# HG changeset patch
# User Cezary Kaliszyk <kaliszyk@in.tum.de>
# Date 1272009653 -7200
# Node ID c4f5823f282d5f5eff26544db9c6f56897b883bb
# Parent  8e58a63ac975d736fd8badb56c9b3d87ee3d445c
Finite set theory

diff -r 8e58a63ac975 -r c4f5823f282d src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Thu Apr 22 22:12:12 2010 +0200
+++ b/src/HOL/IsaMakefile	Fri Apr 23 10:00:53 2010 +0200
@@ -1294,6 +1294,7 @@
 HOL-Quotient_Examples: HOL $(LOG)/HOL-Quotient_Examples.gz
 
 $(LOG)/HOL-Quotient_Examples.gz: $(OUT)/HOL				\
+  Quotient_Examples/FSet.thy                                            \
   Quotient_Examples/LarryInt.thy Quotient_Examples/LarryDatatype.thy
 	@$(ISABELLE_TOOL) usedir $(OUT)/HOL Quotient_Examples
 
diff -r 8e58a63ac975 -r c4f5823f282d src/HOL/Quotient_Examples/FSet.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Quotient_Examples/FSet.thy	Fri Apr 23 10:00:53 2010 +0200
@@ -0,0 +1,1113 @@
+(*  Title:      Quotient.thy
+    Author:     Cezary Kaliszyk 
+    Author:     Christian Urban
+
+    provides a reasoning infrastructure for the type of finite sets
+*)
+theory FSet
+imports Quotient Quotient_List List
+begin
+
+text {* Definiton of List relation and the quotient type *}
+
+fun
+  list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)
+where
+  "list_eq xs ys = (\<forall>x. x \<in> set xs \<longleftrightarrow> x \<in> set ys)"
+
+lemma list_eq_equivp:
+  shows "equivp list_eq"
+  unfolding equivp_reflp_symp_transp
+  unfolding reflp_def symp_def transp_def
+  by auto
+
+quotient_type
+  'a fset = "'a list" / "list_eq"
+  by (rule list_eq_equivp)
+
+text {* Raw definitions *}
+
+definition
+  memb :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
+where
+  "memb x xs \<equiv> x \<in> set xs"
+
+definition
+  sub_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
+where
+  "sub_list xs ys \<equiv> (\<forall>x. x \<in> set xs \<longrightarrow> x \<in> set ys)"
+
+fun
+  fcard_raw :: "'a list \<Rightarrow> nat"
+where
+  fcard_raw_nil:  "fcard_raw [] = 0"
+| fcard_raw_cons: "fcard_raw (x # xs) = (if memb x xs then fcard_raw xs else Suc (fcard_raw xs))"
+
+primrec
+  finter_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+  "finter_raw [] l = []"
+| "finter_raw (h # t) l =
+     (if memb h l then h # (finter_raw t l) else finter_raw t l)"
+
+fun
+  delete_raw :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list"
+where
+  "delete_raw [] x = []"
+| "delete_raw (a # A) x = (if (a = x) then delete_raw A x else a # (delete_raw A x))"
+
+definition
+  rsp_fold
+where
+  "rsp_fold f = (\<forall>u v w. (f u (f v w) = f v (f u w)))"
+
+primrec
+  ffold_raw :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
+where
+  "ffold_raw f z [] = z"
+| "ffold_raw f z (a # A) =
+     (if (rsp_fold f) then
+       if memb a A then ffold_raw f z A
+       else f a (ffold_raw f z A)
+     else z)"
+
+text {* Composition Quotient *}
+
+lemma list_rel_refl:
+  shows "(list_rel op \<approx>) r r"
+  by (rule list_rel_refl) (metis equivp_def fset_equivp)
+
+lemma compose_list_refl:
+  shows "(list_rel op \<approx> OOO op \<approx>) r r"
+proof
+  show c: "list_rel op \<approx> r r" by (rule list_rel_refl)
+  have d: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
+  show b: "(op \<approx> OO list_rel op \<approx>) r r" by (rule pred_compI) (rule d, rule c)
+qed
+
+lemma Quotient_fset_list:
+  shows "Quotient (list_rel op \<approx>) (map abs_fset) (map rep_fset)"
+  by (fact list_quotient[OF Quotient_fset])
+
+lemma set_in_eq: "(\<forall>e. ((e \<in> A) \<longleftrightarrow> (e \<in> B))) \<equiv> A = B"
+  by (rule eq_reflection) auto
+
+lemma map_rel_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"
+  unfolding list_eq.simps
+  by (simp only: set_map set_in_eq)
+
+lemma quotient_compose_list[quot_thm]:
+  shows  "Quotient ((list_rel op \<approx>) OOO (op \<approx>))
+    (abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"
+  unfolding Quotient_def comp_def
+proof (intro conjI allI)
+  fix a r s
+  show "abs_fset (map abs_fset (map rep_fset (rep_fset a))) = a"
+    by (simp add: abs_o_rep[OF Quotient_fset] Quotient_abs_rep[OF Quotient_fset] map_id)
+  have b: "list_rel op \<approx> (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
+    by (rule list_rel_refl)
+  have c: "(op \<approx> OO list_rel op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
+    by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
+  show "(list_rel op \<approx> OOO op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
+    by (rule, rule list_rel_refl) (rule c)
