(* Author: Florian Haftmann, TU Muenchen *)
header {* A dedicated set type which is executable on its finite part *}
theory Cset
imports More_Set More_List
begin
subsection {* Lifting *}
typedef (open) 'a set = "UNIV :: 'a set set"
morphisms set_of Set by rule+
hide_type (open) set
lemma set_of_Set [simp]:
"set_of (Set A) = A"
by (rule Set_inverse) rule
lemma Set_set_of [simp]:
"Set (set_of A) = A"
by (fact set_of_inverse)
definition member :: "'a Cset.set \<Rightarrow> 'a \<Rightarrow> bool" where
"member A x \<longleftrightarrow> x \<in> set_of A"
lemma member_set_of:
"set_of = member"
by (rule ext)+ (simp add: member_def mem_def)
lemma member_Set [simp]:
"member (Set A) x \<longleftrightarrow> x \<in> A"
by (simp add: member_def)
lemma Set_inject [simp]:
"Set A = Set B \<longleftrightarrow> A = B"
by (simp add: Set_inject)
lemma set_eq_iff:
"A = B \<longleftrightarrow> member A = member B"
by (auto simp add: fun_eq_iff set_of_inject [symmetric] member_def mem_def)
hide_fact (open) set_eq_iff
lemma set_eqI:
"member A = member B \<Longrightarrow> A = B"
by (simp add: Cset.set_eq_iff)
hide_fact (open) set_eqI
subsection {* Lattice instantiation *}
instantiation Cset.set :: (type) boolean_algebra
begin
definition less_eq_set :: "'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where
[simp]: "A \<le> B \<longleftrightarrow> set_of A \<subseteq> set_of B"
definition less_set :: "'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where
[simp]: "A < B \<longleftrightarrow> set_of A \<subset> set_of B"
definition inf_set :: "'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where
[simp]: "inf A B = Set (set_of A \<inter> set_of B)"
definition sup_set :: "'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where
[simp]: "sup A B = Set (set_of A \<union> set_of B)"
definition bot_set :: "'a Cset.set" where
[simp]: "bot = Set {}"
definition top_set :: "'a Cset.set" where
[simp]: "top = Set UNIV"
definition uminus_set :: "'a Cset.set \<Rightarrow> 'a Cset.set" where
[simp]: "- A = Set (- (set_of A))"
definition minus_set :: "'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where
[simp]: "A - B = Set (set_of A - set_of B)"
instance proof
qed (auto intro!: Cset.set_eqI simp add: member_def mem_def)
end
instantiation Cset.set :: (type) complete_lattice
begin
definition Inf_set :: "'a Cset.set set \<Rightarrow> 'a Cset.set" where
[simp]: "Inf_set As = Set (Inf (image set_of As))"
definition Sup_set :: "'a Cset.set set \<Rightarrow> 'a Cset.set" where
[simp]: "Sup_set As = Set (Sup (image set_of As))"
instance proof
qed (auto simp add: le_fun_def)
end
instance Cset.set :: (type) complete_boolean_algebra proof
qed (unfold INF_def SUP_def, auto)
subsection {* Basic operations *}
abbreviation empty :: "'a Cset.set" where "empty \<equiv> bot"
hide_const (open) empty
abbreviation UNIV :: "'a Cset.set" where "UNIV \<equiv> top"
hide_const (open) UNIV
definition is_empty :: "'a Cset.set \<Rightarrow> bool" where
[simp]: "is_empty A \<longleftrightarrow> More_Set.is_empty (set_of A)"
definition insert :: "'a \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where
[simp]: "insert x A = Set (Set.insert x (set_of A))"
definition remove :: "'a \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where
[simp]: "remove x A = Set (More_Set.remove x (set_of A))"
definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a Cset.set \<Rightarrow> 'b Cset.set" where
