diff -r db158e995bfc -r 9df6144e281b Quot/Examples/FSet3.thy
--- a/Quot/Examples/FSet3.thy	Thu Feb 25 07:48:57 2010 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,714 +0,0 @@
-theory FSet3
-imports "../Quotient" "../Quotient_List" List
-begin
-
-ML {*
-structure QuotientRules = Named_Thms
-  (val name = "quot_thm"
-   val description = "Quotient theorems.")
-*}
-
-ML {*
-open QuotientRules
-*}
-
-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 reflp_def symp_def transp_def
-by auto
-
-(* FIXME-TODO: because of beta-reduction, one cannot give the *)
-(* FIXME-TODO: relation as a term or abbreviation             *) 
-quotient_type 
-  'a fset = "'a list" / "list_eq"
-by (rule list_eq_equivp)
-
-
-section {* empty fset, finsert and membership *} 
-
-quotient_definition
-  fempty  ("{||}")
-where
-  "fempty :: 'a fset"
-is "[]::'a list"
-
-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 {||}"
-
-definition 
-  memb :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
-where
-  "memb x xs \<equiv> x \<in> set xs"
-
-quotient_definition
-  fin ("_ |\<in>| _" [50, 51] 50)
-where
-  "fin :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool"
-is "memb"
-
-abbreviation
-  fnotin :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" ("_ |\<notin>| _" [50, 51] 50)
-where
-  "a |\<notin>| S \<equiv> \<not>(a |\<in>| S)"
-
-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
-
-
-section {* Augmenting a set -- @{const finsert} *}
-
-text {* raw section *}
-
-lemma nil_not_cons:
-  shows "\<not>[] \<approx> x # xs"
-  by auto
-
-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_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 memb_absorb:
-  shows "memb x xs \<Longrightarrow> x # xs \<approx> xs"
-  by (induct xs) (auto simp add: memb_def id_simps)
-
-text {* lifted section *}
-
-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)
-
-
-section {* Singletons *}
-
-text {* raw section *}
-
-lemma singleton_list_eq:
-  shows "[x] \<approx> [y] \<longleftrightarrow> x = y"
-  by (simp add: id_simps) auto
-
-text {* lifted section *}
-
-lemma fempty_not_finsert[simp]:
-  shows "{||} \<noteq> finsert x S"
-  by (lifting nil_not_cons)
-
-lemma fsingleton_eq[simp]:
-  shows "{|x|} = {|y|} \<longleftrightarrow> x = y"
-  by (lifting singleton_list_eq)
-
-section {* Union *}
-
-quotient_definition
-  funion  (infixl "|\<union>|" 65)
-where
-  "funion :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
-is
-  "op @"
-
-section {* Cardinality of finite sets *}
-
-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))"
-
-quotient_definition
-  "fcard :: 'a fset \<Rightarrow> nat" 
-is "fcard_raw"
-
-text {* raw section *}
-
-lemma fcard_raw_ge_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:
-  "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_ge_0 simp add: memb_def)
-
-lemma fcard_raw_rsp_aux:
-  assumes a: "a \<approx> b"
-  shows "fcard_raw a = fcard_raw b"
-using a
-apply(induct a arbitrary: b)
-apply(auto simp add: memb_def)
-apply(metis)
-apply(drule_tac x="[x \<leftarrow> b. x \<noteq> a1]" in meta_spec)
-apply(simp add: fcard_raw_delete_one)
-apply(metis Suc_pred' fcard_raw_ge_0 fcard_raw_delete_one memb_def)
-done
-
-lemma fcard_raw_rsp[quot_respect]:
-  "(op \<approx> ===> op =) fcard_raw fcard_raw"
-  by (simp add: fcard_raw_rsp_aux)
-
-text {* lifted section *}
-
-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)
-
-
-section {* Induction and Cases rules for finite sets *}
-