+  show "(list_rel op \<approx> OOO op \<approx>) r s = ((list_rel op \<approx> OOO op \<approx>) r r \<and>
+        (list_rel op \<approx> OOO op \<approx>) s s \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s))"
+  proof (intro iffI conjI)
+    show "(list_rel op \<approx> OOO op \<approx>) r r" by (rule compose_list_refl)
+    show "(list_rel op \<approx> OOO op \<approx>) s s" by (rule compose_list_refl)
+  next
+    assume a: "(list_rel op \<approx> OOO op \<approx>) r s"
+    then have b: "map abs_fset r \<approx> map abs_fset s" proof (elim pred_compE)
+      fix b ba
+      assume c: "list_rel op \<approx> r b"
+      assume d: "b \<approx> ba"
+      assume e: "list_rel op \<approx> ba s"
+      have f: "map abs_fset r = map abs_fset b"
+        using Quotient_rel[OF Quotient_fset_list] c by blast
+      have "map abs_fset ba = map abs_fset s"
+        using Quotient_rel[OF Quotient_fset_list] e by blast
+      then have g: "map abs_fset s = map abs_fset ba" by simp
+      then show "map abs_fset r \<approx> map abs_fset s" using d f map_rel_cong by simp
+    qed
+    then show "abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"
+      using Quotient_rel[OF Quotient_fset] by blast
+  next
+    assume a: "(list_rel op \<approx> OOO op \<approx>) r r \<and> (list_rel op \<approx> OOO op \<approx>) s s
+      \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"
+    then have s: "(list_rel op \<approx> OOO op \<approx>) s s" by simp
+    have d: "map abs_fset r \<approx> map abs_fset s"
+      by (subst Quotient_rel[OF Quotient_fset]) (simp add: a)
+    have b: "map rep_fset (map abs_fset r) \<approx> map rep_fset (map abs_fset s)"
+      by (rule map_rel_cong[OF d])
+    have y: "list_rel op \<approx> (map rep_fset (map abs_fset s)) s"
+      by (fact rep_abs_rsp_left[OF Quotient_fset_list, OF list_rel_refl[of s]])
+    have c: "(op \<approx> OO list_rel op \<approx>) (map rep_fset (map abs_fset r)) s"
+      by (rule pred_compI) (rule b, rule y)
+    have z: "list_rel op \<approx> r (map rep_fset (map abs_fset r))"
+      by (fact rep_abs_rsp[OF Quotient_fset_list, OF list_rel_refl[of r]])
+    then show "(list_rel op \<approx> OOO op \<approx>) r s"
+      using a c pred_compI by simp
+  qed
+qed
+
+text {* Respectfullness *}
+
+lemma [quot_respect]:
+  shows "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @"
+  by auto
+
+lemma [quot_respect]:
+  shows "(op \<approx> ===> op \<approx> ===> op =) sub_list sub_list"
+  by (auto simp add: sub_list_def)
+
+lemma memb_rsp[quot_respect]:
+  shows "(op = ===> op \<approx> ===> op =) memb memb"
+  by (auto simp add: memb_def)
+
+lemma nil_rsp[quot_respect]:
+  shows "[] \<approx> []"
+  by simp
+
+lemma cons_rsp[quot_respect]:
+  shows "(op = ===> op \<approx> ===> op \<approx>) op # op #"
+  by simp
+
+lemma map_rsp[quot_respect]:
+  shows "(op = ===> op \<approx> ===> op \<approx>) map map"
+  by auto
+
+lemma set_rsp[quot_respect]:
+  "(op \<approx> ===> op =) set set"
+  by auto
+
+lemma list_equiv_rsp[quot_respect]:
+  shows "(op \<approx> ===> op \<approx> ===> op =) op \<approx> op \<approx>"
+  by auto
+
+lemma not_memb_nil:
+  shows "\<not> memb x []"
+  by (simp add: memb_def)
+
+lemma memb_cons_iff:
+  shows "memb x (y # xs) = (x = y \<or> memb x xs)"
+  by (induct xs) (auto simp add: memb_def)
+
+lemma memb_finter_raw:
+  "memb x (finter_raw xs ys) \<longleftrightarrow> memb x xs \<and> memb x ys"
+  by (induct xs) (auto simp add: not_memb_nil memb_cons_iff)
+
+lemma [quot_respect]:
+  "(op \<approx> ===> op \<approx> ===> op \<approx>) finter_raw finter_raw"
+  by (simp add: memb_def[symmetric] memb_finter_raw)
+
+lemma memb_delete_raw:
+  "memb x (delete_raw xs y) = (memb x xs \<and> x \<noteq> y)"
+  by (induct xs arbitrary: x y) (auto simp add: memb_def)
+
+lemma [quot_respect]:
+  "(op \<approx> ===> op = ===> op \<approx>) delete_raw delete_raw"
+  by (simp add: memb_def[symmetric] memb_delete_raw)
+
+lemma fcard_raw_gt_0:
+  assumes a: "x \<in> set xs"
+  shows "0 < fcard_raw xs"
+  using a by (induct xs) (auto simp add: memb_def)
+
+lemma fcard_raw_delete_one:
+  shows "fcard_raw ([x \<leftarrow> xs. x \<noteq> y]) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)"
+  by (induct xs) (auto dest: fcard_raw_gt_0 simp add: memb_def)
+
+lemma fcard_raw_rsp_aux:
+  assumes a: "xs \<approx> ys"
+  shows "fcard_raw xs = fcard_raw ys"
+  using a
+  apply (induct xs arbitrary: ys)
+  apply (auto simp add: memb_def)
+  apply (subgoal_tac "\<forall>x. (x \<in> set xs) = (x \<in> set ys)")
+  apply (auto)
+  apply (drule_tac x="x" in spec)
+  apply (blast)
