(*  Title:      HOL/Quotient_Examples/FSet.thy

    Author:     Cezary Kaliszyk, TU Munich

    Author:     Christian Urban, TU Munich

    Type of finite sets.

*)

theory FSet

imports Quotient_List

begin

text {* Definiton of the list equivalence relation *}

fun

  list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)

where

  "list_eq xs ys = (set xs = set ys)"

lemma list_eq_equivp:

  shows "equivp list_eq"

  unfolding equivp_reflp_symp_transp

  unfolding reflp_def symp_def transp_def

  by auto

text {* Fset type *}

quotient_type

  'a fset = "'a list" / "list_eq"

  by (rule list_eq_equivp)

text {* 

  Definitions of membership, sublist, cardinality, intersection, 

  difference and respectful fold over lists

*}

definition

  "memb x xs \<equiv> x \<in> set xs"

definition

  "sub_list xs ys \<equiv> set xs \<subseteq> set ys"

definition

  "card_list xs = card (set xs)"

definition

  "inter_list xs ys = [x \<leftarrow> xs. x \<in> set xs \<and> x \<in> set ys]"

definition

  "diff_list xs ys \<equiv> [x \<leftarrow> xs. x\<notin>set ys]"

definition

  rsp_fold

where

  "rsp_fold f \<equiv> \<forall>u v w. (f u (f v w) = f v (f u w))"

primrec

where

     (if (rsp_fold f) then

     else z)"

section {* Quotient composition lemmas *}

lemma list_all2_refl1:

  shows "(list_all2 op \<approx>) r r"

  by (rule list_all2_refl) (metis equivp_def fset_equivp)

lemma compose_list_refl:

  shows "(list_all2 op \<approx> OOO op \<approx>) r r"

proof

  have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])

  show "list_all2 op \<approx> r r" by (rule list_all2_refl1)

  with * show "(op \<approx> OO list_all2 op \<approx>) r r" ..

qed

lemma Quotient_fset_list:

  shows "Quotient (list_all2 op \<approx>) (map abs_fset) (map rep_fset)"

  by (fact list_quotient[OF Quotient_fset])

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

  unfolding list_eq.simps

  by (simp only: set_map)

lemma quotient_compose_list[quot_thm]:

  shows  "Quotient ((list_all2 op \<approx>) OOO (op \<approx>))

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

  unfolding Quotient_def comp_def

proof (intro conjI allI)

  fix a r s

  show "abs_fset (map abs_fset (map rep_fset (rep_fset a))) = a"

    by (simp add: abs_o_rep[OF Quotient_fset] Quotient_abs_rep[OF Quotient_fset] map_id)

  have b: "list_all2 op \<approx> (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"

    by (rule list_all2_refl1)

  have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"

    by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)

  show "(list_all2 op \<approx> OOO op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"

    by (rule, rule list_all2_refl1) (rule c)

  show "(list_all2 op \<approx> OOO op \<approx>) r s = ((list_all2 op \<approx> OOO op \<approx>) r r \<and>

        (list_all2 op \<approx> OOO op \<approx>) s s \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s))"

  proof (intro iffI conjI)

    show "(list_all2 op \<approx> OOO op \<approx>) r r" by (rule compose_list_refl)

    show "(list_all2 op \<approx> OOO op \<approx>) s s" by (rule compose_list_refl)

  next

    assume a: "(list_all2 op \<approx> OOO op \<approx>) r s"

    then have b: "map abs_fset r \<approx> map abs_fset s"

    proof (elim pred_compE)

      fix b ba

      assume c: "list_all2 op \<approx> r b"

      assume d: "b \<approx> ba"

      assume e: "list_all2 op \<approx> ba s"

      have f: "map abs_fset r = map abs_fset b"

        using Quotient_rel[OF Quotient_fset_list] c by blast

      have "map abs_fset ba = map abs_fset s"

        using Quotient_rel[OF Quotient_fset_list] e by blast

      then have g: "map abs_fset s = map abs_fset ba" by simp

      then show "map abs_fset r \<approx> map abs_fset s" using d f map_rel_cong by simp

    qed

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

      using Quotient_rel[OF Quotient_fset] by blast

  next

    assume a: "(list_all2 op \<approx> OOO op \<approx>) r r \<and> (list_all2 op \<approx> OOO op \<approx>) s s