-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[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_induct2[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)
-  case fempty
-  show "P {||}" by (rule prem1)
-next
-  case (finsert x S)
-  have asm: "P S" by fact
-  show "P (finsert x S)"
-  proof(cases "x |\<in>| S")
-    case True
-    have "x |\<in>| S" by fact
-    then show "P (finsert x S)" using asm by simp
-  next
-    case False
-    have "x |\<notin>| S" by fact
-    then show "P (finsert x S)" using prem2 asm by simp
-  qed
-qed
-
-
-section {* fmap and fset comprehension *}
-
-quotient_definition
-  "fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
-is
- "map"
-
-quotient_definition
-  "fconcat :: ('a fset) fset => 'a fset"
-is
- "concat"
-
-(*lemma fconcat_rsp[quot_respect]:
-  shows "((list_rel op \<approx>) ===> op \<approx>) concat concat"
-apply(auto)
-sorry
-*)
-
-(* PROBLEM: these lemmas needs to be restated, since  *)
-(* concat.simps(1) and concat.simps(2) contain the    *)
-(* type variables ?'a1.0 (which are turned into frees *)
-(* 'a_1                                               *)
-lemma concat1:
-  shows "concat [] \<approx> []"
-by (simp add: id_simps)
-
-lemma concat2:
-  shows "concat (x # xs) \<approx> x @ concat xs"
-by (simp add: id_simps)
-
-lemma concat_rsp[quot_respect]:
-  shows "(list_rel op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"
-sorry
-
-lemma nil_rsp2[quot_respect]: "(list_rel op \<approx> OOO op \<approx>) [] []"
-  apply (metis FSet3.nil_rsp list_rel.simps(1) pred_comp.intros)
-  done
-
-lemma set_in_eq: "(\<forall>e. ((e \<in> A) \<longleftrightarrow> (e \<in> B))) \<equiv> A = B"
-  apply (rule eq_reflection)
-  apply auto
-  done
-
-lemma map_rel_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"
-  unfolding list_eq.simps
-  apply(simp only: set_map set_in_eq)
-  done
-
-lemma quotient_compose_list_pre:
-  "(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))"
-  apply rule
-  apply rule
-  apply rule
-  apply (rule list_rel_refl)
-  apply (metis equivp_def fset_equivp)
-  apply rule
-  apply (rule equivp_reflp[OF fset_equivp])
-  apply (rule list_rel_refl)
-  apply (metis equivp_def fset_equivp)
-  apply(rule)
-  apply rule
-  apply (rule list_rel_refl)
-  apply (metis equivp_def fset_equivp)
-  apply rule
-  apply (rule equivp_reflp[OF fset_equivp])
-  apply (rule list_rel_refl)
-  apply (metis equivp_def fset_equivp)
-  apply (subgoal_tac "map abs_fset r \<approx> map abs_fset s")
-  apply (metis Quotient_rel[OF Quotient_fset])
-  apply (auto simp only:)[1]
-  apply (subgoal_tac "map abs_fset r = map abs_fset b")
-  prefer 2
-  apply (metis Quotient_rel[OF list_quotient[OF Quotient_fset]])
-  apply (subgoal_tac "map abs_fset s = map abs_fset ba")
-  prefer 2
-  apply (metis Quotient_rel[OF list_quotient[OF Quotient_fset]])
-  apply (simp only: map_rel_cong)
-  apply rule
-  apply (rule rep_abs_rsp[of "list_rel op \<approx>" "map abs_fset"])
-  apply (tactic {* Quotient_Tacs.quotient_tac @{context} 1 *})
-  apply (rule list_rel_refl)
-  apply (metis equivp_def fset_equivp)
-  apply rule
-  prefer 2
-  apply (rule rep_abs_rsp_left[of "list_rel op \<approx>" "map abs_fset"])
-  apply (tactic {* Quotient_Tacs.quotient_tac @{context} 1 *})
-  apply (rule list_rel_refl)
-  apply (metis equivp_def fset_equivp)
-  apply (erule conjE)+
-  apply (subgoal_tac "map abs_fset r \<approx> map abs_fset s")
-  prefer 2
-  apply (metis Quotient_def Quotient_fset equivp_reflp fset_equivp)
-  apply (rule map_rel_cong)