+  apply (drule_tac x="[x \<leftarrow> ys. x \<noteq> a]" in meta_spec)
+  apply (simp add: fcard_raw_delete_one memb_def)
+  apply (case_tac "a \<in> set ys")
+  apply (simp only: if_True)
+  apply (subgoal_tac "\<forall>x. (x \<in> set xs) = (x \<in> set ys \<and> x \<noteq> a)")
+  apply (drule Suc_pred'[OF fcard_raw_gt_0])
+  apply (auto)
+  done
+
+lemma fcard_raw_rsp[quot_respect]:
+  shows "(op \<approx> ===> op =) fcard_raw fcard_raw"
+  by (simp add: fcard_raw_rsp_aux)
+
+lemma memb_absorb:
+  shows "memb x xs \<Longrightarrow> x # xs \<approx> xs"
+  by (induct xs) (auto simp add: memb_def)
+
+lemma none_memb_nil:
+  "(\<forall>x. \<not> memb x xs) = (xs \<approx> [])"
+  by (simp add: memb_def)
+
+lemma not_memb_delete_raw_ident:
+  shows "\<not> memb x xs \<Longrightarrow> delete_raw xs x = xs"
+  by (induct xs) (auto simp add: memb_def)
+
+lemma memb_commute_ffold_raw:
+  "rsp_fold f \<Longrightarrow> memb h b \<Longrightarrow> ffold_raw f z b = f h (ffold_raw f z (delete_raw b h))"
+  apply (induct b)
+  apply (simp_all add: not_memb_nil)
+  apply (auto)
+  apply (simp_all add: memb_delete_raw not_memb_delete_raw_ident rsp_fold_def  memb_cons_iff)
+  done
+
+lemma ffold_raw_rsp_pre:
+  "\<forall>e. memb e a = memb e b \<Longrightarrow> ffold_raw f z a = ffold_raw f z b"
+  apply (induct a arbitrary: b)
+  apply (simp add: memb_absorb memb_def none_memb_nil)
+  apply (simp)
+  apply (rule conjI)
+  apply (rule_tac [!] impI)
+  apply (rule_tac [!] conjI)
+  apply (rule_tac [!] impI)
+  apply (subgoal_tac "\<forall>e. memb e a2 = memb e b")
+  apply (simp)
+  apply (simp add: memb_cons_iff memb_def)
+  apply (auto)[1]
+  apply (drule_tac x="e" in spec)
+  apply (blast)
+  apply (case_tac b)
+  apply (simp_all)
+  apply (subgoal_tac "ffold_raw f z b = f a1 (ffold_raw f z (delete_raw b a1))")
+  apply (simp only:)
+  apply (rule_tac f="f a1" in arg_cong)
+  apply (subgoal_tac "\<forall>e. memb e a2 = memb e (delete_raw b a1)")
+  apply (simp)
+  apply (simp add: memb_delete_raw)
+  apply (auto simp add: memb_cons_iff)[1]
+  apply (erule memb_commute_ffold_raw)
+  apply (drule_tac x="a1" in spec)
+  apply (simp add: memb_cons_iff)
+  apply (simp add: memb_cons_iff)
+  apply (case_tac b)
+  apply (simp_all)
+  done
+
+lemma [quot_respect]:
+  "(op = ===> op = ===> op \<approx> ===> op =) ffold_raw ffold_raw"
+  by (simp add: memb_def[symmetric] ffold_raw_rsp_pre)
+
+lemma concat_rsp_pre:
+  assumes a: "list_rel op \<approx> x x'"
+  and     b: "x' \<approx> y'"
+  and     c: "list_rel 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_rel_find_element[OF e a])
+  then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto
+  have j: "ya \<in> set y'" using b h by simp
+  have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" by (rule list_rel_find_element[OF j c])
+  then show ?thesis using f i by auto
+qed
+
+lemma [quot_respect]:
+  shows "(list_rel op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"
+proof (rule fun_relI, elim pred_compE)
+  fix a b ba bb
+  assume a: "list_rel op \<approx> a ba"
+  assume b: "ba \<approx> bb"
+  assume c: "list_rel op \<approx> bb b"
+  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"
+      have a': "list_rel op \<approx> ba a" by (rule list_rel_symp[OF list_eq_equivp, OF a])
+      have b': "bb \<approx> ba" by (rule equivp_symp[OF list_eq_equivp, OF b])
+      have c': "list_rel op \<approx> b bb" by (rule list_rel_symp[OF list_eq_equivp, OF c])
+      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 simp
+qed
+
+text {* Distributive lattice with bot *}
+
+lemma sub_list_not_eq:
+  "(sub_list x y \<and> \<not> list_eq x y) = (sub_list x y \<and> \<not> sub_list y x)"
+  by (auto simp add: sub_list_def)
+
+lemma sub_list_refl:
+  "sub_list x x"
+  by (simp add: sub_list_def)
+
+lemma sub_list_trans:
+  "sub_list x y \<Longrightarrow> sub_list y z \<Longrightarrow> sub_list x z"
+  by (simp add: sub_list_def)
+
+lemma sub_list_empty:
+  "sub_list [] x"
+  by (simp add: sub_list_def)
+
+lemma sub_list_append_left:
+  "sub_list x (x @ y)"
+  by (simp add: sub_list_def)
+
+lemma sub_list_append_right:
+  "sub_list y (x @ y)"
+  by (simp add: sub_list_def)
+
+lemma sub_list_inter_left:
+  shows "sub_list (finter_raw x y) x"
+  by (simp add: sub_list_def memb_def[symmetric] memb_finter_raw)
+
+lemma sub_list_inter_right:
+  shows "sub_list (finter_raw x y) y"
+  by (simp add: sub_list_def memb_def[symmetric] memb_finter_raw)
+
+lemma sub_list_list_eq:
+  "sub_list x y \<Longrightarrow> sub_list y x \<Longrightarrow> list_eq x y"
+  unfolding sub_list_def list_eq.simps by blast
+
