src/HOL/Quotient_Examples/Quotient_FSet.thy

+(*  Title:      HOL/Quotient_Examples/Quotient_FSet.thy
+    Author:     Cezary Kaliszyk, TU Munich
+    Author:     Christian Urban, TU Munich
+
+Type of finite sets.
+*)
+
+(********************************************************************
+  WARNING: There is a formalization of 'a fset as a subtype of sets in
+  HOL/Library/FSet.thy using Lifting/Transfer. The user should use
+  that file rather than this file unless there are some very specific
+  reasons.
+*********************************************************************)
+
+theory Quotient_FSet
+imports "~~/src/HOL/Library/Multiset" "~~/src/HOL/Library/Quotient_List"
+begin
+
+text \<open>
+  The type of finite sets is created by a quotient construction
+  over lists. The definition of the equivalence:
+\<close>
+
+definition
+  list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)
+where
+  [simp]: "xs \<approx> ys \<longleftrightarrow> set xs = set ys"
+
+lemma list_eq_reflp:
+  "reflp list_eq"
+  by (auto intro: reflpI)
+
+lemma list_eq_symp:
+  "symp list_eq"
+  by (auto intro: sympI)
+
+lemma list_eq_transp:
+  "transp list_eq"
+  by (auto intro: transpI)
+
+lemma list_eq_equivp:
+  "equivp list_eq"
+  by (auto intro: equivpI list_eq_reflp list_eq_symp list_eq_transp)
+
+text \<open>The \<open>fset\<close> type\<close>
+
+quotient_type
+  'a fset = "'a list" / "list_eq"
+  by (rule list_eq_equivp)
+
+text \<open>
+  Definitions for sublist, cardinality, 
+  intersection, difference and respectful fold over 
+  lists.
+\<close>
+
+declare List.member_def [simp]
+
+definition
+  sub_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
+where 
+  [simp]: "sub_list xs ys \<longleftrightarrow> set xs \<subseteq> set ys"
+
+definition
+  card_list :: "'a list \<Rightarrow> nat"
+where
+  [simp]: "card_list xs = card (set xs)"
+
+definition
+  inter_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+  [simp]: "inter_list xs ys = [x \<leftarrow> xs. x \<in> set xs \<and> x \<in> set ys]"
+
+definition
+  diff_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+  [simp]: "diff_list xs ys = [x \<leftarrow> xs. x \<notin> set ys]"
+
+definition
+  rsp_fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool"
+where
+  "rsp_fold f \<longleftrightarrow> (\<forall>u v. f u \<circ> f v = f v \<circ> f u)"
+
+lemma rsp_foldI:
+  "(\<And>u v. f u \<circ> f v = f v \<circ> f u) \<Longrightarrow> rsp_fold f"
+  by (simp add: rsp_fold_def)
+
+lemma rsp_foldE:
+  assumes "rsp_fold f"
+  obtains "f u \<circ> f v = f v \<circ> f u"
+  using assms by (simp add: rsp_fold_def)
+
+definition
+  fold_once :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
+where
+  "fold_once f xs = (if rsp_fold f then fold f (remdups xs) else id)"
+
+lemma fold_once_default [simp]:
+  "\<not> rsp_fold f \<Longrightarrow> fold_once f xs = id"
+  by (simp add: fold_once_def)
+
+lemma fold_once_fold_remdups:
+  "rsp_fold f \<Longrightarrow> fold_once f xs = fold f (remdups xs)"
+  by (simp add: fold_once_def)
+
+
+section \<open>Quotient composition lemmas\<close>
+
+lemma list_all2_refl':
+  assumes q: "equivp R"
+  shows "(list_all2 R) r r"
+  by (rule list_all2_refl) (metis equivp_def q)
+
+lemma compose_list_refl:
+  assumes q: "equivp R"
+  shows "(list_all2 R OOO op \<approx>) r r"
+proof
+  have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
+  show "list_all2 R r r" by (rule list_all2_refl'[OF q])
+  with * show "(op \<approx> OO list_all2 R) r r" ..
+qed
+
+lemma map_list_eq_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"
+  by (simp only: list_eq_def set_map)
+
+lemma quotient_compose_list_g:
+  assumes q: "Quotient3 R Abs Rep"
+  and     e: "equivp R"
+  shows  "Quotient3 ((list_all2 R) OOO (op \<approx>))
+    (abs_fset \<circ> (map Abs)) ((map Rep) \<circ> rep_fset)"
+  unfolding Quotient3_def comp_def
+proof (intro conjI allI)
+  fix a r s
+  show "abs_fset (map Abs (map Rep (rep_fset a))) = a"
+    by (simp add: abs_o_rep[OF q] Quotient3_abs_rep[OF Quotient3_fset] List.map.id)
+  have b: "list_all2 R (map Rep (rep_fset a)) (map Rep (rep_fset a))"
+    by (rule list_all2_refl'[OF e])
+  have c: "(op \<approx> OO list_all2 R) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
+    by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
+  show "(list_all2 R OOO op \<approx>) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
+    by (rule, rule list_all2_refl'[OF e]) (rule c)
+  show "(list_all2 R OOO op \<approx>) r s = ((list_all2 R OOO op \<approx>) r r \<and>
+        (list_all2 R OOO op \<approx>) s s \<and> abs_fset (map Abs r) = abs_fset (map Abs s))"
+  proof (intro iffI conjI)
+    show "(list_all2 R OOO op \<approx>) r r" by (rule compose_list_refl[OF e])
+    show "(list_all2 R OOO op \<approx>) s s" by (rule compose_list_refl[OF e])
+  next
+    assume a: "(list_all2 R OOO op \<approx>) r s"
+    then have b: "map Abs r \<approx> map Abs s"
+    proof (elim relcomppE)
+      fix b ba
+      assume c: "list_all2 R r b"