-  apply (assumption)
-  done
-
-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
-  apply (rule)+
-  apply (simp add: abs_o_rep[OF Quotient_fset] id_simps Quotient_abs_rep[OF Quotient_fset])
-  apply (rule)
-  apply (rule)
-  apply (rule)
-  apply (rule list_rel_refl)
-  apply (metis equivp_def fset_equivp)
-  apply (rule)
-  apply (rule equivp_reflp[OF fset_equivp])
-  apply (rule list_rel_refl)
-  apply (metis equivp_def fset_equivp)
-  apply rule
-  apply rule
-  apply(rule quotient_compose_list_pre)
-  done
-
-lemma fconcat_empty:
-  shows "fconcat {||} = {||}"
-apply(lifting concat1)
-apply(cleaning)
-apply(simp add: comp_def)
-apply(fold fempty_def[simplified id_simps])
-apply(rule refl)
-done
-
-(* Should be true *)
-
-lemma insert_rsp2[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)
-sorry
-
-lemma append_rsp[quot_respect]:
-  "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @"
-  by (auto)
-
-lemma fconcat_insert:
-  shows "fconcat (finsert x S) = x |\<union>| fconcat S"
-apply(lifting concat2)
-apply(cleaning)
-apply (simp add: finsert_def fconcat_def comp_def)
-apply cleaning
-done
-
-text {* raw section *}
-
-lemma map_rsp_aux:
-  assumes a: "a \<approx> b"
-  shows "map f a \<approx> map f b"
-  using a
-apply(induct a arbitrary: b)
-apply(auto)
-apply(metis rev_image_eqI)
-done
-
-lemma map_rsp[quot_respect]:
-  shows "(op = ===> op \<approx> ===> op \<approx>) map map"
-by (auto simp add: map_rsp_aux)
-
-
-text {* lifted section *}
-
-(* TBD *)
-
-text {* syntax for fset comprehensions (adapted from lists) *}
-
-nonterminals fsc_qual fsc_quals
-
-syntax
-"_fsetcompr" :: "'a \<Rightarrow> fsc_qual \<Rightarrow> fsc_quals \<Rightarrow> 'a fset"  ("{|_ . __")
-"_fsc_gen" :: "'a \<Rightarrow> 'a fset \<Rightarrow> fsc_qual" ("_ <- _")
-"_fsc_test" :: "bool \<Rightarrow> fsc_qual" ("_")
-"_fsc_end" :: "fsc_quals" ("|}")
-"_fsc_quals" :: "fsc_qual \<Rightarrow> fsc_quals \<Rightarrow> fsc_quals" (", __")
-"_fsc_abs" :: "'a => 'b fset => 'b fset"
-
-syntax (xsymbols)
-"_fsc_gen" :: "'a \<Rightarrow> 'a fset \<Rightarrow> fsc_qual" ("_ \<leftarrow> _")
-syntax (HTML output)
-"_fsc_gen" :: "'a \<Rightarrow> 'a fset \<Rightarrow> fsc_qual" ("_ \<leftarrow> _")
-
-parse_translation (advanced) {*
-let
-  val femptyC = Syntax.const @{const_name fempty};
-  val finsertC = Syntax.const @{const_name finsert};
-  val fmapC = Syntax.const @{const_name fmap};
-  val fconcatC = Syntax.const @{const_name fconcat};
-  val IfC = Syntax.const @{const_name If};
-  fun fsingl x = finsertC $ x $ femptyC;
-
-  fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
-    let
-      val x = Free (Name.variant (fold Term.add_free_names [p, e] []) "x", dummyT);
-      val e = if opti then fsingl e else e;
-      val case1 = Syntax.const "_case1" $ p $ e;
-      val case2 = Syntax.const "_case1" $ Syntax.const Term.dummy_patternN
-                                        $ femptyC;
-      val cs = Syntax.const "_case2" $ case1 $ case2
-      val ft = Datatype_Case.case_tr false Datatype.info_of_constr
-                 ctxt [x, cs]
-    in lambda x ft end;
-
-  fun abs_tr ctxt (p as Free(s,T)) e opti =
-        let val thy = ProofContext.theory_of ctxt;
-            val s' = Sign.intern_const thy s
-        in if Sign.declared_const thy s'
-           then (pat_tr ctxt p e opti, false)
-           else (lambda p e, true)
-        end
-    | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
-
-  fun fsc_tr ctxt [e, Const("_fsc_test",_) $ b, qs] =
-        let 
