(* Author: Florian Haftmann, TU Muenchen *)
header {* implementation of Cset.sets based on lists *}
theory List_Cset
imports Cset
begin
code_datatype Cset.set Cset.coset
lemma member_code [code]:
"member (Cset.set xs) = List.member xs"
"member (Cset.coset xs) = Not \<circ> List.member xs"
by (simp_all add: fun_eq_iff List.member_def)
definition (in term_syntax)
csetify :: "'a\<Colon>typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
\<Rightarrow> 'a Cset.set \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
[code_unfold]: "csetify 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 (csetify 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 (simp add: is_empty_set null_def)
hide_fact (open) is_empty_set
lemma empty_set [code]:
"Cset.empty = Cset.set []"
by simp
hide_fact (open) empty_set
lemma UNIV_set [code]:
"top = Cset.coset []"
by (simp add: Cset.coset_def)
hide_fact (open) UNIV_set
lemma remove_set [code]:
"Cset.remove x (Cset.set xs) = Cset.set (removeAll x xs)"
"Cset.remove x (Cset.coset xs) = Cset.coset (List.insert x xs)"
by (simp_all add: Cset.set_def Cset.coset_def Compl_insert)
lemma insert_set [code]:
"Cset.insert x (Cset.set xs) = Cset.set (List.insert x xs)"
"Cset.insert x (Cset.coset xs) = Cset.coset (removeAll x xs)"
by (simp_all add: Cset.set_def Cset.coset_def)
declare compl_set [code]
declare compl_coset [code]
declare subtract_remove [code]
lemma map_set [code]:
"Cset.map f (Cset.set xs) = Cset.set (remdups (List.map f xs))"
by (simp add: Cset.set_def)
lemma filter_set [code]:
"Cset.filter P (Cset.set xs) = Cset.set (List.filter P xs)"
by (simp add: Cset.set_def)
lemma forall_set [code]:
"Cset.forall P (Cset.set xs) \<longleftrightarrow> list_all P xs"
by (simp add: Cset.set_def list_all_iff)
lemma exists_set [code]:
"Cset.exists P (Cset.set xs) \<longleftrightarrow> list_ex P xs"
by (simp add: Cset.set_def list_ex_iff)
lemma card_set [code]:
"Cset.card (Cset.set xs) = length (remdups xs)"
proof -
have "Finite_Set.card (set (remdups xs)) = length (remdups xs)"
by (rule distinct_card) simp
then show ?thesis by (simp add: Cset.set_def)
qed
context complete_lattice
begin
declare Infimum_inf [code]
declare Supremum_sup [code]
end
declare Cset.single_code [code del]
lemma single_set [code]:
"Cset.single a = Cset.set [a]"
by(simp add: Cset.single_code)
hide_fact (open) single_set
declare Cset.bind_set [code]
declare Cset.pred_of_cset_set [code]
subsection {* Derived operations *}
lemma subset_eq_forall [code]:
"A \<le> B \<longleftrightarrow> Cset.forall (member B) A"
by (simp add: subset_eq member_def)
lemma subset_subset_eq [code]:
"A < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> (A :: 'a Cset.set)"
by (fact less_le_not_le)
instantiation Cset.set :: (type) equal
begin
definition [code]:
"HOL.equal A B \<longleftrightarrow> A \<le> B \<and> B \<le> (A :: 'a Cset.set)"
instance proof
qed (auto simp add: equal_set_def Cset.set_eq_iff Cset.member_def fun_eq_iff)
end
lemma [code nbe]:
"HOL.equal (A :: 'a Cset.set) A \<longleftrightarrow> True"
by (fact equal_refl)
subsection {* Functorial operations *}
declare inter_project [code]
declare union_insert [code]
end