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 a set -- @{const finsert} *}

lemma nil_not_cons:

  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)

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_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_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)

  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"

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 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 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[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

  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)"

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"

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)

section {* lifted part *}

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]:

  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 *}

  "fset_to_set {||} = {}"

  "fset_to_set (finsert (h :: 'b) t) = insert h (fset_to_set t)"

  by (lifting list2set_thm)

lemma in_fset_to_set:

  "x \<in> fset_to_set xs \<equiv> x |\<in>| xs"

  by (lifting memb_def[symmetric])

lemma none_in_fempty:

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

  by (lifting none_mem_nil)

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_gt_0: "x \<in> fset_to_set xs \<Longrightarrow> 0 < fcard xs"

  by (lifting fcard_raw_gt_0)

text {* funion *}

lemma funion_simps[simp]:

  "{||} |\<union>| ys = ys"

  "finsert x xs |\<union>| ys = finsert x (xs |\<union>| ys)"

  by (lifting append.simps)

lemma funion_sym:

  "a |\<union>| b = b |\<union>| a"

  by (lifting funion_sym_pre)

lemma funion_assoc:

 "x |\<union>| xa |\<union>| xb = x |\<union>| (xa |\<union>| xb)"

  by (lifting append_assoc)

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

lemma fset_strong_cases:

  "X = {||} \<or> (\<exists>a Y. a \<notin> fset_to_set Y \<and> X = finsert a Y)"

  by (lifting finite_set_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)

  shows "\<lbrakk>P {||}; \<And>x S. P S \<Longrightarrow> P (finsert x S)\<rbrakk> \<Longrightarrow> P S"

  by (lifting list.induct)

  assumes prem1: "P {||}" 

  and     prem2: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"

  shows "P S"

  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

end