src/HOL/Library/RBT_Set.thy
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(*  Title:      HOL/Library/RBT_Set.thy
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    Author:     Ondrej Kuncar
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*)
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header {* Implementation of sets using RBT trees *}
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theory RBT_Set
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imports RBT Product_ord
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begin
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(*
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  Users should be aware that by including this file all code equations
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  outside of List.thy using 'a list as an implenentation of sets cannot be
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  used for code generation. If such equations are not needed, they can be
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  deleted from the code generator. Otherwise, a user has to provide their 
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  own equations using RBT trees. 
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*)
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section {* Definition of code datatype constructors *}
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definition Set :: "('a\<Colon>linorder, unit) rbt \<Rightarrow> 'a set" 
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  where "Set t = {x . lookup t x = Some ()}"
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definition Coset :: "('a\<Colon>linorder, unit) rbt \<Rightarrow> 'a set" 
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  where [simp]: "Coset t = - Set t"
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section {* Deletion of already existing code equations *}
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lemma [code, code del]:
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  "Set.empty = Set.empty" ..
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lemma [code, code del]:
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  "Set.is_empty = Set.is_empty" ..
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lemma [code, code del]:
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  "uminus_set_inst.uminus_set = uminus_set_inst.uminus_set" ..
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lemma [code, code del]:
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  "Set.member = Set.member" ..
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lemma [code, code del]:
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  "Set.insert = Set.insert" ..
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lemma [code, code del]:
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  "Set.remove = Set.remove" ..
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lemma [code, code del]:
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  "UNIV = UNIV" ..
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lemma [code, code del]:
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  "Set.filter = Set.filter" ..
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lemma [code, code del]:
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  "image = image" ..
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lemma [code, code del]:
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  "Set.subset_eq = Set.subset_eq" ..
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lemma [code, code del]:
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  "Ball = Ball" ..
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lemma [code, code del]:
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  "Bex = Bex" ..
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term can_select
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lemma [code, code del]:
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  "can_select = can_select" ..
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lemma [code, code del]:
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  "Set.union = Set.union" ..
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lemma [code, code del]:
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  "minus_set_inst.minus_set = minus_set_inst.minus_set" ..
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lemma [code, code del]:
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  "Set.inter = Set.inter" ..
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lemma [code, code del]:
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  "card = card" ..
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lemma [code, code del]:
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  "the_elem = the_elem" ..
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lemma [code, code del]:
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  "Pow = Pow" ..
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lemma [code, code del]:
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  "setsum = setsum" ..
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lemma [code, code del]:
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  "Product_Type.product = Product_Type.product"  ..
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lemma [code, code del]:
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  "Id_on = Id_on" ..
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lemma [code, code del]:
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  "Image = Image" ..
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lemma [code, code del]:
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  "trancl = trancl" ..
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lemma [code, code del]:
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  "relcomp = relcomp" ..
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lemma [code, code del]:
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  "wf = wf" ..
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lemma [code, code del]:
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  "Min = Min" ..
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lemma [code, code del]:
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  "Inf_fin = Inf_fin" ..
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lemma [code, code del]:
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  "INFI = INFI" ..
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lemma [code, code del]:
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  "Max = Max" ..
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lemma [code, code del]:
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  "Sup_fin = Sup_fin" ..
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lemma [code, code del]:
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  "SUPR = SUPR" ..
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lemma [code, code del]:
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  "(Inf :: 'a set set \<Rightarrow> 'a set) = Inf" ..
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lemma [code, code del]:
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  "(Sup :: 'a set set \<Rightarrow> 'a set) = Sup" ..
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lemma [code, code del]:
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  "sorted_list_of_set = sorted_list_of_set" ..
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lemma [code, code del]: 
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  "List.map_project = List.map_project" ..
