src/HOL/Library/RBT_Impl.thy
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(*  Title:      HOL/Library/RBT_Impl.thy
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    Author:     Markus Reiter, TU Muenchen
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    Author:     Alexander Krauss, TU Muenchen
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*)
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section \<open>Implementation of Red-Black Trees\<close>
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theory RBT_Impl
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imports Main
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begin
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text \<open>
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  For applications, you should use theory @{text RBT} which defines
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  an abstract type of red-black tree obeying the invariant.
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\<close>
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subsection \<open>Datatype of RB trees\<close>
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datatype color = R | B
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datatype ('a, 'b) rbt = Empty | Branch color "('a, 'b) rbt" 'a 'b "('a, 'b) rbt"
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lemma rbt_cases:
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  obtains (Empty) "t = Empty" 
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  | (Red) l k v r where "t = Branch R l k v r" 
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  | (Black) l k v r where "t = Branch B l k v r"
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proof (cases t)
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  case Empty with that show thesis by blast
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next
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  case (Branch c) with that show thesis by (cases c) blast+
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qed
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subsection \<open>Tree properties\<close>
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subsubsection \<open>Content of a tree\<close>
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primrec entries :: "('a, 'b) rbt \<Rightarrow> ('a \<times> 'b) list"
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where 
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  "entries Empty = []"
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| "entries (Branch _ l k v r) = entries l @ (k,v) # entries r"
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abbreviation (input) entry_in_tree :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
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where
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  "entry_in_tree k v t \<equiv> (k, v) \<in> set (entries t)"
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definition keys :: "('a, 'b) rbt \<Rightarrow> 'a list" where
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  "keys t = map fst (entries t)"
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lemma keys_simps [simp, code]:
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  "keys Empty = []"
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  "keys (Branch c l k v r) = keys l @ k # keys r"
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  by (simp_all add: keys_def)
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lemma entry_in_tree_keys:
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  assumes "(k, v) \<in> set (entries t)"
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  shows "k \<in> set (keys t)"
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proof -
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  from assms have "fst (k, v) \<in> fst ` set (entries t)" by (rule imageI)
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  then show ?thesis by (simp add: keys_def)
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qed
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lemma keys_entries:
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  "k \<in> set (keys t) \<longleftrightarrow> (\<exists>v. (k, v) \<in> set (entries t))"
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  by (auto intro: entry_in_tree_keys) (auto simp add: keys_def)
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lemma non_empty_rbt_keys: 
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  "t \<noteq> rbt.Empty \<Longrightarrow> keys t \<noteq> []"
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  by (cases t) simp_all
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subsubsection \<open>Search tree properties\<close>
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context ord begin
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definition rbt_less :: "'a \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
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where
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  rbt_less_prop: "rbt_less k t \<longleftrightarrow> (\<forall>x\<in>set (keys t). x < k)"
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abbreviation rbt_less_symbol (infix "|\<guillemotleft>" 50)
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where "t |\<guillemotleft> x \<equiv> rbt_less x t"
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definition rbt_greater :: "'a \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool" (infix "\<guillemotleft>|" 50) 
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where
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  rbt_greater_prop: "rbt_greater k t = (\<forall>x\<in>set (keys t). k < x)"
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lemma rbt_less_simps [simp]:
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  "Empty |\<guillemotleft> k = True"
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  "Branch c lt kt v rt |\<guillemotleft> k \<longleftrightarrow> kt < k \<and> lt |\<guillemotleft> k \<and> rt |\<guillemotleft> k"
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  by (auto simp add: rbt_less_prop)
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lemma rbt_greater_simps [simp]:
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  "k \<guillemotleft>| Empty = True"
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  "k \<guillemotleft>| (Branch c lt kt v rt) \<longleftrightarrow> k < kt \<and> k \<guillemotleft>| lt \<and> k \<guillemotleft>| rt"
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  by (auto simp add: rbt_greater_prop)
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lemmas rbt_ord_props = rbt_less_prop rbt_greater_prop
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lemmas rbt_greater_nit = rbt_greater_prop entry_in_tree_keys
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lemmas rbt_less_nit = rbt_less_prop entry_in_tree_keys
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lemma (in order)
