src/HOL/Quotient_Examples/FSet.thy
author Cezary Kaliszyk <kaliszyk@in.tum.de>
Fri Oct 15 21:50:26 2010 +0900 (2010-10-15)
changeset 39996 c02078ff8691
parent 39995 849578dd6127
child 40030 9f8dcf6ef563
permissions -rw-r--r--
FSet: definition changes propagated from Nominal and more use of 'descending' tactic
     1 (*  Title:      HOL/Quotient_Examples/FSet.thy
     2     Author:     Cezary Kaliszyk, TU Munich
     3     Author:     Christian Urban, TU Munich
     4 
     5 A reasoning infrastructure for the type of finite sets.
     6 *)
     7 
     8 theory FSet
     9 imports Quotient_List
    10 begin
    11 
    12 text {* Definiton of List relation and the quotient type *}
    13 
    14 fun
    15   list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)
    16 where
    17   "list_eq xs ys = (set xs = set ys)"
    18 
    19 lemma list_eq_equivp:
    20   shows "equivp list_eq"
    21   unfolding equivp_reflp_symp_transp
    22   unfolding reflp_def symp_def transp_def
    23   by auto
    24 
    25 quotient_type
    26   'a fset = "'a list" / "list_eq"
    27   by (rule list_eq_equivp)
    28 
    29 text {* Raw definitions of membership, sublist, cardinality,
    30   intersection
    31 *}
    32 
    33 definition
    34   memb :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
    35 where
    36   "memb x xs \<equiv> x \<in> set xs"
    37 
    38 definition
    39   sub_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
    40 where
    41   "sub_list xs ys \<equiv> set xs \<subseteq> set ys"
    42 
    43 definition
    44   fcard_raw :: "'a list \<Rightarrow> nat"
    45 where
    46   "fcard_raw xs = card (set xs)"
    47 
    48 primrec
    49   finter_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
    50 where
    51   "finter_raw [] ys = []"
    52 | "finter_raw (x # xs) ys =
    53     (if x \<in> set ys then x # (finter_raw xs ys) else finter_raw xs ys)"
    54 
    55 primrec
    56   fminus_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
    57 where
    58   "fminus_raw ys [] = ys"
    59 | "fminus_raw ys (x # xs) = fminus_raw (removeAll x ys) xs"
    60 
    61 definition
    62   rsp_fold
    63 where
    64   "rsp_fold f = (\<forall>u v w. (f u (f v w) = f v (f u w)))"
    65 
    66 primrec
    67   ffold_raw :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
    68 where
    69   "ffold_raw f z [] = z"
    70 | "ffold_raw f z (a # xs) =
    71      (if (rsp_fold f) then
    72        if a \<in> set xs then ffold_raw f z xs
    73        else f a (ffold_raw f z xs)
    74      else z)"
    75 
    76 text {* Composition Quotient *}
    77 
    78 lemma list_all2_refl1:
    79   shows "(list_all2 op \<approx>) r r"
    80   by (rule list_all2_refl) (metis equivp_def fset_equivp)
    81 
    82 lemma compose_list_refl:
    83   shows "(list_all2 op \<approx> OOO op \<approx>) r r"
    84 proof
    85   have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
    86   show "list_all2 op \<approx> r r" by (rule list_all2_refl1)
    87   with * show "(op \<approx> OO list_all2 op \<approx>) r r" ..
    88 qed
    89 
    90 lemma Quotient_fset_list:
    91   shows "Quotient (list_all2 op \<approx>) (map abs_fset) (map rep_fset)"
    92   by (fact list_quotient[OF Quotient_fset])
    93 
    94 lemma map_rel_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"
    95   unfolding list_eq.simps
    96   by (simp only: set_map)
    97 
    98 lemma quotient_compose_list[quot_thm]:
    99   shows  "Quotient ((list_all2 op \<approx>) OOO (op \<approx>))
   100     (abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"
   101   unfolding Quotient_def comp_def
   102 proof (intro conjI allI)
   103   fix a r s
   104   show "abs_fset (map abs_fset (map rep_fset (rep_fset a))) = a"
   105     by (simp add: abs_o_rep[OF Quotient_fset] Quotient_abs_rep[OF Quotient_fset] map_id)
   106   have b: "list_all2 op \<approx> (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
   107     by (rule list_all2_refl1)
   108   have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
   109     by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
   110   show "(list_all2 op \<approx> OOO op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
   111     by (rule, rule list_all2_refl1) (rule c)
   112   show "(list_all2 op \<approx> OOO op \<approx>) r s = ((list_all2 op \<approx> OOO op \<approx>) r r \<and>
   113         (list_all2 op \<approx> OOO op \<approx>) s s \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s))"
   114   proof (intro iffI conjI)
   115     show "(list_all2 op \<approx> OOO op \<approx>) r r" by (rule compose_list_refl)
   116     show "(list_all2 op \<approx> OOO op \<approx>) s s" by (rule compose_list_refl)
   117   next
   118     assume a: "(list_all2 op \<approx> OOO op \<approx>) r s"
   119     then have b: "map abs_fset r \<approx> map abs_fset s"
   120     proof (elim pred_compE)
   121       fix b ba
   122       assume c: "list_all2 op \<approx> r b"
   123       assume d: "b \<approx> ba"
   124       assume e: "list_all2 op \<approx> ba s"
   125       have f: "map abs_fset r = map abs_fset b"
   126         using Quotient_rel[OF Quotient_fset_list] c by blast
   127       have "map abs_fset ba = map abs_fset s"
   128         using Quotient_rel[OF Quotient_fset_list] e by blast
   129       then have g: "map abs_fset s = map abs_fset ba" by simp
   130       then show "map abs_fset r \<approx> map abs_fset s" using d f map_rel_cong by simp
   131     qed
   132     then show "abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"
   133       using Quotient_rel[OF Quotient_fset] by blast
   134   next
   135     assume a: "(list_all2 op \<approx> OOO op \<approx>) r r \<and> (list_all2 op \<approx> OOO op \<approx>) s s
   136       \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"
   137     then have s: "(list_all2 op \<approx> OOO op \<approx>) s s" by simp
   138     have d: "map abs_fset r \<approx> map abs_fset s"
   139       by (subst Quotient_rel[OF Quotient_fset]) (simp add: a)
   140     have b: "map rep_fset (map abs_fset r) \<approx> map rep_fset (map abs_fset s)"