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section {* Lemmas *}
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subsection {* Auxiliary lemmas *}
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lemma [simp]: "x \<noteq> Some () \<longleftrightarrow> x = None"
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by (auto simp: not_Some_eq[THEN iffD1])
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lemma Set_set_keys: "Set x = dom (lookup x)" 
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by (auto simp: Set_def)
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lemma finite_Set [simp, intro!]: "finite (Set x)"
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by (simp add: Set_set_keys)
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lemma set_keys: "Set t = set(keys t)"
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by (simp add: Set_set_keys lookup_keys)
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subsection {* fold and filter *}
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lemma finite_fold_rbt_fold_eq:
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  assumes "comp_fun_commute f" 
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  shows "Finite_Set.fold f A (set(entries t)) = fold (curry f) t A"
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proof -
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  have *: "remdups (entries t) = entries t"
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    using distinct_entries distinct_map by (auto intro: distinct_remdups_id)
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  show ?thesis using assms by (auto simp: fold_def_alt comp_fun_commute.fold_set_fold_remdups *)
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qed
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definition fold_keys :: "('a :: linorder \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a, _) rbt \<Rightarrow> 'b \<Rightarrow> 'b" 
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  where [code_unfold]:"fold_keys f t A = fold (\<lambda>k _ t. f k t) t A"
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lemma fold_keys_def_alt:
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  "fold_keys f t s = List.fold f (keys t) s"
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by (auto simp: fold_map o_def split_def fold_def_alt keys_def_alt fold_keys_def)
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lemma finite_fold_fold_keys:
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  assumes "comp_fun_commute f"
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  shows "Finite_Set.fold f A (Set t) = fold_keys f t A"
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using assms
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proof -
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  interpret comp_fun_commute f by fact
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  have "set (keys t) = fst ` (set (entries t))" by (auto simp: fst_eq_Domain keys_entries)
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  moreover have "inj_on fst (set (entries t))" using distinct_entries distinct_map by auto
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  ultimately show ?thesis 
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    by (auto simp add: set_keys fold_keys_def curry_def fold_image finite_fold_rbt_fold_eq 
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      comp_comp_fun_commute)
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qed
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definition rbt_filter :: "('a :: linorder \<Rightarrow> bool) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> 'a set" where
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  "rbt_filter P t = fold (\<lambda>k _ A'. if P k then Set.insert k A' else A') t {}"
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49758
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lemma Set_filter_rbt_filter:
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  "Set.filter P (Set t) = rbt_filter P t"
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by (simp add: fold_keys_def Set_filter_fold rbt_filter_def 
48623
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  finite_fold_fold_keys[OF comp_fun_commute_filter_fold])
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subsection {* foldi and Ball *}
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lemma Ball_False: "RBT_Impl.fold (\<lambda>k v s. s \<and> P k) t False = False"
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by (induction t) auto
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lemma rbt_foldi_fold_conj: 
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  "RBT_Impl.foldi (\<lambda>s. s = True) (\<lambda>k v s. s \<and> P k) t val = RBT_Impl.fold (\<lambda>k v s. s \<and> P k) t val"
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proof (induction t arbitrary: val) 
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  case (Branch c t1) then show ?case
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    by (cases "RBT_Impl.fold (\<lambda>k v s. s \<and> P k) t1 True") (simp_all add: Ball_False) 
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qed simp
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lemma foldi_fold_conj: "foldi (\<lambda>s. s = True) (\<lambda>k v s. s \<and> P k) t val = fold_keys (\<lambda>k s. s \<and> P k) t val"
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unfolding fold_keys_def by transfer (rule rbt_foldi_fold_conj)
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   211
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subsection {* foldi and Bex *}
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lemma Bex_True: "RBT_Impl.fold (\<lambda>k v s. s \<or> P k) t True = True"
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by (induction t) auto
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lemma rbt_foldi_fold_disj: 
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  "RBT_Impl.foldi (\<lambda>s. s = False) (\<lambda>k v s. s \<or> P k) t val = RBT_Impl.fold (\<lambda>k v s. s \<or> P k) t val"
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proof (induction t arbitrary: val) 
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  case (Branch c t1) then show ?case
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    by (cases "RBT_Impl.fold (\<lambda>k v s. s \<or> P k) t1 False") (simp_all add: Bex_True) 
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qed simp
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lemma foldi_fold_disj: "foldi (\<lambda>s. s = False) (\<lambda>k v s. s \<or> P k) t val = fold_keys (\<lambda>k s. s \<or> P k) t val"
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unfolding fold_keys_def by transfer (rule rbt_foldi_fold_disj)
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   227
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subsection {* folding over non empty trees and selecting the minimal and maximal element *}
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(** concrete **)
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(* The concrete part is here because it's probably not general enough to be moved to RBT_Impl *)
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definition rbt_fold1_keys :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a::linorder, 'b) RBT_Impl.rbt \<Rightarrow> 'a" 
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  where "rbt_fold1_keys f t = List.fold f (tl(RBT_Impl.keys t)) (hd(RBT_Impl.keys t))"
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(* minimum *)
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definition rbt_min :: "('a::linorder, unit) RBT_Impl.rbt \<Rightarrow> 'a" 
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  where "rbt_min t = rbt_fold1_keys min t"
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lemma key_le_right: "rbt_sorted (Branch c lt k v rt) \<Longrightarrow> (\<And>x. x \<in>set (RBT_Impl.keys rt) \<Longrightarrow> k \<le> x)"
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by  (auto simp: rbt_greater_prop less_imp_le)
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lemma left_le_key: "rbt_sorted (Branch c lt k v rt) \<Longrightarrow> (\<And>x. x \<in>set (RBT_Impl.keys lt) \<Longrightarrow> x \<le> k)"
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by (auto simp: rbt_less_prop less_imp_le)
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   248
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   249
lemma fold_min_triv:
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  fixes k :: "_ :: linorder"
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  shows "(\<forall>x\<in>set xs. k \<le> x) \<Longrightarrow> List.fold min xs k = k" 
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by (induct xs) (auto simp add: min_def)
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   253
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lemma rbt_min_simps:
