow> 'a list \<Rightarrow> bool)" by simp
210 abbreviation
211   subset_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)
212 where
213   "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
215 definition
216   less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool"
217 where
218   "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
220 abbreviation
221   psubset_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)
222 where
223   "xs |\<subset>| ys \<equiv> xs < ys"
225 quotient_definition
226   "sup :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
227   is "append :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by simp
229 abbreviation
230   union_fset (infixl "|\<union>|" 65)
231 where
232   "xs |\<union>| ys \<equiv> sup xs (ys::'a fset)"
234 quotient_definition
235   "inf :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
236   is "inter_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by simp
238 abbreviation
239   inter_fset (infixl "|\<inter>|" 65)
240 where
241   "xs |\<inter>| ys \<equiv> inf xs (ys::'a fset)"
243 quotient_definition
244   "minus :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
245   is "diff_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by fastforce
247 instance
248 proof
249   fix x y z :: "'a fset"
250   show "x |\<subset>| y \<longleftrightarrow> x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x"
251     by (unfold less_fset_def, descending) auto
252   show "x |\<subseteq>| x" by (descending) (simp)
253   show "{||} |\<subseteq>| x" by (descending) (simp)
254   show "x |\<subseteq>| x |\<union>| y" by (descending) (simp)
255   show "y |\<subseteq>| x |\<union>| y" by (descending) (simp)
256   show "x |\<inter>| y |\<subseteq>| x" by (descending) (auto)
257   show "x |\<inter>| y |\<subseteq>| y" by (descending) (auto)
258   show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)"
259     by (descending) (auto)
260 next
261   fix x y z :: "'a fset"
262   assume a: "x |\<subseteq>| y"
263   assume b: "y |\<subseteq>| z"
264   show "x |\<subseteq>| z" using a b by (descending) (simp)
265 next
266   fix x y :: "'a fset"
267   assume a: "x |\<subseteq>| y"
268   assume b: "y |\<subseteq>| x"
269   show "x = y" using a b by (descending) (auto)
270 next
271   fix x y z :: "'a fset"
272   assume a: "y |\<subseteq>| x"
273   assume b: "z |\<subseteq>| x"
274   show "y |\<union>| z |\<subseteq>| x" using a b by (descending) (simp)
275 next
276   fix x y z :: "'a fset"
277   assume a: "x |\<subseteq>| y"
278   assume b: "x |\<subseteq>| z"
279   show "x |\<subseteq>| y |\<inter>| z" using a b by (descending) (auto)
280 qed
282 end
285 subsection \<open>Other constants for fsets\<close>
287 quotient_definition
288   "insert_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
289   is "Cons" by auto
291 syntax
292   "_insert_fset"     :: "args => 'a fset"  ("{|(_)|}")
294 translations
295   "{|x, xs|}" == "CONST insert_fset x {|xs|}"
296   "{|x|}"     == "CONST insert_fset x {||}"
298 quotient_definition
299   fset_member
300 where
301   "fset_member :: 'a fset \<Rightarrow> 'a \<Rightarrow> bool" is "List.member" by fastforce
303 abbreviation
304   in_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50)
305 where
306   "x |\<in>| S \<equiv> fset_member S x"
308 abbreviation
309   notin_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)
310 where
311   "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
314 subsection \<open>Other constants on the Quotient Type\<close>
316 quotient_definition
317   "card_fset :: 'a fset \<Rightarrow> nat"
318   is card_list by simp
320 quotient_definition
321   "map_fset :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
322   is map by simp
324 quotient_definition
325   "remove_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
326   is removeAll by simp
328 quotient_definition
329   "fset :: 'a fset \<Rightarrow> 'a set"
330   is "set" by simp
332 lemma fold_once_set_equiv:
333   assumes "xs \<approx> ys"
334   shows "fold_once f xs = fold_once f ys"
335 proof (cases "rsp_fold f")
336   case False then show ?thesis by simp
337 next
338   case True
339   then have "\<And>x y. x \<in> set (remdups xs) \<Longrightarrow> y \<in> set (remdups xs) \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
340     by (rule rsp_foldE)
341   moreover from assms have "mset (remdups xs) = mset (remdups ys)"