+lemma sub_list_append:
+  "sub_list y x \<Longrightarrow> sub_list z x \<Longrightarrow> sub_list (y @ z) x"
+  unfolding sub_list_def by auto
+
+lemma sub_list_inter:
+  "sub_list x y \<Longrightarrow> sub_list x z \<Longrightarrow> sub_list x (finter_raw y z)"
+  by (simp add: sub_list_def memb_def[symmetric] memb_finter_raw)
+
+lemma append_inter_distrib:
+  "x @ (finter_raw y z) \<approx> finter_raw (x @ y) (x @ z)"
+  apply (induct x)
+  apply (simp_all add: memb_def)
+  apply (simp add: memb_def[symmetric] memb_finter_raw)
+  by (auto simp add: memb_def)
+
+instantiation fset :: (type) "{bot,distrib_lattice}"
+begin
+
+quotient_definition
+  "bot :: 'a fset" is "[] :: 'a list"
+
+abbreviation
+  fempty  ("{||}")
+where
+  "{||} \<equiv> bot :: 'a fset"
+
+quotient_definition
+  "less_eq_fset \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> bool)"
+is
+  "sub_list \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> bool)"
+
+abbreviation
+  f_subset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)
+where
+  "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
+
+definition
+  less_fset:
+  "(xs :: 'a fset) < ys \<equiv> xs \<le> ys \<and> xs \<noteq> ys"
+
+abbreviation
+  f_subset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)
+where
+  "xs |\<subset>| ys \<equiv> xs < ys"
+
+quotient_definition
+  "sup  \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset)"
+is
+  "(op @) \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> 'a list)"
+
+abbreviation
+  funion  (infixl "|\<union>|" 65)
+where
+  "xs |\<union>| ys \<equiv> sup (xs :: 'a fset) ys"
+
+quotient_definition
+  "inf  \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset)"
+is
+  "finter_raw \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> 'a list)"
+
+abbreviation
+  finter (infixl "|\<inter>|" 65)
+where
+  "xs |\<inter>| ys \<equiv> inf (xs :: 'a fset) ys"
+
+instance
+proof
+  fix x y z :: "'a fset"
+  show "(x |\<subset>| y) = (x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x)"
+    unfolding less_fset by (lifting sub_list_not_eq)
+  show "x |\<subseteq>| x" by (lifting sub_list_refl)
+  show "{||} |\<subseteq>| x" by (lifting sub_list_empty)
+  show "x |\<subseteq>| x |\<union>| y" by (lifting sub_list_append_left)
+  show "y |\<subseteq>| x |\<union>| y" by (lifting sub_list_append_right)
+  show "x |\<inter>| y |\<subseteq>| x" by (lifting sub_list_inter_left)
+  show "x |\<inter>| y |\<subseteq>| y" by (lifting sub_list_inter_right)
+  show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)" by (lifting append_inter_distrib)
+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 (lifting sub_list_trans)
+next
+  fix x y :: "'a fset"
+  assume a: "x |\<subseteq>| y"
+  assume b: "y |\<subseteq>| x"
+  show "x = y" using a b by (lifting sub_list_list_eq)
+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 (lifting sub_list_append)
+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 (lifting sub_list_inter)
+qed
+
+end
+
+section {* Finsert and Membership *}
+
+quotient_definition
+  "finsert :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+is "op #"
+
+syntax
+  "@Finset"     :: "args => 'a fset"  ("{|(_)|}")
+
+translations
+  "{|x, xs|}" == "CONST finsert x {|xs|}"
+  "{|x|}"     == "CONST finsert x {||}"
+
+quotient_definition
+  fin (infix "|\<in>|" 50)
+where
+  "fin :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool" is "memb"
+
+abbreviation
+  fnotin :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)
+where
+  "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
+
+section {* Other constants on the Quotient Type *} 
+
+quotient_definition
+  "fcard :: 'a fset \<Rightarrow> nat" 
+is
+  "fcard_raw"
+
+quotient_definition
+  "fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
+is
+ "map"
+
+quotient_definition
+  "fdelete :: 'a fset \<Rightarrow> 'a \<Rightarrow> 'a fset" 
+  is "delete_raw"
+
+quotient_definition
+  "fset_to_set :: 'a fset \<Rightarrow> 'a set" 
+  is "set"
+
+quotient_definition
+  "ffold :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b"
+  is "ffold_raw"
+
+quotient_definition
+  "fconcat :: ('a fset) fset \<Rightarrow> 'a fset"
+is
+  "concat"
+
+text {* Compositional Respectfullness and Preservation *}
+
+lemma [quot_respect]: "(list_rel op \<approx> OOO op \<approx>) [] []"
+  by (fact compose_list_refl)
+
+lemma [quot_preserve]: "(abs_fset \<circ> map f) [] = abs_fset []"
+  by simp
+
+lemma [quot_respect]:
+  "(op \<approx> ===> list_rel op \<approx> OOO op \<approx> ===> list_rel op \<approx> OOO op \<approx>) op # op #"
+  apply auto