+      assume d: "b \<approx> ba"
+      assume e: "list_all2 R ba s"
+      have f: "map Abs r = map Abs b"
+        using Quotient3_rel[OF list_quotient3[OF q]] c by blast
+      have "map Abs ba = map Abs s"
+        using Quotient3_rel[OF list_quotient3[OF q]] e by blast
+      then have g: "map Abs s = map Abs ba" by simp
+      then show "map Abs r \<approx> map Abs s" using d f map_list_eq_cong by simp
+    qed
+    then show "abs_fset (map Abs r) = abs_fset (map Abs s)"
+      using Quotient3_rel[OF Quotient3_fset] by blast
+  next
+    assume a: "(list_all2 R OOO op \<approx>) r r \<and> (list_all2 R OOO op \<approx>) s s
+      \<and> abs_fset (map Abs r) = abs_fset (map Abs s)"
+    then have s: "(list_all2 R OOO op \<approx>) s s" by simp
+    have d: "map Abs r \<approx> map Abs s"
+      by (subst Quotient3_rel [OF Quotient3_fset, symmetric]) (simp add: a)
+    have b: "map Rep (map Abs r) \<approx> map Rep (map Abs s)"
+      by (rule map_list_eq_cong[OF d])
+    have y: "list_all2 R (map Rep (map Abs s)) s"
+      by (fact rep_abs_rsp_left[OF list_quotient3[OF q], OF list_all2_refl'[OF e, of s]])
+    have c: "(op \<approx> OO list_all2 R) (map Rep (map Abs r)) s"
+      by (rule relcomppI) (rule b, rule y)
+    have z: "list_all2 R r (map Rep (map Abs r))"
+      by (fact rep_abs_rsp[OF list_quotient3[OF q], OF list_all2_refl'[OF e, of r]])
+    then show "(list_all2 R OOO op \<approx>) r s"
+      using a c relcomppI by simp
+  qed
+qed
+
+lemma quotient_compose_list[quot_thm]:
+  shows  "Quotient3 ((list_all2 op \<approx>) OOO (op \<approx>))
+    (abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"
+  by (rule quotient_compose_list_g, rule Quotient3_fset, rule list_eq_equivp)
+
+
+section \<open>Quotient definitions for fsets\<close>
+
+
+subsection \<open>Finite sets are a bounded, distributive lattice with minus\<close>
+
+instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
+begin
+
+quotient_definition
+  "bot :: 'a fset" 
+  is "Nil :: 'a list" done
+
+abbreviation
+  empty_fset  ("{||}")
+where
+  "{||} \<equiv> bot :: 'a fset"
+
+quotient_definition
+  "less_eq_fset :: ('a fset \<Rightarrow> 'a fset \<Rightarrow> bool)"
+  is "sub_list :: ('a list \<Rightarrow> 'a list \<Rightarrow> bool)" by simp
+
+abbreviation
+  subset_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)
+where
+  "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
+
+definition
+  less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool"
+where  
+  "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
+
+abbreviation
+  psubset_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)
+where
+  "xs |\<subset>| ys \<equiv> xs < ys"
+
+quotient_definition
+  "sup :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+  is "append :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by simp
+
+abbreviation
+  union_fset (infixl "|\<union>|" 65)
+where
+  "xs |\<union>| ys \<equiv> sup xs (ys::'a fset)"
+
+quotient_definition
+  "inf :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+  is "inter_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by simp
+
+abbreviation
+  inter_fset (infixl "|\<inter>|" 65)
+where
+  "xs |\<inter>| ys \<equiv> inf xs (ys::'a fset)"
+
+quotient_definition
+  "minus :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+  is "diff_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by fastforce
+
+instance
+proof
+  fix x y z :: "'a fset"
+  show "x |\<subset>| y \<longleftrightarrow> x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x"
+    by (unfold less_fset_def, descending) auto
+  show "x |\<subseteq>| x" by (descending) (simp)
+  show "{||} |\<subseteq>| x" by (descending) (simp)
+  show "x |\<subseteq>| x |\<union>| y" by (descending) (simp)
+  show "y |\<subseteq>| x |\<union>| y" by (descending) (simp)
+  show "x |\<inter>| y |\<subseteq>| x" by (descending) (auto)
+  show "x |\<inter>| y |\<subseteq>| y" by (descending) (auto)
+  show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)" 
+    by (descending) (auto)
+next
+  fix x y z :: "'a fset"
+  assume a: "x |\<subseteq>| y"
+  assume b: "y |\<subseteq>| z"
+  show "x |\<subseteq>| z" using a b by (descending) (simp)
+next
+  fix x y :: "'a fset"
+  assume a: "x |\<subseteq>| y"
+  assume b: "y |\<subseteq>| x"
+  show "x = y" using a b by (descending) (auto)
+next
+  fix x y z :: "'a fset"
+  assume a: "y |\<subseteq>| x"
+  assume b: "z |\<subseteq>| x"
+  show "y |\<union>| z |\<subseteq>| x" using a b by (descending) (simp)
+next
+  fix x y z :: "'a fset"
+  assume a: "x |\<subseteq>| y"
+  assume b: "x |\<subseteq>| z"
+  show "x |\<subseteq>| y |\<inter>| z" using a b by (descending) (auto)
+qed
+
+end
+
+
+subsection \<open>Other constants for fsets\<close>
+
+quotient_definition
+  "insert_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+  is "Cons" by auto
+
+syntax
+  "_insert_fset"     :: "args => 'a fset"  ("{|(_)|}")
+
+translations
+  "{|x, xs|}" == "CONST insert_fset x {|xs|}"