343   ultimately have "fold f (remdups xs) = fold f (remdups ys)"
344     by (rule fold_multiset_equiv)
345   with True show ?thesis by (simp add: fold_once_fold_remdups)
346 qed
348 quotient_definition
349   "fold_fset :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b \<Rightarrow> 'b"
350   is fold_once by (rule fold_once_set_equiv)
352 lemma concat_rsp_pre:
353   assumes a: "list_all2 op \<approx> x x'"
354   and     b: "x' \<approx> y'"
355   and     c: "list_all2 op \<approx> y' y"
356   and     d: "\<exists>x\<in>set x. xa \<in> set x"
357   shows "\<exists>x\<in>set y. xa \<in> set x"
358 proof -
359   obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto
360   have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a])
361   then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto
362   have "ya \<in> set y'" using b h by simp
363   then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_all2_find_element)
364   then show ?thesis using f i by auto
365 qed
367 quotient_definition
368   "concat_fset :: ('a fset) fset \<Rightarrow> 'a fset"
369   is concat
370 proof (elim relcomppE)
371 fix a b ba bb
372   assume a: "list_all2 op \<approx> a ba"
373   with list_symp [OF list_eq_symp] have a': "list_all2 op \<approx> ba a" by (rule sympE)
374   assume b: "ba \<approx> bb"
375   with list_eq_symp have b': "bb \<approx> ba" by (rule sympE)
376   assume c: "list_all2 op \<approx> bb b"
377   with list_symp [OF list_eq_symp] have c': "list_all2 op \<approx> b bb" by (rule sympE)
378   have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)"
379   proof
380     fix x
381     show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)"
382     proof
383       assume d: "\<exists>xa\<in>set a. x \<in> set xa"
384       show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])
385     next
386       assume e: "\<exists>xa\<in>set b. x \<in> set xa"
387       show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])
388     qed
389   qed
390   then show "concat a \<approx> concat b" by auto
391 qed
393 quotient_definition
394   "filter_fset :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
395   is filter by force
398 subsection \<open>Compositional respectfulness and preservation lemmas\<close>
400 lemma Nil_rsp2 [quot_respect]:
401   shows "(list_all2 op \<approx> OOO op \<approx>) Nil Nil"
402   by (rule compose_list_refl, rule list_eq_equivp)
404 lemma Cons_rsp2 [quot_respect]:
405   shows "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) Cons Cons"
406   apply (auto intro!: rel_funI)
407   apply (rule_tac b="x # b" in relcomppI)
408   apply auto
409   apply (rule_tac b="x # ba" in relcomppI)
410   apply auto
411   done
413 lemma Nil_prs2 [quot_preserve]:
414   assumes "Quotient3 R Abs Rep"
415   shows "(Abs \<circ> map f) [] = Abs []"
416   by simp
418 lemma Cons_prs2 [quot_preserve]:
419   assumes q: "Quotient3 R1 Abs1 Rep1"
420   and     r: "Quotient3 R2 Abs2 Rep2"
421   shows "(Rep1 ---> (map Rep1 \<circ> Rep2) ---> (Abs2 \<circ> map Abs1)) (op #) = (id ---> Rep2 ---> Abs2) (op #)"
422   by (auto simp add: fun_eq_iff comp_def Quotient3_abs_rep [OF q])
424 lemma append_prs2 [quot_preserve]:
425   assumes q: "Quotient3 R1 Abs1 Rep1"
426   and     r: "Quotient3 R2 Abs2 Rep2"
427   shows "((map Rep1 \<circ> Rep2) ---> (map Rep1 \<circ> Rep2) ---> (Abs2 \<circ> map Abs1)) op @ =
428     (Rep2 ---> Rep2 ---> Abs2) op @"
429   by (simp add: fun_eq_iff abs_o_rep[OF q] List.map.id)
431 lemma list_all2_app_l:
432   assumes a: "reflp R"
433   and b: "list_all2 R l r"
434   shows "list_all2 R (z @ l) (z @ r)"
435   using a b by (induct z) (auto elim: reflpE)
437 lemma append_rsp2_pre0:
438   assumes a:"list_all2 op \<approx> x x'"
439   shows "list_all2 op \<approx> (x @ z) (x' @ z)"
440   using a apply (induct x x' rule: list_induct2')
441   by simp_all (rule list_all2_refl'[OF list_eq_equivp])
443 lemma append_rsp2_pre1:
444   assumes a:"list_all2 op \<approx> x x'"
445   shows "list_all2 op \<approx> (z @ x) (z @ x')"
446   using a apply (induct x x' arbitrary: z rule: list_induct2')
447   apply (rule list_all2_refl'[OF list_eq_equivp])
448   apply (simp_all del: list_eq_def)
449   apply (rule list_all2_app_l)
451   done
453 lemma append_rsp2_pre:
454   assumes "list_all2 op \<approx> x x'"
455     and "list_all2 op \<approx> z z'"