[simp]: "map f A = Set (image f (set_of A))"
enriched_type map: map
by (simp_all add: fun_eq_iff image_compose)
definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> 'a Cset.set" where
[simp]: "filter P A = Set (More_Set.project P (set_of A))"
definition forall :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where
[simp]: "forall P A \<longleftrightarrow> Ball (set_of A) P"
definition exists :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where
[simp]: "exists P A \<longleftrightarrow> Bex (set_of A) P"
definition card :: "'a Cset.set \<Rightarrow> nat" where
[simp]: "card A = Finite_Set.card (set_of A)"
context complete_lattice
begin
definition Infimum :: "'a Cset.set \<Rightarrow> 'a" where
[simp]: "Infimum A = Inf (set_of A)"
definition Supremum :: "'a Cset.set \<Rightarrow> 'a" where
[simp]: "Supremum A = Sup (set_of A)"
end
subsection {* More operations *}
text {* conversion from @{typ "'a list"} *}
definition set :: "'a list \<Rightarrow> 'a Cset.set" where
"set xs = Set (List.set xs)"
hide_const (open) set
definition coset :: "'a list \<Rightarrow> 'a Cset.set" where
"coset xs = Set (- List.set xs)"
hide_const (open) coset
text {* conversion from @{typ "'a Predicate.pred"} *}
definition pred_of_cset :: "'a Cset.set \<Rightarrow> 'a Predicate.pred" where
[code del]: "pred_of_cset = Predicate.Pred \<circ> Cset.member"
definition of_pred :: "'a Predicate.pred \<Rightarrow> 'a Cset.set" where
"of_pred = Cset.Set \<circ> Collect \<circ> Predicate.eval"
definition of_seq :: "'a Predicate.seq \<Rightarrow> 'a Cset.set" where
"of_seq = of_pred \<circ> Predicate.pred_of_seq"
text {* monad operations *}
definition single :: "'a \<Rightarrow> 'a Cset.set" where
"single a = Set {a}"
definition bind :: "'a Cset.set \<Rightarrow> ('a \<Rightarrow> 'b Cset.set) \<Rightarrow> 'b Cset.set" (infixl "\<guillemotright>=" 70) where
"A \<guillemotright>= f = (SUP x : set_of A. f x)"
subsection {* Simplified simprules *}
lemma empty_simp [simp]: "member Cset.empty = bot"
by (simp add: fun_eq_iff bot_apply)
lemma UNIV_simp [simp]: "member Cset.UNIV = top"
by (simp add: fun_eq_iff top_apply)
lemma is_empty_simp [simp]:
"is_empty A \<longleftrightarrow> set_of A = {}"
by (simp add: More_Set.is_empty_def)
declare is_empty_def [simp del]
lemma remove_simp [simp]:
"remove x A = Set (set_of A - {x})"
by (simp add: More_Set.remove_def)
declare remove_def [simp del]
lemma filter_simp [simp]:
"filter P A = Set {x \<in> set_of A. P x}"
by (simp add: More_Set.project_def)
declare filter_def [simp del]
lemma set_of_set [simp]:
"set_of (Cset.set xs) = set xs"
by (simp add: set_def)
hide_fact (open) set_def
lemma member_set [simp]:
"member (Cset.set xs) = (\<lambda>x. x \<in> set xs)"
by (simp add: fun_eq_iff member_def)
hide_fact (open) member_set
lemma set_of_coset [simp]:
"set_of (Cset.coset xs) = - set xs"
by (simp add: coset_def)
hide_fact (open) coset_def
lemma member_coset [simp]:
"member (Cset.coset xs) = (\<lambda>x. x \<in> - set xs)"
by (simp add: fun_eq_iff member_def)
hide_fact (open) member_coset
lemma set_simps [simp]:
"Cset.set [] = Cset.empty"
"Cset.set (x # xs) = insert x (Cset.set xs)"
by(simp_all add: Cset.set_def)
lemma member_SUP [simp]:
"member (SUPR A f) = SUPR A (member \<circ> f)"
by (auto simp add: fun_eq_iff SUP_apply member_def, unfold SUP_def, auto)
lemma member_bind [simp]:
"member (P \<guillemotright>= f) = SUPR (set_of P) (member \<circ> f)"