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  shows rbt_less_eq_trans: "l |\<guillemotleft> u \<Longrightarrow> u \<le> v \<Longrightarrow> l |\<guillemotleft> v"
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  and rbt_less_trans: "t |\<guillemotleft> x \<Longrightarrow> x < y \<Longrightarrow> t |\<guillemotleft> y"
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  and rbt_greater_eq_trans: "u \<le> v \<Longrightarrow> v \<guillemotleft>| r \<Longrightarrow> u \<guillemotleft>| r"
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  and rbt_greater_trans: "x < y \<Longrightarrow> y \<guillemotleft>| t \<Longrightarrow> x \<guillemotleft>| t"
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  by (auto simp: rbt_ord_props)
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primrec rbt_sorted :: "('a, 'b) rbt \<Rightarrow> bool"
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where
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  "rbt_sorted Empty = True"
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| "rbt_sorted (Branch c l k v r) = (l |\<guillemotleft> k \<and> k \<guillemotleft>| r \<and> rbt_sorted l \<and> rbt_sorted r)"
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end
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context linorder begin
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lemma rbt_sorted_entries:
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  "rbt_sorted t \<Longrightarrow> List.sorted (map fst (entries t))"
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by (induct t) 
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  (force simp: sorted_append sorted_Cons rbt_ord_props 
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      dest!: entry_in_tree_keys)+
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lemma distinct_entries:
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  "rbt_sorted t \<Longrightarrow> distinct (map fst (entries t))"
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by (induct t) 
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  (force simp: sorted_append sorted_Cons rbt_ord_props 
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      dest!: entry_in_tree_keys)+
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lemma distinct_keys:
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  "rbt_sorted t \<Longrightarrow> distinct (keys t)"
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  by (simp add: distinct_entries keys_def)
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subsubsection \<open>Tree lookup\<close>
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primrec (in ord) rbt_lookup :: "('a, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"
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where
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  "rbt_lookup Empty k = None"
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| "rbt_lookup (Branch _ l x y r) k = 
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   (if k < x then rbt_lookup l k else if x < k then rbt_lookup r k else Some y)"
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lemma rbt_lookup_keys: "rbt_sorted t \<Longrightarrow> dom (rbt_lookup t) = set (keys t)"
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  by (induct t) (auto simp: dom_def rbt_greater_prop rbt_less_prop)
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lemma dom_rbt_lookup_Branch: 
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  "rbt_sorted (Branch c t1 k v t2) \<Longrightarrow> 
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    dom (rbt_lookup (Branch c t1 k v t2)) 
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    = Set.insert k (dom (rbt_lookup t1) \<union> dom (rbt_lookup t2))"
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proof -
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  assume "rbt_sorted (Branch c t1 k v t2)"
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  then show ?thesis by (simp add: rbt_lookup_keys)
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qed
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   151
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lemma finite_dom_rbt_lookup [simp, intro!]: "finite (dom (rbt_lookup t))"
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proof (induct t)
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  case Empty then show ?case by simp
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next
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  case (Branch color t1 a b t2)
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  let ?A = "Set.insert a (dom (rbt_lookup t1) \<union> dom (rbt_lookup t2))"
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  have "dom (rbt_lookup (Branch color t1 a b t2)) \<subseteq> ?A" by (auto split: split_if_asm)
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  moreover from Branch have "finite (insert a (dom (rbt_lookup t1) \<union> dom (rbt_lookup t2)))" by simp
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  ultimately show ?case by (rule finite_subset)
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qed 
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end
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context ord begin
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lemma rbt_lookup_rbt_less[simp]: "t |\<guillemotleft> k \<Longrightarrow> rbt_lookup t k = None" 
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by (induct t) auto
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lemma rbt_lookup_rbt_greater[simp]: "k \<guillemotleft>| t \<Longrightarrow> rbt_lookup t k = None"
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by (induct t) auto
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lemma rbt_lookup_Empty: "rbt_lookup Empty = empty"
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by (rule ext) simp
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end
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context linorder begin
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lemma map_of_entries:
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  "rbt_sorted t \<Longrightarrow> map_of (entries t) = rbt_lookup t"
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proof (induct t)
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  case Empty thus ?case by (simp add: rbt_lookup_Empty)
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next
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  case (Branch c t1 k v t2)
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  have "rbt_lookup (Branch c t1 k v t2) = rbt_lookup t2 ++ [k\<mapsto>v] ++ rbt_lookup t1"
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  proof (rule ext)
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    fix x
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    from Branch have RBT_SORTED: "rbt_sorted (Branch c t1 k v t2)" by simp
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    let ?thesis = "rbt_lookup (Branch c t1 k v t2) x = (rbt_lookup t2 ++ [k \<mapsto> v] ++ rbt_lookup t1) x"