456   shows "list_all2 op \<approx> (x @ z) (x' @ z')"
457   using assms by (rule list_all2_appendI)
459 lemma compositional_rsp3:
460   assumes "(R1 ===> R2 ===> R3) C C" and "(R4 ===> R5 ===> R6) C C"
461   shows "(R1 OOO R4 ===> R2 OOO R5 ===> R3 OOO R6) C C"
462   by (auto intro!: rel_funI)
463      (metis (full_types) assms rel_funE relcomppI)
465 lemma append_rsp2 [quot_respect]:
466   "(list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) append append"
467   by (intro compositional_rsp3)
468      (auto intro!: rel_funI simp add: append_rsp2_pre)
470 lemma map_rsp2 [quot_respect]:
471   "((op \<approx> ===> op \<approx>) ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) map map"
472 proof (auto intro!: rel_funI)
473   fix f f' :: "'a list \<Rightarrow> 'b list"
474   fix xa ya x y :: "'a list list"
475   assume fs: "(op \<approx> ===> op \<approx>) f f'" and x: "list_all2 op \<approx> xa x" and xy: "set x = set y" and y: "list_all2 op \<approx> y ya"
476   have a: "(list_all2 op \<approx>) (map f xa) (map f x)"
477     using x
478     by (induct xa x rule: list_induct2')
479        (simp_all, metis fs rel_funE list_eq_def)
480   have b: "set (map f x) = set (map f y)"
481     using xy fs
482     by (induct x y rule: list_induct2')
483        (simp_all, metis image_insert)
484   have c: "(list_all2 op \<approx>) (map f y) (map f' ya)"
485     using y fs
486     by (induct y ya rule: list_induct2')
487        (simp_all, metis apply_rsp' list_eq_def)
488   show "(list_all2 op \<approx> OOO op \<approx>) (map f xa) (map f' ya)"
489     by (metis a b c list_eq_def relcomppI)
490 qed
492 lemma map_prs2 [quot_preserve]:
493   shows "((abs_fset ---> rep_fset) ---> (map rep_fset \<circ> rep_fset) ---> abs_fset \<circ> map abs_fset) map = (id ---> rep_fset ---> abs_fset) map"
494   by (auto simp add: fun_eq_iff)
495      (simp only: map_map[symmetric] map_prs_aux[OF Quotient3_fset Quotient3_fset])
497 section \<open>Lifted theorems\<close>
499 subsection \<open>fset\<close>
501 lemma fset_simps [simp]:
502   shows "fset {||} = {}"
503   and   "fset (insert_fset x S) = insert x (fset S)"
504   by (descending, simp)+
506 lemma finite_fset [simp]:
507   shows "finite (fset S)"
508   by (descending) (simp)
510 lemma fset_cong:
511   shows "fset S = fset T \<longleftrightarrow> S = T"
512   by (descending) (simp)
514 lemma filter_fset [simp]:
515   shows "fset (filter_fset P xs) = Collect P \<inter> fset xs"
516   by (descending) (auto)
518 lemma remove_fset [simp]:
519   shows "fset (remove_fset x xs) = fset xs - {x}"
520   by (descending) (simp)
522 lemma inter_fset [simp]:
523   shows "fset (xs |\<inter>| ys) = fset xs \<inter> fset ys"
524   by (descending) (auto)
526 lemma union_fset [simp]:
527   shows "fset (xs |\<union>| ys) = fset xs \<union> fset ys"
528   by (lifting set_append)
530 lemma minus_fset [simp]:
531   shows "fset (xs - ys) = fset xs - fset ys"
532   by (descending) (auto)
535 subsection \<open>in_fset\<close>
537 lemma in_fset:
538   shows "x |\<in>| S \<longleftrightarrow> x \<in> fset S"
539   by descending simp
541 lemma notin_fset:
542   shows "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S"
545 lemma notin_empty_fset:
546   shows "x |\<notin>| {||}"
549 lemma fset_eq_iff:
550   shows "S = T \<longleftrightarrow> (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"
551   by descending auto
553 lemma none_in_empty_fset:
554   shows "(\<forall>x. x |\<notin>| S) \<longleftrightarrow> S = {||}"
555   by descending simp
558 subsection \<open>insert_fset\<close>
560 lemma in_insert_fset_iff [simp]:
561   shows "x |\<in>| insert_fset y S \<longleftrightarrow> x = y \<or> x |\<in>| S"
562   by descending simp
564 lemma
565   shows insert_fsetI1: "x |\<in>| insert_fset x S"
566   and   insert_fsetI2: "x |\<in>| S \<Longrightarrow> x |\<in>| insert_fset y S"
567   by simp_all
569 lemma insert_absorb_fset [simp]:
570   shows "x |\<in>| S \<Longrightarrow> insert_fset x S = S"
571   by (descending) (auto)
573 lemma empty_not_insert_fset[simp]:
574   shows "{||} \<noteq> insert_fset x S"
575   and   "insert_fset x S \<noteq> {||}"
576   by (descending, simp)+
578 lemma insert_fset_left_comm:
579   shows "insert_fset x (insert_fset y S) = insert_fset y (insert_fset x S)"
580   by (descending) (auto)
582 lemma insert_fset_left_idem:
583   shows "insert_fset x (insert_fset x S) = insert_fset x S"
584   by (descending) (auto)
586 lemma singleton_fset_eq[simp]:
587   shows "{|x|} = {|y|} \<longleftrightarrow> x = y"
588   by (descending) (auto)
590 lemma in_fset_mdef:
591   shows "x |\<in>| F \<longleftrightarrow> x |\<notin>| (F - {|x|}) \<and> F = insert_fset x (F - {|x|})"
592   by (descending) (auto)
595 subsection \<open>union_fset\<close>
597 lemmas [simp] =
598   sup_bot_left[where 'a="'a fset"]
599   sup_bot_right[where 'a="'a fset"]
601 lemma union_insert_fset [simp]:
602   shows "insert_fset x S |\<union>| T = insert_fset x (S |\<union>| T)"
603   by (lifting append.simps(2))
605 lemma singleton_union_fset_left:
606   shows "{|a|} |\<union>| S = insert_fset a S"
607   by simp
609 lemma singleton_union_fset_right:
610   shows "S |\<union>| {|a|} = insert_fset a S"
611   by (subst sup.commute) simp
613 lemma in_union_fset:
614   shows "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"
615   by (descending) (simp)
618 subsection \<open>minus_fset\<close>
620 lemma minus_in_fset:
621   shows "x |\<in>| (xs - ys) \<longleftrightarrow> x |\<in>| xs \<and> x |\<notin>| ys"
622   by (descending) (simp)
624 lemma minus_insert_fset:
625   shows "insert_fset x xs - ys = (if x |\<in>| ys then xs - ys else insert_fset x (xs - ys))"
626   by (descending) (auto)
628 lemma minus_insert_in_fset[simp]:
629   shows "x |\<in>| ys \<Longrightarrow> insert_fset x xs - ys = xs - ys"
632 lemma minus_insert_notin_fset[simp]:
633   shows "x |\<notin>| ys \<Longrightarrow> insert_fset x xs - ys = insert_fset x (xs - ys)"
636 lemma in_minus_fset:
637   shows "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S"
638   unfolding in_fset minus_fset
639   by blast
641 lemma notin_minus_fset:
642   shows "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S"
643   unfolding in_fset minus_fset
644   by blast
647 subsection \<open>remove_fset\<close>
649 lemma in_remove_fset:
650   shows "x |\<in>| remove_fset y S \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"
651   by (descending) (simp)
653 lemma notin_remove_fset:
654   shows "x |\<notin>| remove_fset x S"
655   by (descending) (simp)
657 lemma notin_remove_ident_fset:
658   shows "x |\<notin>| S \<Longrightarrow> remove_fset x S = S"
659   by (descending) (simp)
661 lemma remove_fset_cases:
662   shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = insert_fset x (remove_fset x S))"
663   by (descending) (auto simp add: insert_absorb)
666 subsection \<open>inter_fset\<close>
668 lemma inter_empty_fset_l:
669   shows "{||} |\<inter>| S = {||}"
670   by simp
672 lemma inter_empty_fset_r:
673   shows "S |\<inter>| {||} = {||}"
674   by simp
676 lemma inter_insert_fset:
677   shows "insert_fset x S |\<inter>| T = (if x |\<in>| T then insert_fset x (S |\<inter>| T) else S |\<inter>| T)"
678   by (descending) (auto)
680 lemma in_inter_fset:
681   shows "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"
682   by (descending) (simp)
685 subsection \<open>subset_fset and psubset_fset\<close>
687 lemma subset_fset:
688   shows "xs |\<subseteq>| ys \<longleftrightarrow> fset xs \<subseteq> fset ys"
689   by (descending) (simp)
691 lemma psubset_fset:
692   shows "xs |\<subset>| ys \<longleftrightarrow> fset xs \<subset> fset ys"
693   unfolding less_fset_def
694   by (descending) (auto)
696 lemma subset_insert_fset:
697   shows "(insert_fset x xs) |\<subseteq>| ys \<longleftrightarrow> x |\<in>| ys \<and> xs |\<subseteq>| ys"
698   by (descending) (simp)
700 lemma subset_in_fset:
701   shows "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"
702   by (descending) (auto)
704 lemma subset_empty_fset:
705   shows "xs |\<subseteq>| {||} \<longleftrightarrow> xs = {||}"
706   by (descending) (simp)
708 lemma not_psubset_empty_fset:
709   shows "\<not> xs |\<subset>| {||}"
710   by (metis fset_simps(1) psubset_fset not_psubset_empty)
713 subsection \<open>map_fset\<close>
715 lemma map_fset_simps [simp]:
716    shows "map_fset f {||} = {||}"
717   and   "map_fset f (insert_fset x S) = insert_fset (f x) (map_fset f S)"
718   by (descending, simp)+
720 lemma map_fset_image [simp]:
721   shows "fset (map_fset f S) = f ` (fset S)"
722   by (descending) (simp)
724 lemma inj_map_fset_cong:
725   shows "inj f \<Longrightarrow> map_fset f S = map_fset f T \<longleftrightarrow> S = T"