by (simp add: bind_def Cset.set_eq_iff)
lemma member_single [simp]:
"member (single a) = (\<lambda>x. x \<in> {a})"
by (simp add: single_def fun_eq_iff)
lemma single_sup_simps [simp]:
shows single_sup: "sup (single a) A = insert a A"
and sup_single: "sup A (single a) = insert a A"
by (auto simp add: Cset.set_eq_iff single_def)
lemma single_bind [simp]:
"single a \<guillemotright>= B = B a"
by (simp add: Cset.set_eq_iff SUP_insert single_def)
lemma bind_bind:
"(A \<guillemotright>= B) \<guillemotright>= C = A \<guillemotright>= (\<lambda>x. B x \<guillemotright>= C)"
by (simp add: bind_def, simp only: SUP_def image_image, simp)
lemma bind_single [simp]:
"A \<guillemotright>= single = A"
by (simp add: Cset.set_eq_iff SUP_apply fun_eq_iff single_def member_def)
lemma bind_const: "A \<guillemotright>= (\<lambda>_. B) = (if Cset.is_empty A then Cset.empty else B)"
by (auto simp add: Cset.set_eq_iff fun_eq_iff)
lemma empty_bind [simp]:
"Cset.empty \<guillemotright>= f = Cset.empty"
by (simp add: Cset.set_eq_iff fun_eq_iff bot_apply)
lemma member_of_pred [simp]:
"member (of_pred P) = (\<lambda>x. x \<in> {x. Predicate.eval P x})"
by (simp add: of_pred_def fun_eq_iff)
lemma member_of_seq [simp]:
"member (of_seq xq) = (\<lambda>x. x \<in> {x. Predicate.member xq x})"
by (simp add: of_seq_def eval_member)
lemma eval_pred_of_cset [simp]:
"Predicate.eval (pred_of_cset A) = Cset.member A"
by (simp add: pred_of_cset_def)
subsection {* Default implementations *}
lemma set_code [code]:
"Cset.set = (\<lambda>xs. fold insert xs Cset.empty)"
proof (rule ext, rule Cset.set_eqI)
fix xs :: "'a list"
show "member (Cset.set xs) = member (fold insert xs Cset.empty)"
by (simp add: fold_commute_apply [symmetric, where ?h = Set and ?g = Set.insert]
fun_eq_iff Cset.set_def union_set [symmetric])
qed
lemma single_code [code]:
"single a = insert a Cset.empty"
by (simp add: Cset.single_def)
lemma compl_set [simp]:
"- Cset.set xs = Cset.coset xs"
by (simp add: Cset.set_def Cset.coset_def)
lemma compl_coset [simp]:
"- Cset.coset xs = Cset.set xs"
by (simp add: Cset.set_def Cset.coset_def)
lemma member_cset_of:
"member = set_of"
by (rule ext)+ (simp add: member_def mem_def)
lemma inter_project:
"inf A (Cset.set xs) = Cset.set (List.filter (Cset.member A) xs)"
"inf A (Cset.coset xs) = foldr Cset.remove xs A"
proof -
show "inf A (Cset.set xs) = Cset.set (List.filter (member A) xs)"
by (simp add: inter project_def Cset.set_def member_def)
have *: "\<And>x::'a. Cset.remove = (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member)"
by (simp add: fun_eq_iff More_Set.remove_def member_cset_of)
have "member \<circ> fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs =
fold More_Set.remove xs \<circ> member"
by (rule fold_commute) (simp add: fun_eq_iff mem_def)
then have "fold More_Set.remove xs (member A) =
member (fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs A)"
by (simp add: fun_eq_iff)
then have "inf A (Cset.coset xs) = fold Cset.remove xs A"
by (simp add: Diff_eq [symmetric] minus_set * member_cset_of)
moreover have "\<And>x y :: 'a. Cset.remove y \<circ> Cset.remove x = Cset.remove x \<circ> Cset.remove y"
by (auto simp add: More_Set.remove_def * member_cset_of)
ultimately show "inf A (Cset.coset xs) = foldr Cset.remove xs A"
by (simp add: foldr_fold)
qed
lemma subtract_remove:
"A - Cset.set xs = foldr Cset.remove xs A"
"A - Cset.coset xs = Cset.set (List.filter (member A) xs)"