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   191
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    have DOM_T1: "!!k'. k'\<in>dom (rbt_lookup t1) \<Longrightarrow> k>k'"
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   193
    proof -
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      fix k'
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      from RBT_SORTED have "t1 |\<guillemotleft> k" by simp
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      with rbt_less_prop have "\<forall>k'\<in>set (keys t1). k>k'" by auto
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      moreover assume "k'\<in>dom (rbt_lookup t1)"
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      ultimately show "k>k'" using rbt_lookup_keys RBT_SORTED by auto
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    qed
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   200
    
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    have DOM_T2: "!!k'. k'\<in>dom (rbt_lookup t2) \<Longrightarrow> k<k'"
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   202
    proof -
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      fix k'
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      from RBT_SORTED have "k \<guillemotleft>| t2" by simp
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      with rbt_greater_prop have "\<forall>k'\<in>set (keys t2). k<k'" by auto
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      moreover assume "k'\<in>dom (rbt_lookup t2)"
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      ultimately show "k<k'" using rbt_lookup_keys RBT_SORTED by auto
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    qed
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    {
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      assume C: "x<k"
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      hence "rbt_lookup (Branch c t1 k v t2) x = rbt_lookup t1 x" by simp
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      moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
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      moreover have "x \<notin> dom (rbt_lookup t2)"
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      proof
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        assume "x \<in> dom (rbt_lookup t2)"
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        with DOM_T2 have "k<x" by blast
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        with C show False by simp
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      qed
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      ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
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    } moreover {
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      assume [simp]: "x=k"
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      hence "rbt_lookup (Branch c t1 k v t2) x = [k \<mapsto> v] x" by simp
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      moreover have "x \<notin> dom (rbt_lookup t1)" 
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      proof
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        assume "x \<in> dom (rbt_lookup t1)"
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        with DOM_T1 have "k>x" by blast
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        thus False by simp
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   229
      qed
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      ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
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   231
    } moreover {
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      assume C: "x>k"
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      hence "rbt_lookup (Branch c t1 k v t2) x = rbt_lookup t2 x" by (simp add: less_not_sym[of k x])
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      moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
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      moreover have "x\<notin>dom (rbt_lookup t1)" proof
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        assume "x\<in>dom (rbt_lookup t1)"
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        with DOM_T1 have "k>x" by simp
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        with C show False by simp
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   239
      qed
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   240
      ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
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    } ultimately show ?thesis using less_linear by blast
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   242
  qed
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   243
  also from Branch 
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  have "rbt_lookup t2 ++ [k \<mapsto> v] ++ rbt_lookup t1 = map_of (entries (Branch c t1 k v t2))" by simp
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   245
  finally show ?case by simp
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   246
qed
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   247
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lemma rbt_lookup_in_tree: "rbt_sorted t \<Longrightarrow> rbt_lookup t k = Some v \<longleftrightarrow> (k, v) \<in> set (entries t)"
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  by (simp add: map_of_entries [symmetric] distinct_entries)
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lemma set_entries_inject:
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  assumes rbt_sorted: "rbt_sorted t1" "rbt_sorted t2" 
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   253
  shows "set (entries t1) = set (entries t2) \<longleftrightarrow> entries t1 = entries t2"
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   254
proof -
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  from rbt_sorted have "distinct (map fst (entries t1))"
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   256
    "distinct (map fst (entries t2))"
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    by (auto intro: distinct_entries)
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  with rbt_sorted show ?thesis
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    by (auto intro: map_sorted_distinct_set_unique rbt_sorted_entries simp add: distinct_map)
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   260
qed
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   261
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lemma entries_eqI:
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  assumes rbt_sorted: "rbt_sorted t1" "rbt_sorted t2" 
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   264
  assumes rbt_lookup: "rbt_lookup t1 = rbt_lookup t2"
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   265
  shows "entries t1 = entries t2"
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   266
proof -
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   267
  from rbt_sorted rbt_lookup have "map_of (entries t1) = map_of (entries t2)"
