(*  Title:      HOL/Quotient_Examples/List_Cset.thy

     Author:     Florian Haftmann, Alexander Krauss, TU Muenchen

 *)

 header {* Implementation of type Cset.set based on lists. Code equations obtained via quotient lifting. *}

 theory List_Cset

 imports Cset

 begin

 lemma [quot_respect]: "((op = ===> set_eq ===> set_eq) ===> op = ===> set_eq ===> set_eq)

   foldr foldr"

 by (simp add: fun_rel_eq)

 lemma [quot_preserve]: "((id ---> abs_set ---> rep_set) ---> id ---> rep_set ---> abs_set) foldr = foldr"

 apply (rule ext)+

 by (induct_tac xa) (auto simp: Quotient_abs_rep[OF Quotient_set])

 subsection {* Relationship to lists *}

 (*FIXME: maybe define on sets first and then lift -> more canonical*)

 definition coset :: "'a list \<Rightarrow> 'a Cset.set" where

   "coset xs = Cset.uminus (Cset.set xs)"

 code_datatype Cset.set List_Cset.coset

 lemma member_code [code]:

   "member x (Cset.set xs) \<longleftrightarrow> List.member xs x"

   "member x (coset xs) \<longleftrightarrow> \<not> List.member xs x"

 unfolding coset_def

 apply (lifting in_set_member)

 by descending (simp add: in_set_member)

 definition (in term_syntax)

   setify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)

     \<Rightarrow> 'a Cset.set \<times> (unit \<Rightarrow> Code_Evaluation.term)" where

   [code_unfold]: "setify xs = Code_Evaluation.valtermify Cset.set {\<cdot>} xs"

 notation fcomp (infixl "\<circ>>" 60)

 notation scomp (infixl "\<circ>\<rightarrow>" 60)

 instantiation Cset.set :: (random) random

 begin

 definition

   "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (setify xs))"

 instance ..

 end

 no_notation fcomp (infixl "\<circ>>" 60)

 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)

 subsection {* Basic operations *}

 lemma is_empty_set [code]:

   "Cset.is_empty (Cset.set xs) \<longleftrightarrow> List.null xs"

   by (lifting is_empty_set)

 hide_fact (open) is_empty_set

 lemma empty_set [code]:

   "Cset.empty = Cset.set []"

   by (lifting set.simps(1)[symmetric])

 hide_fact (open) empty_set

 lemma UNIV_set [code]:

   "Cset.UNIV = coset []"

   unfolding coset_def by descending simp

 hide_fact (open) UNIV_set

 lemma remove_set [code]:

   "Cset.remove x (Cset.set xs) = Cset.set (removeAll x xs)"

   "Cset.remove x (coset xs) = coset (List.insert x xs)"

 unfolding coset_def

 apply descending

 apply (simp add: More_Set.remove_def)

 apply descending

 by (simp add: remove_set_compl)

 lemma insert_set [code]:

   "Cset.insert x (Cset.set xs) = Cset.set (List.insert x xs)"

   "Cset.insert x (coset xs) = coset (removeAll x xs)"

 unfolding coset_def

 apply (lifting set_insert[symmetric])

 by descending simp

 lemma map_set [code]:

   "Cset.map f (Cset.set xs) = Cset.set (remdups (List.map f xs))"

 by descending simp

 lemma filter_set [code]:

   "Cset.filter P (Cset.set xs) = Cset.set (List.filter P xs)"

 by descending (simp add: project_set)

 lemma forall_set [code]:

   "Cset.forall (Cset.set xs) P \<longleftrightarrow> list_all P xs"

 (* FIXME: why does (lifting Ball_set_list_all) fail? *)

 by descending (fact Ball_set_list_all)

 lemma exists_set [code]:

   "Cset.exists (Cset.set xs) P \<longleftrightarrow> list_ex P xs"

 by descending (fact Bex_set_list_ex)

 lemma card_set [code]:

   "Cset.card (Cset.set xs) = length (remdups xs)"

 by (lifting length_remdups_card_conv[symmetric])

 lemma compl_set [simp, code]:

   "Cset.uminus (Cset.set xs) = coset xs"

 unfolding coset_def by descending simp

 lemma compl_coset [simp, code]:

   "Cset.uminus (coset xs) = Cset.set xs"

 unfolding coset_def by descending simp

 lemma Inf_inf [code]:

   "Cset.Inf (Cset.set (xs\<Colon>'a\<Colon>complete_lattice list)) = foldr inf xs top"

   "Cset.Inf (coset ([]\<Colon>'a\<Colon>complete_lattice list)) = bot"

   unfolding List_Cset.UNIV_set[symmetric]

   by (lifting Inf_set_foldr Inf_UNIV)

 lemma Sup_sup [code]:

   "Cset.Sup (Cset.set (xs\<Colon>'a\<Colon>complete_lattice list)) = foldr sup xs bot"

   "Cset.Sup (coset ([]\<Colon>'a\<Colon>complete_lattice list)) = top"

   unfolding List_Cset.UNIV_set[symmetric]

   by (lifting Sup_set_foldr Sup_UNIV)

 subsection {* Derived operations *}

 lemma subset_eq_forall [code]:

   "Cset.subset A B \<longleftrightarrow> Cset.forall A (\<lambda>x. member x B)"

 by descending blast

 lemma subset_subset_eq [code]:

   "Cset.psubset A B \<longleftrightarrow> Cset.subset A B \<and> \<not> Cset.subset B A"

 by descending blast

 instantiation Cset.set :: (type) equal

 begin

 definition [code]:

   "HOL.equal A B \<longleftrightarrow> Cset.subset A B \<and> Cset.subset B A"

 instance

 apply intro_classes

 unfolding equal_set_def

 by descending auto

 end

 lemma [code nbe]:

   "HOL.equal (A :: 'a Cset.set) A \<longleftrightarrow> True"

   by (fact equal_refl)

 subsection {* Functorial operations *}

 lemma inter_project [code]:

   "Cset.inter A (Cset.set xs) = Cset.set (List.filter (\<lambda>x. Cset.member x A) xs)"

   "Cset.inter A (coset xs) = foldr Cset.remove xs A"

 apply descending

 apply auto

 unfolding coset_def

 apply descending

 apply simp

 by (metis diff_eq minus_set_foldr)

 lemma subtract_remove [code]:

   "Cset.minus A (Cset.set xs) = foldr Cset.remove xs A"

   "Cset.minus A (coset xs) = Cset.set (List.filter (\<lambda>x. member x A) xs)"

 unfolding coset_def

 apply (lifting minus_set_foldr)

 by descending auto

 lemma union_insert [code]:

   "Cset.union (Cset.set xs) A = foldr Cset.insert xs A"

   "Cset.union (coset xs) A = coset (List.filter (\<lambda>x. \<not> member x A) xs)"

 unfolding coset_def

 apply (lifting union_set_foldr)

 by descending auto

 lemma UNION_code [code]:

   "Cset.UNION (Cset.set []) f = Cset.set []"

   "Cset.UNION (Cset.set (x#xs)) f =

      Cset.union (f x) (Cset.UNION (Cset.set xs) f)"

   by (descending, simp)+

 end