   141       by (rule map_rel_cong[OF d])
   142     have y: "list_all2 op \<approx> (map rep_fset (map abs_fset s)) s"
   143       by (fact rep_abs_rsp_left[OF Quotient_fset_list, OF list_all2_refl1[of s]])
   144     have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (map abs_fset r)) s"
   145       by (rule pred_compI) (rule b, rule y)
   146     have z: "list_all2 op \<approx> r (map rep_fset (map abs_fset r))"
   147       by (fact rep_abs_rsp[OF Quotient_fset_list, OF list_all2_refl1[of r]])
   148     then show "(list_all2 op \<approx> OOO op \<approx>) r s"
   149       using a c pred_compI by simp
   150   qed
   151 qed
   152 
   153 
   154 lemma set_finter_raw[simp]:
   155   "set (finter_raw xs ys) = set xs \<inter> set ys"
   156   by (induct xs) (auto simp add: memb_def)
   157 
   158 lemma set_fminus_raw[simp]: 
   159   "set (fminus_raw xs ys) = (set xs - set ys)"
   160   by (induct ys arbitrary: xs) (auto)
   161 
   162 
   163 text {* Respectfullness *}
   164 
   165 lemma append_rsp[quot_respect]:
   166   shows "(op \<approx> ===> op \<approx> ===> op \<approx>) append append"
   167   by (simp)
   168 
   169 lemma sub_list_rsp[quot_respect]:
   170   shows "(op \<approx> ===> op \<approx> ===> op =) sub_list sub_list"
   171   by (auto simp add: sub_list_def)
   172 
   173 lemma memb_rsp[quot_respect]:
   174   shows "(op = ===> op \<approx> ===> op =) memb memb"
   175   by (auto simp add: memb_def)
   176 
   177 lemma nil_rsp[quot_respect]:
   178   shows "(op \<approx>) Nil Nil"
   179   by simp
   180 
   181 lemma cons_rsp[quot_respect]:
   182   shows "(op = ===> op \<approx> ===> op \<approx>) Cons Cons"
   183   by simp
   184 
   185 lemma map_rsp[quot_respect]:
   186   shows "(op = ===> op \<approx> ===> op \<approx>) map map"
   187   by auto
   188 
   189 lemma set_rsp[quot_respect]:
   190   "(op \<approx> ===> op =) set set"
   191   by auto
   192 
   193 lemma list_equiv_rsp[quot_respect]:
   194   shows "(op \<approx> ===> op \<approx> ===> op =) op \<approx> op \<approx>"
   195   by auto
   196 
   197 lemma finter_raw_rsp[quot_respect]:
   198   shows "(op \<approx> ===> op \<approx> ===> op \<approx>) finter_raw finter_raw"
   199   by simp
   200 
   201 lemma removeAll_rsp[quot_respect]:
   202   shows "(op = ===> op \<approx> ===> op \<approx>) removeAll removeAll"
   203   by simp
   204 
   205 lemma fminus_raw_rsp[quot_respect]:
   206   shows "(op \<approx> ===> op \<approx> ===> op \<approx>) fminus_raw fminus_raw"
   207   by simp
   208 
   209 lemma fcard_raw_rsp[quot_respect]:
   210   shows "(op \<approx> ===> op =) fcard_raw fcard_raw"
   211   by (simp add: fcard_raw_def)
   212 
   213 
   214 
   215 lemma not_memb_nil:
   216   shows "\<not> memb x []"
   217   by (simp add: memb_def)
   218 
   219 lemma memb_cons_iff:
   220   shows "memb x (y # xs) = (x = y \<or> memb x xs)"
   221   by (induct xs) (auto simp add: memb_def)
   222 
   223 lemma memb_absorb:
   224   shows "memb x xs \<Longrightarrow> x # xs \<approx> xs"
   225   by (induct xs) (auto simp add: memb_def)
   226 
   227 lemma none_memb_nil:
   228   "(\<forall>x. \<not> memb x xs) = (xs \<approx> [])"
   229   by (simp add: memb_def)
   230 
   231 
   232 lemma memb_commute_ffold_raw:
   233   "rsp_fold f \<Longrightarrow> h \<in> set b \<Longrightarrow> ffold_raw f z b = f h (ffold_raw f z (removeAll h b))"
   234   apply (induct b)
   235   apply (auto simp add: rsp_fold_def)
   236   done
   237 
   238 lemma ffold_raw_rsp_pre:
   239   "set a = set b \<Longrightarrow> ffold_raw f z a = ffold_raw f z b"
   240   apply (induct a arbitrary: b)
   241   apply (simp)
   242   apply (simp (no_asm_use))
   243   apply (rule conjI)
   244   apply (rule_tac [!] impI)
   245   apply (rule_tac [!] conjI)
   246   apply (rule_tac [!] impI)
   247   apply (metis insert_absorb)
   248   apply (metis List.insert_def List.set.simps(2) List.set_insert ffold_raw.simps(2))
   249   apply (metis Diff_insert_absorb insertI1 memb_commute_ffold_raw set_removeAll)
   250   apply(drule_tac x="removeAll a1 b" in meta_spec)
   251   apply(auto)
   252   apply(drule meta_mp)
   253   apply(blast)
   254   by (metis List.set.simps(2) emptyE ffold_raw.simps(2) in_listsp_conv_set listsp.simps mem_def)
   255 
   256 lemma ffold_raw_rsp[quot_respect]:
   257   shows "(op = ===> op = ===> op \<approx> ===> op =) ffold_raw ffold_raw"
   258   unfolding fun_rel_def
   259   by(auto intro: ffold_raw_rsp_pre)
   260 
   261 lemma concat_rsp_pre:
   262   assumes a: "list_all2 op \<approx> x x'"
   263   and     b: "x' \<approx> y'"
   264   and     c: "list_all2 op \<approx> y' y"
   265   and     d: "\<exists>x\<in>set x. xa \<in> set x"
   266   shows "\<exists>x\<in>set y. xa \<in> set x"
   267 proof -
   268   obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto
   269   have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a])
   270   then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto
   271   have "ya \<in> set y'" using b h by simp
   272   then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_all2_find_element)
   273   then show ?thesis using f i by auto
   274 qed
   275 
   276 lemma concat_rsp[quot_respect]:
   277   shows "(list_all2 op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"
   278 proof (rule fun_relI, elim pred_compE)
   279   fix a b ba bb
   280   assume a: "list_all2 op \<approx> a ba"
   281   assume b: "ba \<approx> bb"
   282   assume c: "list_all2 op \<approx> bb b"
   283   have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" 
   284   proof
   285     fix x
   286     show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" 
   287     proof
   288       assume d: "\<exists>xa\<in>set a. x \<in> set xa"
   289       show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])
   290     next
   291       assume e: "\<exists>xa\<in>set b. x \<in> set xa"
   292       have a': "list_all2 op \<approx> ba a" by (rule list_all2_symp[OF list_eq_equivp, OF a])
   293       have b': "bb \<approx> ba" by (rule equivp_symp[OF list_eq_equivp, OF b])