      \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"

    then have s: "(list_all2 op \<approx> OOO op \<approx>) s s" by simp

    have d: "map abs_fset r \<approx> map abs_fset s"

      by (subst Quotient_rel[OF Quotient_fset]) (simp add: a)

    have b: "map rep_fset (map abs_fset r) \<approx> map rep_fset (map abs_fset s)"

      by (rule map_rel_cong[OF d])

    have y: "list_all2 op \<approx> (map rep_fset (map abs_fset s)) s"

      by (fact rep_abs_rsp_left[OF Quotient_fset_list, OF list_all2_refl1[of s]])

    have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (map abs_fset r)) s"

      by (rule pred_compI) (rule b, rule y)

    have z: "list_all2 op \<approx> r (map rep_fset (map abs_fset r))"

      by (fact rep_abs_rsp[OF Quotient_fset_list, OF list_all2_refl1[of r]])

    then show "(list_all2 op \<approx> OOO op \<approx>) r s"

      using a c pred_compI by simp

  qed

qed

subsection {* Respectfulness lemmas for list operations *}

lemma list_equiv_rsp [quot_respect]:

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

  by auto

lemma append_rsp [quot_respect]:

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

  by simp

lemma sub_list_rsp [quot_respect]:

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

  by (auto simp add: sub_list_def)

lemma memb_rsp [quot_respect]:

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

  by (auto simp add: memb_def)

lemma nil_rsp [quot_respect]:

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

  by simp

lemma cons_rsp [quot_respect]:

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

  by simp

lemma map_rsp [quot_respect]:

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

  by auto

lemma set_rsp [quot_respect]:

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

  by auto

lemma inter_list_rsp [quot_respect]:

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

  by (simp add: inter_list_def)

lemma removeAll_rsp [quot_respect]:

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

  by simp

lemma diff_list_rsp [quot_respect]:

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

  by (simp add: diff_list_def)

lemma card_list_rsp [quot_respect]:

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

  by (simp add: card_list_def)

lemma filter_rsp [quot_respect]:

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

  by simp

  apply (induct b)

  apply (auto simp add: rsp_fold_def)

  done

  apply (induct a arbitrary: b)

  apply (simp)

  apply (simp (no_asm_use))

  apply (rule conjI)

  apply (rule_tac [!] impI)

  apply (rule_tac [!] conjI)

  apply (rule_tac [!] impI)

  apply (metis insert_absorb)

  apply(drule_tac x="removeAll a1 b" in meta_spec)

  apply(auto)

  apply(drule meta_mp)

  apply(blast)

  unfolding fun_rel_def

lemma concat_rsp_pre:

  assumes a: "list_all2 op \<approx> x x'"

  and     b: "x' \<approx> y'"

  and     c: "list_all2 op \<approx> y' y"

  and     d: "\<exists>x\<in>set x. xa \<in> set x"

  shows "\<exists>x\<in>set y. xa \<in> set x"

proof -

  obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto

  have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a])

  then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto

  have "ya \<in> set y'" using b h by simp

  then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_all2_find_element)

  then show ?thesis using f i by auto

qed

lemma concat_rsp [quot_respect]:

  shows "(list_all2 op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"

proof (rule fun_relI, elim pred_compE)

  fix a b ba bb

  assume a: "list_all2 op \<approx> a ba"

  assume b: "ba \<approx> bb"

  assume c: "list_all2 op \<approx> bb b"

  have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" 

  proof

    fix x

    show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)" 

    proof

      assume d: "\<exists>xa\<in>set a. x \<in> set xa"

      show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])

    next

      assume e: "\<exists>xa\<in>set b. x \<in> set xa"

      have a': "list_all2 op \<approx> ba a" by (rule list_all2_symp[OF list_eq_equivp, OF a])

      have b': "bb \<approx> ba" by (rule equivp_symp[OF list_eq_equivp, OF b])

      have c': "list_all2 op \<approx> b bb" by (rule list_all2_symp[OF list_eq_equivp, OF c])

      show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])

    qed

  qed

  then show "concat a \<approx> concat b" by auto

qed

subsection {* Finite sets are a bounded, distributive lattice with minus *}

instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"

begin

quotient_definition

  "bot :: 'a fset" 

  is "Nil :: 'a list"

abbreviation

  fempty  ("{||}")

where

  "{||} \<equiv> bot :: 'a fset"

quotient_definition

  "less_eq_fset :: ('a fset \<Rightarrow> 'a fset \<Rightarrow> bool)"

  is "sub_list :: ('a list \<Rightarrow> 'a list \<Rightarrow> bool)"

abbreviation

  f_subset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)

where

  "xs |\<subseteq>| ys \<equiv> xs \<le> ys"

definition

  less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool"

where  

  "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"

abbreviation

  fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)

where

  "xs |\<subset>| ys \<equiv> xs < ys"

quotient_definition

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

  is "append :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"

abbreviation

  funion (infixl "|\<union>|" 65)

where

  "xs |\<union>| ys \<equiv> sup xs (ys::'a fset)"

quotient_definition

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

  is "inter_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"

abbreviation

  finter (infixl "|\<inter>|" 65)

where

  "xs |\<inter>| ys \<equiv> inf xs (ys::'a fset)"

