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