   294       have c': "list_all2 op \<approx> b bb" by (rule list_all2_symp[OF list_eq_equivp, OF c])
   295       show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])
   296     qed
   297   qed
   298   then show "concat a \<approx> concat b" by auto
   299 qed
   300 
   301 lemma [quot_respect]:
   302   shows "((op =) ===> op \<approx> ===> op \<approx>) filter filter"
   303   by auto
   304 
   305 text {* Distributive lattice with bot *}
   306 
   307 lemma append_inter_distrib:
   308   "x @ (finter_raw y z) \<approx> finter_raw (x @ y) (x @ z)"
   309   apply (induct x)
   310   apply (auto)
   311   done
   312 
   313 instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
   314 begin
   315 
   316 quotient_definition
   317   "bot :: 'a fset" is "[] :: 'a list"
   318 
   319 abbreviation
   320   fempty  ("{||}")
   321 where
   322   "{||} \<equiv> bot :: 'a fset"
   323 
   324 quotient_definition
   325   "less_eq_fset \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> bool)"
   326 is
   327   "sub_list \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> bool)"
   328 
   329 abbreviation
   330   f_subset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)
   331 where
   332   "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
   333 
   334 definition
   335   less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool"
   336 where  
   337   "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
   338 
   339 abbreviation
   340   fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)
   341 where
   342   "xs |\<subset>| ys \<equiv> xs < ys"
   343 
   344 quotient_definition
   345   "sup :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
   346 is
   347   "append :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
   348 
   349 abbreviation
   350   funion (infixl "|\<union>|" 65)
   351 where
   352   "xs |\<union>| ys \<equiv> sup (xs :: 'a fset) ys"
   353 
   354 quotient_definition
   355   "inf :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
   356 is
   357   "finter_raw :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
   358 
   359 abbreviation
   360   finter (infixl "|\<inter>|" 65)
   361 where
   362   "xs |\<inter>| ys \<equiv> inf (xs :: 'a fset) ys"
   363 
   364 quotient_definition
   365   "minus :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
   366 is
   367   "fminus_raw :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
   368 
   369 instance
   370 proof
   371   fix x y z :: "'a fset"
   372   show "x |\<subset>| y \<longleftrightarrow> x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x"
   373     unfolding less_fset_def 
   374     by (descending) (auto simp add: sub_list_def)
   375   show "x |\<subseteq>| x"  by (descending) (simp add: sub_list_def)
   376   show "{||} |\<subseteq>| x" by (descending) (simp add: sub_list_def)
   377   show "x |\<subseteq>| x |\<union>| y" by (descending) (simp add: sub_list_def)
   378   show "y |\<subseteq>| x |\<union>| y" by (descending) (simp add: sub_list_def)
   379   show "x |\<inter>| y |\<subseteq>| x"
   380     by (descending) (simp add: sub_list_def memb_def[symmetric])
   381   show "x |\<inter>| y |\<subseteq>| y" 
   382     by (descending) (simp add: sub_list_def memb_def[symmetric])
   383   show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)" 
   384     by (descending) (rule append_inter_distrib)
   385 next
   386   fix x y z :: "'a fset"
   387   assume a: "x |\<subseteq>| y"
   388   assume b: "y |\<subseteq>| z"
   389   show "x |\<subseteq>| z" using a b 
   390     by (descending) (simp add: sub_list_def)
   391 next
   392   fix x y :: "'a fset"
   393   assume a: "x |\<subseteq>| y"
   394   assume b: "y |\<subseteq>| x"
   395   show "x = y" using a b 
   396     by (descending) (unfold sub_list_def list_eq.simps, blast)
   397 next
   398   fix x y z :: "'a fset"
   399   assume a: "y |\<subseteq>| x"
   400   assume b: "z |\<subseteq>| x"
   401   show "y |\<union>| z |\<subseteq>| x" using a b 
   402     by (descending) (simp add: sub_list_def)
   403 next
   404   fix x y z :: "'a fset"
   405   assume a: "x |\<subseteq>| y"
   406   assume b: "x |\<subseteq>| z"
   407   show "x |\<subseteq>| y |\<inter>| z" using a b 
   408     by (descending) (simp add: sub_list_def memb_def[symmetric])
   409 qed
   410 
   411 end
   412 
   413 section {* Finsert and Membership *}
   414 
   415 quotient_definition
   416   "finsert :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
   417 is "Cons"
   418 
   419 syntax
   420   "@Finset"     :: "args => 'a fset"  ("{|(_)|}")
   421 
   422 translations
   423   "{|x, xs|}" == "CONST finsert x {|xs|}"
   424   "{|x|}"     == "CONST finsert x {||}"
   425 
   426 quotient_definition
   427   fin (infix "|\<in>|" 50)
   428 where
   429   "fin :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool" is "memb"
   430 
   431 abbreviation
   432   fnotin :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)
   433 where
   434   "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
   435 
   436 section {* Other constants on the Quotient Type *}
   437 
   438 quotient_definition
   439   "fcard :: 'a fset \<Rightarrow> nat"
   440 is
   441   fcard_raw
   442 
   443 quotient_definition
   444   "fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
   445 is
   446   map
   447 
   448 quotient_definition
   449   "fdelete :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
   450   is removeAll
   451 
   452 quotient_definition
   453   "fset :: 'a fset \<Rightarrow> 'a set"
   454   is "set"
   455 
   456 quotient_definition
   457   "ffold :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b"
   458   is "ffold_raw"
   459 
   460 quotient_definition
   461   "fconcat :: ('a fset) fset \<Rightarrow> 'a fset"
   462 is
   463   "concat"
   464 
   465 quotient_definition
   466   "ffilter :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
   467 is
   468   "filter"
   469 
   470 text {* Compositional Respectfullness and Preservation *}
   471 
   472 lemma [quot_respect]: "(list_all2 op \<approx> OOO op \<approx>) [] []"
   473   by (fact compose_list_refl)