726   by (descending) (metis inj_vimage_image_eq list_eq_def set_map)
728 lemma map_union_fset:
729   shows "map_fset f (S |\<union>| T) = map_fset f S |\<union>| map_fset f T"
730   by (descending) (simp)
732 lemma in_fset_map_fset[simp]: "a |\<in>| map_fset f X = (\<exists>b. b |\<in>| X \<and> a = f b)"
733   by descending auto
736 subsection \<open>card_fset\<close>
738 lemma card_fset:
739   shows "card_fset xs = card (fset xs)"
740   by (descending) (simp)
742 lemma card_insert_fset_iff [simp]:
743   shows "card_fset (insert_fset x S) = (if x |\<in>| S then card_fset S else Suc (card_fset S))"
744   by (descending) (simp add: insert_absorb)
746 lemma card_fset_0[simp]:
747   shows "card_fset S = 0 \<longleftrightarrow> S = {||}"
748   by (descending) (simp)
750 lemma card_empty_fset[simp]:
751   shows "card_fset {||} = 0"
754 lemma card_fset_1:
755   shows "card_fset S = 1 \<longleftrightarrow> (\<exists>x. S = {|x|})"
756   by (descending) (auto simp add: card_Suc_eq)
758 lemma card_fset_gt_0:
759   shows "x \<in> fset S \<Longrightarrow> 0 < card_fset S"
760   by (descending) (auto simp add: card_gt_0_iff)
762 lemma card_notin_fset:
763   shows "(x |\<notin>| S) = (card_fset (insert_fset x S) = Suc (card_fset S))"
764   by simp
766 lemma card_fset_Suc:
767   shows "card_fset S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = insert_fset x T \<and> card_fset T = n"
768   apply(descending)
769   apply(auto dest!: card_eq_SucD)
770   by (metis Diff_insert_absorb set_removeAll)
772 lemma card_remove_fset_iff [simp]:
773   shows "card_fset (remove_fset y S) = (if y |\<in>| S then card_fset S - 1 else card_fset S)"
774   by (descending) (simp)
776 lemma card_Suc_exists_in_fset:
777   shows "card_fset S = Suc n \<Longrightarrow> \<exists>a. a |\<in>| S"
778   by (drule card_fset_Suc) (auto)
780 lemma in_card_fset_not_0:
781   shows "a |\<in>| A \<Longrightarrow> card_fset A \<noteq> 0"
782   by (descending) (auto)
784 lemma card_fset_mono:
785   shows "xs |\<subseteq>| ys \<Longrightarrow> card_fset xs \<le> card_fset ys"
786   unfolding card_fset psubset_fset
787   by (simp add: card_mono subset_fset)
789 lemma card_subset_fset_eq:
790   shows "xs |\<subseteq>| ys \<Longrightarrow> card_fset ys \<le> card_fset xs \<Longrightarrow> xs = ys"
791   unfolding card_fset subset_fset
792   by (auto dest: card_seteq[OF finite_fset] simp add: fset_cong)
794 lemma psubset_card_fset_mono:
795   shows "xs |\<subset>| ys \<Longrightarrow> card_fset xs < card_fset ys"
796   unfolding card_fset subset_fset
797   by (metis finite_fset psubset_fset psubset_card_mono)
799 lemma card_union_inter_fset:
800   shows "card_fset xs + card_fset ys = card_fset (xs |\<union>| ys) + card_fset (xs |\<inter>| ys)"
801   unfolding card_fset union_fset inter_fset
802   by (rule card_Un_Int[OF finite_fset finite_fset])
804 lemma card_union_disjoint_fset:
805   shows "xs |\<inter>| ys = {||} \<Longrightarrow> card_fset (xs |\<union>| ys) = card_fset xs + card_fset ys"
806   unfolding card_fset union_fset
807   apply (rule card_Un_disjoint[OF finite_fset finite_fset])
808   by (metis inter_fset fset_simps(1))
810 lemma card_remove_fset_less1:
811   shows "x |\<in>| xs \<Longrightarrow> card_fset (remove_fset x xs) < card_fset xs"
812   unfolding card_fset in_fset remove_fset
813   by (rule card_Diff1_less[OF finite_fset])
815 lemma card_remove_fset_less2:
816   shows "x |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> card_fset (remove_fset y (remove_fset x xs)) < card_fset xs"
817   unfolding card_fset remove_fset in_fset
818   by (rule card_Diff2_less[OF finite_fset])
820 lemma card_remove_fset_le1:
821   shows "card_fset (remove_fset x xs) \<le> card_fset xs"
822   unfolding remove_fset card_fset
823   by (rule card_Diff1_le[OF finite_fset])
825 lemma card_psubset_fset:
826   shows "ys |\<subseteq>| xs \<Longrightarrow> card_fset ys < card_fset xs \<Longrightarrow> ys |\<subset>| xs"
827   unfolding card_fset psubset_fset subset_fset
828   by (rule card_psubset[OF finite_fset])
830 lemma card_map_fset_le:
831   shows "card_fset (map_fset f xs) \<le> card_fset xs"
832   unfolding card_fset map_fset_image
833   by (rule card_image_le[OF finite_fset])
835 lemma card_minus_insert_fset[simp]:
836   assumes "a |\<in>| A" and "a |\<notin>| B"
837   shows "card_fset (A - insert_fset a B) = card_fset (A - B) - 1"
838   using assms