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  "is_rbt (Branch c RBT_Impl.Empty k v rt) \<Longrightarrow> rbt_min (Branch c RBT_Impl.Empty k v rt) = k"
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   256
by (auto intro: fold_min_triv dest: key_le_right is_rbt_rbt_sorted simp: rbt_fold1_keys_def rbt_min_def)
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   257
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fun rbt_min_opt where
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  "rbt_min_opt (Branch c RBT_Impl.Empty k v rt) = k" |
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  "rbt_min_opt (Branch c (Branch lc llc lk lv lrt) k v rt) = rbt_min_opt (Branch lc llc lk lv lrt)"
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lemma rbt_min_opt_Branch:
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  "t1 \<noteq> rbt.Empty \<Longrightarrow> rbt_min_opt (Branch c t1 k () t2) = rbt_min_opt t1" 
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by (cases t1) auto
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   265
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lemma rbt_min_opt_induct [case_names empty left_empty left_non_empty]:
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  fixes t :: "('a :: linorder, unit) RBT_Impl.rbt"
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   268
  assumes "P rbt.Empty"
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  assumes "\<And>color t1 a b t2. P t1 \<Longrightarrow> P t2 \<Longrightarrow> t1 = rbt.Empty \<Longrightarrow> P (Branch color t1 a b t2)"
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   270
  assumes "\<And>color t1 a b t2. P t1 \<Longrightarrow> P t2 \<Longrightarrow> t1 \<noteq> rbt.Empty \<Longrightarrow> P (Branch color t1 a b t2)"
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   271
  shows "P t"
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using assms
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  apply (induction t)
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  apply simp
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  apply (case_tac "t1 = rbt.Empty")
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  apply simp_all
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   277
done
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   278
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   279
lemma rbt_min_opt_in_set: 
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  fixes t :: "('a :: linorder, unit) RBT_Impl.rbt"
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   281
  assumes "t \<noteq> rbt.Empty"
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   282
  shows "rbt_min_opt t \<in> set (RBT_Impl.keys t)"
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   283
using assms by (induction t rule: rbt_min_opt.induct) (auto)
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   284
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   285
lemma rbt_min_opt_is_min:
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  fixes t :: "('a :: linorder, unit) RBT_Impl.rbt"
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   287
  assumes "rbt_sorted t"
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   288
  assumes "t \<noteq> rbt.Empty"
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   289
  shows "\<And>y. y \<in> set (RBT_Impl.keys t) \<Longrightarrow> y \<ge> rbt_min_opt t"
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   290
using assms 
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proof (induction t rule: rbt_min_opt_induct)
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  case empty
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    then show ?case by simp
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   294
next
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   295
  case left_empty
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   296
    then show ?case by (auto intro: key_le_right simp del: rbt_sorted.simps)
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   297
next
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   298
  case (left_non_empty c t1 k v t2 y)
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   299
    then have "y = k \<or> y \<in> set (RBT_Impl.keys t1) \<or> y \<in> set (RBT_Impl.keys t2)" by auto
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   300
    with left_non_empty show ?case 
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   301
    proof(elim disjE)
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   302
      case goal1 then show ?case 
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   303
        by (auto simp add: rbt_min_opt_Branch intro: left_le_key rbt_min_opt_in_set)
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   304
    next
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parents:
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   305
      case goal2 with left_non_empty show ?case by (auto simp add: rbt_min_opt_Branch)
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   306
    next 
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   307
      case goal3 show ?case
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   308
      proof -
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   309
        from goal3 have "rbt_min_opt t1 \<le> k" by (simp add: left_le_key rbt_min_opt_in_set)
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        moreover from goal3 have "k \<le> y" by (simp add: key_le_right)
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   311
        ultimately show ?thesis using goal3 by (simp add: rbt_min_opt_Branch)
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      qed
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    qed
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   314
qed
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   315
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lemma rbt_min_eq_rbt_min_opt:
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  assumes "t \<noteq> RBT_Impl.Empty"
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  assumes "is_rbt t"
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diff changeset
   319
  shows "rbt_min t = rbt_min_opt t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   320
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   321
  interpret ab_semigroup_idem_mult "(min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a)" using ab_semigroup_idem_mult_min
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   322
    unfolding class.ab_semigroup_idem_mult_def by blast
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   323
  show ?thesis
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   324
    by (simp add: Min_eqI rbt_min_opt_is_min rbt_min_opt_in_set assms Min_def[symmetric]
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   325
      non_empty_rbt_keys fold1_set_fold[symmetric] rbt_min_def rbt_fold1_keys_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   326
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   327
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   328
(* maximum *)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   329
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   330
definition rbt_max :: "('a::linorder, unit) RBT_Impl.rbt \<Rightarrow> 'a" 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   331
  where "rbt_max t = rbt_fold1_keys max t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   332
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   333
lemma fold_max_triv:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   334
  fixes k :: "_ :: linorder"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   335
  shows "(\<forall>x\<in>set xs. x \<le> k) \<Longrightarrow> List.fold max xs k = k" 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   336
by (induct xs) (auto simp add: max_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   337
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   338
lemma fold_max_rev_eq:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   339
  fixes xs :: "('a :: linorder) list"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   340
  assumes "xs \<noteq> []"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   341
  shows "List.fold max (tl xs) (hd xs) = List.fold max (tl (rev xs)) (hd (rev xs))" 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   342
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   343
  interpret ab_semigroup_idem_mult "(max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a)" using ab_semigroup_idem_mult_max
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   344
    unfolding class.ab_semigroup_idem_mult_def by blast
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   345
  show ?thesis
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   346
  using assms by (auto simp add: fold1_set_fold[symmetric])
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   347
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   348