   474 
   475 lemma [quot_preserve]: "(abs_fset \<circ> map f) [] = abs_fset []"
   476   by simp
   477 
   478 lemma [quot_respect]:
   479   shows "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) Cons Cons"
   480   apply auto
   481   apply (rule_tac b="x # b" in pred_compI)
   482   apply auto
   483   apply (rule_tac b="x # ba" in pred_compI)
   484   apply auto
   485   done
   486 
   487 lemma [quot_preserve]:
   488   "(rep_fset ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op # = finsert"
   489   by (simp add: fun_eq_iff Quotient_abs_rep[OF Quotient_fset]
   490       abs_o_rep[OF Quotient_fset] map_id finsert_def)
   491 
   492 lemma [quot_preserve]:
   493   "((map rep_fset \<circ> rep_fset) ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op @ = funion"
   494   by (simp add: fun_eq_iff Quotient_abs_rep[OF Quotient_fset]
   495       abs_o_rep[OF Quotient_fset] map_id sup_fset_def)
   496 
   497 lemma list_all2_app_l:
   498   assumes a: "reflp R"
   499   and b: "list_all2 R l r"
   500   shows "list_all2 R (z @ l) (z @ r)"
   501   by (induct z) (simp_all add: b rev_iffD1[OF a meta_eq_to_obj_eq[OF reflp_def]])
   502 
   503 lemma append_rsp2_pre0:
   504   assumes a:"list_all2 op \<approx> x x'"
   505   shows "list_all2 op \<approx> (x @ z) (x' @ z)"
   506   using a apply (induct x x' rule: list_induct2')
   507   by simp_all (rule list_all2_refl1)
   508 
   509 lemma append_rsp2_pre1:
   510   assumes a:"list_all2 op \<approx> x x'"
   511   shows "list_all2 op \<approx> (z @ x) (z @ x')"
   512   using a apply (induct x x' arbitrary: z rule: list_induct2')
   513   apply (rule list_all2_refl1)
   514   apply (simp_all del: list_eq.simps)
   515   apply (rule list_all2_app_l)
   516   apply (simp_all add: reflp_def)
   517   done
   518 
   519 lemma append_rsp2_pre:
   520   assumes a:"list_all2 op \<approx> x x'"
   521   and     b: "list_all2 op \<approx> z z'"
   522   shows "list_all2 op \<approx> (x @ z) (x' @ z')"
   523   apply (rule list_all2_transp[OF fset_equivp])
   524   apply (rule append_rsp2_pre0)
   525   apply (rule a)
   526   using b apply (induct z z' rule: list_induct2')
   527   apply (simp_all only: append_Nil2)
   528   apply (rule list_all2_refl1)
   529   apply simp_all
   530   apply (rule append_rsp2_pre1)
   531   apply simp
   532   done
   533 
   534 lemma [quot_respect]:
   535   "(list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) op @ op @"
   536 proof (intro fun_relI, elim pred_compE)
   537   fix x y z w x' z' y' w' :: "'a list list"
   538   assume a:"list_all2 op \<approx> x x'"
   539   and b:    "x' \<approx> y'"
   540   and c:    "list_all2 op \<approx> y' y"
   541   assume aa: "list_all2 op \<approx> z z'"
   542   and bb:   "z' \<approx> w'"
   543   and cc:   "list_all2 op \<approx> w' w"
   544   have a': "list_all2 op \<approx> (x @ z) (x' @ z')" using a aa append_rsp2_pre by auto
   545   have b': "x' @ z' \<approx> y' @ w'" using b bb by simp
   546   have c': "list_all2 op \<approx> (y' @ w') (y @ w)" using c cc append_rsp2_pre by auto
   547   have d': "(op \<approx> OO list_all2 op \<approx>) (x' @ z') (y @ w)"
   548     by (rule pred_compI) (rule b', rule c')
   549   show "(list_all2 op \<approx> OOO op \<approx>) (x @ z) (y @ w)"
   550     by (rule pred_compI) (rule a', rule d')
   551 qed
   552 
   553 text {* Raw theorems. Finsert, memb, singleron, sub_list *}
   554 
   555 lemma nil_not_cons:
   556   shows "\<not> ([] \<approx> x # xs)"
   557   and   "\<not> (x # xs \<approx> [])"
   558   by auto
   559 
   560 lemma no_memb_nil:
   561   "(\<forall>x. \<not> memb x xs) = (xs = [])"
   562   by (simp add: memb_def)
   563 
   564 lemma memb_consI1:
   565   shows "memb x (x # xs)"
   566   by (simp add: memb_def)
   567 
   568 lemma memb_consI2:
   569   shows "memb x xs \<Longrightarrow> memb x (y # xs)"
   570   by (simp add: memb_def)
   571 
   572 lemma singleton_list_eq:
   573   shows "[x] \<approx> [y] \<longleftrightarrow> x = y"
   574   by (simp)
   575 
   576 lemma sub_list_cons:
   577   "sub_list (x # xs) ys = (memb x ys \<and> sub_list xs ys)"
   578   by (auto simp add: memb_def sub_list_def)
   579 
   580 lemma fminus_raw_red: 
   581   "fminus_raw (x # xs) ys = (if x \<in> set ys then fminus_raw xs ys else x # (fminus_raw xs ys))"
   582   by (induct ys arbitrary: xs x) (simp_all)
   583 
   584 text {* Cardinality of finite sets *}
   585 
   586 lemma fcard_raw_0:
   587   shows "fcard_raw xs = 0 \<longleftrightarrow> xs \<approx> []"
   588   unfolding fcard_raw_def
   589   by (induct xs) (auto)
   590 
   591 lemma memb_card_not_0:
   592   assumes a: "memb a A"
   593   shows "\<not>(fcard_raw A = 0)"
   594 proof -
   595   have "\<not>(\<forall>x. \<not> memb x A)" using a by auto
   596   then have "\<not>A \<approx> []" using none_memb_nil[of A] by simp
   597   then show ?thesis using fcard_raw_0[of A] by simp
   598 qed
   599 
   600 text {* fmap *}
   601 
   602 lemma map_append:
   603   "map f (xs @ ys) \<approx> (map f xs) @ (map f ys)"
   604   by simp
   605 
   606 lemma memb_append:
   607   "memb x (xs @ ys) \<longleftrightarrow> memb x xs \<or> memb x ys"
   608   by (induct xs) (simp_all add: not_memb_nil memb_cons_iff)
   609 
   610 lemma fset_raw_strong_cases:
   611   obtains "xs = []"
   612     | x ys where "\<not> memb x ys" and "xs \<approx> x # ys"
   613 proof (induct xs arbitrary: x ys)
   614   case Nil
   615   then show thesis by simp
   616 next
   617   case (Cons a xs)
   618   have a: "\<lbrakk>xs = [] \<Longrightarrow> thesis; \<And>x ys. \<lbrakk>\<not> memb x ys; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis" by fact
   619   have b: "\<And>x' ys'. \<lbrakk>\<not> memb x' ys'; a # xs \<approx> x' # ys'\<rbrakk> \<Longrightarrow> thesis" by fact
   620   have c: "xs = [] \<Longrightarrow> thesis" by (metis no_memb_nil singleton_list_eq b)
   621   have "\<And>x ys. \<lbrakk>\<not> memb x ys; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis"
   622   proof -
   623     fix x :: 'a
   624     fix ys :: "'a list"
   625     assume d:"\<not> memb x ys"
   626     assume e:"xs \<approx> x # ys"