839   unfolding in_fset card_fset minus_fset
840   by (simp add: card_Diff_insert[OF finite_fset])
842 lemma card_minus_subset_fset:
843   assumes "B |\<subseteq>| A"
844   shows "card_fset (A - B) = card_fset A - card_fset B"
845   using assms
846   unfolding subset_fset card_fset minus_fset
847   by (rule card_Diff_subset[OF finite_fset])
849 lemma card_minus_fset:
850   shows "card_fset (A - B) = card_fset A - card_fset (A |\<inter>| B)"
851   unfolding inter_fset card_fset minus_fset
852   by (rule card_Diff_subset_Int) (simp)
855 subsection \<open>concat_fset\<close>
857 lemma concat_empty_fset [simp]:
858   shows "concat_fset {||} = {||}"
859   by descending simp
861 lemma concat_insert_fset [simp]:
862   shows "concat_fset (insert_fset x S) = x |\<union>| concat_fset S"
863   by descending simp
865 lemma concat_union_fset [simp]:
866   shows "concat_fset (xs |\<union>| ys) = concat_fset xs |\<union>| concat_fset ys"
867   by descending simp
869 lemma map_concat_fset:
870   shows "map_fset f (concat_fset xs) = concat_fset (map_fset (map_fset f) xs)"
871   by (lifting map_concat)
873 subsection \<open>filter_fset\<close>
875 lemma subset_filter_fset:
876   "filter_fset P xs |\<subseteq>| filter_fset Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"
877   by descending auto
879 lemma eq_filter_fset:
880   "(filter_fset P xs = filter_fset Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"
881   by descending auto
883 lemma psubset_filter_fset:
884   "(\<And>x. x |\<in>| xs \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| xs & \<not> P x & Q x) \<Longrightarrow>
885     filter_fset P xs |\<subset>| filter_fset Q xs"
886   unfolding less_fset_def by (auto simp add: subset_filter_fset eq_filter_fset)
889 subsection \<open>fold_fset\<close>
891 lemma fold_empty_fset:
892   "fold_fset f {||} = id"
893   by descending (simp add: fold_once_def)
895 lemma fold_insert_fset: "fold_fset f (insert_fset a A) =
896   (if rsp_fold f then if a |\<in>| A then fold_fset f A else fold_fset f A \<circ> f a else id)"
897   by descending (simp add: fold_once_fold_remdups)
899 lemma remdups_removeAll:
900   "remdups (removeAll x xs) = remove1 x (remdups xs)"
901   by (induct xs) auto
903 lemma member_commute_fold_once:
904   assumes "rsp_fold f"
905     and "x \<in> set xs"
906   shows "fold_once f xs = fold_once f (removeAll x xs) \<circ> f x"
907 proof -
908   from assms have "fold f (remdups xs) = fold f (remove1 x (remdups xs)) \<circ> f x"
909     by (auto intro!: fold_remove1_split elim: rsp_foldE)
910   then show ?thesis using \<open>rsp_fold f\<close> by (simp add: fold_once_fold_remdups remdups_removeAll)
911 qed
913 lemma in_commute_fold_fset:
914   "rsp_fold f \<Longrightarrow> h |\<in>| b \<Longrightarrow> fold_fset f b = fold_fset f (remove_fset h b) \<circ> f h"
915   by descending (simp add: member_commute_fold_once)
918 subsection \<open>Choice in fsets\<close>
920 lemma fset_choice:
921   assumes a: "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
922   shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
923   using a
924   apply(descending)
925   using finite_set_choice
926   by (auto simp add: Ball_def)
929 section \<open>Induction and Cases rules for fsets\<close>
931 lemma fset_exhaust [case_names empty insert, cases type: fset]:
932   assumes empty_fset_case: "S = {||} \<Longrightarrow> P"
933   and     insert_fset_case: "\<And>x S'. S = insert_fset x S' \<Longrightarrow> P"
934   shows "P"
935   using assms by (lifting list.exhaust)
937 lemma fset_induct [case_names empty insert]:
938   assumes empty_fset_case: "P {||}"
939   and     insert_fset_case: "\<And>x S. P S \<Longrightarrow> P (insert_fset x S)"
940   shows "P S"
941   using assms
942   by (descending) (blast intro: list.induct)
944 lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
945   assumes empty_fset_case: "P {||}"
946   and     insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (insert_fset x S)"
947   shows "P S"
948 proof(induct S rule: fset_induct)
949   case empty
950   show "P {||}" using empty_fset_case by simp
951 next
952   case (insert x S)
953   have "P S" by fact
954   then show "P (insert_fset x S)" using insert_fset_case
955     by (cases "x |\<in>| S") (simp_all)
956 qed
958 lemma fset_card_induct:
959   assumes empty_fset_case: "P {||}"