+  "{|x|}"     == "CONST insert_fset x {||}"
+
+quotient_definition
+  fset_member
+where
+  "fset_member :: 'a fset \<Rightarrow> 'a \<Rightarrow> bool" is "List.member" by fastforce
+
+abbreviation
+  in_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50)
+where
+  "x |\<in>| S \<equiv> fset_member S x"
+
+abbreviation
+  notin_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)
+where
+  "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
+
+
+subsection \<open>Other constants on the Quotient Type\<close>
+
+quotient_definition
+  "card_fset :: 'a fset \<Rightarrow> nat"
+  is card_list by simp
+
+quotient_definition
+  "map_fset :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
+  is map by simp
+
+quotient_definition
+  "remove_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+  is removeAll by simp
+
+quotient_definition
+  "fset :: 'a fset \<Rightarrow> 'a set"
+  is "set" by simp
+
+lemma fold_once_set_equiv:
+  assumes "xs \<approx> ys"
+  shows "fold_once f xs = fold_once f ys"
+proof (cases "rsp_fold f")
+  case False then show ?thesis by simp
+next
+  case True
+  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"
+    by (rule rsp_foldE)
+  moreover from assms have "mset (remdups xs) = mset (remdups ys)"
+    by (simp add: set_eq_iff_mset_remdups_eq)
+  ultimately have "fold f (remdups xs) = fold f (remdups ys)"
+    by (rule fold_multiset_equiv)
+  with True show ?thesis by (simp add: fold_once_fold_remdups)
+qed
+
+quotient_definition
+  "fold_fset :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b \<Rightarrow> 'b"
+  is fold_once by (rule fold_once_set_equiv)
+
+lemma concat_rsp_pre:
+  assumes a: "list_all2 op \<approx> x x'"
+  and     b: "x' \<approx> y'"
+  and     c: "list_all2 op \<approx> y' y"
+  and     d: "\<exists>x\<in>set x. xa \<in> set x"
+  shows "\<exists>x\<in>set y. xa \<in> set x"
+proof -
+  obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto
+  have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a])
+  then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto
+  have "ya \<in> set y'" using b h by simp
+  then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_all2_find_element)
+  then show ?thesis using f i by auto
+qed
+
+quotient_definition
+  "concat_fset :: ('a fset) fset \<Rightarrow> 'a fset"
+  is concat 
+proof (elim relcomppE)
+fix a b ba bb
+  assume a: "list_all2 op \<approx> a ba"
+  with list_symp [OF list_eq_symp] have a': "list_all2 op \<approx> ba a" by (rule sympE)
+  assume b: "ba \<approx> bb"
+  with list_eq_symp have b': "bb \<approx> ba" by (rule sympE)
+  assume c: "list_all2 op \<approx> bb b"
+  with list_symp [OF list_eq_symp] have c': "list_all2 op \<approx> b bb" by (rule sympE)
+  have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" 
+  proof
+    fix x
+    show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" 
+    proof
+      assume d: "\<exists>xa\<in>set a. x \<in> set xa"
+      show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])
+    next
+      assume e: "\<exists>xa\<in>set b. x \<in> set xa"
+      show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])
+    qed
+  qed
+  then show "concat a \<approx> concat b" by auto
+qed
+
+quotient_definition
+  "filter_fset :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+  is filter by force
+
+
+subsection \<open>Compositional respectfulness and preservation lemmas\<close>
+
+lemma Nil_rsp2 [quot_respect]: 
+  shows "(list_all2 op \<approx> OOO op \<approx>) Nil Nil"
+  by (rule compose_list_refl, rule list_eq_equivp)
+
+lemma Cons_rsp2 [quot_respect]:
+  shows "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) Cons Cons"
+  apply (auto intro!: rel_funI)
+  apply (rule_tac b="x # b" in relcomppI)
+  apply auto
+  apply (rule_tac b="x # ba" in relcomppI)
+  apply auto
+  done
+
+lemma Nil_prs2 [quot_preserve]:
+  assumes "Quotient3 R Abs Rep"
+  shows "(Abs \<circ> map f) [] = Abs []"
+  by simp
+
+lemma Cons_prs2 [quot_preserve]:
+  assumes q: "Quotient3 R1 Abs1 Rep1"
+  and     r: "Quotient3 R2 Abs2 Rep2"
+  shows "(Rep1 ---> (map Rep1 \<circ> Rep2) ---> (Abs2 \<circ> map Abs1)) (op #) = (id ---> Rep2 ---> Abs2) (op #)"
+  by (auto simp add: fun_eq_iff comp_def Quotient3_abs_rep [OF q])
+
+lemma append_prs2 [quot_preserve]:
+  assumes q: "Quotient3 R1 Abs1 Rep1"
+  and     r: "Quotient3 R2 Abs2 Rep2"
+  shows "((map Rep1 \<circ> Rep2) ---> (map Rep1 \<circ> Rep2) ---> (Abs2 \<circ> map Abs1)) op @ =
+    (Rep2 ---> Rep2 ---> Abs2) op @"
+  by (simp add: fun_eq_iff abs_o_rep[OF q] List.map.id)
+
+lemma list_all2_app_l:
+  assumes a: "reflp R"
+  and b: "list_all2 R l r"
+  shows "list_all2 R (z @ l) (z @ r)"
+  using a b by (induct z) (auto elim: reflpE)
+
+lemma append_rsp2_pre0:
+  assumes a:"list_all2 op \<approx> x x'"
+  shows "list_all2 op \<approx> (x @ z) (x' @ z)"
+  using a apply (induct x x' rule: list_induct2')
+  by simp_all (rule list_all2_refl'[OF list_eq_equivp])
+
+lemma append_rsp2_pre1:
+  assumes a:"list_all2 op \<approx> x x'"
+  shows "list_all2 op \<approx> (z @ x) (z @ x')"
+  using a apply (induct x x' arbitrary: z rule: list_induct2')
+  apply (rule list_all2_refl'[OF list_eq_equivp])
+  apply (simp_all del: list_eq_def)
+  apply (rule list_all2_app_l)
+  apply (simp_all add: reflpI)
+  done
+
+lemma append_rsp2_pre:
+  assumes "list_all2 op \<approx> x x'"
+    and "list_all2 op \<approx> z z'"
+  shows "list_all2 op \<approx> (x @ z) (x' @ z')"
+  using assms by (rule list_all2_appendI)
+
+lemma compositional_rsp3:
+  assumes "(R1 ===> R2 ===> R3) C C" and "(R4 ===> R5 ===> R6) C C"
+  shows "(R1 OOO R4 ===> R2 OOO R5 ===> R3 OOO R6) C C"
+  by (auto intro!: rel_funI)
+     (metis (full_types) assms rel_funE relcomppI)
+
+lemma append_rsp2 [quot_respect]:
+  "(list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) append append"
+  by (intro compositional_rsp3)
+     (auto intro!: rel_funI simp add: append_rsp2_pre)
+
+lemma map_rsp2 [quot_respect]:
+  "((op \<approx> ===> op \<approx>) ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) map map"
+proof (auto intro!: rel_funI)
+  fix f f' :: "'a list \<Rightarrow> 'b list"
+  fix xa ya x y :: "'a list list"
+  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"
+  have a: "(list_all2 op \<approx>) (map f xa) (map f x)"
+    using x
+    by (induct xa x rule: list_induct2')
+       (simp_all, metis fs rel_funE list_eq_def)
+  have b: "set (map f x) = set (map f y)"
+    using xy fs
+    by (induct x y rule: list_induct2')
+       (simp_all, metis image_insert)
+  have c: "(list_all2 op \<approx>) (map f y) (map f' ya)"
+    using y fs
+    by (induct y ya rule: list_induct2')
+       (simp_all, metis apply_rsp' list_eq_def)
+  show "(list_all2 op \<approx> OOO op \<approx>) (map f xa) (map f' ya)"
+    by (metis a b c list_eq_def relcomppI)
+qed
+
+lemma map_prs2 [quot_preserve]:
+  shows "((abs_fset ---> rep_fset) ---> (map rep_fset \<circ> rep_fset) ---> abs_fset \<circ> map abs_fset) map = (id ---> rep_fset ---> abs_fset) map"
+  by (auto simp add: fun_eq_iff)
+     (simp only: map_map[symmetric] map_prs_aux[OF Quotient3_fset Quotient3_fset])
+
+section \<open>Lifted theorems\<close>
+
+subsection \<open>fset\<close>
+
+lemma fset_simps [simp]:
+  shows "fset {||} = {}"
+  and   "fset (insert_fset x S) = insert x (fset S)"
+  by (descending, simp)+
+
+lemma finite_fset [simp]: 
+  shows "finite (fset S)"
+  by (descending) (simp)
+
+lemma fset_cong:
+  shows "fset S = fset T \<longleftrightarrow> S = T"
+  by (descending) (simp)
+
+lemma filter_fset [simp]:
+  shows "fset (filter_fset P xs) = Collect P \<inter> fset xs"
+  by (descending) (auto)
+
+lemma remove_fset [simp]: 
+  shows "fset (remove_fset x xs) = fset xs - {x}"
+  by (descending) (simp)
+
+lemma inter_fset [simp]: 
+  shows "fset (xs |\<inter>| ys) = fset xs \<inter> fset ys"
+  by (descending) (auto)
+
+lemma union_fset [simp]: 
+  shows "fset (xs |\<union>| ys) = fset xs \<union> fset ys"
+  by (lifting set_append)
+
+lemma minus_fset [simp]: 
+  shows "fset (xs - ys) = fset xs - fset ys"
+  by (descending) (auto)
+
+
+subsection \<open>in_fset\<close>
+
+lemma in_fset: 
+  shows "x |\<in>| S \<longleftrightarrow> x \<in> fset S"
+  by descending simp
+
+lemma notin_fset: 
+  shows "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S"
+  by (simp add: in_fset)
+
+lemma notin_empty_fset: 
+  shows "x |\<notin>| {||}"
+  by (simp add: in_fset)
+
+lemma fset_eq_iff:
+  shows "S = T \<longleftrightarrow> (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"
+  by descending auto
+
+lemma none_in_empty_fset:
+  shows "(\<forall>x. x |\<notin>| S) \<longleftrightarrow> S = {||}"
+  by descending simp
+
+
+subsection \<open>insert_fset\<close>
+
+lemma in_insert_fset_iff [simp]:
+  shows "x |\<in>| insert_fset y S \<longleftrightarrow> x = y \<or> x |\<in>| S"
+  by descending simp
+
+lemma
+  shows insert_fsetI1: "x |\<in>| insert_fset x S"
+  and   insert_fsetI2: "x |\<in>| S \<Longrightarrow> x |\<in>| insert_fset y S"
+  by simp_all
+
+lemma insert_absorb_fset [simp]:
+  shows "x |\<in>| S \<Longrightarrow> insert_fset x S = S"
+  by (descending) (auto)
+
+lemma empty_not_insert_fset[simp]:
+  shows "{||} \<noteq> insert_fset x S"
+  and   "insert_fset x S \<noteq> {||}"
+  by (descending, simp)+
+
+lemma insert_fset_left_comm:
+  shows "insert_fset x (insert_fset y S) = insert_fset y (insert_fset x S)"
+  by (descending) (auto)
+
+lemma insert_fset_left_idem:
+  shows "insert_fset x (insert_fset x S) = insert_fset x S"
+  by (descending) (auto)
+
+lemma singleton_fset_eq[simp]:
+  shows "{|x|} = {|y|} \<longleftrightarrow> x = y"
+  by (descending) (auto)
+
+lemma in_fset_mdef:
+  shows "x |\<in>| F \<longleftrightarrow> x |\<notin>| (F - {|x|}) \<and> F = insert_fset x (F - {|x|})"
+  by (descending) (auto)
+
+
+subsection \<open>union_fset\<close>
+
+lemmas [simp] =
+  sup_bot_left[where 'a="'a fset"]
+  sup_bot_right[where 'a="'a fset"]
+
+lemma union_insert_fset [simp]:
+  shows "insert_fset x S |\<union>| T = insert_fset x (S |\<union>| T)"
+  by (lifting append.simps(2))
+
+lemma singleton_union_fset_left:
+  shows "{|a|} |\<union>| S = insert_fset a S"
+  by simp
+
+lemma singleton_union_fset_right:
+  shows "S |\<union>| {|a|} = insert_fset a S"
+  by (subst sup.commute) simp
+
+lemma in_union_fset:
+  shows "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"
+  by (descending) (simp)
+
+
+subsection \<open>minus_fset\<close>
+
+lemma minus_in_fset: 
+  shows "x |\<in>| (xs - ys) \<longleftrightarrow> x |\<in>| xs \<and> x |\<notin>| ys"
+  by (descending) (simp)
+
+lemma minus_insert_fset: 
+  shows "insert_fset x xs - ys = (if x |\<in>| ys then xs - ys else insert_fset x (xs - ys))"
+  by (descending) (auto)
+
+lemma minus_insert_in_fset[simp]: 
+  shows "x |\<in>| ys \<Longrightarrow> insert_fset x xs - ys = xs - ys"
+  by (simp add: minus_insert_fset)
+
+lemma minus_insert_notin_fset[simp]: 
+  shows "x |\<notin>| ys \<Longrightarrow> insert_fset x xs - ys = insert_fset x (xs - ys)"
+  by (simp add: minus_insert_fset)
+
+lemma in_minus_fset: 
+  shows "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S"
+  unfolding in_fset minus_fset
+  by blast
+
+lemma notin_minus_fset: 
+  shows "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S"
+  unfolding in_fset minus_fset
+  by blast
+
+
+subsection \<open>remove_fset\<close>
+
+lemma in_remove_fset:
+  shows "x |\<in>| remove_fset y S \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"
+  by (descending) (simp)
+
+lemma notin_remove_fset:
+  shows "x |\<notin>| remove_fset x S"
+  by (descending) (simp)
+
+lemma notin_remove_ident_fset:
+  shows "x |\<notin>| S \<Longrightarrow> remove_fset x S = S"
+  by (descending) (simp)
+
+lemma remove_fset_cases:
+  shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = insert_fset x (remove_fset x S))"
+  by (descending) (auto simp add: insert_absorb)
+  
+
+subsection \<open>inter_fset\<close>
+
+lemma inter_empty_fset_l:
+  shows "{||} |\<inter>| S = {||}"
+  by simp
+
+lemma inter_empty_fset_r:
+  shows "S |\<inter>| {||} = {||}"
+  by simp
+
+lemma inter_insert_fset:
+  shows "insert_fset x S |\<inter>| T = (if x |\<in>| T then insert_fset x (S |\<inter>| T) else S |\<inter>| T)"
+  by (descending) (auto)
+
+lemma in_inter_fset:
+  shows "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"
+  by (descending) (simp)
+
+
+subsection \<open>subset_fset and psubset_fset\<close>
+
+lemma subset_fset: 
+  shows "xs |\<subseteq>| ys \<longleftrightarrow> fset xs \<subseteq> fset ys"
+  by (descending) (simp)
+
+lemma psubset_fset: 
+  shows "xs |\<subset>| ys \<longleftrightarrow> fset xs \<subset> fset ys"
+  unfolding less_fset_def 
+  by (descending) (auto)
+
+lemma subset_insert_fset:
+  shows "(insert_fset x xs) |\<subseteq>| ys \<longleftrightarrow> x |\<in>| ys \<and> xs |\<subseteq>| ys"
+  by (descending) (simp)
+
+lemma subset_in_fset: 
+  shows "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"
+  by (descending) (auto)
+
+lemma subset_empty_fset:
+  shows "xs |\<subseteq>| {||} \<longleftrightarrow> xs = {||}"
+  by (descending) (simp)
+
+lemma not_psubset_empty_fset: 
+  shows "\<not> xs |\<subset>| {||}"
+  by (metis fset_simps(1) psubset_fset not_psubset_empty)
+
+
+subsection \<open>map_fset\<close>
+
+lemma map_fset_simps [simp]:
+   shows "map_fset f {||} = {||}"
+  and   "map_fset f (insert_fset x S) = insert_fset (f x) (map_fset f S)"
+  by (descending, simp)+
+
+lemma map_fset_image [simp]:
+  shows "fset (map_fset f S) = f ` (fset S)"
+  by (descending) (simp)
+
+lemma inj_map_fset_cong:
+  shows "inj f \<Longrightarrow> map_fset f S = map_fset f T \<longleftrightarrow> S = T"
+  by (descending) (metis inj_vimage_image_eq list_eq_def set_map)
+
+lemma map_union_fset: 
+  shows "map_fset f (S |\<union>| T) = map_fset f S |\<union>| map_fset f T"
+  by (descending) (simp)
+
+lemma in_fset_map_fset[simp]: "a |\<in>| map_fset f X = (\<exists>b. b |\<in>| X \<and> a = f b)"
+  by descending auto
+
+
+subsection \<open>card_fset\<close>
+
+lemma card_fset: 
+  shows "card_fset xs = card (fset xs)"
+  by (descending) (simp)
+
+lemma card_insert_fset_iff [simp]:
+  shows "card_fset (insert_fset x S) = (if x |\<in>| S then card_fset S else Suc (card_fset S))"
+  by (descending) (simp add: insert_absorb)
+
+lemma card_fset_0[simp]:
+  shows "card_fset S = 0 \<longleftrightarrow> S = {||}"
+  by (descending) (simp)
+
+lemma card_empty_fset[simp]:
+  shows "card_fset {||} = 0"
+  by (simp add: card_fset)
+
+lemma card_fset_1:
+  shows "card_fset S = 1 \<longleftrightarrow> (\<exists>x. S = {|x|})"
+  by (descending) (auto simp add: card_Suc_eq)
+
+lemma card_fset_gt_0:
+  shows "x \<in> fset S \<Longrightarrow> 0 < card_fset S"
+  by (descending) (auto simp add: card_gt_0_iff)
+  
+lemma card_notin_fset:
+  shows "(x |\<notin>| S) = (card_fset (insert_fset x S) = Suc (card_fset S))"
+  by simp
+
+lemma card_fset_Suc: 