quotient_definition

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

  is "diff_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"

instance

proof

  fix x y z :: "'a fset"

  show "x |\<subset>| y \<longleftrightarrow> x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x"

    unfolding less_fset_def 

    by (descending) (auto simp add: sub_list_def)

  show "x |\<subseteq>| x"  by (descending) (simp add: sub_list_def)

  show "{||} |\<subseteq>| x" by (descending) (simp add: sub_list_def)

  show "x |\<subseteq>| x |\<union>| y" by (descending) (simp add: sub_list_def)

  show "y |\<subseteq>| x |\<union>| y" by (descending) (simp add: sub_list_def)

  show "x |\<inter>| y |\<subseteq>| x"

    by (descending) (auto simp add: inter_list_def sub_list_def memb_def)

  show "x |\<inter>| y |\<subseteq>| y" 

    by (descending) (auto simp add: inter_list_def sub_list_def memb_def)

  show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)" 

    by (descending) (auto simp add: inter_list_def)

next

  fix x y z :: "'a fset"

  assume a: "x |\<subseteq>| y"

  assume b: "y |\<subseteq>| z"

  show "x |\<subseteq>| z" using a b 

    by (descending) (simp add: sub_list_def)

next

  fix x y :: "'a fset"

  assume a: "x |\<subseteq>| y"

  assume b: "y |\<subseteq>| x"

  show "x = y" using a b 

    by (descending) (unfold sub_list_def list_eq.simps, blast)

next

  fix x y z :: "'a fset"

  assume a: "y |\<subseteq>| x"

  assume b: "z |\<subseteq>| x"

  show "y |\<union>| z |\<subseteq>| x" using a b 

    by (descending) (simp add: sub_list_def)

next

  fix x y z :: "'a fset"

  assume a: "x |\<subseteq>| y"

  assume b: "x |\<subseteq>| z"

  show "x |\<subseteq>| y |\<inter>| z" using a b 

    by (descending) (auto simp add: inter_list_def sub_list_def memb_def)

qed

end

quotient_definition

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

  is "Cons"

syntax

  "@Finset"     :: "args => 'a fset"  ("{|(_)|}")

translations

  "{|x, xs|}" == "CONST finsert x {|xs|}"

  "{|x|}"     == "CONST finsert x {||}"

quotient_definition

  fin (infix "|\<in>|" 50)

where

  "fin :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool" is "memb"

abbreviation

  fnotin :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)

where

  "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"

section {* Other constants on the Quotient Type *}

quotient_definition

  "fcard :: 'a fset \<Rightarrow> nat"

  is card_list

quotient_definition

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

  is map

quotient_definition

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

  is removeAll

quotient_definition

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

  is "set"

quotient_definition

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

quotient_definition

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

  is concat

quotient_definition

  "ffilter :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"

  is filter

  by (fact compose_list_refl)

  shows "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) Cons Cons"

  apply auto

  apply (rule_tac b="x # b" in pred_compI)

  apply auto

  apply (rule_tac b="x # ba" in pred_compI)

  apply auto

  done

  "(rep_fset ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op # = finsert"

  by (simp add: fun_eq_iff Quotient_abs_rep[OF Quotient_fset]

      abs_o_rep[OF Quotient_fset] map_id finsert_def)

  "((map rep_fset \<circ> rep_fset) ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op @ = funion"

  by (simp add: fun_eq_iff Quotient_abs_rep[OF Quotient_fset]

      abs_o_rep[OF Quotient_fset] map_id sup_fset_def)

lemma list_all2_app_l:

  assumes a: "reflp R"

  and b: "list_all2 R l r"

  shows "list_all2 R (z @ l) (z @ r)"

  by (induct z) (simp_all add: b rev_iffD1[OF a meta_eq_to_obj_eq[OF reflp_def]])

lemma append_rsp2_pre0:

  assumes a:"list_all2 op \<approx> x x'"

  shows "list_all2 op \<approx> (x @ z) (x' @ z)"

  using a apply (induct x x' rule: list_induct2')

  by simp_all (rule list_all2_refl1)

lemma append_rsp2_pre1:

  assumes a:"list_all2 op \<approx> x x'"

  shows "list_all2 op \<approx> (z @ x) (z @ x')"

  using a apply (induct x x' arbitrary: z rule: list_induct2')

  apply (rule list_all2_refl1)

  apply (simp_all del: list_eq.simps)

  apply (rule list_all2_app_l)

  apply (simp_all add: reflp_def)

  done

lemma append_rsp2_pre:

  assumes a:"list_all2 op \<approx> x x'"

  and     b: "list_all2 op \<approx> z z'"

  shows "list_all2 op \<approx> (x @ z) (x' @ z')"

  apply (rule list_all2_transp[OF fset_equivp])