960   and     card_fset_Suc_case: "\<And>S T. Suc (card_fset S) = (card_fset T) \<Longrightarrow> P S \<Longrightarrow> P T"
961   shows "P S"
962 proof (induct S)
963   case empty
964   show "P {||}" by (rule empty_fset_case)
965 next
966   case (insert x S)
967   have h: "P S" by fact
968   have "x |\<notin>| S" by fact
969   then have "Suc (card_fset S) = card_fset (insert_fset x S)"
970     using card_fset_Suc by auto
971   then show "P (insert_fset x S)"
972     using h card_fset_Suc_case by simp
973 qed
975 lemma fset_raw_strong_cases:
976   obtains "xs = []"
977     | ys x where "\<not> List.member ys x" and "xs \<approx> x # ys"
978 proof (induct xs)
979   case Nil
980   then show thesis by simp
981 next
982   case (Cons a xs)
983   have a: "\<lbrakk>xs = [] \<Longrightarrow> thesis; \<And>x ys. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis"
984     by (rule Cons(1))
985   have b: "\<And>x' ys'. \<lbrakk>\<not> List.member ys' x'; a # xs \<approx> x' # ys'\<rbrakk> \<Longrightarrow> thesis" by fact
986   have c: "xs = [] \<Longrightarrow> thesis" using b
987     apply(simp)
988     by (metis list.set(1) emptyE empty_subsetI)
989   have "\<And>x ys. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis"
990   proof -
991     fix x :: 'a
992     fix ys :: "'a list"
993     assume d:"\<not> List.member ys x"
994     assume e:"xs \<approx> x # ys"
995     show thesis
996     proof (cases "x = a")
997       assume h: "x = a"
998       then have f: "\<not> List.member ys a" using d by simp
999       have g: "a # xs \<approx> a # ys" using e h by auto
1000       show thesis using b f g by simp
1001     next
1002       assume h: "x \<noteq> a"
1003       then have f: "\<not> List.member (a # ys) x" using d by auto
1004       have g: "a # xs \<approx> x # (a # ys)" using e h by auto
1005       show thesis using b f g by (simp del: List.member_def)
1006     qed
1007   qed
1008   then show thesis using a c by blast
1009 qed
1012 lemma fset_strong_cases:
1013   obtains "xs = {||}"
1014     | ys x where "x |\<notin>| ys" and "xs = insert_fset x ys"
1015   by (lifting fset_raw_strong_cases)
1018 lemma fset_induct2:
1019   "P {||} {||} \<Longrightarrow>
1020   (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (insert_fset x xs) {||}) \<Longrightarrow>
1021   (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (insert_fset y ys)) \<Longrightarrow>
1022   (\<And>x xs y ys. \<lbrakk>P xs ys; x |\<notin>| xs; y |\<notin>| ys\<rbrakk> \<Longrightarrow> P (insert_fset x xs) (insert_fset y ys)) \<Longrightarrow>
1023   P xsa ysa"
1024   apply (induct xsa arbitrary: ysa)
1025   apply (induct_tac x rule: fset_induct_stronger)
1026   apply simp_all
1027   apply (induct_tac xa rule: fset_induct_stronger)
1028   apply simp_all
1029   done
1031 text \<open>Extensionality\<close>
1033 lemma fset_eqI:
1034   assumes "\<And>x. x \<in> fset A \<longleftrightarrow> x \<in> fset B"
1035   shows "A = B"
1036 using assms proof (induct A arbitrary: B)
1037   case empty then show ?case
1038     by (auto simp add: in_fset none_in_empty_fset [symmetric] sym)
1039 next
1040   case (insert x A)
1041   from insert.prems insert.hyps(1) have "\<And>z. z \<in> fset A \<longleftrightarrow> z \<in> fset (B - {|x|})"
1042     by (auto simp add: in_fset)
1043   then have A: "A = B - {|x|}" by (rule insert.hyps(2))
1044   with insert.prems [symmetric, of x] have "x |\<in>| B" by (simp add: in_fset)
1045   with A show ?case by (metis in_fset_mdef)
1046 qed
1048 subsection \<open>alternate formulation with a different decomposition principle
1049   and a proof of equivalence\<close>
1051 inductive
1052   list_eq2 :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<approx>2 _")
1053 where
1054   "(a # b # xs) \<approx>2 (b # a # xs)"
1055 | "[] \<approx>2 []"
1056 | "xs \<approx>2 ys \<Longrightarrow> ys \<approx>2 xs"
1057 | "(a # a # xs) \<approx>2 (a # xs)"
1058 | "xs \<approx>2 ys \<Longrightarrow> (a # xs) \<approx>2 (a # ys)"
1059 | "xs1 \<approx>2 xs2 \<Longrightarrow> xs2 \<approx>2 xs3 \<Longrightarrow> xs1 \<approx>2 xs3"
1061 lemma list_eq2_refl:
1062   shows "xs \<approx>2 xs"
1063   by (induct xs) (auto intro: list_eq2.intros)
1065 lemma cons_delete_list_eq2:
1066   shows "(a # (removeAll a A)) \<approx>2 (if List.member A a then A else a # A)"
1067   apply (induct A)
1069   apply (case_tac "List.member (aa # A) a")
1070   apply (simp_all)
1071   apply (case_tac [!] "a = aa")