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   349
lemma rbt_max_simps:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   350
  assumes "is_rbt (Branch c lt k v RBT_Impl.Empty)" 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   351
  shows "rbt_max (Branch c lt k v RBT_Impl.Empty) = k"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   352
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   353
  have "List.fold max (tl (rev(RBT_Impl.keys lt @ [k]))) (hd (rev(RBT_Impl.keys lt @ [k]))) = k"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   354
    using assms by (auto intro!: fold_max_triv dest!: left_le_key is_rbt_rbt_sorted)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   355
  then show ?thesis by (auto simp add: rbt_max_def rbt_fold1_keys_def fold_max_rev_eq)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   356
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   357
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   358
fun rbt_max_opt where
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   359
  "rbt_max_opt (Branch c lt k v RBT_Impl.Empty) = k" |
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   360
  "rbt_max_opt (Branch c lt k v (Branch rc rlc rk rv rrt)) = rbt_max_opt (Branch rc rlc rk rv rrt)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   361
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   362
lemma rbt_max_opt_Branch:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   363
  "t2 \<noteq> rbt.Empty \<Longrightarrow> rbt_max_opt (Branch c t1 k () t2) = rbt_max_opt t2" 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   364
by (cases t2) auto
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   365
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   366
lemma rbt_max_opt_induct [case_names empty right_empty right_non_empty]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   367
  fixes t :: "('a :: linorder, unit) RBT_Impl.rbt"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   368
  assumes "P rbt.Empty"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   369
  assumes "\<And>color t1 a b t2. P t1 \<Longrightarrow> P t2 \<Longrightarrow> t2 = rbt.Empty \<Longrightarrow> P (Branch color t1 a b t2)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   370
  assumes "\<And>color t1 a b t2. P t1 \<Longrightarrow> P t2 \<Longrightarrow> t2 \<noteq> rbt.Empty \<Longrightarrow> P (Branch color t1 a b t2)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   371
  shows "P t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   372
using assms
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   373
  apply (induction t)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   374
  apply simp
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   375
  apply (case_tac "t2 = rbt.Empty")
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   376
  apply simp_all
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   377
done
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   378
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   379
lemma rbt_max_opt_in_set: 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   380
  fixes t :: "('a :: linorder, unit) RBT_Impl.rbt"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   381
  assumes "t \<noteq> rbt.Empty"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   382
  shows "rbt_max_opt t \<in> set (RBT_Impl.keys t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   383
using assms by (induction t rule: rbt_max_opt.induct) (auto)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   384
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   385
lemma rbt_max_opt_is_max:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   386
  fixes t :: "('a :: linorder, unit) RBT_Impl.rbt"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   387
  assumes "rbt_sorted t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   388
  assumes "t \<noteq> rbt.Empty"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   389
  shows "\<And>y. y \<in> set (RBT_Impl.keys t) \<Longrightarrow> y \<le> rbt_max_opt t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   390
using assms 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   391
proof (induction t rule: rbt_max_opt_induct)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   392
  case empty
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   393
    then show ?case by simp
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   394
next
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   395
  case right_empty
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   396
    then show ?case by (auto intro: left_le_key simp del: rbt_sorted.simps)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   397
next
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   398
  case (right_non_empty c t1 k v t2 y)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   399
    then have "y = k \<or> y \<in> set (RBT_Impl.keys t2) \<or> y \<in> set (RBT_Impl.keys t1)" by auto
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   400
    with right_non_empty show ?case 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   401
    proof(elim disjE)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   402
      case goal1 then show ?case 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   403
        by (auto simp add: rbt_max_opt_Branch intro: key_le_right rbt_max_opt_in_set)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   404
    next
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   405
      case goal2 with right_non_empty show ?case by (auto simp add: rbt_max_opt_Branch)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   406
    next 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   407
      case goal3 show ?case
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   408
      proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   409
        from goal3 have "rbt_max_opt t2 \<ge> k" by (simp add: key_le_right rbt_max_opt_in_set)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   410
        moreover from goal3 have "y \<le> k" by (simp add: left_le_key)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   411
        ultimately show ?thesis using goal3 by (simp add: rbt_max_opt_Branch)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   412
      qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   413
    qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   414
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   415
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   416
lemma rbt_max_eq_rbt_max_opt:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   417
  assumes "t \<noteq> RBT_Impl.Empty"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   418
  assumes "is_rbt t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   419
  shows "rbt_max t = rbt_max_opt t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   420
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   421
  interpret ab_semigroup_idem_mult "(max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a)" using ab_semigroup_idem_mult_max
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   422
    unfolding class.ab_semigroup_idem_mult_def by blast
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   423
  show ?thesis
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   424
    by (simp add: Max_eqI rbt_max_opt_is_max rbt_max_opt_in_set assms Max_def[symmetric]
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   425
      non_empty_rbt_keys fold1_set_fold[symmetric] rbt_max_def rbt_fold1_keys_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   426
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   427
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   428
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   429
(** abstract **)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   430
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   431
lift_definition fold1_keys :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a::linorder, 'b) rbt \<Rightarrow> 'a"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   432
  is rbt_fold1_keys by simp
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   433
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   434
lemma fold1_keys_def_alt:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   435
  "fold1_keys f t = List.fold f (tl(keys t)) (hd(keys t))"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   436
  by transfer (simp add: rbt_fold1_keys_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   437
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   438
lemma finite_fold1_fold1_keys:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   439
  assumes "class.ab_semigroup_mult f"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   440
  assumes "\<not> (is_empty t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   441
  shows "Finite_Set.fold1 f (Set t) = fold1_keys f t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   442