+  apply (simp add: set_in_eq)
+  apply (rule_tac b="x # b" in pred_compI)
+  apply auto
+  apply (rule_tac b="x # ba" in pred_compI)
+  apply auto
+  done
+
+lemma [quot_preserve]:
+  "(rep_fset ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op # = finsert"
+  by (simp add: expand_fun_eq Quotient_abs_rep[OF Quotient_fset]
+      abs_o_rep[OF Quotient_fset] map_id finsert_def)
+
+lemma [quot_preserve]:
+  "((map rep_fset \<circ> rep_fset) ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op @ = funion"
+  by (simp add: expand_fun_eq Quotient_abs_rep[OF Quotient_fset]
+      abs_o_rep[OF Quotient_fset] map_id sup_fset_def)
+
+lemma list_rel_app_l:
+  assumes a: "reflp R"
+  and b: "list_rel R l r"
+  shows "list_rel R (z @ l) (z @ r)"
+  by (induct z) (simp_all add: b rev_iffD1[OF a meta_eq_to_obj_eq[OF reflp_def]])
+
+lemma append_rsp2_pre0:
+  assumes a:"list_rel op \<approx> x x'"
+  shows "list_rel op \<approx> (x @ z) (x' @ z)"
+  using a apply (induct x x' rule: list_induct2')
+  by simp_all (rule list_rel_refl)
+
+lemma append_rsp2_pre1:
+  assumes a:"list_rel op \<approx> x x'"
+  shows "list_rel op \<approx> (z @ x) (z @ x')"
+  using a apply (induct x x' arbitrary: z rule: list_induct2')
+  apply (rule list_rel_refl)
+  apply (simp_all del: list_eq.simps)
+  apply (rule list_rel_app_l)
+  apply (simp_all add: reflp_def)
+  done
+
+lemma append_rsp2_pre:
+  assumes a:"list_rel op \<approx> x x'"
+  and     b: "list_rel op \<approx> z z'"
+  shows "list_rel op \<approx> (x @ z) (x' @ z')"
+  apply (rule list_rel_transp[OF fset_equivp])
+  apply (rule append_rsp2_pre0)
+  apply (rule a)
+  using b apply (induct z z' rule: list_induct2')
+  apply (simp_all only: append_Nil2)
+  apply (rule list_rel_refl)
+  apply simp_all
+  apply (rule append_rsp2_pre1)
+  apply simp
+  done
+
+lemma [quot_respect]:
+  "(list_rel op \<approx> OOO op \<approx> ===> list_rel op \<approx> OOO op \<approx> ===> list_rel op \<approx> OOO op \<approx>) op @ op @"
+proof (intro fun_relI, elim pred_compE)
+  fix x y z w x' z' y' w' :: "'a list list"
+  assume a:"list_rel op \<approx> x x'"
+  and b:    "x' \<approx> y'"
+  and c:    "list_rel op \<approx> y' y"
+  assume aa: "list_rel op \<approx> z z'"
+  and bb:   "z' \<approx> w'"
+  and cc:   "list_rel op \<approx> w' w"
+  have a': "list_rel op \<approx> (x @ z) (x' @ z')" using a aa append_rsp2_pre by auto
+  have b': "x' @ z' \<approx> y' @ w'" using b bb by simp
+  have c': "list_rel op \<approx> (y' @ w') (y @ w)" using c cc append_rsp2_pre by auto
+  have d': "(op \<approx> OO list_rel op \<approx>) (x' @ z') (y @ w)"
+    by (rule pred_compI) (rule b', rule c')
+  show "(list_rel op \<approx> OOO op \<approx>) (x @ z) (y @ w)"
+    by (rule pred_compI) (rule a', rule d')
+qed
+
+text {* Raw theorems. Finsert, memb, singleron, sub_list *}
+
+lemma nil_not_cons:
+  shows "\<not> ([] \<approx> x # xs)"
+  and   "\<not> (x # xs \<approx> [])"
+  by auto
+
+lemma no_memb_nil:
+  "(\<forall>x. \<not> memb x xs) = (xs = [])"
+  by (simp add: memb_def)
+
+lemma memb_consI1:
+  shows "memb x (x # xs)"
+  by (simp add: memb_def)
+
+lemma memb_consI2:
+  shows "memb x xs \<Longrightarrow> memb x (y # xs)"
+  by (simp add: memb_def)
+
+lemma singleton_list_eq:
+  shows "[x] \<approx> [y] \<longleftrightarrow> x = y"
+  by (simp add: id_simps) auto
+
+lemma sub_list_cons:
+  "sub_list (x # xs) ys = (memb x ys \<and> sub_list xs ys)"
+  by (auto simp add: memb_def sub_list_def)
+
+text {* Cardinality of finite sets *}
+
+lemma fcard_raw_0:
+  shows "fcard_raw xs = 0 \<longleftrightarrow> xs \<approx> []"
+  by (induct xs) (auto simp add: memb_def)
+
+lemma fcard_raw_not_memb:
+  shows "\<not> memb x xs \<longleftrightarrow> fcard_raw (x # xs) = Suc (fcard_raw xs)"
+  by auto
+
+lemma fcard_raw_suc:
+  assumes a: "fcard_raw xs = Suc n"
+  shows "\<exists>x ys. \<not> (memb x ys) \<and> xs \<approx> (x # ys) \<and> fcard_raw ys = n"
+  using a
+  by (induct xs) (auto simp add: memb_def split: if_splits)
+
+lemma singleton_fcard_1:
+  shows "set xs = {x} \<Longrightarrow> fcard_raw xs = 1"
+  by (induct xs) (auto simp add: memb_def subset_insert)
+
+lemma fcard_raw_1:
+  shows "fcard_raw xs = 1 \<longleftrightarrow> (\<exists>x. xs \<approx> [x])"
+  apply (auto dest!: fcard_raw_suc)
+  apply (simp add: fcard_raw_0)
+  apply (rule_tac x="x" in exI)
+  apply simp
+  apply (subgoal_tac "set xs = {x}")
+  apply (drule singleton_fcard_1)
+  apply auto
+  done
+
+lemma fcard_raw_suc_memb:
+  assumes a: "fcard_raw A = Suc n"
+  shows "\<exists>a. memb a A"
+  using a
+  by (induct A) (auto simp add: memb_def)