-          val res = case qs of 
-                      Const("_fsc_end",_) => fsingl e
-                    | Const("_fsc_quals",_)$ q $ qs => fsc_tr ctxt [e, q, qs];
-        in 
-          IfC $ b $ res $ femptyC 
-        end
-
-    | fsc_tr ctxt [e, Const("_fsc_gen",_) $ p $ es, Const("_fsc_end",_)] =
-         (case abs_tr ctxt p e true of
-            (f,true) => fmapC $ f $ es
-          | (f, false) => fconcatC $ (fmapC $ f $ es))
-       
-    | fsc_tr ctxt [e, Const("_fsc_gen",_) $ p $ es, Const("_fsc_quals",_) $ q $ qs] =
-        let
-          val e' = fsc_tr ctxt [e, q, qs];
-        in 
-          fconcatC $ (fmapC $ (fst (abs_tr ctxt p e' false)) $ es) 
-        end
-
-in [("_fsetcompr", fsc_tr)] end
-*}
-
-(* examles *)
-term "{|(x,y,z). b|}"
-term "{|x. x \<leftarrow> xs|}"
-term "{|(x,y,z). x\<leftarrow>xs|}"
-term "{|e x y. x\<leftarrow>xs, y\<leftarrow>ys|}"
-term "{|(x,y,z). x<a, x>b|}"
-term "{|(x,y,z). x\<leftarrow>xs, x>b|}"
-term "{|(x,y,z). x<a, x\<leftarrow>xs|}"
-term "{|(x,y). Cons True x \<leftarrow> xs|}"
-term "{|(x,y,z). Cons x [] \<leftarrow> xs|}"
-term "{|(x,y,z). x<a, x>b, x=d|}"
-term "{|(x,y,z). x<a, x>b, y\<leftarrow>ys|}"
-term "{|(x,y,z). x<a, x\<leftarrow>xs,y>b|}"
-term "{|(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys|}"
-term "{|(x,y,z). x\<leftarrow>xs, x>b, y<a|}"
-term "{|(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys|}"
-term "{|(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x|}"
-term "{|(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs|}"
-
-
-(* BELOW CONSTRUCTION SITE *)
-
-
-lemma no_mem_nil: 
-  "(\<forall>a. a \<notin> set A) = (A = [])"
-by (induct A) (auto)
-
-lemma none_mem_nil: 
-  "(\<forall>a. a \<notin> set A) = (A \<approx> [])"
-by simp
-
-lemma mem_cons: 
-  "a \<in> set A \<Longrightarrow> a # A \<approx> A"
-by auto
-
-lemma cons_left_comm:
-  "x # y # A \<approx> y # x # A"
-by (auto simp add: id_simps)
-
-lemma cons_left_idem:
-  "x # x # A \<approx> x # A"
-by (auto simp add: id_simps)
-
-lemma finite_set_raw_strong_cases:
-  "(X = []) \<or> (\<exists>a Y. ((a \<notin> set Y) \<and> (X \<approx> a # Y)))"
-  apply (induct X)
-  apply (simp)
-  apply (rule disjI2)
-  apply (erule disjE)
-  apply (rule_tac x="a" in exI)
-  apply (rule_tac x="[]" in exI)
-  apply (simp)
-  apply (erule exE)+
-  apply (case_tac "a = aa")
-  apply (rule_tac x="a" in exI)
-  apply (rule_tac x="Y" in exI)
-  apply (simp)
-  apply (rule_tac x="aa" in exI)
-  apply (rule_tac x="a # Y" in exI)
-  apply (auto)
-  done
-
-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))"
-
-lemma mem_delete_raw:
-  "x \<in> set (delete_raw A a) = (x \<in> set A \<and> \<not>(x = a))"
-  by (induct A arbitrary: x a) (auto)
-
-lemma mem_delete_raw_ident:
-  "\<not>(a \<in> set (delete_raw A a))"
-by (induct A) (auto)
-
-lemma not_mem_delete_raw_ident:
-  "b \<notin> set A \<Longrightarrow> (delete_raw A b = A)"
-by (induct A) (auto)
-
-lemma delete_raw_RSP:
-  "A \<approx> B \<Longrightarrow> delete_raw A a \<approx> delete_raw B a"
-apply(induct A arbitrary: B a)
-apply(auto)
-sorry
-
-lemma cons_delete_raw:
-  "a # (delete_raw A a) \<approx> (if a \<in> set A then A else (a # A))"
-sorry
-
-lemma mem_cons_delete_raw:
-    "a \<in> set A \<Longrightarrow> a # (delete_raw A a) \<approx> A"
-sorry
-
-lemma finite_set_raw_delete_raw_cases:
-    "X = [] \<or> (\<exists>a. a mem X \<and> X \<approx> a # delete_raw X a)"
-  by (induct X) (auto)
-
-
-
-
-
-lemma list2set_thm:
-  shows "set [] = {}"