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   443
  interpret ab_semigroup_mult f by fact
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   444
  show ?thesis using assms 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   445
    by (auto simp: fold1_keys_def_alt set_keys fold_def_alt fold1_distinct_set_fold non_empty_keys)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   446
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   447
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   448
(* minimum *)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   449
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   450
lift_definition r_min :: "('a :: linorder, unit) rbt \<Rightarrow> 'a" is rbt_min by simp
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   451
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   452
lift_definition r_min_opt :: "('a :: linorder, unit) rbt \<Rightarrow> 'a" is rbt_min_opt by simp
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   453
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   454
lemma r_min_alt_def: "r_min t = fold1_keys min t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   455
by transfer (simp add: rbt_min_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   456
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   457
lemma r_min_eq_r_min_opt:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   458
  assumes "\<not> (is_empty t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   459
  shows "r_min t = r_min_opt t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   460
using assms unfolding is_empty_empty by transfer (auto intro: rbt_min_eq_rbt_min_opt)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   461
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   462
lemma fold_keys_min_top_eq:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   463
  fixes t :: "('a :: {linorder, bounded_lattice_top}, unit) rbt"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   464
  assumes "\<not> (is_empty t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   465
  shows "fold_keys min t top = fold1_keys min t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   466
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   467
  have *: "\<And>t. RBT_Impl.keys t \<noteq> [] \<Longrightarrow> List.fold min (RBT_Impl.keys t) top = 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   468
    List.fold min (hd(RBT_Impl.keys t) # tl(RBT_Impl.keys t)) top"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   469
    by (simp add: hd_Cons_tl[symmetric])
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   470
  { fix x :: "_ :: {linorder, bounded_lattice_top}" and xs
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   471
    have "List.fold min (x#xs) top = List.fold min xs x"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   472
    by (simp add: inf_min[symmetric])
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   473
  } note ** = this
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   474
  show ?thesis using assms
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   475
    unfolding fold_keys_def_alt fold1_keys_def_alt is_empty_empty
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   476
    apply transfer 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   477
    apply (case_tac t) 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   478
    apply simp 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   479
    apply (subst *)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   480
    apply simp
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   481
    apply (subst **)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   482
    apply simp
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   483
  done
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   484
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   485
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   486
(* maximum *)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   487
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   488
lift_definition r_max :: "('a :: linorder, unit) rbt \<Rightarrow> 'a" is rbt_max by simp
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   489
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   490
lift_definition r_max_opt :: "('a :: linorder, unit) rbt \<Rightarrow> 'a" is rbt_max_opt by simp
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   491
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   492
lemma r_max_alt_def: "r_max t = fold1_keys max t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   493
by transfer (simp add: rbt_max_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   494
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   495
lemma r_max_eq_r_max_opt:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   496
  assumes "\<not> (is_empty t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   497
  shows "r_max t = r_max_opt t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   498
using assms unfolding is_empty_empty by transfer (auto intro: rbt_max_eq_rbt_max_opt)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   499
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   500
lemma fold_keys_max_bot_eq:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   501
  fixes t :: "('a :: {linorder, bounded_lattice_bot}, unit) rbt"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   502
  assumes "\<not> (is_empty t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   503
  shows "fold_keys max t bot = fold1_keys max t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   504
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   505
  have *: "\<And>t. RBT_Impl.keys t \<noteq> [] \<Longrightarrow> List.fold max (RBT_Impl.keys t) bot = 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   506
    List.fold max (hd(RBT_Impl.keys t) # tl(RBT_Impl.keys t)) bot"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   507
    by (simp add: hd_Cons_tl[symmetric])
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   508
  { fix x :: "_ :: {linorder, bounded_lattice_bot}" and xs
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   509
    have "List.fold max (x#xs) bot = List.fold max xs x"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   510
    by (simp add: sup_max[symmetric])
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   511
  } note ** = this
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   512
  show ?thesis using assms
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   513
    unfolding fold_keys_def_alt fold1_keys_def_alt is_empty_empty
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   514
    apply transfer 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   515
    apply (case_tac t) 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   516
    apply simp 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   517
    apply (subst *)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   518
    apply simp
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   519
    apply (subst **)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   520
    apply simp
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   521
  done
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   522
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   523
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   524
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   525
section {* Code equations *}
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   526
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   527
code_datatype Set Coset
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   528
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   529
lemma empty_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   530
  "Set.empty = Set RBT.empty"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   531
by (auto simp: Set_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   532
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   533
lemma UNIV_Coset [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   534
  "UNIV = Coset RBT.empty"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   535
by (auto simp: Set_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   536
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   537
lemma is_empty_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   538
  "Set.is_empty (Set t) = RBT.is_empty t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   539
  unfolding Set.is_empty_def by (auto simp: fun_eq_iff Set_def intro: lookup_empty_empty[THEN iffD1])
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   540
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   541
lemma compl_code [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   542
  "- Set xs = Coset xs"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   543
  "- Coset xs = Set xs"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   544