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   268
    by (simp add: map_of_entries)
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  with rbt_sorted have "set (entries t1) = set (entries t2)"
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    by (simp add: map_of_inject_set distinct_entries)
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  with rbt_sorted show ?thesis by (simp add: set_entries_inject)
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qed
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   273
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lemma entries_rbt_lookup:
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  assumes "rbt_sorted t1" "rbt_sorted t2" 
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   276
  shows "entries t1 = entries t2 \<longleftrightarrow> rbt_lookup t1 = rbt_lookup t2"
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  using assms by (auto intro: entries_eqI simp add: map_of_entries [symmetric])
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   278
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lemma rbt_lookup_from_in_tree: 
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  assumes "rbt_sorted t1" "rbt_sorted t2" 
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  and "\<And>v. (k, v) \<in> set (entries t1) \<longleftrightarrow> (k, v) \<in> set (entries t2)" 
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  shows "rbt_lookup t1 k = rbt_lookup t2 k"
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   283
proof -
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  from assms have "k \<in> dom (rbt_lookup t1) \<longleftrightarrow> k \<in> dom (rbt_lookup t2)"
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   285
    by (simp add: keys_entries rbt_lookup_keys)
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   286
  with assms show ?thesis by (auto simp add: rbt_lookup_in_tree [symmetric])
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qed
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   289
end
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60500
903bb1495239 isabelle update_cartouches;
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   291
subsubsection \<open>Red-black properties\<close>
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   292
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primrec color_of :: "('a, 'b) rbt \<Rightarrow> color"
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where
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  "color_of Empty = B"
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| "color_of (Branch c _ _ _ _) = c"
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primrec bheight :: "('a,'b) rbt \<Rightarrow> nat"
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where
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  "bheight Empty = 0"
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| "bheight (Branch c lt k v rt) = (if c = B then Suc (bheight lt) else bheight lt)"
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primrec inv1 :: "('a, 'b) rbt \<Rightarrow> bool"
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where
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  "inv1 Empty = True"
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| "inv1 (Branch c lt k v rt) \<longleftrightarrow> inv1 lt \<and> inv1 rt \<and> (c = B \<or> color_of lt = B \<and> color_of rt = B)"
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primrec inv1l :: "('a, 'b) rbt \<Rightarrow> bool" -- \<open>Weaker version\<close>
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where
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  "inv1l Empty = True"
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| "inv1l (Branch c l k v r) = (inv1 l \<and> inv1 r)"
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lemma [simp]: "inv1 t \<Longrightarrow> inv1l t" by (cases t) simp+
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primrec inv2 :: "('a, 'b) rbt \<Rightarrow> bool"
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where
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  "inv2 Empty = True"
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| "inv2 (Branch c lt k v rt) = (inv2 lt \<and> inv2 rt \<and> bheight lt = bheight rt)"
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context ord begin
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definition is_rbt :: "('a, 'b) rbt \<Rightarrow> bool" where
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  "is_rbt t \<longleftrightarrow> inv1 t \<and> inv2 t \<and> color_of t = B \<and> rbt_sorted t"
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   323
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lemma is_rbt_rbt_sorted [simp]:
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  "is_rbt t \<Longrightarrow> rbt_sorted t" by (simp add: is_rbt_def)
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theorem Empty_is_rbt [simp]:
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  "is_rbt Empty" by (simp add: is_rbt_def)
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   330
end
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903bb1495239 isabelle update_cartouches;
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subsection \<open>Insertion\<close>
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52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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fun (* slow, due to massive case splitting *)
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  balance :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
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   336
where
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  "balance (Branch R a w x b) s t (Branch R c y z d) = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
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   338
  "balance (Branch R (Branch R a w x b) s t c) y z d = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
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   339
  "balance (Branch R a w x (Branch R b s t c)) y z d = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
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   340
  "balance a w x (Branch R b s t (Branch R c y z d)) = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
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   341
  "balance a w x (Branch R (Branch R b s t c) y z d) = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
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   342
  "balance a s t b = Branch B a s t b"
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52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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lemma balance_inv1: "\<lbrakk>inv1l l; inv1l r\<rbrakk> \<Longrightarrow> inv1 (balance l k v r)" 
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  by (induct l k v r rule: balance.induct) auto
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   346
