(*  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

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)

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

  "x |\<notin>| S \<equiv> \<not>(x |\<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 xs :: "'a list"

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

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

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"

  shows "\<not>(memb x xs) \<longleftrightarrow> fcard_raw (x # xs) = Suc (fcard_raw xs)"

  by auto

lemma fcard_raw_suc:

  fixes xs :: "'a list"

  assumes c: "fcard_raw xs = Suc n"

  shows "\<exists>x ys. \<not>(memb x ys) \<and> xs \<approx> (x # 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

    next

      case False

      then show ?thesis using f1 f2 apply -

        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: 

  shows "set xs = {x} \<Longrightarrow> fcard_raw xs = Suc 0"

  apply (induct xs)

  apply simp_all

  apply auto

  apply (subgoal_tac "set xs = {x}")

  apply simp

  apply (simp add: memb_def)

  apply auto

  apply (subgoal_tac "set xs = {}")

  apply simp

  by (metis memb_def subset_empty subset_insert)

lemma fcard_raw_1:

  fixes a :: "'a list"

  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 (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: "xs \<approx> ys"

  shows "fcard_raw xs = fcard_raw ys"

  using a

  apply(induct xs arbitrary: ys)

  apply(auto simp add: memb_def)

  apply(metis)

  apply(drule_tac x="[x \<leftarrow> ys. x \<noteq> a]" 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 (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)

text {* raw section *}

lemma map_rsp[quot_respect]:

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

  by auto

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 none_memb_nil:

  "(\<forall>x. \<not> memb x xs) = (xs \<approx> [])"

  by (simp add: memb_def)

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

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 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 memb_delete_raw_ident:

  "\<not> memb x (delete_raw xs x)"

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

lemma not_memb_delete_raw_ident:

  "\<not> memb x xs \<Longrightarrow> delete_raw xs x = xs"

  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 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

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: hd_in_set memb_absorb memb_def none_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 add: in_set_code memb_cons_iff memb_def)

  apply (metis)

  apply (metis Nitpick.list_size_simp(2) ffold_raw.simps(2) 

    length_Suc_conv memb_absorb memb_cons_iff nil_not_cons(2))

  defer

  apply (metis Nitpick.list_size_simp(2) ffold_raw.simps(2) 

    length_Suc_conv memb_absorb memb_cons_iff nil_not_cons(2))

  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 (metis memb_cons_iff)

  apply (erule memb_commute_ffold_raw)

  apply (drule_tac x="a1" in spec)

  apply (simp add: memb_cons_iff)

  done

lemma [quot_respect]:

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

  by (simp add: memb_def[symmetric] ffold_raw_rsp_pre)

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 x (finter_raw xs ys) \<longleftrightarrow> memb x xs \<and> memb x ys"

  apply (induct xs)

  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:

  "xs @ ys \<approx> ys @ xs"

  by auto

lemma append_rsp[quot_respect]:

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

  by auto

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)

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 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)

thm memb_def[symmetric, THEN meta_eq_to_obj_eq]

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 *}