by (simp_all add: Set_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   545
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   546
lemma member_code [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   547
  "x \<in> (Set t) = (RBT.lookup t x = Some ())"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   548
  "x \<in> (Coset t) = (RBT.lookup t x = None)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   549
by (simp_all add: Set_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   550
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   551
lemma insert_code [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   552
  "Set.insert x (Set t) = Set (RBT.insert x () t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   553
  "Set.insert x (Coset t) = Coset (RBT.delete x t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   554
by (auto simp: Set_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   555
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   556
lemma remove_code [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   557
  "Set.remove x (Set t) = Set (RBT.delete x t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   558
  "Set.remove x (Coset t) = Coset (RBT.insert x () t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   559
by (auto simp: Set_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   560
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   561
lemma union_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   562
  "Set t \<union> A = fold_keys Set.insert t A"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   563
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   564
  interpret comp_fun_idem Set.insert
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   565
    by (fact comp_fun_idem_insert)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   566
  from finite_fold_fold_keys[OF `comp_fun_commute Set.insert`]
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   567
  show ?thesis by (auto simp add: union_fold_insert)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   568
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   569
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   570
lemma inter_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   571
  "A \<inter> Set t = rbt_filter (\<lambda>k. k \<in> A) t"
49758
718f10c8bbfc use Set.filter instead of Finite_Set.filter, which is removed then
kuncar
parents: 49757
diff changeset
   572
by (simp add: inter_Set_filter Set_filter_rbt_filter)
48623
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   573
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   574
lemma minus_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   575
  "A - Set t = fold_keys Set.remove t A"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   576
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   577
  interpret comp_fun_idem Set.remove
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   578
    by (fact comp_fun_idem_remove)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   579
  from finite_fold_fold_keys[OF `comp_fun_commute Set.remove`]
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   580
  show ?thesis by (auto simp add: minus_fold_remove)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   581
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   582
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   583
lemma union_Coset [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   584
  "Coset t \<union> A = - rbt_filter (\<lambda>k. k \<notin> A) t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   585
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   586
  have *: "\<And>A B. (-A \<union> B) = -(-B \<inter> A)" by blast
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   587
  show ?thesis by (simp del: boolean_algebra_class.compl_inf add: * inter_Set)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   588
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   589
 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   590
lemma union_Set_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   591
  "Set t1 \<union> Set t2 = Set (union t1 t2)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   592
by (auto simp add: lookup_union map_add_Some_iff Set_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   593
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   594
lemma inter_Coset [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   595
  "A \<inter> Coset t = fold_keys Set.remove t A"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   596
by (simp add: Diff_eq [symmetric] minus_Set)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   597
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   598
lemma inter_Coset_Coset [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   599
  "Coset t1 \<inter> Coset t2 = Coset (union t1 t2)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   600
by (auto simp add: lookup_union map_add_Some_iff Set_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   601
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   602
lemma minus_Coset [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   603
  "A - Coset t = rbt_filter (\<lambda>k. k \<in> A) t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   604
by (simp add: inter_Set[simplified Int_commute])
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   605
49757
73ab6d4a9236 rename Set.project to Set.filter - more appropriate name
kuncar
parents: 48623
diff changeset
   606
lemma filter_Set [code]:
73ab6d4a9236 rename Set.project to Set.filter - more appropriate name
kuncar
parents: 48623
diff changeset
   607
  "Set.filter P (Set t) = (rbt_filter P t)"
49758
718f10c8bbfc use Set.filter instead of Finite_Set.filter, which is removed then
kuncar
parents: 49757
diff changeset
   608
by (auto simp add: Set_filter_rbt_filter)
48623
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   609
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   610
lemma image_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   611
  "image f (Set t) = fold_keys (\<lambda>k A. Set.insert (f k) A) t {}"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   612
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   613
  have "comp_fun_commute (\<lambda>k. Set.insert (f k))" by default auto
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   614
  then show ?thesis by (auto simp add: image_fold_insert intro!: finite_fold_fold_keys)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   615
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   616
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   617
lemma Ball_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   618
  "Ball (Set t) P \<longleftrightarrow> foldi (\<lambda>s. s = True) (\<lambda>k v s. s \<and> P k) t True"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   619
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   620
  have "comp_fun_commute (\<lambda>k s. s \<and> P k)" by default auto
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   621
  then show ?thesis 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   622
    by (simp add: foldi_fold_conj[symmetric] Ball_fold finite_fold_fold_keys)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   623
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   624
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   625
lemma Bex_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   626
  "Bex (Set t) P \<longleftrightarrow> foldi (\<lambda>s. s = False) (\<lambda>k v s. s \<or> P k) t False"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   627
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   628
  have "comp_fun_commute (\<lambda>k s. s \<or> P k)" by default auto
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   629
  then show ?thesis 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   630
    by (simp add: foldi_fold_disj[symmetric] Bex_fold finite_fold_fold_keys)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   631
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   632
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   633
lemma subset_code [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   634
  "Set t \<le> B \<longleftrightarrow> (\<forall>x\<in>Set t. x \<in> B)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   635
  "A \<le> Coset t \<longleftrightarrow> (\<forall>y\<in>Set t. y \<notin> A)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   636
by auto
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   637
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   638
definition non_empty_trees where [code del]: "non_empty_trees t1 t2 \<longleftrightarrow> Coset t1 \<le> Set t2"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   639
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   640
code_abort non_empty_trees
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   641
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   642
