(*  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)"
  and   "\<not> (x # xs \<approx> [])"
  by auto

lemma not_memb_nil:
  shows "\<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 {* Unions *}

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:
  shows "fcard_raw xs = 0 \<longleftrightarrow> 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:
  shows "\<not> memb x xs \<longleftrightarrow> fcard_raw (x # xs) = Suc (fcard_raw xs)"
  by auto

lemma fcard_raw_suc:
  assumes a: "fcard_raw xs = Suc n"
  shows "\<exists>x ys. \<not> (memb x ys) \<and> xs \<approx> (x # ys) \<and> fcard_raw ys = n"
  using a
  by (induct xs) (auto simp add: memb_def split: if_splits)

lemma singleton_fcard_1: 
  shows "set xs = {x} \<Longrightarrow> fcard_raw xs = 1"
  by (induct xs) (auto simp add: memb_def subset_insert)

lemma fcard_raw_1:
  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 (drule singleton_fcard_1)
  apply auto
  done

lemma fcard_raw_delete_one:
  shows "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]:
  shows "(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

section {* deletion *}

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 delete_raw_rsp:
  "xs \<approx> ys \<Longrightarrow> delete_raw xs x \<approx> delete_raw ys x"
  by (simp add: memb_def[symmetric] memb_delete_raw)

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:
  shows "\<not> memb x (delete_raw xs x)"
  by (induct xs) (auto simp add: memb_def)

lemma not_memb_delete_raw_ident:
  shows "\<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)

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 *}

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)

lemma fcard_0: "(fcard S = 0) = (S = {||})"
  by (lifting fcard_raw_0)

lemma fcard_1:
  shows "(fcard S = 1) = (\<exists>x. S = {|x|})"
  by (lifting fcard_raw_1)

lemma fcard_gt_0: 
  shows "x \<in> fset_to_set S \<Longrightarrow> 0 < fcard S"
  by (lifting fcard_raw_gt_0)

lemma fcard_not_fin: 
  shows "(x |\<notin>| S) = (fcard (finsert x S) = Suc (fcard S))"
  by (lifting fcard_raw_not_memb)

lemma fcard_suc: "fcard S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = finsert x T \<and> fcard T = n"
  by (lifting fcard_raw_suc)

lemma fcard_delete:
  "fcard (fdelete S y) = (if y |\<in>| S then fcard S - 1 else fcard S)"
  by (lifting fcard_raw_delete)

text {* funion *}

lemma funion_simps[simp]:
  shows "{||} |\<union>| S = S"
  and   "finsert x S |\<union>| T = finsert x (S |\<union>| T)"
  by (lifting append.simps)

lemma funion_sym:
  shows "S |\<union>| T = T |\<union>| S"
  by (lifting funion_sym_pre)

lemma funion_assoc:
  shows "S |\<union>| T |\<union>| U = S |\<union>| (T |\<union>| U)"
  by (lifting append_assoc)

section {* Induction and Cases rules for finite sets *}

lemma fset_strong_cases:
  "S = {||} \<or> (\<exists>x T. x |\<notin>| T \<and> S = finsert x T)"
  by (lifting fset_raw_strong_cases)

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_weak[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_induct[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_weak)
  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

lemma fset_induct2:
  "P {||} {||} \<Longrightarrow>
  (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
  (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
  (\<And>x xs y ys. \<lbrakk>P xs ys; x |\<notin>| xs; y |\<notin>| ys\<rbrakk> \<Longrightarrow> P (finsert x xs) (finsert y ys)) \<Longrightarrow>
  P xsa ysa"
  apply (induct xsa arbitrary: ysa)
  apply (induct_tac x rule: fset_induct)
  apply simp_all
  apply (induct_tac xa rule: fset_induct)
  apply simp_all
  done

text {* fmap *}

lemma fmap_simps[simp]:
  "fmap (f :: 'a \<Rightarrow> 'b) {||} = {||}"
  "fmap f (finsert x S) = finsert (f x) (fmap f S)"
  by (lifting map.simps)

lemma fmap_set_image:
  "fset_to_set (fmap f S) = f ` (fset_to_set S)"
  by (induct S) (simp_all)

lemma inj_fmap_eq_iff:
  "inj f \<Longrightarrow> (fmap f S = fmap f T) = (S = T)"
  by (lifting inj_map_eq_iff)

lemma fmap_funion: "fmap f (S |\<union>| T) = fmap f S |\<union>| fmap f T"
  by (lifting map_append)

lemma fin_funion:
  "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"
  by (lifting memb_append)

text {* ffold *}

lemma ffold_nil: "ffold f z {||} = z"
  by (lifting ffold_raw.simps(1)[where 'a="'b" and 'b="'a"])

lemma ffold_finsert: "ffold f z (finsert a A) =
  (if rsp_fold f then if a |\<in>| A then ffold f z A else f a (ffold f z A) else z)"
  by (lifting ffold_raw.simps(2)[where 'a="'b" and 'b="'a"])

lemma fin_commute_ffold:
  "\<lbrakk>rsp_fold f; h |\<in>| b\<rbrakk> \<Longrightarrow> ffold f z b = f h (ffold f z (fdelete b h))"
  by (lifting memb_commute_ffold_raw)

text {* fdelete *}

lemma fin_fdelete: 
  shows "x |\<in>| fdelete S y \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"
  by (lifting memb_delete_raw)

lemma fin_fdelete_ident: 
  shows "x |\<notin>| fdelete S x"
  by (lifting memb_delete_raw_ident)

lemma not_memb_fdelete_ident: 
  shows "x |\<notin>| S \<Longrightarrow> fdelete S x = S"
  by (lifting not_memb_delete_raw_ident)

lemma fset_fdelete_cases:
  shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = finsert x (fdelete S x))"
  by (lifting fset_raw_delete_raw_cases)

text {* inter *}

lemma finter_empty_l: "({||} |\<inter>| S) = {||}"
  by (lifting finter_raw.simps(1))

lemma finter_empty_r: "(S |\<inter>| {||}) = {||}"
  by (lifting finter_raw_empty)

lemma finter_finsert:
  "finsert x S |\<inter>| T = (if x |\<in>| T then finsert x (S |\<inter>| T) else S |\<inter>| T)"
  by (lifting finter_raw.simps(2))

lemma fin_finter:
  "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"
  by (lifting memb_finter_raw)

lemma expand_fset_eq:
  "(S = T) = (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"
  by (lifting list_eq.simps[simplified memb_def[symmetric]])


ML {*
fun dest_fsetT (Type ("FSet.fset", [T])) = T
  | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
*}

no_notation
  list_eq (infix "\<approx>" 50)

end