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lemma balance_bheight: "bheight l = bheight r \<Longrightarrow> bheight (balance l k v r) = Suc (bheight l)"
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  by (induct l k v r rule: balance.induct) auto
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   349
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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lemma balance_inv2: 
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  assumes "inv2 l" "inv2 r" "bheight l = bheight r"
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  shows "inv2 (balance l k v r)"
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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   353
  using assms
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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   354
  by (induct l k v r rule: balance.induct) auto
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   355
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context ord begin
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   357
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   358
lemma balance_rbt_greater[simp]: "(v \<guillemotleft>| balance a k x b) = (v \<guillemotleft>| a \<and> v \<guillemotleft>| b \<and> v < k)" 
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52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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   359
  by (induct a k x b rule: balance.induct) auto
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   360
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lemma balance_rbt_less[simp]: "(balance a k x b |\<guillemotleft> v) = (a |\<guillemotleft> v \<and> b |\<guillemotleft> v \<and> k < v)"
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52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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   362
  by (induct a k x b rule: balance.induct) auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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   363
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   364
end
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   365
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   366
lemma (in linorder) balance_rbt_sorted: 
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   367
  fixes k :: "'a"
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   368
  assumes "rbt_sorted l" "rbt_sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
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   369
  shows "rbt_sorted (balance l k v r)"
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52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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   370
using assms proof (induct l k v r rule: balance.induct)
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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   371
  case ("2_2" a x w b y t c z s va vb vd vc)
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   372
  hence "y < z \<and> z \<guillemotleft>| Branch B va vb vd vc" 
47450
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   373
    by (auto simp add: rbt_ord_props)
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diff changeset
   374
  hence "y \<guillemotleft>| (Branch B va vb vd vc)" by (blast dest: rbt_greater_trans)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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   375
  with "2_2" show ?case by simp
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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diff changeset
   376
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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   377
  case ("3_2" va vb vd vc x w b y s c z)
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diff changeset
   378
  from "3_2" have "x < y \<and> Branch B va vb vd vc |\<guillemotleft> x" 
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diff changeset
   379
    by simp
47450
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diff changeset
   380
  hence "Branch B va vb vd vc |\<guillemotleft> y" by (blast dest: rbt_less_trans)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
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   381
  with "3_2" show ?case by simp
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
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diff changeset
   382
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
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diff changeset
   383
  case ("3_3" x w b y s c z t va vb vd vc)
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diff changeset
   384
  from "3_3" have "y < z \<and> z \<guillemotleft>| Branch B va vb vd vc" by simp
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parents: 47397
diff changeset
   385
  hence "y \<guillemotleft>| Branch B va vb vd vc" by (blast dest: rbt_greater_trans)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   386
  with "3_3" show ?case by simp
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   387
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   388
  case ("3_4" vd ve vg vf x w b y s c z t va vb vii vc)
47450
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parents: 47397
diff changeset
   389
  hence "x < y \<and> Branch B vd ve vg vf |\<guillemotleft> x" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   390
  hence 1: "Branch B vd ve vg vf |\<guillemotleft> y" by (blast dest: rbt_less_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   391
  from "3_4" have "y < z \<and> z \<guillemotleft>| Branch B va vb vii vc" by simp
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Andreas Lochbihler
parents: 47397
diff changeset
   392
  hence "y \<guillemotleft>| Branch B va vb vii vc" by (blast dest: rbt_greater_trans)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   393
  with 1 "3_4" show ?case by simp
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   394
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   395
  case ("4_2" va vb vd vc x w b y s c z t dd)
47450
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parents: 47397
diff changeset
   396
  hence "x < y \<and> Branch B va vb vd vc |\<guillemotleft> x" by simp
2ada2be850cb move RBT implementation into type class contexts
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parents: 47397
diff changeset
   397
  hence "Branch B va vb vd vc |\<guillemotleft> y" by (blast dest: rbt_less_trans)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   398
  with "4_2" show ?case by simp
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   399
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   400
  case ("5_2" x w b y s c z t va vb vd vc)
47450
2ada2be850cb move RBT implementation into type class contexts
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parents: 47397
diff changeset
   401
  hence "y < z \<and> z \<guillemotleft>| Branch B va vb vd vc" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   402