lemma subset_Coset_empty_Set_empty [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   643
  "Coset t1 \<le> Set t2 \<longleftrightarrow> (case (impl_of t1, impl_of t2) of 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   644
    (rbt.Empty, rbt.Empty) => False |
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   645
    (_, _) => non_empty_trees t1 t2)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   646
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   647
  have *: "\<And>t. impl_of t = rbt.Empty \<Longrightarrow> t = RBT rbt.Empty"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   648
    by (subst(asm) RBT_inverse[symmetric]) (auto simp: impl_of_inject)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   649
  have **: "Lifting.invariant is_rbt rbt.Empty rbt.Empty" unfolding Lifting.invariant_def by simp
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   650
  show ?thesis  
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   651
    by (auto simp: Set_def lookup.abs_eq[OF **] dest!: * split: rbt.split simp: non_empty_trees_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   652
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   653
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   654
text {* A frequent case – avoid intermediate sets *}
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   655
lemma [code_unfold]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   656
  "Set t1 \<subseteq> Set t2 \<longleftrightarrow> foldi (\<lambda>s. s = True) (\<lambda>k v s. s \<and> k \<in> Set t2) t1 True"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   657
by (simp add: subset_code Ball_Set)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   658
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   659
lemma card_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   660
  "card (Set t) = fold_keys (\<lambda>_ n. n + 1) t 0"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   661
by (auto simp add: card_def fold_image_def intro!: finite_fold_fold_keys) (default, simp) 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   662
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   663
lemma setsum_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   664
  "setsum f (Set xs) = fold_keys (plus o f) xs 0"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   665
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   666
  have "comp_fun_commute (\<lambda>x. op + (f x))" by default (auto simp: add_ac)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   667
  then show ?thesis 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   668
    by (auto simp add: setsum_def fold_image_def finite_fold_fold_keys o_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   669
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   670
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   671
definition not_a_singleton_tree  where [code del]: "not_a_singleton_tree x y = x y"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   672
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   673
code_abort not_a_singleton_tree
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   674
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   675
lemma the_elem_set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   676
  fixes t :: "('a :: linorder, unit) rbt"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   677
  shows "the_elem (Set t) = (case impl_of t of 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   678
    (Branch RBT_Impl.B RBT_Impl.Empty x () RBT_Impl.Empty) \<Rightarrow> x
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   679
    | _ \<Rightarrow> not_a_singleton_tree the_elem (Set t))"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   680
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   681
  {
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   682
    fix x :: "'a :: linorder"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   683
    let ?t = "Branch RBT_Impl.B RBT_Impl.Empty x () RBT_Impl.Empty" 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   684
    have *:"?t \<in> {t. is_rbt t}" unfolding is_rbt_def by auto
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   685
    then have **:"Lifting.invariant is_rbt ?t ?t" unfolding Lifting.invariant_def by auto
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   686
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   687
    have "impl_of t = ?t \<Longrightarrow> the_elem (Set t) = x" 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   688
      by (subst(asm) RBT_inverse[symmetric, OF *])
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   689
        (auto simp: Set_def the_elem_def lookup.abs_eq[OF **] impl_of_inject)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   690
  }
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   691
  then show ?thesis unfolding not_a_singleton_tree_def
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   692
    by(auto split: rbt.split unit.split color.split)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   693
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   694
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   695
lemma Pow_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   696
  "Pow (Set t) = fold_keys (\<lambda>x A. A \<union> Set.insert x ` A) t {{}}"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   697
by (simp add: Pow_fold finite_fold_fold_keys[OF comp_fun_commute_Pow_fold])
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   698
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   699
lemma product_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   700
  "Product_Type.product (Set t1) (Set t2) = 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   701
    fold_keys (\<lambda>x A. fold_keys (\<lambda>y. Set.insert (x, y)) t2 A) t1 {}"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   702
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   703
  have *:"\<And>x. comp_fun_commute (\<lambda>y. Set.insert (x, y))" by default auto
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   704
  show ?thesis using finite_fold_fold_keys[OF comp_fun_commute_product_fold, of "Set t2" "{}" "t1"]  
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   705
    by (simp add: product_fold Product_Type.product_def finite_fold_fold_keys[OF *])
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   706
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   707
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   708
lemma Id_on_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   709
  "Id_on (Set t) =  fold_keys (\<lambda>x. Set.insert (x, x)) t {}"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   710
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   711
  have "comp_fun_commute (\<lambda>x. Set.insert (x, x))" by default auto
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   712
  then show ?thesis
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   713
    by (auto simp add: Id_on_fold intro!: finite_fold_fold_keys)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   714
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   715
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   716
lemma Image_Set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   717
  "(Set t) `` S = fold_keys (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A) t {}"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   718
by (auto simp add: Image_fold finite_fold_fold_keys[OF comp_fun_commute_Image_fold])
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   719
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   720
lemma trancl_set_ntrancl [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   721
  "trancl (Set t) = ntrancl (card (Set t) - 1) (Set t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   722
by (simp add: finite_trancl_ntranl)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   723
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   724
lemma relcomp_Set[code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   725
  "(Set t1) O (Set t2) = fold_keys 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   726
    (\<lambda>(x,y) A. fold_keys (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') t2 A) t1 {}"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   727
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   728
  interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   729
  have *: "\<And>x y. comp_fun_commute (\<lambda>(w, z) A'. if y = w then Set.insert (x, z) A' else A')"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   730
    by default (auto simp add: fun_eq_iff)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   731
  show ?thesis using finite_fold_fold_keys[OF comp_fun_commute_relcomp_fold, of "Set t2" "{}" t1]
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   732
    by (simp add: relcomp_fold finite_fold_fold_keys[OF *])
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   733