   627     show thesis
   628     proof (cases "x = a")
   629       assume h: "x = a"
   630       then have f: "\<not> memb a ys" using d by simp
   631       have g: "a # xs \<approx> a # ys" using e h by auto
   632       show thesis using b f g by simp
   633     next
   634       assume h: "x \<noteq> a"
   635       then have f: "\<not> memb x (a # ys)" using d unfolding memb_def by auto
   636       have g: "a # xs \<approx> x # (a # ys)" using e h by auto
   637       show thesis using b f g by simp
   638     qed
   639   qed
   640   then show thesis using a c by blast
   641 qed
   642 
   643 section {* deletion *}
   644 
   645 
   646 lemma fset_raw_removeAll_cases:
   647   "xs = [] \<or> (\<exists>x. memb x xs \<and> xs \<approx> x # removeAll x xs)"
   648   by (induct xs) (auto simp add: memb_def)
   649 
   650 lemma fremoveAll_filter:
   651   "removeAll y xs = [x \<leftarrow> xs. x \<noteq> y]"
   652   by (induct xs) simp_all
   653 
   654 lemma fcard_raw_delete:
   655   "fcard_raw (removeAll y xs) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)"
   656   by (auto simp add: fcard_raw_def memb_def)
   657 
   658 lemma set_cong:
   659   shows "(x \<approx> y) = (set x = set y)"
   660   by auto
   661 
   662 lemma inj_map_eq_iff:
   663   "inj f \<Longrightarrow> (map f l \<approx> map f m) = (l \<approx> m)"
   664   by (simp add: set_eq_iff[symmetric] inj_image_eq_iff)
   665 
   666 text {* alternate formulation with a different decomposition principle
   667   and a proof of equivalence *}
   668 
   669 inductive
   670   list_eq2
   671 where
   672   "list_eq2 (a # b # xs) (b # a # xs)"
   673 | "list_eq2 [] []"
   674 | "list_eq2 xs ys \<Longrightarrow> list_eq2 ys xs"
   675 | "list_eq2 (a # a # xs) (a # xs)"
   676 | "list_eq2 xs ys \<Longrightarrow> list_eq2 (a # xs) (a # ys)"
   677 | "\<lbrakk>list_eq2 xs1 xs2; list_eq2 xs2 xs3\<rbrakk> \<Longrightarrow> list_eq2 xs1 xs3"
   678 
   679 lemma list_eq2_refl:
   680   shows "list_eq2 xs xs"
   681   by (induct xs) (auto intro: list_eq2.intros)
   682 
   683 lemma cons_delete_list_eq2:
   684   shows "list_eq2 (a # (removeAll a A)) (if memb a A then A else a # A)"
   685   apply (induct A)
   686   apply (simp add: memb_def list_eq2_refl)
   687   apply (case_tac "memb a (aa # A)")
   688   apply (simp_all only: memb_cons_iff)
   689   apply (case_tac [!] "a = aa")
   690   apply (simp_all)
   691   apply (case_tac "memb a A")
   692   apply (auto simp add: memb_def)[2]
   693   apply (metis list_eq2.intros(3) list_eq2.intros(4) list_eq2.intros(5) list_eq2.intros(6))
   694   apply (metis list_eq2.intros(1) list_eq2.intros(5) list_eq2.intros(6))
   695   apply (auto simp add: list_eq2_refl memb_def)
   696   done
   697 
   698 lemma memb_delete_list_eq2:
   699   assumes a: "memb e r"
   700   shows "list_eq2 (e # removeAll e r) r"
   701   using a cons_delete_list_eq2[of e r]
   702   by simp
   703 
   704 lemma list_eq2_equiv:
   705   "(l \<approx> r) \<longleftrightarrow> (list_eq2 l r)"
   706 proof
   707   show "list_eq2 l r \<Longrightarrow> l \<approx> r" by (induct rule: list_eq2.induct) auto
   708 next
   709   {
   710     fix n
   711     assume a: "fcard_raw l = n" and b: "l \<approx> r"
   712     have "list_eq2 l r"
   713       using a b
   714     proof (induct n arbitrary: l r)
   715       case 0
   716       have "fcard_raw l = 0" by fact
   717       then have "\<forall>x. \<not> memb x l" using memb_card_not_0[of _ l] by auto
   718       then have z: "l = []" using no_memb_nil by auto
   719       then have "r = []" using `l \<approx> r` by simp
   720       then show ?case using z list_eq2_refl by simp
   721     next
   722       case (Suc m)
   723       have b: "l \<approx> r" by fact
   724       have d: "fcard_raw l = Suc m" by fact
   725       then have "\<exists>a. memb a l" 
   726 	apply(simp add: fcard_raw_def memb_def)
   727 	apply(drule card_eq_SucD)
   728 	apply(blast)
   729 	done
   730       then obtain a where e: "memb a l" by auto
   731       then have e': "memb a r" using list_eq.simps[simplified memb_def[symmetric], of l r] b 
   732 	unfolding memb_def by auto
   733       have f: "fcard_raw (removeAll a l) = m" using fcard_raw_delete[of a l] e d by simp
   734       have g: "removeAll a l \<approx> removeAll a r" using removeAll_rsp b by simp
   735       have "list_eq2 (removeAll a l) (removeAll a r)" by (rule Suc.hyps[OF f g])
   736       then have h: "list_eq2 (a # removeAll a l) (a # removeAll a r)" by (rule list_eq2.intros(5))
   737       have i: "list_eq2 l (a # removeAll a l)"
   738         by (rule list_eq2.intros(3)[OF memb_delete_list_eq2[OF e]])
   739       have "list_eq2 l (a # removeAll a r)" by (rule list_eq2.intros(6)[OF i h])
   740       then show ?case using list_eq2.intros(6)[OF _ memb_delete_list_eq2[OF e']] by simp
   741     qed
   742     }
   743   then show "l \<approx> r \<Longrightarrow> list_eq2 l r" by blast
   744 qed
   745 
   746 text {* Lifted theorems *}
   747 
   748 lemma not_fin_fnil: "x |\<notin>| {||}"
   749   by (descending) (simp add: memb_def)
   750 
   751 lemma fin_finsert_iff[simp]:
   752   "x |\<in>| finsert y S \<longleftrightarrow> x = y \<or> x |\<in>| S"
   753   by (descending) (simp add: memb_def)
   754 
   755 lemma
   756   shows finsertI1: "x |\<in>| finsert x S"
   757   and   finsertI2: "x |\<in>| S \<Longrightarrow> x |\<in>| finsert y S"
   758   by (lifting memb_consI1 memb_consI2)
   759 
   760 lemma finsert_absorb[simp]:
   761   shows "x |\<in>| S \<Longrightarrow> finsert x S = S"
   762   by (descending) (auto simp add: memb_def)
   763 
   764 lemma fempty_not_finsert[simp]:
   765   "{||} \<noteq> finsert x S"
   766   "finsert x S \<noteq> {||}"
   767   by (lifting nil_not_cons)
   768 
   769 lemma finsert_left_comm:
   770   "finsert x (finsert y S) = finsert y (finsert x S)"
   771   by (descending) (auto)
   772 
   773 lemma finsert_left_idem:
   774   "finsert x (finsert x S) = finsert x S"
   775   by (descending) (auto)
   776 
   777 lemma fsingleton_eq[simp]:
   778   shows "{|x|} = {|y|} \<longleftrightarrow> x = y"
