theory FSet3

imports "../Quotient" "../Quotient_List" List

begin

ML {*

structure QuotientRules = Named_Thms

  (val name = "quot_thm"

   val description = "Quotient theorems.")

*}

ML {*

open QuotientRules

*}

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

(* FIXME-TODO: because of beta-reduction, one cannot give the *)

(* FIXME-TODO: relation as a term or abbreviation             *) 

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} *}

text {* raw section *}

lemma nil_not_cons:

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

  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)

text {* lifted section *}

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)

section {* Singletons *}

text {* raw section *}

lemma singleton_list_eq:

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

  by (simp add: id_simps) auto

text {* lifted section *}

lemma fempty_not_finsert[simp]:

  shows "{||} \<noteq> finsert x S"

  by (lifting nil_not_cons)

lemma fsingleton_eq[simp]:

  shows "{|x|} = {|y|} \<longleftrightarrow> x = y"

  by (lifting singleton_list_eq)

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"

text {* raw section *}

lemma fcard_raw_ge_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_ge_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' fcard_raw_ge_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)

text {* lifted section *}

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)

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

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[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_induct2[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)

  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

section {* fmap and fset comprehension *}

quotient_definition

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

is

 "map"

quotient_definition

  "fconcat :: ('a fset) fset => 'a fset"

is

 "concat"

(*lemma fconcat_rsp[quot_respect]:

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

apply(auto)

sorry

*)

(* PROBLEM: these lemmas needs to be restated, since  *)

(* concat.simps(1) and concat.simps(2) contain the    *)

(* type variables ?'a1.0 (which are turned into frees *)

(* 'a_1                                               *)

lemma concat1:

  shows "concat [] \<approx> []"

by (simp add: id_simps)

lemma concat2:

  shows "concat (x # xs) \<approx> x @ concat xs"

by (simp add: id_simps)

lemma concat_rsp[quot_respect]:

  shows "(list_rel op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"

sorry

lemma nil_rsp2[quot_respect]: "(list_rel op \<approx> OOO op \<approx>) [] []"

  apply (metis FSet3.nil_rsp list_rel.simps(1) pred_comp.intros)

  done

lemma set_in_eq: "(\<forall>e. ((e \<in> A) \<longleftrightarrow> (e \<in> B))) \<equiv> A = B"

  apply (rule eq_reflection)

  apply auto

  done

lemma map_rel_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"

  unfolding list_eq.simps

  apply(simp only: set_map set_in_eq)

  done

lemma quotient_compose_list_pre:

  "(list_rel op \<approx> OOO op \<approx>) r s =

  ((list_rel op \<approx> OOO op \<approx>) r r \<and> (list_rel op \<approx> OOO op \<approx>) s s \<and>

  abs_fset (map abs_fset r) = abs_fset (map abs_fset s))"

  apply rule

  apply rule

  apply rule

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply rule

  apply (rule equivp_reflp[OF fset_equivp])

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply(rule)

  apply rule

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply rule

  apply (rule equivp_reflp[OF fset_equivp])

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply (subgoal_tac "map abs_fset r \<approx> map abs_fset s")

  apply (metis Quotient_rel[OF Quotient_fset])

  apply (auto simp only:)[1]

  apply (subgoal_tac "map abs_fset r = map abs_fset b")

  prefer 2

  apply (metis Quotient_rel[OF list_quotient[OF Quotient_fset]])

  apply (subgoal_tac "map abs_fset s = map abs_fset ba")

  prefer 2

  apply (metis Quotient_rel[OF list_quotient[OF Quotient_fset]])

  apply (simp only: map_rel_cong)

  apply rule

  apply (rule rep_abs_rsp[of "list_rel op \<approx>" "map abs_fset"])

  apply (tactic {* Quotient_Tacs.quotient_tac @{context} 1 *})

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply rule

  prefer 2

  apply (rule rep_abs_rsp_left[of "list_rel op \<approx>" "map abs_fset"])

  apply (tactic {* Quotient_Tacs.quotient_tac @{context} 1 *})

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply (erule conjE)+

  apply (subgoal_tac "map abs_fset r \<approx> map abs_fset s")

  prefer 2

  apply (metis Quotient_def Quotient_fset equivp_reflp fset_equivp)

  apply (rule map_rel_cong)

  apply (assumption)

  done

lemma quotient_compose_list[quot_thm]:

  shows  "Quotient ((list_rel op \<approx>) OOO (op \<approx>))

    (abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"

  unfolding Quotient_def comp_def

  apply (rule)+

  apply (simp add: abs_o_rep[OF Quotient_fset] id_simps Quotient_abs_rep[OF Quotient_fset])

  apply (rule)

  apply (rule)

  apply (rule)

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply (rule)

  apply (rule equivp_reflp[OF fset_equivp])

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply rule

  apply rule

  apply(rule quotient_compose_list_pre)

  done

lemma fconcat_empty:

  shows "fconcat {||} = {||}"

apply(lifting concat1)

apply(cleaning)

apply(simp add: comp_def)

apply(fold fempty_def[simplified id_simps])

apply(rule refl)

done

(* Should be true *)

lemma insert_rsp2[quot_respect]:

  "(op \<approx> ===> list_rel op \<approx> OOO op \<approx> ===> list_rel op \<approx> OOO op \<approx>) op # op #"

apply auto

apply (simp add: set_in_eq)

sorry

lemma append_rsp[quot_respect]:

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

  by (auto)

lemma fconcat_insert:

  shows "fconcat (finsert x S) = x |\<union>| fconcat S"

apply(lifting concat2)

apply(cleaning)

apply (simp add: finsert_def fconcat_def comp_def)

apply cleaning

done

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)