+
+lemma memb_card_not_0:
+  assumes a: "memb a A"
+  shows "\<not>(fcard_raw A = 0)"
+proof -
+  have "\<not>(\<forall>x. \<not> memb x A)" using a by auto
+  then have "\<not>A \<approx> []" using none_memb_nil[of A] by simp
+  then show ?thesis using fcard_raw_0[of A] by simp
+qed
+
+text {* fmap *}
+
+lemma map_append:
+  "map f (xs @ ys) \<approx> (map f xs) @ (map f ys)"
+  by simp
+
+lemma memb_append:
+  "memb x (xs @ ys) \<longleftrightarrow> memb x xs \<or> memb x ys"
+  by (induct xs) (simp_all add: not_memb_nil memb_cons_iff)
+
+lemma cons_left_comm:
+  "x # y # xs \<approx> y # x # xs"
+  by auto
+
+lemma cons_left_idem:
+  "x # x # xs \<approx> x # xs"
+  by auto
+
+lemma fset_raw_strong_cases:
+  "(xs = []) \<or> (\<exists>x ys. ((\<not> memb x ys) \<and> (xs \<approx> x # ys)))"
+  apply (induct xs)
+  apply (simp)
+  apply (rule disjI2)
+  apply (erule disjE)
+  apply (rule_tac x="a" in exI)
+  apply (rule_tac x="[]" in exI)
+  apply (simp add: memb_def)
+  apply (erule exE)+
+  apply (case_tac "x = a")
+  apply (rule_tac x="a" in exI)
+  apply (rule_tac x="ys" in exI)
+  apply (simp)
+  apply (rule_tac x="x" in exI)
+  apply (rule_tac x="a # ys" in exI)
+  apply (auto simp add: memb_def)
+  done
+
+section {* deletion *}
+
+lemma memb_delete_raw_ident:
+  shows "\<not> memb x (delete_raw xs x)"
+  by (induct xs) (auto simp add: memb_def)
+
+lemma fset_raw_delete_raw_cases:
+  "xs = [] \<or> (\<exists>x. memb x xs \<and> xs \<approx> x # delete_raw xs x)"
+  by (induct xs) (auto simp add: memb_def)
+
+lemma fdelete_raw_filter:
+  "delete_raw xs y = [x \<leftarrow> xs. x \<noteq> y]"
+  by (induct xs) simp_all
+
+lemma fcard_raw_delete:
+  "fcard_raw (delete_raw xs y) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)"
+  by (simp add: fdelete_raw_filter fcard_raw_delete_one)
+
+lemma finter_raw_empty:
+  "finter_raw l [] = []"
+  by (induct l) (simp_all add: not_memb_nil)
+
+lemma set_cong: 
+  shows "(set x = set y) = (x \<approx> y)"
+  by auto
+
+lemma inj_map_eq_iff:
+  "inj f \<Longrightarrow> (map f l \<approx> map f m) = (l \<approx> m)"
+  by (simp add: expand_set_eq[symmetric] inj_image_eq_iff)
+
+text {* alternate formulation with a different decomposition principle
+  and a proof of equivalence *}
+
+inductive
+  list_eq2
+where
+  "list_eq2 (a # b # xs) (b # a # xs)"
+| "list_eq2 [] []"
+| "list_eq2 xs ys \<Longrightarrow> list_eq2 ys xs"
+| "list_eq2 (a # a # xs) (a # xs)"
+| "list_eq2 xs ys \<Longrightarrow> list_eq2 (a # xs) (a # ys)"
+| "\<lbrakk>list_eq2 xs1 xs2; list_eq2 xs2 xs3\<rbrakk> \<Longrightarrow> list_eq2 xs1 xs3"
+
+lemma list_eq2_refl:
+  shows "list_eq2 xs xs"
+  by (induct xs) (auto intro: list_eq2.intros)
+
+lemma cons_delete_list_eq2:
+  shows "list_eq2 (a # (delete_raw A a)) (if memb a A then A else a # A)"
+  apply (induct A)
+  apply (simp add: memb_def list_eq2_refl)
+  apply (case_tac "memb a (aa # A)")
+  apply (simp_all only: memb_cons_iff)
+  apply (case_tac [!] "a = aa")
+  apply (simp_all)
+  apply (case_tac "memb a A")
+  apply (auto simp add: memb_def)[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 not_memb_delete_raw_ident)
+  done
+
+lemma memb_delete_list_eq2:
+  assumes a: "memb e r"
+  shows "list_eq2 (e # delete_raw r e) r"
+  using a cons_delete_list_eq2[of e r]
+  by simp
+
+lemma delete_raw_rsp:
+  "xs \<approx> ys \<Longrightarrow> delete_raw xs x \<approx> delete_raw ys x"
+  by (simp add: memb_def[symmetric] memb_delete_raw)
+
+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: "fcard_raw l = n" and b: "l \<approx> r"
+    have "list_eq2 l r"
+      using a b
+    proof (induct n arbitrary: l r)
+      case 0
+      have "fcard_raw l = 0" by fact
+      then have "\<forall>x. \<not> memb x l" using memb_card_not_0[of _ l] by auto
+      then have z: "l = []" using no_memb_nil by auto
+      then have "r = []" using `l \<approx> r` 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: "fcard_raw l = Suc m" by fact
+      have "\<exists>a. memb a l" by (rule fcard_raw_suc_memb[OF d])
+      then obtain a where e: "memb a l" by auto
+      then have e': "memb a r" using list_eq.simps[simplified memb_def[symmetric], of l r] b by auto
+      have f: "fcard_raw (delete_raw l a) = m" using fcard_raw_delete[of l a] e d by simp
+      have g: "delete_raw l a \<approx> delete_raw r a" using delete_raw_rsp[OF b] by simp
+      have g': "list_eq2 (delete_raw l a) (delete_raw r a)" by (rule Suc.hyps[OF f g])
+      have h: "list_eq2 (a # delete_raw l a) (a # delete_raw r a)" by (rule list_eq2.intros(5)[OF g'])
+      have i: "list_eq2 l (a # delete_raw l a)"
