# HG changeset patch
# User haftmann
# Date 1314385883 -7200
# Node ID da75ffe3d988bce2d6a876326279c0ac7353460f
# Parent  a24b97aeec0cb73d7a0cfa7bed63cb5074919719
separating predicates and sets syntactically

diff -r a24b97aeec0c -r da75ffe3d988 src/HOL/Library/Cset.thy
--- a/src/HOL/Library/Cset.thy	Fri Aug 26 18:24:22 2011 +0200
+++ b/src/HOL/Library/Cset.thy	Fri Aug 26 21:11:23 2011 +0200
@@ -10,16 +10,27 @@
 subsection {* Lifting *}
 
 typedef (open) 'a set = "UNIV :: 'a set set"
-  morphisms member Set by rule+
+  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) = A"
-  by (rule Set_inverse) rule
-
-lemma Set_member [simp]:
-  "Set (member A) = A"
-  by (fact member_inverse)
+  "member (Set A) x \<longleftrightarrow> x \<in> A"
+  by (simp add: member_def)
 
 lemma Set_inject [simp]:
   "Set A = Set B \<longleftrightarrow> A = B"
@@ -27,7 +38,7 @@
 
 lemma set_eq_iff:
   "A = B \<longleftrightarrow> member A = member B"
-  by (simp add: member_inject)
+  by (auto simp add: fun_eq_iff set_of_inject [symmetric] member_def mem_def)
 hide_fact (open) set_eq_iff
 
 lemma set_eqI:
@@ -41,16 +52,16 @@
 begin
 
 definition less_eq_set :: "'a Cset.set \<Rightarrow> 'a Cset.set \<Rightarrow> bool" where
-  [simp]: "A \<le> B \<longleftrightarrow> member A \<subseteq> member B"
+  [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> member A \<subset> member B"
+  [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 (member A \<inter> member B)"
+  [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 (member A \<union> member B)"
+  [simp]: "sup A B = Set (set_of A \<union> set_of B)"
 
 definition bot_set :: "'a Cset.set" where
   [simp]: "bot = Set {}"
@@ -59,13 +70,13 @@
   [simp]: "top = Set UNIV"
 
 definition uminus_set :: "'a Cset.set \<Rightarrow> 'a Cset.set" where
-  [simp]: "- A = Set (- (member A))"
+  [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 (member A - member B)"
+  [simp]: "A - B = Set (set_of A - set_of B)"
 
 instance proof
-qed (auto intro: Cset.set_eqI)
+qed (auto intro!: Cset.set_eqI simp add: member_def mem_def)
 
 end
 
@@ -73,16 +84,19 @@
 begin
 
 definition Inf_set :: "'a Cset.set set \<Rightarrow> 'a Cset.set" where
-  [simp]: "Inf_set As = Set (Inf (image member As))"
+  [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 member As))"
+  [simp]: "Sup_set As = Set (Sup (image set_of As))"
 
 instance proof
-qed (auto simp add: le_fun_def le_bool_def)
+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 *}
 
@@ -93,40 +107,40 @@
 hide_const (open) UNIV
 
 definition is_empty :: "'a Cset.set \<Rightarrow> bool" where
-  [simp]: "is_empty A \<longleftrightarrow> More_Set.is_empty (member A)"
+  [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 (member A))"
+  [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 (member A))"
+  [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 (member A))"
+  [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 (member A))"
+  [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 (member A) P"
+  [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 (member A) P"
+  [simp]: "exists P A \<longleftrightarrow> Bex (set_of A) P"
 
 definition card :: "'a Cset.set \<Rightarrow> nat" where
-  [simp]: "card A = Finite_Set.card (member A)"
+  [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 (member A)"
+  [simp]: "Infimum A = Inf (set_of A)"
 
 definition Supremum :: "'a Cset.set \<Rightarrow> 'a" where
-  [simp]: "Supremum A = Sup (member A)"
+  [simp]: "Supremum A = Sup (set_of A)"
 
 end
 
@@ -140,134 +154,138 @@
 
 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 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_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"
+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))"
+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 = {}"
-  by(simp)
+lemma empty_simp [simp]: "member Cset.empty = bot"
+  by (simp add: fun_eq_iff bot_apply)
 
-lemma UNIV_simp [simp]: "member Cset.UNIV = UNIV"
-  by simp
+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> member A = {}"
+  "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 (member A - {x})"
+  "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> member A. P x}"
+  "filter P A = Set {x \<in> set_of A. P x}"
   by (simp add: More_Set.project_def)
 declare filter_def [simp del]
 
-lemma member_set [simp]:
-  "member (Cset.set xs) = set xs"
+lemma set_of_set [simp]:
+  "set_of (Cset.set xs) = set xs"
   by (simp add: set_def)
-hide_fact (open) member_set set_def
+hide_fact (open) 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]:
+lemma member_SUP [simp]:
   "member (SUPR A f) = SUPR A (member \<circ> f)"
-unfolding SUPR_def by simp
+  by (auto simp add: fun_eq_iff SUP_apply member_def, unfold SUP_def, auto)
 
 lemma member_bind [simp]:
-  "member (P \<guillemotright>= f) = member (SUPR (member P) f)"
-by (simp add: bind_def Cset.set_eq_iff)
+  "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) = {a}"
-by(simp add: single_def)
+  "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)
+  by (auto simp add: Cset.set_eq_iff single_def)
 
 lemma single_bind [simp]:
   "single a \<guillemotright>= B = B a"
-by(simp add: bind_def single_def)
+  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)
-
+  by (simp add: bind_def, simp only: SUP_def image_image, simp)
+ 
 lemma bind_single [simp]:
   "A \<guillemotright>= single = A"
-by(simp add: bind_def single_def)
+  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)
+  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)
+  by (simp add: Cset.set_eq_iff fun_eq_iff bot_apply)
 
 lemma member_of_pred [simp]:
-  "member (of_pred P) = {x. Predicate.eval P x}"
-by(simp add: of_pred_def Collect_def)
+  "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) = {x. Predicate.member xq x}"
-by(simp add: of_seq_def eval_member)
+  "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)
+  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)
+  "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)
+  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)
+  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)"
-by(auto simp add: of_seq_def of_pred_def Cset.set_eq_iff)
-
-declare mem_def [simp del]
+  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
 
 no_notation bind (infixl "\<guillemotright>=" 70)
 
@@ -275,7 +293,7 @@
   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 
+  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