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   734
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   735
lemma wf_set [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   736
  "wf (Set t) = acyclic (Set t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   737
by (simp add: wf_iff_acyclic_if_finite)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   738
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   739
definition not_non_empty_tree  where [code del]: "not_non_empty_tree x y = x y"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   740
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   741
code_abort not_non_empty_tree
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   742
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   743
lemma Min_fin_set_fold [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   744
  "Min (Set t) = (if is_empty t then not_non_empty_tree Min (Set t) else r_min_opt t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   745
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   746
  have *:"(class.ab_semigroup_mult (min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a))" using ab_semigroup_idem_mult_min
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   747
    unfolding class.ab_semigroup_idem_mult_def by blast
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   748
  show ?thesis
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   749
    by (auto simp add: Min_def not_non_empty_tree_def finite_fold1_fold1_keys[OF *] r_min_alt_def 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   750
      r_min_eq_r_min_opt[symmetric])  
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   751
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   752
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   753
lemma Inf_fin_set_fold [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   754
  "Inf_fin (Set t) = Min (Set t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   755
by (simp add: inf_min Inf_fin_def Min_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   756
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   757
lemma Inf_Set_fold:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   758
  fixes t :: "('a :: {linorder, complete_lattice}, unit) rbt"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   759
  shows "Inf (Set t) = (if is_empty t then top else r_min_opt t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   760
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   761
  have "comp_fun_commute (min :: 'a \<Rightarrow> 'a \<Rightarrow> 'a)" by default (simp add: fun_eq_iff ac_simps)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   762
  then have "t \<noteq> empty \<Longrightarrow> Finite_Set.fold min top (Set t) = fold1_keys min t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   763
    by (simp add: finite_fold_fold_keys fold_keys_min_top_eq)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   764
  then show ?thesis 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   765
    by (auto simp add: Inf_fold_inf inf_min empty_Set[symmetric] r_min_eq_r_min_opt[symmetric] r_min_alt_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   766
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   767
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   768
definition Inf' :: "'a :: {linorder, complete_lattice} set \<Rightarrow> 'a" where [code del]: "Inf' x = Inf x"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   769
declare Inf'_def[symmetric, code_unfold]
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   770
declare Inf_Set_fold[folded Inf'_def, code]
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   771
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   772
lemma INFI_Set_fold [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   773
  "INFI (Set t) f = fold_keys (inf \<circ> f) t top"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   774
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   775
  have "comp_fun_commute ((inf :: 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<circ> f)" 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   776
    by default (auto simp add: fun_eq_iff ac_simps)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   777
  then show ?thesis
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   778
    by (auto simp: INF_fold_inf finite_fold_fold_keys)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   779
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   780
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   781
lemma Max_fin_set_fold [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   782
  "Max (Set t) = (if is_empty t then not_non_empty_tree Max (Set t) else r_max_opt t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   783
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   784
  have *:"(class.ab_semigroup_mult (max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a))" using ab_semigroup_idem_mult_max
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   785
    unfolding class.ab_semigroup_idem_mult_def by blast
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   786
  show ?thesis
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   787
    by (auto simp add: Max_def not_non_empty_tree_def finite_fold1_fold1_keys[OF *] r_max_alt_def 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   788
      r_max_eq_r_max_opt[symmetric])  
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   789
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   790
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   791
lemma Sup_fin_set_fold [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   792
  "Sup_fin (Set t) = Max (Set t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   793
by (simp add: sup_max Sup_fin_def Max_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   794
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   795
lemma Sup_Set_fold:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   796
  fixes t :: "('a :: {linorder, complete_lattice}, unit) rbt"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   797
  shows "Sup (Set t) = (if is_empty t then bot else r_max_opt t)"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   798
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   799
  have "comp_fun_commute (max :: 'a \<Rightarrow> 'a \<Rightarrow> 'a)" by default (simp add: fun_eq_iff ac_simps)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   800
  then have "t \<noteq> empty \<Longrightarrow> Finite_Set.fold max bot (Set t) = fold1_keys max t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   801
    by (simp add: finite_fold_fold_keys fold_keys_max_bot_eq)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   802
  then show ?thesis 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   803
    by (auto simp add: Sup_fold_sup sup_max empty_Set[symmetric] r_max_eq_r_max_opt[symmetric] r_max_alt_def)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   804
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   805
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   806
definition Sup' :: "'a :: {linorder, complete_lattice} set \<Rightarrow> 'a" where [code del]: "Sup' x = Sup x"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   807
declare Sup'_def[symmetric, code_unfold]
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   808
declare Sup_Set_fold[folded Sup'_def, code]
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   809
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   810
lemma SUPR_Set_fold [code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   811
  "SUPR (Set t) f = fold_keys (sup \<circ> f) t bot"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   812
proof -
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   813
  have "comp_fun_commute ((sup :: 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<circ> f)" 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   814
    by default (auto simp add: fun_eq_iff ac_simps)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   815
  then show ?thesis
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   816
    by (auto simp: SUP_fold_sup finite_fold_fold_keys)
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   817
qed
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   818
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   819
lemma sorted_list_set[code]:
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   820
  "sorted_list_of_set (Set t) = keys t"
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   821
by (auto simp add: set_keys intro: sorted_distinct_set_unique) 
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   822
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   823
hide_const (open) RBT_Set.Set RBT_Set.Coset
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   824
bea613f2543d implementation of sets by RBT trees for the code generator
kuncar
parents:
diff changeset
   825
end