+  shows "card_fset S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = insert_fset x T \<and> card_fset T = n"
+  apply(descending)
+  apply(auto dest!: card_eq_SucD)
+  by (metis Diff_insert_absorb set_removeAll)
+
+lemma card_remove_fset_iff [simp]:
+  shows "card_fset (remove_fset y S) = (if y |\<in>| S then card_fset S - 1 else card_fset S)"
+  by (descending) (simp)
+
+lemma card_Suc_exists_in_fset: 
+  shows "card_fset S = Suc n \<Longrightarrow> \<exists>a. a |\<in>| S"
+  by (drule card_fset_Suc) (auto)
+
+lemma in_card_fset_not_0: 
+  shows "a |\<in>| A \<Longrightarrow> card_fset A \<noteq> 0"
+  by (descending) (auto)
+
+lemma card_fset_mono: 
+  shows "xs |\<subseteq>| ys \<Longrightarrow> card_fset xs \<le> card_fset ys"
+  unfolding card_fset psubset_fset
+  by (simp add: card_mono subset_fset)
+
+lemma card_subset_fset_eq: 
+  shows "xs |\<subseteq>| ys \<Longrightarrow> card_fset ys \<le> card_fset xs \<Longrightarrow> xs = ys"
+  unfolding card_fset subset_fset
+  by (auto dest: card_seteq[OF finite_fset] simp add: fset_cong)
+
+lemma psubset_card_fset_mono: 
+  shows "xs |\<subset>| ys \<Longrightarrow> card_fset xs < card_fset ys"
+  unfolding card_fset subset_fset
+  by (metis finite_fset psubset_fset psubset_card_mono)
+
+lemma card_union_inter_fset: 
+  shows "card_fset xs + card_fset ys = card_fset (xs |\<union>| ys) + card_fset (xs |\<inter>| ys)"
+  unfolding card_fset union_fset inter_fset
+  by (rule card_Un_Int[OF finite_fset finite_fset])
+
+lemma card_union_disjoint_fset: 
+  shows "xs |\<inter>| ys = {||} \<Longrightarrow> card_fset (xs |\<union>| ys) = card_fset xs + card_fset ys"
+  unfolding card_fset union_fset 
+  apply (rule card_Un_disjoint[OF finite_fset finite_fset])
+  by (metis inter_fset fset_simps(1))
+
+lemma card_remove_fset_less1: 
+  shows "x |\<in>| xs \<Longrightarrow> card_fset (remove_fset x xs) < card_fset xs"
+  unfolding card_fset in_fset remove_fset 
+  by (rule card_Diff1_less[OF finite_fset])
+
+lemma card_remove_fset_less2: 
+  shows "x |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> card_fset (remove_fset y (remove_fset x xs)) < card_fset xs"
+  unfolding card_fset remove_fset in_fset
+  by (rule card_Diff2_less[OF finite_fset])
+
+lemma card_remove_fset_le1: 
+  shows "card_fset (remove_fset x xs) \<le> card_fset xs"
+  unfolding remove_fset card_fset
+  by (rule card_Diff1_le[OF finite_fset])
+
+lemma card_psubset_fset: 
+  shows "ys |\<subseteq>| xs \<Longrightarrow> card_fset ys < card_fset xs \<Longrightarrow> ys |\<subset>| xs"
+  unfolding card_fset psubset_fset subset_fset
+  by (rule card_psubset[OF finite_fset])
+
+lemma card_map_fset_le: 
+  shows "card_fset (map_fset f xs) \<le> card_fset xs"
+  unfolding card_fset map_fset_image
+  by (rule card_image_le[OF finite_fset])
+
+lemma card_minus_insert_fset[simp]:
+  assumes "a |\<in>| A" and "a |\<notin>| B"
+  shows "card_fset (A - insert_fset a B) = card_fset (A - B) - 1"
+  using assms 
+  unfolding in_fset card_fset minus_fset
+  by (simp add: card_Diff_insert[OF finite_fset])
+
+lemma card_minus_subset_fset:
+  assumes "B |\<subseteq>| A"
+  shows "card_fset (A - B) = card_fset A - card_fset B"
+  using assms 
+  unfolding subset_fset card_fset minus_fset
+  by (rule card_Diff_subset[OF finite_fset])
+
+lemma card_minus_fset:
+  shows "card_fset (A - B) = card_fset A - card_fset (A |\<inter>| B)"
+  unfolding inter_fset card_fset minus_fset
+  by (rule card_Diff_subset_Int) (simp)
+
+
+subsection \<open>concat_fset\<close>
+
+lemma concat_empty_fset [simp]:
+  shows "concat_fset {||} = {||}"
+  by descending simp
+
+lemma concat_insert_fset [simp]:
+  shows "concat_fset (insert_fset x S) = x |\<union>| concat_fset S"
+  by descending simp
+
+lemma concat_union_fset [simp]:
+  shows "concat_fset (xs |\<union>| ys) = concat_fset xs |\<union>| concat_fset ys"
+  by descending simp
+
+lemma map_concat_fset:
+  shows "map_fset f (concat_fset xs) = concat_fset (map_fset (map_fset f) xs)"
+  by (lifting map_concat)
+
+subsection \<open>filter_fset\<close>
+
+lemma subset_filter_fset: 
+  "filter_fset P xs |\<subseteq>| filter_fset Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"
+  by descending auto
+
+lemma eq_filter_fset: 
+  "(filter_fset P xs = filter_fset Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"
+  by descending auto
+
+lemma psubset_filter_fset:
+  "(\<And>x. x |\<in>| xs \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| xs & \<not> P x & Q x) \<Longrightarrow> 
+    filter_fset P xs |\<subset>| filter_fset Q xs"
+  unfolding less_fset_def by (auto simp add: subset_filter_fset eq_filter_fset)
+
+
+subsection \<open>fold_fset\<close>
+
+lemma fold_empty_fset: 
+  "fold_fset f {||} = id"
+  by descending (simp add: fold_once_def)
+
+lemma fold_insert_fset: "fold_fset f (insert_fset a A) =
+  (if rsp_fold f then if a |\<in>| A then fold_fset f A else fold_fset f A \<circ> f a else id)"
+  by descending (simp add: fold_once_fold_remdups)
+
+lemma remdups_removeAll:
+  "remdups (removeAll x xs) = remove1 x (remdups xs)"