   779   by (descending) (auto)
   780 
   781 
   782 text {* fset *}
   783 
   784 lemma fset_simps[simp]:
   785   "fset {||} = ({} :: 'a set)"
   786   "fset (finsert (h :: 'a) t) = insert h (fset t)"
   787   by (lifting set.simps)
   788 
   789 lemma in_fset:
   790   "x \<in> fset S \<equiv> x |\<in>| S"
   791   by (lifting memb_def[symmetric])
   792 
   793 lemma none_fin_fempty:
   794   "(\<forall>x. x |\<notin>| S) \<longleftrightarrow> S = {||}"
   795   by (lifting none_memb_nil)
   796 
   797 lemma fset_cong:
   798   "S = T \<longleftrightarrow> fset S = fset T"
   799   by (lifting set_cong)
   800 
   801 
   802 text {* fcard *}
   803 
   804 lemma fcard_finsert_if [simp]:
   805   shows "fcard (finsert x S) = (if x |\<in>| S then fcard S else Suc (fcard S))"
   806   by (descending) (auto simp add: fcard_raw_def memb_def insert_absorb)
   807 
   808 lemma fcard_0[simp]:
   809   shows "fcard S = 0 \<longleftrightarrow> S = {||}"
   810   by (descending) (simp add: fcard_raw_def)
   811 
   812 lemma fcard_fempty[simp]:
   813   shows "fcard {||} = 0"
   814   by (simp add: fcard_0)
   815 
   816 lemma fcard_1:
   817   shows "fcard S = 1 \<longleftrightarrow> (\<exists>x. S = {|x|})"
   818   by (descending) (auto simp add: fcard_raw_def card_Suc_eq)
   819 
   820 lemma fcard_gt_0:
   821   shows "x \<in> fset S \<Longrightarrow> 0 < fcard S"
   822   by (descending) (auto simp add: fcard_raw_def card_gt_0_iff)
   823   
   824 lemma fcard_not_fin:
   825   shows "(x |\<notin>| S) = (fcard (finsert x S) = Suc (fcard S))"
   826   by (descending) (auto simp add: memb_def fcard_raw_def insert_absorb)
   827 
   828 lemma fcard_suc: "fcard S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = finsert x T \<and> fcard T = n"
   829   apply descending
   830   apply(simp add: fcard_raw_def memb_def)
   831   apply(drule card_eq_SucD)
   832   apply(auto)
   833   apply(rule_tac x="b" in exI)
   834   apply(rule_tac x="removeAll b S" in exI)
   835   apply(auto)
   836   done
   837 
   838 lemma fcard_delete:
   839   "fcard (fdelete y S) = (if y |\<in>| S then fcard S - 1 else fcard S)"
   840   by (lifting fcard_raw_delete)
   841 
   842 lemma fcard_suc_memb: 
   843   shows "fcard A = Suc n \<Longrightarrow> \<exists>a. a |\<in>| A"
   844   apply(descending)
   845   apply(simp add: fcard_raw_def memb_def)
   846   apply(drule card_eq_SucD)
   847   apply(auto)
   848   done
   849 
   850 lemma fin_fcard_not_0: 
   851   shows "a |\<in>| A \<Longrightarrow> fcard A \<noteq> 0"
   852   by (descending) (auto simp add: fcard_raw_def memb_def)
   853 
   854 
   855 text {* funion *}
   856 
   857 lemmas [simp] =
   858   sup_bot_left[where 'a="'a fset", standard]
   859   sup_bot_right[where 'a="'a fset", standard]
   860 
   861 lemma funion_finsert[simp]:
   862   shows "finsert x S |\<union>| T = finsert x (S |\<union>| T)"
   863   by (lifting append.simps(2))
   864 
   865 lemma singleton_union_left:
   866   shows "{|a|} |\<union>| S = finsert a S"
   867   by simp
   868 
   869 lemma singleton_union_right:
   870   shows "S |\<union>| {|a|} = finsert a S"
   871   by (subst sup.commute) simp
   872 
   873 
   874 section {* Induction and Cases rules for fsets *}
   875 
   876 lemma fset_strong_cases:
   877   obtains "xs = {||}"
   878     | x ys where "x |\<notin>| ys" and "xs = finsert x ys"
   879   by (lifting fset_raw_strong_cases)
   880 
   881 lemma fset_exhaust[case_names fempty finsert, cases type: fset]:
   882   shows "\<lbrakk>S = {||} \<Longrightarrow> P; \<And>x S'. S = finsert x S' \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
   883   by (lifting list.exhaust)
   884 
   885 lemma fset_induct_weak[case_names fempty finsert]:
   886   shows "\<lbrakk>P {||}; \<And>x S. P S \<Longrightarrow> P (finsert x S)\<rbrakk> \<Longrightarrow> P S"
   887   by (lifting list.induct)
   888 
   889 lemma fset_induct[case_names fempty finsert, induct type: fset]:
   890   assumes prem1: "P {||}"
   891   and     prem2: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
   892   shows "P S"
   893 proof(induct S rule: fset_induct_weak)
   894   case fempty
   895   show "P {||}" by (rule prem1)
   896 next
   897   case (finsert x S)
   898   have asm: "P S" by fact
   899   show "P (finsert x S)"
   900     by (cases "x |\<in>| S") (simp_all add: asm prem2)
   901 qed
   902 
   903 lemma fset_induct2:
   904   "P {||} {||} \<Longrightarrow>
   905   (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
   906   (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
   907   (\<And>x xs y ys. \<lbrakk>P xs ys; x |\<notin>| xs; y |\<notin>| ys\<rbrakk> \<Longrightarrow> P (finsert x xs) (finsert y ys)) \<Longrightarrow>
   908   P xsa ysa"
   909   apply (induct xsa arbitrary: ysa)
   910   apply (induct_tac x rule: fset_induct)
   911   apply simp_all
   912   apply (induct_tac xa rule: fset_induct)
   913   apply simp_all
   914   done
   915 
   916 lemma fset_fcard_induct:
   917   assumes a: "P {||}"
   918   and     b: "\<And>xs ys. Suc (fcard xs) = (fcard ys) \<Longrightarrow> P xs \<Longrightarrow> P ys"
   919   shows "P zs"
   920 proof (induct zs)
   921   show "P {||}" by (rule a)
   922 next
   923   fix x :: 'a and zs :: "'a fset"
   924   assume h: "P zs"
   925   assume "x |\<notin>| zs"
   926   then have H1: "Suc (fcard zs) = fcard (finsert x zs)" using fcard_suc by auto
   927   then show "P (finsert x zs)" using b h by simp
   928 qed
   929 
   930 text {* fmap *}
   931 
   932 lemma fmap_simps[simp]:
   933   fixes f::"'a \<Rightarrow> 'b"
   934   shows "fmap f {||} = {||}"
   935   and   "fmap f (finsert x S) = finsert (f x) (fmap f S)"
   936   by (lifting map.simps)
   937 
   938 lemma fmap_set_image:
   939   "fset (fmap f S) = f ` (fset S)"
   940   by (induct S) simp_all
   941 
   942 lemma inj_fmap_eq_iff:
   943   "inj f \<Longrightarrow> fmap f S = fmap f T \<longleftrightarrow> S = T"
   944   by (lifting inj_map_eq_iff)
   945 
   946 lemma fmap_funion: 
   947   shows "fmap f (S |\<union>| T) = fmap f S |\<union>| fmap f T"
   948   by (lifting map_append)
   949 
   950 lemma fin_funion:
