theory FSet3
imports "../QuotMain" List
begin

definition
  list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)
where
  "list_eq x y = (\<forall>e. e mem x = e mem y)"

lemma list_eq_reflp:
  shows "xs \<approx> xs"
  by (metis list_eq_def)

lemma list_eq_equivp:
  shows "equivp list_eq"
  unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def
  apply (auto intro: list_eq_def)
  apply (metis list_eq_def)
  apply (metis list_eq_def)
  sorry

quotient fset = "'a list" / "list_eq"
  by (rule list_eq_equivp)

lemma not_nil_equiv_cons: "\<not>[] \<approx> a # A" sorry

(* The 2 lemmas below are different from the ones in QuotList *)
lemma nil_rsp[quot_respect]:
  shows "[] \<approx> []"
by (rule list_eq_reflp)

lemma cons_rsp[quot_respect]: (* Better then the one from QuotList *)
  " (op = ===> op \<approx> ===> op \<approx>) op # op #"
  sorry

lemma mem_rsp[quot_respect]:
  "(op = ===> op \<approx> ===> op =) (op mem) (op mem)"
  sorry

lemma no_mem_nil: "(\<forall>a. \<not>(a mem A)) = (A = [])"
sorry

lemma none_mem_nil: "(\<forall>a. \<not>(a mem A)) = (A \<approx> [])"
sorry

lemma mem_cons: "a mem A \<Longrightarrow> a # A \<approx> A"
sorry

lemma cons_left_comm: "x # y # A \<approx> y # x # A"
sorry

lemma cons_left_idem: "x # x # A \<approx> x # A"
sorry

lemma finite_set_raw_strong_cases:
  "(X = []) \<or> (\<exists>a. \<exists> Y. (~(a mem Y) \<and> (X \<approx> a # Y)))"
  apply (induct X)
  apply (simp)
  sorry

primrec
  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 mem (delete_raw A a) = x mem A \<and> \<not>(x = a)"
sorry

lemma mem_delete_raw_ident:
  "\<not>(MEM a (A delete_raw a))"
sorry

lemma not_mem_delete_raw_ident:
  "\<not>(b mem A) \<Longrightarrow> (delete_raw A b = A)"
sorry

lemma delete_raw_RSP:
  "A \<approx> B \<Longrightarrow> delete_raw A a \<approx> delete_raw B a"
sorry

lemma cons_delete_raw:
  "a # (delete_raw A a) \<approx> (if a mem A then A else (a # A))"
sorry

lemma mem_cons_delete_raw:
    "a mem A \<Longrightarrow> a # (delete_raw A a) \<approx> A"
sorry

lemma finite_set_raw_delete_raw_cases1:
    "X = [] \<or> (\<exists>a. X \<approx> a # delete_raw X a)"
sorry

lemma finite_set_raw_delete_raw_cases:
    "X = [] \<or> (\<exists>a. a mem X \<and> X \<approx> a # delete_raw X a)"
sorry

fun
  card_raw :: "'a list \<Rightarrow> nat"
where
  card_raw_nil: "card_raw [] = 0"
| card_raw_cons: "card_raw (x # xs) = (if x mem xs then card_raw xs else Suc (card_raw xs))"

lemma not_mem_card_raw:
  fixes x :: "'a"
  fixes xs :: "'a list"
  shows "(\<not>(x mem xs)) = (card_raw (x # xs) = Suc (card_raw xs))"
  sorry

lemma card_raw_suc:
  fixes xs :: "'a list"
  fixes n :: "nat"
  assumes c: "card_raw xs = Suc n"
  shows "\<exists>a ys. \<not>(a mem ys) \<and> xs \<approx> (a # ys)"
  using c
apply(induct xs)
apply(metis mem_delete_raw)
apply(metis mem_delete_raw)
done

lemma mem_card_raw_not_0:
  "a mem A \<Longrightarrow> \<not>(card_raw A = 0)"
sorry

lemma card_raw_cons_gt_0:
  "0 < card_raw (a # A)"
sorry

lemma card_raw_delete_raw:
  "card_raw (delete_raw A a) = (if a mem A then card_raw A - 1 else card_raw A)"
sorry

lemma card_raw_rsp_aux:
  assumes e: "a \<approx> b"
  shows "card_raw a = card_raw b"
  using e sorry

lemma card_raw_rsp[quot_respect]:
  "(op \<approx> ===> op =) card_raw card_raw"
  by (simp add: card_raw_rsp_aux)

lemma card_raw_0:
  "(card_raw A = 0) = (A = [])"
sorry

lemma list2set_thm:
  shows "set [] = {}"
  and "set (h # t) = insert h (set t)"
sorry

lemma list2set_RSP:
  "A \<approx> B \<Longrightarrow> set A = set B"
sorry

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)
  apply (case_tac "rsp_fold x")
sorry

lemma append_rsp[quot_respect]:
  "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @"
  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 *)

quotient_def
  "Empty :: 'a fset" as "[]::'a list"

quotient_def
  "Insert :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" as "op #"

quotient_def
  "In :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool" as "op mem"

quotient_def
  "Card :: 'a fset \<Rightarrow> nat" as "card_raw"

quotient_def
  "Delete :: 'a fset \<Rightarrow> 'a \<Rightarrow> 'a fset" as "delete_raw"

quotient_def
  "funion :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" as "op @"

quotient_def
  "finter :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" as "inter_raw"

quotient_def
  "ffold :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" as "fold_raw"

quotient_def
  "fset_to_set :: 'a fset \<Rightarrow> 'a set" as "set"


(* LIFTING THMS *)

thm list.cases (* ??? *)

thm cons_left_comm
lemma "Insert a (Insert b x) = Insert b (Insert a x)"
by (lifting cons_left_comm)

thm cons_left_idem
lemma "Insert a (Insert a x) = Insert a x"
by (lifting cons_left_idem)

(* thm MEM:
  MEM x [] = F
  MEM x (h::t) = (x=h) \/ MEM x t *)
thm none_mem_nil
thm mem_cons
thm finite_set_raw_strong_cases
thm card_raw.simps
thm not_mem_card_raw
thm card_raw_suc
lemma "Card x = Suc n \<Longrightarrow> (\<exists>a b. \<not> In a b & x = Insert a b)"
by (lifting card_raw_suc)

thm card_raw_cons_gt_0
thm mem_card_raw_not_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 Empty; \<And>a x. P x \<Longrightarrow> P (Insert 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_def
  "fset_case :: 'a \<Rightarrow> ('b \<Rightarrow> 'b fset \<Rightarrow> 'a) \<Rightarrow> 'b fset \<Rightarrow> 'a"
as
  "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 (Insert a xa) = f2 a xa"
apply (lifting list.cases(2))
done


end