--- a/src/HOL/Library/Cset.thy Mon Jul 25 14:10:12 2011 +0200
+++ b/src/HOL/Library/Cset.thy Mon Jul 25 16:55:48 2011 +0200
@@ -86,6 +86,12 @@
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 (member A)"
@@ -124,9 +130,42 @@
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
+
+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> 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 = Set (\<Union>x \<in> member A. member (f x))"
subsection {* Simplified simprules *}
+lemma empty_simp [simp]: "member Cset.empty = {}"
+ by(simp)
+
+lemma UNIV_simp [simp]: "member Cset.UNIV = UNIV"
+ by simp
+
lemma is_empty_simp [simp]:
"is_empty A \<longleftrightarrow> member A = {}"
by (simp add: More_Set.is_empty_def)
@@ -142,10 +181,103 @@
by (simp add: More_Set.project_def)
declare filter_def [simp del]
+lemma member_set [simp]:
+ "member (Cset.set xs) = set xs"
+ by (simp add: set_def)
+hide_fact (open) member_set set_def
+
+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_SUPR [simp]:
+ "member (SUPR A f) = SUPR A (member \<circ> f)"
+unfolding SUPR_def by simp
+
+lemma member_bind [simp]:
+ "member (P \<guillemotright>= f) = member (SUPR (member P) f)"
+by (simp add: bind_def Cset.set_eq_iff)
+
+lemma member_single [simp]:
+ "member (single a) = {a}"
+by(simp add: single_def)
+
+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)
+
+lemma single_bind [simp]:
+ "single a \<guillemotright>= B = B a"
+by(simp add: bind_def single_def)
+
+lemma bind_bind:
+ "(A \<guillemotright>= B) \<guillemotright>= C = A \<guillemotright>= (\<lambda>x. B x \<guillemotright>= C)"
+by(simp add: bind_def)
+
+lemma bind_single [simp]:
+ "A \<guillemotright>= single = A"
+by(simp add: bind_def single_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)
+
+lemma empty_bind [simp]:
+ "Cset.empty \<guillemotright>= f = Cset.empty"
+by(simp add: Cset.set_eq_iff)
+
+lemma member_of_pred [simp]:
+ "member (of_pred P) = {x. Predicate.eval P x}"
+by(simp add: of_pred_def Collect_def)
+
+lemma member_of_seq [simp]:
+ "member (of_seq xq) = {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 = foldl (\<lambda>A x. insert x A) Cset.empty"
+proof(rule ext, rule Cset.set_eqI)
+ fix xs
+ show "member (Cset.set xs) = member (foldl (\<lambda>A x. insert x A) Cset.empty xs)"
+ by(induct xs rule: rev_induct)(simp_all)
+qed
+
+lemma single_code [code]:
+ "single a = insert a Cset.empty"
+by(simp add: Cset.single_def)
+
+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))"
+by(auto split: seq.split
+ simp add: Predicate.Seq_def of_pred_def eval_member Cset.set_eq_iff)
+
+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)"
+by(auto simp add: of_seq_def of_pred_def Cset.set_eq_iff)
+
declare mem_def [simp del]
+no_notation bind (infixl "\<guillemotright>=" 70)
hide_const (open) is_empty insert remove map filter forall exists card
- Inter Union
+ 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 member_set set_simps member_SUPR 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
--- a/src/HOL/Library/Dlist_Cset.thy Mon Jul 25 14:10:12 2011 +0200
+++ b/src/HOL/Library/Dlist_Cset.thy Mon Jul 25 16:55:48 2011 +0200
@@ -7,7 +7,7 @@
begin
definition Set :: "'a dlist \<Rightarrow> 'a Cset.set" where
- "Set dxs = List_Cset.set (list_of_dlist dxs)"
+ "Set dxs = Cset.set (list_of_dlist dxs)"
definition Coset :: "'a dlist \<Rightarrow> 'a Cset.set" where
"Coset dxs = List_Cset.coset (list_of_dlist dxs)"
@@ -27,6 +27,9 @@
declare forall_set [code del]
declare exists_set [code del]
declare card_set [code del]
+declare List_Cset.single_set [code del]
+declare List_Cset.bind_set [code del]
+declare List_Cset.pred_of_cset_set [code del]
declare inter_project [code del]
declare subtract_remove [code del]
