
diff -r e72018e0dd75 -r 2b002c6b0f7d src/HOL/Quotient_Examples/List_Quotient_Set.thy
--- a/src/HOL/Quotient_Examples/List_Quotient_Set.thy	Mon Oct 31 19:12:41 2011 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,191 +0,0 @@
-(*  Title:      HOL/Quotient_Examples/List_Quotient_Set.thy
-    Author:     Florian Haftmann, Alexander Krauss, TU Muenchen
-*)
-
-header {* Implementation of type Quotient_Set.set based on lists. Code equations obtained via quotient lifting. *}
-
-theory List_Quotient_Set
-imports Quotient_Set
-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 Quotient_Set.set" where
-  "coset xs = Quotient_Set.uminus (Quotient_Set.set xs)"
-
-code_datatype Quotient_Set.set List_Quotient_Set.coset
-
-lemma member_code [code]:
-  "member x (Quotient_Set.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 Quotient_Set.set \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
-  [code_unfold]: "setify xs = Code_Evaluation.valtermify Quotient_Set.set {\<cdot>} xs"
-
-notation fcomp (infixl "\<circ>>" 60)
-notation scomp (infixl "\<circ>\<rightarrow>" 60)
-
-instantiation Quotient_Set.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]:
-  "Quotient_Set.is_empty (Quotient_Set.set xs) \<longleftrightarrow> List.null xs"
-  by (lifting is_empty_set)
-hide_fact (open) is_empty_set
-
-lemma empty_set [code]:
-  "Quotient_Set.empty = Quotient_Set.set []"
-  by (lifting set.simps(1)[symmetric])
-hide_fact (open) empty_set
-
-lemma UNIV_set [code]:
-  "Quotient_Set.UNIV = coset []"
-  unfolding coset_def by descending simp
-hide_fact (open) UNIV_set
-
-lemma remove_set [code]:
-  "Quotient_Set.remove x (Quotient_Set.set xs) = Quotient_Set.set (removeAll x xs)"
-  "Quotient_Set.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]:
-  "Quotient_Set.insert x (Quotient_Set.set xs) = Quotient_Set.set (List.insert x xs)"
-  "Quotient_Set.insert x (coset xs) = coset (removeAll x xs)"
-unfolding coset_def
-apply (lifting set_insert[symmetric])
-by descending simp
-
-lemma map_set [code]:
-  "Quotient_Set.map f (Quotient_Set.set xs) = Quotient_Set.set (remdups (List.map f xs))"
-by descending simp
-
-lemma filter_set [code]:
-  "Quotient_Set.filter P (Quotient_Set.set xs) = Quotient_Set.set (List.filter P xs)"
-by descending (simp add: project_set)
-
-lemma forall_set [code]:
-  "Quotient_Set.forall (Quotient_Set.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]:
-  "Quotient_Set.exists (Quotient_Set.set xs) P \<longleftrightarrow> list_ex P xs"
-by descending (fact Bex_set_list_ex)
-
-lemma card_set [code]:
-  "Quotient_Set.card (Quotient_Set.set xs) = length (remdups xs)"
-by (lifting length_remdups_card_conv[symmetric])
-
-lemma compl_set [simp, code]:
-  "Quotient_Set.uminus (Quotient_Set.set xs) = coset xs"
-unfolding coset_def by descending simp
-
-lemma compl_coset [simp, code]:
-  "Quotient_Set.uminus (coset xs) = Quotient_Set.set xs"
-unfolding coset_def by descending simp
-
-lemma Inf_inf [code]:
-  "Quotient_Set.Inf (Quotient_Set.set (xs\<Colon>'a\<Colon>complete_lattice list)) = foldr inf xs top"
-  "Quotient_Set.Inf (coset ([]\<Colon>'a\<Colon>complete_lattice list)) = bot"
-  unfolding List_Quotient_Set.UNIV_set[symmetric]
-  by (lifting Inf_set_foldr Inf_UNIV)
-
-lemma Sup_sup [code]:
-  "Quotient_Set.Sup (Quotient_Set.set (xs\<Colon>'a\<Colon>complete_lattice list)) = foldr sup xs bot"
-  "Quotient_Set.Sup (coset ([]\<Colon>'a\<Colon>complete_lattice list)) = top"
-  unfolding List_Quotient_Set.UNIV_set[symmetric]
-  by (lifting Sup_set_foldr Sup_UNIV)
-
-subsection {* Derived operations *}
-
-lemma subset_eq_forall [code]:
-  "Quotient_Set.subset A B \<longleftrightarrow> Quotient_Set.forall A (\<lambda>x. member x B)"
-by descending blast
-
-lemma subset_subset_eq [code]:
-  "Quotient_Set.psubset A B \<longleftrightarrow> Quotient_Set.subset A B \<and> \<not> Quotient_Set.subset B A"
-by descending blast
-
-instantiation Quotient_Set.set :: (type) equal
-begin
-
-definition [code]:
-  "HOL.equal A B \<longleftrightarrow> Quotient_Set.subset A B \<and> Quotient_Set.subset B A"
-
-instance
-apply intro_classes
-unfolding equal_set_def
-by descending auto
-
-end
-
-lemma [code nbe]:
-  "HOL.equal (A :: 'a Quotient_Set.set) A \<longleftrightarrow> True"
-  by (fact equal_refl)
-
-
-subsection {* Functorial operations *}
-
-lemma inter_project [code]:
-  "Quotient_Set.inter A (Quotient_Set.set xs) = Quotient_Set.set (List.filter (\<lambda>x. Quotient_Set.member x A) xs)"
-  "Quotient_Set.inter A (coset xs) = foldr Quotient_Set.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]:
-  "Quotient_Set.minus A (Quotient_Set.set xs) = foldr Quotient_Set.remove xs A"
-  "Quotient_Set.minus A (coset xs) = Quotient_Set.set (List.filter (\<lambda>x. member x A) xs)"
-unfolding coset_def
-apply (lifting minus_set_foldr)
-by descending auto
-
-lemma union_insert [code]:
-  "Quotient_Set.union (Quotient_Set.set xs) A = foldr Quotient_Set.insert xs A"
-  "Quotient_Set.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]:
-  "Quotient_Set.UNION (Quotient_Set.set []) f = Quotient_Set.set []"
-  "Quotient_Set.UNION (Quotient_Set.set (x#xs)) f =
-     Quotient_Set.union (f x) (Quotient_Set.UNION (Quotient_Set.set xs) f)"
-  by (descending, simp)+
-
-
-end