+        by (rule list_eq2.intros(3)[OF memb_delete_list_eq2[OF e]])
+      have "list_eq2 l (a # delete_raw r a)" by (rule list_eq2.intros(6)[OF i h])
+      then show ?case using list_eq2.intros(6)[OF _ memb_delete_list_eq2[OF e']] by simp
+    qed
+    }
+  then show "l \<approx> r \<Longrightarrow> list_eq2 l r" by blast
+qed
+
+text {* Lifted theorems *}
+
+lemma not_fin_fnil: "x |\<notin>| {||}"
+  by (lifting not_memb_nil)
+
+lemma fin_finsert_iff[simp]:
+  "x |\<in>| finsert y S = (x = y \<or> x |\<in>| S)"
+  by (lifting memb_cons_iff)
+
+lemma
+  shows finsertI1: "x |\<in>| finsert x S"
+  and   finsertI2: "x |\<in>| S \<Longrightarrow> x |\<in>| finsert y S"
+  by (lifting memb_consI1, lifting memb_consI2)
+
+lemma finsert_absorb[simp]:
+  shows "x |\<in>| S \<Longrightarrow> finsert x S = S"
+  by (lifting memb_absorb)
+
+lemma fempty_not_finsert[simp]:
+  "{||} \<noteq> finsert x S"
+  "finsert x S \<noteq> {||}"
+  by (lifting nil_not_cons)
+
+lemma finsert_left_comm:
+  "finsert x (finsert y S) = finsert y (finsert x S)"
+  by (lifting cons_left_comm)
+
+lemma finsert_left_idem:
+  "finsert x (finsert x S) = finsert x S"
+  by (lifting cons_left_idem)
+
+lemma fsingleton_eq[simp]:
+  shows "{|x|} = {|y|} \<longleftrightarrow> x = y"
+  by (lifting singleton_list_eq)
+
+text {* fset_to_set *}
+
+lemma fset_to_set_simps[simp]:
+  "fset_to_set {||} = ({} :: 'a set)"
+  "fset_to_set (finsert (h :: 'a) t) = insert h (fset_to_set t)"
+  by (lifting set.simps)
+
+lemma in_fset_to_set:
+  "x \<in> fset_to_set S \<equiv> x |\<in>| S"
+  by (lifting memb_def[symmetric])
+
+lemma none_fin_fempty:
+  "(\<forall>x. x |\<notin>| S) = (S = {||})"
+  by (lifting none_memb_nil)
+
+lemma fset_cong:
+  "(fset_to_set S = fset_to_set T) = (S = T)"
+  by (lifting set_cong)
+
+text {* fcard *}
+
+lemma fcard_fempty [simp]:
+  shows "fcard {||} = 0"
+  by (lifting fcard_raw_nil)
+
+lemma fcard_finsert_if [simp]:
+  shows "fcard (finsert x S) = (if x |\<in>| S then fcard S else Suc (fcard S))"
+  by (lifting fcard_raw_cons)
+
+lemma fcard_0: "(fcard S = 0) = (S = {||})"
+  by (lifting fcard_raw_0)
+
+lemma fcard_1:
+  shows "(fcard S = 1) = (\<exists>x. S = {|x|})"
+  by (lifting fcard_raw_1)
+
+lemma fcard_gt_0: 
+  shows "x \<in> fset_to_set S \<Longrightarrow> 0 < fcard S"
+  by (lifting fcard_raw_gt_0)
+
+lemma fcard_not_fin: 
+  shows "(x |\<notin>| S) = (fcard (finsert x S) = Suc (fcard S))"
+  by (lifting fcard_raw_not_memb)
+
+lemma fcard_suc: "fcard S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = finsert x T \<and> fcard T = n"
+  by (lifting fcard_raw_suc)
+
+lemma fcard_delete:
+  "fcard (fdelete S y) = (if y |\<in>| S then fcard S - 1 else fcard S)"
+  by (lifting fcard_raw_delete)
+
+lemma fcard_suc_memb: "fcard A = Suc n \<Longrightarrow> \<exists>a. a |\<in>| A"
+  by (lifting fcard_raw_suc_memb)
+
+lemma fin_fcard_not_0: "a |\<in>| A \<Longrightarrow> fcard A \<noteq> 0"
+  by (lifting memb_card_not_0)
+
+text {* funion *}
+
+lemma funion_simps[simp]:
+  shows "{||} |\<union>| S = S"
+  and   "finsert x S |\<union>| T = finsert x (S |\<union>| T)"
+  by (lifting append.simps)
+
+lemma funion_empty[simp]:
+  shows "S |\<union>| {||} = S"
+  by (lifting append_Nil2)
+
+lemma singleton_union_left:
+  "{|a|} |\<union>| S = finsert a S"
+  by simp
+
+lemma singleton_union_right:
+  "S |\<union>| {|a|} = finsert a S"
+  by (subst sup.commute) simp
+
+section {* Induction and Cases rules for finite sets *}
+
+lemma fset_strong_cases:
+  "S = {||} \<or> (\<exists>x T. x |\<notin>| T \<and> S = finsert x T)"
+  by (lifting fset_raw_strong_cases)
+
+lemma fset_exhaust[case_names fempty finsert, cases type: fset]:
+  shows "\<lbrakk>S = {||} \<Longrightarrow> P; \<And>x S'. S = finsert x S' \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
+  by (lifting list.exhaust)
+
+lemma fset_induct_weak[case_names fempty finsert]:
+  shows "\<lbrakk>P {||}; \<And>x S. P S \<Longrightarrow> P (finsert x S)\<rbrakk> \<Longrightarrow> P S"
+  by (lifting list.induct)
+
+lemma fset_induct[case_names fempty finsert, induct type: fset]:
+  assumes prem1: "P {||}" 
+  and     prem2: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
+  shows "P S"
+proof(induct S rule: fset_induct_weak)
+  case fempty
+  show "P {||}" by (rule prem1)
+next
+  case (finsert x S)
+  have asm: "P S" by fact
+  show "P (finsert x S)"
+    by (cases "x |\<in>| S") (simp_all add: asm prem2)
+qed
+
+lemma fset_induct2:
+  "P {||} {||} \<Longrightarrow>
+  (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
+  (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