   951   shows "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"
   952   by (lifting memb_append)
   953 
   954 
   955 section {* fset *}
   956 
   957 lemma fin_set: 
   958   shows "x |\<in>| xs \<longleftrightarrow> x \<in> fset xs"
   959   by (lifting memb_def)
   960 
   961 lemma fnotin_set: 
   962   shows "x |\<notin>| xs \<longleftrightarrow> x \<notin> fset xs"
   963   by (simp add: fin_set)
   964 
   965 lemma fcard_set: 
   966   shows "fcard xs = card (fset xs)"
   967   by (lifting fcard_raw_def)
   968 
   969 lemma fsubseteq_set: 
   970   shows "xs |\<subseteq>| ys \<longleftrightarrow> fset xs \<subseteq> fset ys"
   971   by (lifting sub_list_def)
   972 
   973 lemma fsubset_set: 
   974   shows "xs |\<subset>| ys \<longleftrightarrow> fset xs \<subset> fset ys"
   975   unfolding less_fset_def 
   976   by (descending) (auto simp add: sub_list_def)
   977 
   978 lemma ffilter_set [simp]: 
   979   shows "fset (ffilter P xs) = P \<inter> fset xs"
   980   by (descending) (auto simp add: mem_def)
   981 
   982 lemma fdelete_set [simp]: 
   983   shows "fset (fdelete x xs) = fset xs - {x}"
   984   by (lifting set_removeAll)
   985 
   986 lemma finter_set [simp]: 
   987   shows "fset (xs |\<inter>| ys) = fset xs \<inter> fset ys"
   988   by (lifting set_finter_raw)
   989 
   990 lemma funion_set [simp]: 
   991   shows "fset (xs |\<union>| ys) = fset xs \<union> fset ys"
   992   by (lifting set_append)
   993 
   994 lemma fminus_set [simp]: 
   995   shows "fset (xs - ys) = fset xs - fset ys"
   996   by (lifting set_fminus_raw)
   997 
   998 lemmas fset_to_set_trans =
   999   fin_set fnotin_set fcard_set fsubseteq_set fsubset_set
  1000   finter_set funion_set ffilter_set fset_simps
  1001   fset_cong fdelete_set fmap_set_image fminus_set
  1002 
  1003 
  1004 text {* ffold *}
  1005 
  1006 lemma ffold_nil: 
  1007   shows "ffold f z {||} = z"
  1008   by (lifting ffold_raw.simps(1)[where 'a="'b" and 'b="'a"])
  1009 
  1010 lemma ffold_finsert: "ffold f z (finsert a A) =
  1011   (if rsp_fold f then if a |\<in>| A then ffold f z A else f a (ffold f z A) else z)"
  1012   by (descending) (simp add: memb_def)
  1013 
  1014 lemma fin_commute_ffold:
  1015   "\<lbrakk>rsp_fold f; h |\<in>| b\<rbrakk> \<Longrightarrow> ffold f z b = f h (ffold f z (fdelete h b))"
  1016   by (descending) (simp add: memb_def memb_commute_ffold_raw)
  1017 
  1018 
  1019 text {* fdelete *}
  1020 
  1021 lemma fin_fdelete:
  1022   shows "x |\<in>| fdelete y S \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"
  1023   by (descending) (simp add: memb_def)
  1024 
  1025 lemma fnotin_fdelete:
  1026   shows "x |\<notin>| fdelete x S"
  1027   by (descending) (simp add: memb_def)
  1028 
  1029 lemma fnotin_fdelete_ident:
  1030   shows "x |\<notin>| S \<Longrightarrow> fdelete x S = S"
  1031   by (descending) (simp add: memb_def)
  1032 
  1033 lemma fset_fdelete_cases:
  1034   shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = finsert x (fdelete x S))"
  1035   by (lifting fset_raw_removeAll_cases)
  1036 
  1037 text {* finite intersection *}
  1038 
  1039 lemma finter_empty_l:
  1040   shows "{||} |\<inter>| S = {||}"
  1041   by simp
  1042 
  1043 
  1044 lemma finter_empty_r:
  1045   shows "S |\<inter>| {||} = {||}"
  1046   by simp
  1047 
  1048 lemma finter_finsert:
  1049   shows "finsert x S |\<inter>| T = (if x |\<in>| T then finsert x (S |\<inter>| T) else S |\<inter>| T)"
  1050   by (descending) (simp add: memb_def)
  1051 
  1052 lemma fin_finter:
  1053   shows "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"
  1054   by (descending) (simp add: memb_def)
  1055 
  1056 lemma fsubset_finsert:
  1057   shows "finsert x xs |\<subseteq>| ys \<longleftrightarrow> x |\<in>| ys \<and> xs |\<subseteq>| ys"
  1058   by (lifting sub_list_cons)
  1059 
  1060 lemma 
  1061   shows "xs |\<subseteq>| ys \<equiv> \<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys"
  1062   by (descending) (auto simp add: sub_list_def memb_def)
  1063 
  1064 lemma fsubset_fin: 
  1065   shows "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"
  1066   by (descending) (auto simp add: sub_list_def memb_def)
  1067 
  1068 lemma fminus_fin: 
  1069   shows "x |\<in>| xs - ys \<longleftrightarrow> x |\<in>| xs \<and> x |\<notin>| ys"
  1070   by (descending) (simp add: memb_def)
  1071 
  1072 lemma fminus_red: 
  1073   shows "finsert x xs - ys = (if x |\<in>| ys then xs - ys else finsert x (xs - ys))"
  1074   by (descending) (auto simp add: memb_def)
  1075 
  1076 lemma fminus_red_fin [simp]: 
  1077   shows "x |\<in>| ys \<Longrightarrow> finsert x xs - ys = xs - ys"
  1078   by (simp add: fminus_red)
  1079 
  1080 lemma fminus_red_fnotin[simp]: 
  1081   shows "x |\<notin>| ys \<Longrightarrow> finsert x xs - ys = finsert x (xs - ys)"
  1082   by (simp add: fminus_red)
  1083 
  1084 lemma fset_eq_iff:
  1085   shows "S = T \<longleftrightarrow> (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"
  1086   by (descending) (auto simp add: memb_def)
  1087 
  1088 (* We cannot write it as "assumes .. shows" since Isabelle changes
  1089    the quantifiers to schematic variables and reintroduces them in
  1090    a different order *)
  1091 lemma fset_eq_cases:
  1092  "\<lbrakk>a1 = a2;
  1093    \<And>a b xs. \<lbrakk>a1 = finsert a (finsert b xs); a2 = finsert b (finsert a xs)\<rbrakk> \<Longrightarrow> P;
  1094    \<lbrakk>a1 = {||}; a2 = {||}\<rbrakk> \<Longrightarrow> P; \<And>xs ys. \<lbrakk>a1 = ys; a2 = xs; xs = ys\<rbrakk> \<Longrightarrow> P;
  1095    \<And>a xs. \<lbrakk>a1 = finsert a (finsert a xs); a2 = finsert a xs\<rbrakk> \<Longrightarrow> P;
  1096    \<And>xs ys a. \<lbrakk>a1 = finsert a xs; a2 = finsert a ys; xs = ys\<rbrakk> \<Longrightarrow> P;
  1097    \<And>xs1 xs2 xs3. \<lbrakk>a1 = xs1; a2 = xs3; xs1 = xs2; xs2 = xs3\<rbrakk> \<Longrightarrow> P\<rbrakk>
  1098   \<Longrightarrow> P"
  1099   by (lifting list_eq2.cases[simplified list_eq2_equiv[symmetric]])
  1100 
  1101 lemma fset_eq_induct:
  1102   assumes "x1 = x2"
  1103   and "\<And>a b xs. P (finsert a (finsert b xs)) (finsert b (finsert a xs))"
