(* 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
context complete_lattice
begin
(* FIXME: automated lifting fails, since @{term inf} and @{term top}
are variables (???) *)
lemma Infimum_inf [code]:
"Infimum (Cset.set As) = foldr inf As top"
"Infimum (coset []) = bot"
unfolding Infimum_def member_code List.member_def
apply (simp add: mem_def Inf_set_foldr)
apply (simp add: Inf_UNIV[unfolded UNIV_def Collect_def])
done
lemma Supremum_sup [code]:
"Supremum (Cset.set As) = foldr sup As bot"
"Supremum (coset []) = top"
unfolding Supremum_def member_code List.member_def
apply (simp add: mem_def Sup_set_foldr)
apply (simp add: Sup_UNIV[unfolded UNIV_def Collect_def])
done
end
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