+  (\<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>
+  P xsa ysa"
+  apply (induct xsa arbitrary: ysa)
+  apply (induct_tac x rule: fset_induct)
+  apply simp_all
+  apply (induct_tac xa rule: fset_induct)
+  apply simp_all
+  done
+
+text {* fmap *}
+
+lemma fmap_simps[simp]:
+  "fmap (f :: 'a \<Rightarrow> 'b) {||} = {||}"
+  "fmap f (finsert x S) = finsert (f x) (fmap f S)"
+  by (lifting map.simps)
+
+lemma fmap_set_image:
+  "fset_to_set (fmap f S) = f ` (fset_to_set S)"
+  by (induct S) (simp_all)
+
+lemma inj_fmap_eq_iff:
+  "inj f \<Longrightarrow> (fmap f S = fmap f T) = (S = T)"
+  by (lifting inj_map_eq_iff)
+
+lemma fmap_funion: "fmap f (S |\<union>| T) = fmap f S |\<union>| fmap f T"
+  by (lifting map_append)
+
+lemma fin_funion:
+  "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"
+  by (lifting memb_append)
+
+text {* ffold *}
+
+lemma ffold_nil: "ffold f z {||} = z"
+  by (lifting ffold_raw.simps(1)[where 'a="'b" and 'b="'a"])
+
+lemma ffold_finsert: "ffold f z (finsert a A) =
+  (if rsp_fold f then if a |\<in>| A then ffold f z A else f a (ffold f z A) else z)"
+  by (lifting ffold_raw.simps(2)[where 'a="'b" and 'b="'a"])
+
+lemma fin_commute_ffold:
+  "\<lbrakk>rsp_fold f; h |\<in>| b\<rbrakk> \<Longrightarrow> ffold f z b = f h (ffold f z (fdelete b h))"
+  by (lifting memb_commute_ffold_raw)
+
+text {* fdelete *}
+
+lemma fin_fdelete: 
+  shows "x |\<in>| fdelete S y \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"
+  by (lifting memb_delete_raw)
+
+lemma fin_fdelete_ident: 
+  shows "x |\<notin>| fdelete S x"
+  by (lifting memb_delete_raw_ident)
+
+lemma not_memb_fdelete_ident: 
+  shows "x |\<notin>| S \<Longrightarrow> fdelete S x = S"
+  by (lifting not_memb_delete_raw_ident)
+
+lemma fset_fdelete_cases:
+  shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = finsert x (fdelete S x))"
+  by (lifting fset_raw_delete_raw_cases)
+
+text {* inter *}
+
+lemma finter_empty_l: "({||} |\<inter>| S) = {||}"
+  by (lifting finter_raw.simps(1))
+
+lemma finter_empty_r: "(S |\<inter>| {||}) = {||}"
+  by (lifting finter_raw_empty)
+
+lemma finter_finsert:
+  "finsert x S |\<inter>| T = (if x |\<in>| T then finsert x (S |\<inter>| T) else S |\<inter>| T)"
+  by (lifting finter_raw.simps(2))
+
+lemma fin_finter:
+  "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"
+  by (lifting memb_finter_raw)
+
+lemma fsubset_finsert:
+  "(finsert x xs |\<subseteq>| ys) = (x |\<in>| ys \<and> xs |\<subseteq>| ys)"
+  by (lifting sub_list_cons)
+
+lemma "xs |\<subseteq>| ys \<equiv> \<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys"
+  by (lifting sub_list_def[simplified memb_def[symmetric]])
+
+lemma fsubset_fin: "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"
+by (rule meta_eq_to_obj_eq)
+   (lifting sub_list_def[simplified memb_def[symmetric]])
+
+lemma expand_fset_eq:
+  "(S = T) = (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"
+  by (lifting list_eq.simps[simplified memb_def[symmetric]])
+
+(* 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 = finsert a (finsert b xs); a2 = finsert b (finsert 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 = finsert a (finsert a xs); a2 = finsert a xs\<rbrakk> \<Longrightarrow> P;
+   \<And>xs ys a. \<lbrakk>a1 = finsert a xs; a2 = finsert 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 (finsert a (finsert b xs)) (finsert b (finsert a xs))"
+  and "P {||} {||}"
+  and "\<And>xs ys. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P ys xs"
+  and "\<And>a xs. P (finsert a (finsert a xs)) (finsert a xs)"
+  and "\<And>xs ys a. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P (finsert a xs) (finsert 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]])
+
+text {* concat *}
+
+lemma fconcat_empty:
+  shows "fconcat {||} = {||}"
+  by (lifting concat.simps(1))
+
+lemma fconcat_insert:
+  shows "fconcat (finsert x S) = x |\<union>| fconcat S"
+  by (lifting concat.simps(2))
+
+lemma "fconcat (xs |\<union>| ys) = fconcat xs |\<union>| fconcat ys"
+  by (lifting concat_append)
+
+ML {*
+fun dest_fsetT (Type ("FSet.fset", [T])) = T
+  | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
+*}
+
+no_notation
+  list_eq (infix "\<approx>" 50)
+
+end
diff -r 8e58a63ac975 -r c4f5823f282d src/HOL/Quotient_Examples/ROOT.ML
--- a/src/HOL/Quotient_Examples/ROOT.ML	Thu Apr 22 22:12:12 2010 +0200
+++ b/src/HOL/Quotient_Examples/ROOT.ML	Fri Apr 23 10:00:53 2010 +0200
@@ -4,5 +4,5 @@
 Testing the quotient package.
 *)
 
-use_thys ["LarryInt", "LarryDatatype"];
+use_thys ["FSet", "LarryInt", "LarryDatatype"];