theory FSet

imports Quotient Quotient_List List

begin

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

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 an fset -- @{const finsert} *}

lemma nil_not_cons:

  shows

  "\<not>[] \<approx> x # xs"

  "\<not>x # xs \<approx> []"

  by auto

lemma not_memb_nil:

  "\<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_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)

section {* Singletons *}

lemma singleton_list_eq:

  shows "[x] \<approx> [y] \<longleftrightarrow> x = y"

  by (simp add: id_simps) auto

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"

lemma fcard_raw_0:

  fixes a :: "'a list"

  shows "(fcard_raw a = 0) = (a \<approx> [])"

  apply (induct a)

  apply auto

  apply (drule memb_absorb)

  apply (auto simp add: nil_not_cons)

  done

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_not_memb:

  fixes x :: "'a"

  fixes xs :: "'a list"

  shows "(~(memb x xs)) = (fcard_raw (x # xs) = Suc (fcard_raw xs))"

  by auto

lemma fcard_raw_suc:

  fixes xs :: "'a list"

  fixes n :: "nat"

  assumes c: "fcard_raw xs = Suc n"

  shows "\<exists>a ys. ~(memb a ys) \<and> xs \<approx> (a # ys) \<and> fcard_raw ys = n"

  unfolding memb_def

  using c

  proof (induct xs)

    case Nil

    then show ?case by simp

  next

    case (Cons a xs)

    have f1: "fcard_raw xs = Suc n \<Longrightarrow> \<exists>a ys. a \<notin> set ys \<and> xs \<approx> a # ys \<and> fcard_raw ys = n" by fact

    have f2: "fcard_raw (a # xs) = Suc n" by fact

    then show ?case proof (cases "a \<in> set xs")

      case True

      then show ?thesis using f1 f2 apply -

        apply (simp add: memb_def)

        apply clarify

        by metis

      case False

        apply (rule_tac x="a" in exI)

        apply (rule_tac x="xs" in exI)

        apply (simp add: memb_def)

        done

    qed

  qed

lemma singleton_fcard_1: "set a = {b} \<Longrightarrow> fcard_raw a = Suc 0"

  apply (induct a)

  apply simp_all

  apply auto

  apply (subgoal_tac "set a2 = {b}")

  apply simp

  apply (simp add: memb_def)

  apply auto

  apply (subgoal_tac "set a2 = {}")

  apply simp

  by (metis memb_def subset_empty subset_insert)

lemma fcard_raw_1:

  fixes a :: "'a list"

  shows "(fcard_raw a = 1) = (\<exists>b. a \<approx> [b])"

  apply auto

  apply (drule fcard_raw_suc)

  apply clarify

  apply (simp add: fcard_raw_0)

  apply (rule_tac x="aa" in exI)

  apply simp

  apply (subgoal_tac "set a = {b}")

  apply (erule singleton_fcard_1)

  apply auto

  done

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_gt_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'[OF fcard_raw_gt_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)

section {* fmap and fset comprehension *}

quotient_definition

  "fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"

is

 "map"

lemma map_append:

  "(map f (a @ b)) \<approx> (map f a) @ (map f b)"

  by simp

lemma memb_append:

  "memb e (append l r) = (memb e l \<or> memb e r)"

  by (induct l) (simp_all add: not_memb_nil memb_cons_iff)

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)

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 none_mem_nil:

  "(\<forall>a. a \<notin> set A) = (A \<approx> [])"

  by simp

lemma fset_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 memb_delete_raw:

  "memb x (delete_raw A a) = (memb x A \<and> x \<noteq> a)"

  by (induct A arbitrary: x a) (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 memb_delete_raw_ident:

  "\<not> memb a (delete_raw A a)"

  by (induct A) (auto simp add: memb_def)

lemma not_memb_delete_raw_ident:

  "\<not> memb b A \<Longrightarrow> delete_raw A b = A"

  by (induct A) (auto simp add: memb_def)

lemma fset_raw_delete_raw_cases:

  "X = [] \<or> (\<exists>a. memb a X \<and> X \<approx> a # delete_raw X a)"

  by (induct X) (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 set_rsp[quot_respect]:

  "(op \<approx> ===> op =) set set"

  by auto

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)"

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 add: not_memb_nil)

  apply (simp add: ffold_raw.simps)

  apply (rule conjI)

  apply (rule_tac [!] impI)

  apply (rule_tac [!] conjI)

  apply (rule_tac [!] impI)

  apply (simp_all add: memb_delete_raw)

  apply (simp add: memb_cons_iff)

  apply (simp add: not_memb_delete_raw_ident)

  apply (simp add: memb_cons_iff rsp_fold_def)

  done

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)"

lemma finter_raw_empty:

  "finter_raw l [] = []"

  by (induct l) (simp_all add: not_memb_nil)

lemma memb_finter_raw:

  "memb e (finter_raw l r) = (memb e l \<and> memb e r)"

  apply (induct l)

  apply (simp add: not_memb_nil)

  apply (simp add: finter_raw.simps)

  apply (simp add: memb_cons_iff)

  apply auto

  done

lemma [quot_respect]:

  "(op \<approx> ===> op \<approx> ===> op \<approx>) finter_raw finter_raw"

  by (simp add: memb_def[symmetric] memb_finter_raw)

section {* Constants on the Quotient Type *} 

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

  finter (infix "|\<inter>|" 50)

where

  "finter :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"

is "finter_raw"

lemma funion_sym_pre:

  "a @ b \<approx> b @ a"

  by auto

lemma append_rsp[quot_respect]:

  shows "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @"

  by (auto)

lemma set_cong: "(set x = set y) = (x \<approx> y)"

  apply rule

  apply simp_all

  apply (induct x y rule: list_induct2')

  apply simp_all

  apply auto

  done

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)

quotient_definition

  "fconcat :: ('a fset) fset \<Rightarrow> 'a fset"

is

  "concat"

lemma list_equiv_rsp[quot_respect]:

  shows "(op \<approx> ===> op \<approx> ===> op =) op \<approx> op \<approx>"

  by auto

section {* lifted part *}

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 a (finsert b S) = finsert b (finsert a S)"

  by (lifting cons_left_comm)

lemma finsert_left_idem:

  "finsert a (finsert a S) = finsert a 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 xs \<equiv> x |\<in>| xs"

  by (lifting memb_def[symmetric])

lemma none_fin_fempty:

  "(\<forall>a. a \<notin> fset_to_set A) = (A = {||})"

  by (lifting none_mem_nil)

lemma fset_cong:

  "(fset_to_set x = fset_to_set y) = (x = y)"

  by (lifting set_cong)

text {* fcard *}

lemma fcard_fempty [simp]:

  shows "fcard {||} = 0"