  1104   and "P {||} {||}"
  1105   and "\<And>xs ys. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P ys xs"
  1106   and "\<And>a xs. P (finsert a (finsert a xs)) (finsert a xs)"
  1107   and "\<And>xs ys a. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P (finsert a xs) (finsert a ys)"
  1108   and "\<And>xs1 xs2 xs3. \<lbrakk>xs1 = xs2; P xs1 xs2; xs2 = xs3; P xs2 xs3\<rbrakk> \<Longrightarrow> P xs1 xs3"
  1109   shows "P x1 x2"
  1110   using assms
  1111   by (lifting list_eq2.induct[simplified list_eq2_equiv[symmetric]])
  1112 
  1113 section {* fconcat *}
  1114 
  1115 lemma fconcat_empty:
  1116   shows "fconcat {||} = {||}"
  1117   by (lifting concat.simps(1))
  1118 
  1119 lemma fconcat_insert:
  1120   shows "fconcat (finsert x S) = x |\<union>| fconcat S"
  1121   by (lifting concat.simps(2))
  1122 
  1123 lemma 
  1124   shows "fconcat (xs |\<union>| ys) = fconcat xs |\<union>| fconcat ys"
  1125   by (lifting concat_append)
  1126 
  1127 
  1128 section {* ffilter *}
  1129 
  1130 lemma subseteq_filter: 
  1131   shows "ffilter P xs <= ffilter Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"
  1132   by  (descending) (auto simp add: memb_def sub_list_def)
  1133 
  1134 lemma eq_ffilter: 
  1135   shows "(ffilter P xs = ffilter Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"
  1136   by (descending) (auto simp add: memb_def)
  1137 
  1138 lemma subset_ffilter:
  1139   shows "(\<And>x. x |\<in>| xs \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| xs & \<not> P x & Q x) \<Longrightarrow> ffilter P xs < ffilter Q xs"
  1140   unfolding less_fset_def by (auto simp add: subseteq_filter eq_ffilter)
  1141 
  1142 
  1143 section {* lemmas transferred from Finite_Set theory *}
  1144 
  1145 text {* finiteness for finite sets holds *}
  1146 lemma finite_fset [simp]: 
  1147   shows "finite (fset S)"
  1148   by (induct S) auto
  1149 
  1150 lemma fset_choice: 
  1151   shows "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y) \<Longrightarrow> \<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
  1152   unfolding fset_to_set_trans
  1153   by (rule finite_set_choice[simplified Ball_def, OF finite_fset])
  1154 
  1155 lemma fsubseteq_fempty:
  1156   shows "xs |\<subseteq>| {||} \<longleftrightarrow> xs = {||}"
  1157   by (metis finter_empty_r le_iff_inf)
  1158 
  1159 lemma not_fsubset_fnil: 
  1160   shows "\<not> xs |\<subset>| {||}"
  1161   by (metis fset_simps(1) fsubset_set not_psubset_empty)
  1162   
  1163 lemma fcard_mono: 
  1164   shows "xs |\<subseteq>| ys \<Longrightarrow> fcard xs \<le> fcard ys"
  1165   unfolding fset_to_set_trans
  1166   by (rule card_mono[OF finite_fset])
  1167 
  1168 lemma fcard_fseteq: 
  1169   shows "xs |\<subseteq>| ys \<Longrightarrow> fcard ys \<le> fcard xs \<Longrightarrow> xs = ys"
  1170   unfolding fcard_set fsubseteq_set
  1171   by (simp add: card_seteq[OF finite_fset] fset_cong)
  1172 
  1173 lemma psubset_fcard_mono: 
  1174   shows "xs |\<subset>| ys \<Longrightarrow> fcard xs < fcard ys"
  1175   unfolding fset_to_set_trans
  1176   by (rule psubset_card_mono[OF finite_fset])
  1177 
  1178 lemma fcard_funion_finter: 
  1179   shows "fcard xs + fcard ys = fcard (xs |\<union>| ys) + fcard (xs |\<inter>| ys)"
  1180   unfolding fset_to_set_trans
  1181   by (rule card_Un_Int[OF finite_fset finite_fset])
  1182 
  1183 lemma fcard_funion_disjoint: 
  1184   shows "xs |\<inter>| ys = {||} \<Longrightarrow> fcard (xs |\<union>| ys) = fcard xs + fcard ys"
  1185   unfolding fset_to_set_trans
  1186   by (rule card_Un_disjoint[OF finite_fset finite_fset])
  1187 
  1188 lemma fcard_delete1_less: 
  1189   shows "x |\<in>| xs \<Longrightarrow> fcard (fdelete x xs) < fcard xs"
  1190   unfolding fset_to_set_trans
  1191   by (rule card_Diff1_less[OF finite_fset])
  1192 
  1193 lemma fcard_delete2_less: 
  1194   shows "x |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> fcard (fdelete y (fdelete x xs)) < fcard xs"
  1195   unfolding fset_to_set_trans
  1196   by (rule card_Diff2_less[OF finite_fset])
  1197 
  1198 lemma fcard_delete1_le: 
  1199   shows "fcard (fdelete x xs) \<le> fcard xs"
  1200   unfolding fset_to_set_trans
  1201   by (rule card_Diff1_le[OF finite_fset])
  1202 
  1203 lemma fcard_psubset: 
  1204   shows "ys |\<subseteq>| xs \<Longrightarrow> fcard ys < fcard xs \<Longrightarrow> ys |\<subset>| xs"
  1205   unfolding fset_to_set_trans
  1206   by (rule card_psubset[OF finite_fset])
  1207 
  1208 lemma fcard_fmap_le: 
  1209   shows "fcard (fmap f xs) \<le> fcard xs"
  1210   unfolding fset_to_set_trans
  1211   by (rule card_image_le[OF finite_fset])
  1212 
  1213 lemma fin_fminus_fnotin: 
  1214   shows "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S"
  1215   unfolding fset_to_set_trans
  1216   by blast
  1217 
  1218 lemma fin_fnotin_fminus: 
  1219   shows "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S"
  1220   unfolding fset_to_set_trans
  1221   by blast
  1222 
  1223 lemma fin_mdef: 
  1224   "x |\<in>| F \<longleftrightarrow> x |\<notin>| (F - {|x|}) \<and> F = finsert x (F - {|x|})"
  1225   unfolding fset_to_set_trans
  1226   by blast
  1227 
  1228 lemma fcard_fminus_finsert[simp]:
  1229   assumes "a |\<in>| A" and "a |\<notin>| B"
  1230   shows "fcard(A - finsert a B) = fcard(A - B) - 1"
  1231   using assms 
  1232   unfolding fset_to_set_trans
  1233   by (rule card_Diff_insert[OF finite_fset])
  1234 
  1235 lemma fcard_fminus_fsubset:
  1236   assumes "B |\<subseteq>| A"
  1237   shows "fcard (A - B) = fcard A - fcard B"
  1238   using assms unfolding fset_to_set_trans
  1239   by (rule card_Diff_subset[OF finite_fset])
  1240 
  1241 lemma fcard_fminus_subset_finter:
  1242   shows "fcard (A - B) = fcard A - fcard (A |\<inter>| B)"
  1243   unfolding fset_to_set_trans
  1244   by (rule card_Diff_subset_Int) (fold finter_set, rule finite_fset)
  1245 
  1246 
  1247 ML {*
  1248 fun dest_fsetT (Type (@{type_name fset}, [T])) = T
  1249   | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
  1250 *}
  1251 
  1252 no_notation
  1253   list_eq (infix "\<approx>" 50)
  1254 
  1255 end