+  by (induct xs) auto
+
+lemma member_commute_fold_once:
+  assumes "rsp_fold f"
+    and "x \<in> set xs"
+  shows "fold_once f xs = fold_once f (removeAll x xs) \<circ> f x"
+proof -
+  from assms have "fold f (remdups xs) = fold f (remove1 x (remdups xs)) \<circ> f x"
+    by (auto intro!: fold_remove1_split elim: rsp_foldE)
+  then show ?thesis using \<open>rsp_fold f\<close> by (simp add: fold_once_fold_remdups remdups_removeAll)
+qed
+
+lemma in_commute_fold_fset:
+  "rsp_fold f \<Longrightarrow> h |\<in>| b \<Longrightarrow> fold_fset f b = fold_fset f (remove_fset h b) \<circ> f h"
+  by descending (simp add: member_commute_fold_once)
+
+
+subsection \<open>Choice in fsets\<close>
+
+lemma fset_choice: 
+  assumes a: "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
+  shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
+  using a
+  apply(descending)
+  using finite_set_choice
+  by (auto simp add: Ball_def)
+
+
+section \<open>Induction and Cases rules for fsets\<close>
+
+lemma fset_exhaust [case_names empty insert, cases type: fset]:
+  assumes empty_fset_case: "S = {||} \<Longrightarrow> P" 
+  and     insert_fset_case: "\<And>x S'. S = insert_fset x S' \<Longrightarrow> P"
+  shows "P"
+  using assms by (lifting list.exhaust)
+
+lemma fset_induct [case_names empty insert]:
+  assumes empty_fset_case: "P {||}"
+  and     insert_fset_case: "\<And>x S. P S \<Longrightarrow> P (insert_fset x S)"
+  shows "P S"
+  using assms 
+  by (descending) (blast intro: list.induct)
+
+lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
+  assumes empty_fset_case: "P {||}"
+  and     insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (insert_fset x S)"
+  shows "P S"
+proof(induct S rule: fset_induct)
+  case empty
+  show "P {||}" using empty_fset_case by simp
+next
+  case (insert x S)
+  have "P S" by fact
+  then show "P (insert_fset x S)" using insert_fset_case 
+    by (cases "x |\<in>| S") (simp_all)
+qed
+
+lemma fset_card_induct:
+  assumes empty_fset_case: "P {||}"
+  and     card_fset_Suc_case: "\<And>S T. Suc (card_fset S) = (card_fset T) \<Longrightarrow> P S \<Longrightarrow> P T"
+  shows "P S"
+proof (induct S)
+  case empty
+  show "P {||}" by (rule empty_fset_case)
+next
+  case (insert x S)
+  have h: "P S" by fact
+  have "x |\<notin>| S" by fact
+  then have "Suc (card_fset S) = card_fset (insert_fset x S)" 
+    using card_fset_Suc by auto
+  then show "P (insert_fset x S)" 
+    using h card_fset_Suc_case by simp
+qed
+
+lemma fset_raw_strong_cases:
+  obtains "xs = []"
+    | ys x where "\<not> List.member ys x" and "xs \<approx> x # ys"
+proof (induct xs)
+  case Nil
+  then show thesis by simp
+next
+  case (Cons a xs)
+  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"
+    by (rule Cons(1))
+  have b: "\<And>x' ys'. \<lbrakk>\<not> List.member ys' x'; a # xs \<approx> x' # ys'\<rbrakk> \<Longrightarrow> thesis" by fact
+  have c: "xs = [] \<Longrightarrow> thesis" using b 
+    apply(simp)
+    by (metis list.set(1) emptyE empty_subsetI)
+  have "\<And>x ys. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis"
+  proof -
+    fix x :: 'a
+    fix ys :: "'a list"
+    assume d:"\<not> List.member ys x"
+    assume e:"xs \<approx> x # ys"
+    show thesis
+    proof (cases "x = a")
+      assume h: "x = a"
+      then have f: "\<not> List.member ys a" using d by simp
+      have g: "a # xs \<approx> a # ys" using e h by auto
+      show thesis using b f g by simp
+    next
+      assume h: "x \<noteq> a"
+      then have f: "\<not> List.member (a # ys) x" using d by auto
+      have g: "a # xs \<approx> x # (a # ys)" using e h by auto
+      show thesis using b f g by (simp del: List.member_def) 
+    qed
+  qed
+  then show thesis using a c by blast
+qed
+
+
+lemma fset_strong_cases:
+  obtains "xs = {||}"
+    | ys x where "x |\<notin>| ys" and "xs = insert_fset x ys"
+  by (lifting fset_raw_strong_cases)
+
+
+lemma fset_induct2:
+  "P {||} {||} \<Longrightarrow>
+  (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (insert_fset x xs) {||}) \<Longrightarrow>
+  (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (insert_fset y ys)) \<Longrightarrow>
+  (\<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>
+  P xsa ysa"
+  apply (induct xsa arbitrary: ysa)
+  apply (induct_tac x rule: fset_induct_stronger)
+  apply simp_all
+  apply (induct_tac xa rule: fset_induct_stronger)
+  apply simp_all
+  done
+
+text \<open>Extensionality\<close>
+
+lemma fset_eqI:
+  assumes "\<And>x. x \<in> fset A \<longleftrightarrow> x \<in> fset B"
+  shows "A = B"
+using assms proof (induct A arbitrary: B)
+  case empty then show ?case
+    by (auto simp add: in_fset none_in_empty_fset [symmetric] sym)
+next
+  case (insert x A)
+  from insert.prems insert.hyps(1) have "\<And>z. z \<in> fset A \<longleftrightarrow> z \<in> fset (B - {|x|})"
+    by (auto simp add: in_fset)
+  then have A: "A = B - {|x|}" by (rule insert.hyps(2))
+  with insert.prems [symmetric, of x] have "x |\<in>| B" by (simp add: in_fset)