declare union_insert [code del]
@@ -50,7 +53,7 @@
by (simp add: Coset_def member_set not_set_compl)
lemma Set_dlist_of_list [code]:
- "List_Cset.set xs = Set (dlist_of_list xs)"
+ "Cset.set xs = Set (dlist_of_list xs)"
by (rule Cset.set_eqI) simp
lemma Coset_dlist_of_list [code]:
@@ -137,4 +140,16 @@
end
+declare Cset.single_code[code]
+
+lemma bind_set [code]:
+ "Cset.bind (Dlist_Cset.Set xs) f = foldl (\<lambda>A x. sup A (f x)) Cset.empty (list_of_dlist xs)"
+by(simp add: List_Cset.bind_set Dlist_Cset.Set_def)
+hide_fact (open) bind_set
+
+lemma pred_of_cset_set [code]:
+ "pred_of_cset (Dlist_Cset.Set xs) = foldr sup (map Predicate.single (list_of_dlist xs)) bot"
+by(simp add: List_Cset.pred_of_cset_set Dlist_Cset.Set_def)
+hide_fact (open) pred_of_cset_set
+
end
--- a/src/HOL/Library/List_Cset.thy Mon Jul 25 14:10:12 2011 +0200
+++ b/src/HOL/Library/List_Cset.thy Mon Jul 25 16:55:48 2011 +0200
@@ -8,15 +8,7 @@
begin
declare mem_def [simp]
-
-definition set :: "'a list \<Rightarrow> 'a Cset.set" where
- "set xs = Set (List.set xs)"
-hide_const (open) set
-
-lemma member_set [simp]:
- "member (List_Cset.set xs) = set xs"
- by (simp add: set_def)
-hide_fact (open) member_set
+declare Cset.set_code [code del]
definition coset :: "'a list \<Rightarrow> 'a Cset.set" where
"coset xs = Set (- set xs)"
@@ -27,10 +19,10 @@
by (simp add: coset_def)
hide_fact (open) member_coset
-code_datatype List_Cset.set List_Cset.coset
+code_datatype Cset.set List_Cset.coset
lemma member_code [code]:
- "member (List_Cset.set xs) = List.member xs"
+ "member (Cset.set xs) = List.member xs"
"member (List_Cset.coset xs) = Not \<circ> List.member xs"
by (simp_all add: fun_eq_iff member_def fun_Compl_def bool_Compl_def)
@@ -48,7 +40,7 @@
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 List_Cset.set {\<cdot>} xs"
+ [code_unfold]: "setify xs = Code_Evaluation.valtermify Cset.set {\<cdot>} xs"
notation fcomp (infixl "\<circ>>" 60)
notation scomp (infixl "\<circ>\<rightarrow>" 60)
@@ -69,12 +61,12 @@
subsection {* Basic operations *}
lemma is_empty_set [code]:
- "Cset.is_empty (List_Cset.set xs) \<longleftrightarrow> List.null xs"
+ "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]:
- "bot = List_Cset.set []"
+ "Cset.empty = Cset.set []"
by (simp add: set_def)
hide_fact (open) empty_set
@@ -84,63 +76,87 @@
hide_fact (open) UNIV_set
lemma remove_set [code]:
- "Cset.remove x (List_Cset.set xs) = List_Cset.set (removeAll x xs)"
+ "Cset.remove x (Cset.set xs) = Cset.set (removeAll x xs)"
"Cset.remove x (List_Cset.coset xs) = List_Cset.coset (List.insert x xs)"
-by (simp_all add: set_def coset_def)
+by (simp_all add: Cset.set_def coset_def)
(metis List.set_insert More_Set.remove_def remove_set_compl)
lemma insert_set [code]:
- "Cset.insert x (List_Cset.set xs) = List_Cset.set (List.insert x xs)"
+ "Cset.insert x (Cset.set xs) = Cset.set (List.insert x xs)"
"Cset.insert x (List_Cset.coset xs) = List_Cset.coset (removeAll x xs)"
- by (simp_all add: set_def coset_def)
+ by (simp_all add: Cset.set_def coset_def)
lemma map_set [code]:
- "Cset.map f (List_Cset.set xs) = List_Cset.set (remdups (List.map f xs))"
- by (simp add: set_def)
+ "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 (List_Cset.set xs) = List_Cset.set (List.filter P xs)"
- by (simp add: set_def project_set)
+ "Cset.filter P (Cset.set xs) = Cset.set (List.filter P xs)"
+ by (simp add: Cset.set_def project_set)
lemma forall_set [code]:
- "Cset.forall P (List_Cset.set xs) \<longleftrightarrow> list_all P xs"
- by (simp add: set_def list_all_iff)
+ "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 (List_Cset.set xs) \<longleftrightarrow> list_ex P xs"
- by (simp add: set_def list_ex_iff)