1072   apply (simp_all)
1073   apply (case_tac "List.member A a")
1074   apply (auto)[2]
1075   apply (metis list_eq2.intros(3) list_eq2.intros(4) list_eq2.intros(5) list_eq2.intros(6))
1076   apply (metis list_eq2.intros(1) list_eq2.intros(5) list_eq2.intros(6))
1077   apply (auto simp add: list_eq2_refl)
1078   done
1080 lemma member_delete_list_eq2:
1081   assumes a: "List.member r e"
1082   shows "(e # removeAll e r) \<approx>2 r"
1083   using a cons_delete_list_eq2[of e r]
1084   by simp
1086 lemma list_eq2_equiv:
1087   "(l \<approx> r) \<longleftrightarrow> (list_eq2 l r)"
1088 proof
1089   show "list_eq2 l r \<Longrightarrow> l \<approx> r" by (induct rule: list_eq2.induct) auto
1090 next
1091   {
1092     fix n
1093     assume a: "card_list l = n" and b: "l \<approx> r"
1094     have "l \<approx>2 r"
1095       using a b
1096     proof (induct n arbitrary: l r)
1097       case 0
1098       have "card_list l = 0" by fact
1099       then have "\<forall>x. \<not> List.member l x" by auto
1100       then have z: "l = []" by auto
1101       then have "r = []" using \<open>l \<approx> r\<close> by simp
1102       then show ?case using z list_eq2_refl by simp
1103     next
1104       case (Suc m)
1105       have b: "l \<approx> r" by fact
1106       have d: "card_list l = Suc m" by fact
1107       then have "\<exists>a. List.member l a"
1108         apply(simp)
1109         apply(drule card_eq_SucD)
1110         apply(blast)
1111         done
1112       then obtain a where e: "List.member l a" by auto
1113       then have e': "List.member r a" using list_eq_def [simplified List.member_def [symmetric], of l r] b
1114         by auto
1115       have f: "card_list (removeAll a l) = m" using e d by (simp)
1116       have g: "removeAll a l \<approx> removeAll a r" using remove_fset.rsp b by simp
1117       have "(removeAll a l) \<approx>2 (removeAll a r)" by (rule Suc.hyps[OF f g])
1118       then have h: "(a # removeAll a l) \<approx>2 (a # removeAll a r)" by (rule list_eq2.intros(5))
1119       have i: "l \<approx>2 (a # removeAll a l)"
1120         by (rule list_eq2.intros(3)[OF member_delete_list_eq2[OF e]])
1121       have "l \<approx>2 (a # removeAll a r)" by (rule list_eq2.intros(6)[OF i h])
1122       then show ?case using list_eq2.intros(6)[OF _ member_delete_list_eq2[OF e']] by simp
1123     qed
1124     }
1125   then show "l \<approx> r \<Longrightarrow> l \<approx>2 r" by blast
1126 qed
1129 (* We cannot write it as "assumes .. shows" since Isabelle changes
1130    the quantifiers to schematic variables and reintroduces them in
1131    a different order *)
1132 lemma fset_eq_cases:
1133  "\<lbrakk>a1 = a2;
1134    \<And>a b xs. \<lbrakk>a1 = insert_fset a (insert_fset b xs); a2 = insert_fset b (insert_fset a xs)\<rbrakk> \<Longrightarrow> P;
1135    \<lbrakk>a1 = {||}; a2 = {||}\<rbrakk> \<Longrightarrow> P; \<And>xs ys. \<lbrakk>a1 = ys; a2 = xs; xs = ys\<rbrakk> \<Longrightarrow> P;
1136    \<And>a xs. \<lbrakk>a1 = insert_fset a (insert_fset a xs); a2 = insert_fset a xs\<rbrakk> \<Longrightarrow> P;
1137    \<And>xs ys a. \<lbrakk>a1 = insert_fset a xs; a2 = insert_fset a ys; xs = ys\<rbrakk> \<Longrightarrow> P;
1138    \<And>xs1 xs2 xs3. \<lbrakk>a1 = xs1; a2 = xs3; xs1 = xs2; xs2 = xs3\<rbrakk> \<Longrightarrow> P\<rbrakk>
1139   \<Longrightarrow> P"
1140   by (lifting list_eq2.cases[simplified list_eq2_equiv[symmetric]])
1142 lemma fset_eq_induct:
1143   assumes "x1 = x2"
1144   and "\<And>a b xs. P (insert_fset a (insert_fset b xs)) (insert_fset b (insert_fset a xs))"
1145   and "P {||} {||}"
1146   and "\<And>xs ys. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P ys xs"
1147   and "\<And>a xs. P (insert_fset a (insert_fset a xs)) (insert_fset a xs)"
1148   and "\<And>xs ys a. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P (insert_fset a xs) (insert_fset a ys)"
1149   and "\<And>xs1 xs2 xs3. \<lbrakk>xs1 = xs2; P xs1 xs2; xs2 = xs3; P xs2 xs3\<rbrakk> \<Longrightarrow> P xs1 xs3"
1150   shows "P x1 x2"
1151   using assms
1152   by (lifting list_eq2.induct[simplified list_eq2_equiv[symmetric]])
1154 ML \<open>
1155 fun dest_fsetT (Type (@{type_name fset}, [T])) = T
1156   | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
1157 \<close>
1159 no_notation
1160   list_eq (infix "\<approx>" 50) and
1161   list_eq2 (infix "\<approx>2" 50)
1163 end