+  with A show ?case by (metis in_fset_mdef)
+qed
+
+subsection \<open>alternate formulation with a different decomposition principle
+  and a proof of equivalence\<close>
+
+inductive
+  list_eq2 :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<approx>2 _")
+where
+  "(a # b # xs) \<approx>2 (b # a # xs)"
+| "[] \<approx>2 []"
+| "xs \<approx>2 ys \<Longrightarrow> ys \<approx>2 xs"
+| "(a # a # xs) \<approx>2 (a # xs)"
+| "xs \<approx>2 ys \<Longrightarrow> (a # xs) \<approx>2 (a # ys)"
+| "xs1 \<approx>2 xs2 \<Longrightarrow> xs2 \<approx>2 xs3 \<Longrightarrow> xs1 \<approx>2 xs3"
+
+lemma list_eq2_refl:
+  shows "xs \<approx>2 xs"
+  by (induct xs) (auto intro: list_eq2.intros)
+
+lemma cons_delete_list_eq2:
+  shows "(a # (removeAll a A)) \<approx>2 (if List.member A a then A else a # A)"
+  apply (induct A)
+  apply (simp add: list_eq2_refl)
+  apply (case_tac "List.member (aa # A) a")
+  apply (simp_all)
+  apply (case_tac [!] "a = aa")
+  apply (simp_all)
+  apply (case_tac "List.member A a")
+  apply (auto)[2]
+  apply (metis list_eq2.intros(3) list_eq2.intros(4) list_eq2.intros(5) list_eq2.intros(6))
+  apply (metis list_eq2.intros(1) list_eq2.intros(5) list_eq2.intros(6))
+  apply (auto simp add: list_eq2_refl)
+  done
+
+lemma member_delete_list_eq2:
+  assumes a: "List.member r e"
+  shows "(e # removeAll e r) \<approx>2 r"
+  using a cons_delete_list_eq2[of e r]
+  by simp
+
+lemma list_eq2_equiv:
+  "(l \<approx> r) \<longleftrightarrow> (list_eq2 l r)"
+proof
+  show "list_eq2 l r \<Longrightarrow> l \<approx> r" by (induct rule: list_eq2.induct) auto
+next
+  {
+    fix n
+    assume a: "card_list l = n" and b: "l \<approx> r"
+    have "l \<approx>2 r"
+      using a b
+    proof (induct n arbitrary: l r)
+      case 0
+      have "card_list l = 0" by fact
+      then have "\<forall>x. \<not> List.member l x" by auto
+      then have z: "l = []" by auto
+      then have "r = []" using \<open>l \<approx> r\<close> by simp
+      then show ?case using z list_eq2_refl by simp
+    next
+      case (Suc m)
+      have b: "l \<approx> r" by fact
+      have d: "card_list l = Suc m" by fact
+      then have "\<exists>a. List.member l a" 
+        apply(simp)
+        apply(drule card_eq_SucD)
+        apply(blast)
+        done
+      then obtain a where e: "List.member l a" by auto
+      then have e': "List.member r a" using list_eq_def [simplified List.member_def [symmetric], of l r] b 
+        by auto
+      have f: "card_list (removeAll a l) = m" using e d by (simp)
+      have g: "removeAll a l \<approx> removeAll a r" using remove_fset.rsp b by simp
+      have "(removeAll a l) \<approx>2 (removeAll a r)" by (rule Suc.hyps[OF f g])
+      then have h: "(a # removeAll a l) \<approx>2 (a # removeAll a r)" by (rule list_eq2.intros(5))
+      have i: "l \<approx>2 (a # removeAll a l)"
+        by (rule list_eq2.intros(3)[OF member_delete_list_eq2[OF e]])
+      have "l \<approx>2 (a # removeAll a r)" by (rule list_eq2.intros(6)[OF i h])
+      then show ?case using list_eq2.intros(6)[OF _ member_delete_list_eq2[OF e']] by simp
+    qed
+    }
+  then show "l \<approx> r \<Longrightarrow> l \<approx>2 r" by blast
+qed
+
+
+(* We cannot write it as "assumes .. shows" since Isabelle changes
+   the quantifiers to schematic variables and reintroduces them in
+   a different order *)
+lemma fset_eq_cases:
+ "\<lbrakk>a1 = a2;
+   \<And>a b xs. \<lbrakk>a1 = insert_fset a (insert_fset b xs); a2 = insert_fset b (insert_fset a xs)\<rbrakk> \<Longrightarrow> P;
+   \<lbrakk>a1 = {||}; a2 = {||}\<rbrakk> \<Longrightarrow> P; \<And>xs ys. \<lbrakk>a1 = ys; a2 = xs; xs = ys\<rbrakk> \<Longrightarrow> P;
+   \<And>a xs. \<lbrakk>a1 = insert_fset a (insert_fset a xs); a2 = insert_fset a xs\<rbrakk> \<Longrightarrow> P;
+   \<And>xs ys a. \<lbrakk>a1 = insert_fset a xs; a2 = insert_fset a ys; xs = ys\<rbrakk> \<Longrightarrow> P;
+   \<And>xs1 xs2 xs3. \<lbrakk>a1 = xs1; a2 = xs3; xs1 = xs2; xs2 = xs3\<rbrakk> \<Longrightarrow> P\<rbrakk>
+  \<Longrightarrow> P"
+  by (lifting list_eq2.cases[simplified list_eq2_equiv[symmetric]])
+
+lemma fset_eq_induct:
+  assumes "x1 = x2"
+  and "\<And>a b xs. P (insert_fset a (insert_fset b xs)) (insert_fset b (insert_fset a xs))"
+  and "P {||} {||}"
+  and "\<And>xs ys. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P ys xs"
+  and "\<And>a xs. P (insert_fset a (insert_fset a xs)) (insert_fset a xs)"
+  and "\<And>xs ys a. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P (insert_fset a xs) (insert_fset a ys)"
+  and "\<And>xs1 xs2 xs3. \<lbrakk>xs1 = xs2; P xs1 xs2; xs2 = xs3; P xs2 xs3\<rbrakk> \<Longrightarrow> P xs1 xs3"
+  shows "P x1 x2"
+  using assms
+  by (lifting list_eq2.induct[simplified list_eq2_equiv[symmetric]])
+
+ML \<open>
+fun dest_fsetT (Type (@{type_name fset}, [T])) = T
+  | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
+\<close>
+
+no_notation
+  list_eq (infix "\<approx>" 50) and 
+  list_eq2 (infix "\<approx>2" 50)
+
+end