by (simp_all only: diff_eq compl_set compl_coset inter_project)
lemma union_insert:
"sup (Cset.set xs) A = foldr Cset.insert xs A"
"sup (Cset.coset xs) A = Cset.coset (List.filter (Not \<circ> member A) xs)"
proof -
have *: "\<And>x::'a. Cset.insert = (\<lambda>x. Set \<circ> Set.insert x \<circ> member)"
by (simp add: fun_eq_iff member_cset_of)
have "member \<circ> fold (\<lambda>x. Set \<circ> Set.insert x \<circ> member) xs =
fold Set.insert xs \<circ> member"
by (rule fold_commute) (simp add: fun_eq_iff mem_def)
then have "fold Set.insert xs (member A) =
member (fold (\<lambda>x. Set \<circ> Set.insert x \<circ> member) xs A)"
by (simp add: fun_eq_iff)
then have "sup (Cset.set xs) A = fold Cset.insert xs A"
by (simp add: union_set * member_cset_of)
moreover have "\<And>x y :: 'a. Cset.insert y \<circ> Cset.insert x = Cset.insert x \<circ> Cset.insert y"
by (auto simp add: * member_cset_of)
ultimately show "sup (Cset.set xs) A = foldr Cset.insert xs A"
by (simp add: foldr_fold)
show "sup (Cset.coset xs) A = Cset.coset (List.filter (Not \<circ> member A) xs)"
by (auto simp add: Cset.coset_def member_cset_of mem_def)
qed
context complete_lattice
begin
lemma Infimum_inf:
"Infimum (Cset.set As) = foldr inf As top"
"Infimum (Cset.coset []) = bot"
by (simp_all add: Inf_set_foldr)
lemma Supremum_sup:
"Supremum (Cset.set As) = foldr sup As bot"
"Supremum (Cset.coset []) = top"
by (simp_all add: Sup_set_foldr)
end
lemma of_pred_code [code]:
"of_pred (Predicate.Seq f) = (case f () of
Predicate.Empty \<Rightarrow> Cset.empty
| Predicate.Insert x P \<Rightarrow> Cset.insert x (of_pred P)
| Predicate.Join P xq \<Rightarrow> sup (of_pred P) (of_seq xq))"
apply (auto split: seq.split simp add: Predicate.Seq_def of_pred_def Cset.set_eq_iff sup_apply eval_member [symmetric] member_def [symmetric] Collect_def mem_def member_set_of)
apply (unfold Set.insert_def Collect_def sup_apply member_set_of)
apply simp_all
done
lemma of_seq_code [code]:
"of_seq Predicate.Empty = Cset.empty"
"of_seq (Predicate.Insert x P) = Cset.insert x (of_pred P)"
"of_seq (Predicate.Join P xq) = sup (of_pred P) (of_seq xq)"
apply (auto simp add: of_seq_def of_pred_def Cset.set_eq_iff mem_def Collect_def)
apply (unfold Set.insert_def Collect_def sup_apply member_set_of)
apply simp_all
done
lemma bind_set:
"Cset.bind (Cset.set xs) f = fold (sup \<circ> f) xs (Cset.set [])"
by (simp add: Cset.bind_def SUPR_set_fold)
hide_fact (open) bind_set
lemma pred_of_cset_set:
"pred_of_cset (Cset.set xs) = foldr sup (List.map Predicate.single xs) bot"
proof -
have "pred_of_cset (Cset.set xs) = Predicate.Pred (\<lambda>x. x \<in> set xs)"
by (simp add: Cset.pred_of_cset_def member_set)
moreover have "foldr sup (List.map Predicate.single xs) bot = \<dots>"
by (induct xs) (auto simp add: bot_pred_def intro: pred_eqI, simp add: mem_def)
ultimately show ?thesis by simp
qed
hide_fact (open) pred_of_cset_set
no_notation bind (infixl "\<guillemotright>=" 70)
hide_const (open) is_empty insert remove map filter forall exists card
Inter Union bind single of_pred of_seq
hide_fact (open) set_def pred_of_cset_def of_pred_def of_seq_def single_def
bind_def empty_simp UNIV_simp set_simps member_bind
member_single single_sup_simps single_sup sup_single single_bind
bind_bind bind_single bind_const empty_bind member_of_pred member_of_seq
eval_pred_of_cset set_code single_code of_pred_code of_seq_code
end