+ "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 (List_Cset.set xs) = length (remdups xs)"
+ "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: set_def)
+ then show ?thesis by (simp add: Cset.set_def)
qed
lemma compl_set [simp, code]:
- "- List_Cset.set xs = List_Cset.coset xs"
- by (simp add: set_def coset_def)
+ "- Cset.set xs = List_Cset.coset xs"
+ by (simp add: Cset.set_def coset_def)
lemma compl_coset [simp, code]:
- "- List_Cset.coset xs = List_Cset.set xs"
- by (simp add: set_def coset_def)
+ "- List_Cset.coset xs = Cset.set xs"
+ by (simp add: Cset.set_def coset_def)
context complete_lattice
begin
lemma Infimum_inf [code]:
- "Infimum (List_Cset.set As) = foldr inf As top"
+ "Infimum (Cset.set As) = foldr inf As top"
"Infimum (List_Cset.coset []) = bot"
by (simp_all add: Inf_set_foldr Inf_UNIV)
lemma Supremum_sup [code]:
- "Supremum (List_Cset.set As) = foldr sup As bot"
+ "Supremum (Cset.set As) = foldr sup As bot"
"Supremum (List_Cset.coset []) = top"
by (simp_all add: Sup_set_foldr Sup_UNIV)
end
+declare 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
+
+lemma bind_set [code]:
+ "Cset.bind (Cset.set xs) f = foldl (\<lambda>A x. sup A (f x)) (Cset.set []) xs"
+proof(rule sym)
+ show "foldl (\<lambda>A x. sup A (f x)) (Cset.set []) xs = Cset.bind (Cset.set xs) f"
+ by(induct xs rule: rev_induct)(auto simp add: Cset.bind_def Cset.set_def)
+qed
+hide_fact (open) bind_set
+
+lemma pred_of_cset_set [code]:
+ "pred_of_cset (Cset.set xs) = foldr sup (map Predicate.single xs) bot"
+proof -
+ have "pred_of_cset (Cset.set xs) = Predicate.Pred (\<lambda>x. x \<in> set xs)"
+ by(auto simp add: Cset.pred_of_cset_def mem_def)
+ moreover have "foldr sup (map Predicate.single xs) bot = \<dots>"
+ by(induct xs)(auto simp add: bot_pred_def simp del: mem_def intro: pred_eqI, simp)
+ ultimately show ?thesis by(simp)
+qed
+hide_fact (open) pred_of_cset_set
subsection {* Derived operations *}
@@ -171,11 +187,11 @@
subsection {* Functorial operations *}
lemma inter_project [code]:
- "inf A (List_Cset.set xs) = List_Cset.set (List.filter (Cset.member A) xs)"
+ "inf A (Cset.set xs) = Cset.set (List.filter (Cset.member A) xs)"
"inf A (List_Cset.coset xs) = foldr Cset.remove xs A"
proof -
- show "inf A (List_Cset.set xs) = List_Cset.set (List.filter (member A) xs)"
- by (simp add: inter project_def set_def)
+ show "inf A (Cset.set xs) = Cset.set (List.filter (member A) xs)"
+ by (simp add: inter project_def Cset.set_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)
have "member \<circ> fold (\<lambda>x. Set \<circ> More_Set.remove x \<circ> member) xs =
@@ -193,12 +209,12 @@
qed
lemma subtract_remove [code]:
- "A - List_Cset.set xs = foldr Cset.remove xs A"
- "A - List_Cset.coset xs = List_Cset.set (List.filter (member A) xs)"
+ "A - Cset.set xs = foldr Cset.remove xs A"
+ "A - List_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 [code]:
- "sup (List_Cset.set xs) A = foldr Cset.insert xs A"
+ "sup (Cset.set xs) A = foldr Cset.insert xs A"
"sup (List_Cset.coset xs) A = List_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)"
@@ -209,11 +225,11 @@
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 (List_Cset.set xs) A = fold Cset.insert xs A"
+ then have "sup (Cset.set xs) A = fold Cset.insert xs A"
by (simp add: union_set *)
moreover have "\<And>x y :: 'a. Cset.insert y \<circ> Cset.insert x = Cset.insert x \<circ> Cset.insert y"
by (auto simp add: * intro: ext)
- ultimately show "sup (List_Cset.set xs) A = foldr Cset.insert xs A"
+ ultimately show "sup (Cset.set xs) A = foldr Cset.insert xs A"
by (simp add: foldr_fold)
show "sup (List_Cset.coset xs) A = List_Cset.coset (List.filter (Not \<circ> member A) xs)"
by (auto simp add: coset_def)