-  and "set (h # t) = insert h (set t)"
-  by (auto)
-
-lemma list2set_RSP:
-  "A \<approx> B \<Longrightarrow> set A = set B"
-  by auto
-
-definition
-  rsp_fold
-where
-  "rsp_fold f = (\<forall>u v w. (f u (f v w) = f v (f u w)))"
-
-primrec
-  fold_raw :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
-where
-  "fold_raw f z [] = z"
-| "fold_raw f z (a # A) =
-     (if (rsp_fold f) then
-       if a mem A then fold_raw f z A
-       else f a (fold_raw f z A)
-     else z)"
-
-lemma mem_lcommuting_fold_raw:
-  "rsp_fold f \<Longrightarrow> h mem B \<Longrightarrow> fold_raw f z B = f h (fold_raw f z (delete_raw B h))"
-sorry
-
-lemma fold_rsp[quot_respect]:
-  "(op = ===> op = ===> op \<approx> ===> op =) fold_raw fold_raw"
-apply(auto)
-sorry
-
-primrec
-  inter_raw
-where
-  "inter_raw [] B = []"
-| "inter_raw (a # A) B = (if a mem B then a # inter_raw A B else inter_raw A B)"
-
-lemma mem_inter_raw:
-  "x mem (inter_raw A B) = x mem A \<and> x mem B"
-sorry
-
-lemma inter_raw_RSP:
-  "A1 \<approx> A2 \<and> B1 \<approx> B2 \<Longrightarrow> (inter_raw A1 B1) \<approx> (inter_raw A2 B2)"
-sorry
-
-
-(* LIFTING DEFS *)
-
-
-section {* Constants on the Quotient Type *} 
-
-
-quotient_definition
-  "fdelete :: 'a fset \<Rightarrow> 'a \<Rightarrow> 'a fset" 
-  is "delete_raw"
-
-quotient_definition
-  finter ("_ \<inter>f _" [70, 71] 70)
-where
-  "finter :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
-  is "inter_raw"
-
-quotient_definition
-  "ffold :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" 
-  is "fold_raw"
-
-quotient_definition
-  "fset_to_set :: 'a fset \<Rightarrow> 'a set" 
-  is "set"
-
-
-section {* Lifted Theorems *}
-
-thm list.cases (* ??? *)
-
-thm cons_left_comm
-lemma "finsert a (finsert b S) = finsert b (finsert a S)"
-by (lifting cons_left_comm)
-
-thm cons_left_idem
-lemma "finsert a (finsert a S) = finsert a S"
-by (lifting cons_left_idem)
-
-(* thm MEM:
-  MEM x [] = F
-  MEM x (h::t) = (x=h) \/ MEM x t *)
-thm none_mem_nil
-(*lemma "(\<forall>a. a \<notin>f A) = (A = fempty)"*)
-
-thm mem_cons
-thm finite_set_raw_strong_cases
-(*thm card_raw.simps*)
-(*thm not_mem_card_raw*)
-(*thm card_raw_suc*)
-
-lemma "fcard X = Suc n \<Longrightarrow> (\<exists>a S. a \<notin>f S & X = finsert a S)"
-(*by (lifting card_raw_suc)*)
-sorry
-
-(*thm card_raw_cons_gt_0
-thm mem_card_raw_gt_0
-thm not_nil_equiv_cons
-*)
-thm delete_raw.simps
-(*thm mem_delete_raw*)
-(*thm card_raw_delete_raw*)
-thm cons_delete_raw
-thm mem_cons_delete_raw
-thm finite_set_raw_delete_raw_cases
-thm append.simps
-(* MEM_APPEND: MEM e (APPEND l1 l2) = MEM e l1 \/ MEM e l2 *)
-thm inter_raw.simps
-thm mem_inter_raw
-thm fold_raw.simps
-thm list2set_thm
-thm list_eq_def
-thm list.induct
-lemma "\<lbrakk>P fempty; \<And>a x. P x \<Longrightarrow> P (finsert a x)\<rbrakk> \<Longrightarrow> P l"
-by (lifting list.induct)
-
-(* We also have map and some properties of it in FSet *)
-(* and the following which still lifts ok *)
-lemma "funion (funion x xa) xb = funion x (funion xa xb)"
-by (lifting append_assoc)
-
-quotient_definition
-  "fset_case :: 'a \<Rightarrow> ('b \<Rightarrow> 'b fset \<Rightarrow> 'a) \<Rightarrow> 'b fset \<Rightarrow> 'a"
-is
-  "list_case"
-
-(* NOT SURE IF TRUE *)
-lemma list_case_rsp[quot_respect]:
-  "(op = ===> (op = ===> op \<approx> ===> op =) ===> op \<approx> ===> op =) list_case list_case"
-  apply (auto)
-  sorry
-
-lemma "fset_case (f1::'t) f2 (finsert a xa) = f2 a xa"
-apply (lifting list.cases(2))
-done
-
-end