  hence "y \<guillemotleft>| Branch B va vb vd vc" by (blast dest: rbt_greater_trans)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   403
  with "5_2" show ?case by simp
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   404
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   405
  case ("5_3" va vb vd vc x w b y s c z t)
47450
2ada2be850cb move RBT implementation into type class contexts
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parents: 47397
diff changeset
   406
  hence "x < y \<and> Branch B va vb vd vc |\<guillemotleft> x" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   407
  hence "Branch B va vb vd vc |\<guillemotleft> y" by (blast dest: rbt_less_trans)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   408
  with "5_3" show ?case by simp
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   409
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   410
  case ("5_4" va vb vg vc x w b y s c z t vd ve vii vf)
47450
2ada2be850cb move RBT implementation into type class contexts
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parents: 47397
diff changeset
   411
  hence "x < y \<and> Branch B va vb vg vc |\<guillemotleft> x" by simp
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Andreas Lochbihler
parents: 47397
diff changeset
   412
  hence 1: "Branch B va vb vg vc |\<guillemotleft> y" by (blast dest: rbt_less_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   413
  from "5_4" have "y < z \<and> z \<guillemotleft>| Branch B vd ve vii vf" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   414
  hence "y \<guillemotleft>| Branch B vd ve vii vf" by (blast dest: rbt_greater_trans)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   415
  with 1 "5_4" show ?case by simp
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   416
qed simp+
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   417
35550
e2bc7f8d8d51 restructured RBT theory
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diff changeset
   418
lemma entries_balance [simp]:
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   419
  "entries (balance l k v r) = entries l @ (k, v) # entries r"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   420
  by (induct l k v r rule: balance.induct) auto
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   421
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
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diff changeset
   422
lemma keys_balance [simp]: 
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   423
  "keys (balance l k v r) = keys l @ k # keys r"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   424
  by (simp add: keys_def)
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   425
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   426
lemma balance_in_tree:  
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   427
  "entry_in_tree k x (balance l v y r) \<longleftrightarrow> entry_in_tree k x l \<or> k = v \<and> x = y \<or> entry_in_tree k x r"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   428
  by (auto simp add: keys_def)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   429
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   430
lemma (in linorder) rbt_lookup_balance[simp]: 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   431
fixes k :: "'a"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   432
assumes "rbt_sorted l" "rbt_sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   433
shows "rbt_lookup (balance l k v r) x = rbt_lookup (Branch B l k v r) x"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   434
by (rule rbt_lookup_from_in_tree) (auto simp:assms balance_in_tree balance_rbt_sorted)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   435
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   436
primrec paint :: "color \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   437
where
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   438
  "paint c Empty = Empty"
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   439
| "paint c (Branch _ l k v r) = Branch c l k v r"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   440
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   441
lemma paint_inv1l[simp]: "inv1l t \<Longrightarrow> inv1l (paint c t)" by (cases t) auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   442
lemma paint_inv1[simp]: "inv1l t \<Longrightarrow> inv1 (paint B t)" by (cases t) auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   443
lemma paint_inv2[simp]: "inv2 t \<Longrightarrow> inv2 (paint c t)" by (cases t) auto
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   444
lemma paint_color_of[simp]: "color_of (paint B t) = B" by (cases t) auto
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   445
lemma paint_in_tree[simp]: "entry_in_tree k x (paint c t) = entry_in_tree k x t" by (cases t) auto
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   446
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   447
context ord begin
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   448
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   449
lemma paint_rbt_sorted[simp]: "rbt_sorted t \<Longrightarrow> rbt_sorted (paint c t)" by (cases t) auto
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   450
lemma paint_rbt_lookup[simp]: "rbt_lookup (paint c t) = rbt_lookup t" by (rule ext) (cases t, auto)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   451
lemma paint_rbt_greater[simp]: "(v \<guillemotleft>| paint c t) = (v \<guillemotleft>| t)" by (cases t) auto
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   452
lemma paint_rbt_less[simp]: "(paint c t |\<guillemotleft> v) = (t |\<guillemotleft> v)" by (cases t) auto
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   453
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   454
fun
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   455
  rbt_ins :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   456
where
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   457
  "rbt_ins f k v Empty = Branch R Empty k v Empty" |
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   458
  "rbt_ins f k v (Branch B l x y r) = (if k < x then balance (rbt_ins f k v l) x y r
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   459
                                       else if k > x then balance l x y (rbt_ins f k v r)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   460
                                       else Branch B l x (f k y v) r)" |
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   461
  "rbt_ins f k v (Branch R l x y r) = (if k < x then Branch R (rbt_ins f k v l) x y r
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   462
                                       else if k > x then Branch R l x y (rbt_ins f k v r)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   463
                                       else Branch R l x (f k y v) r)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   464
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   465
lemma ins_inv1_inv2: 
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   466
  assumes "inv1 t" "inv2 t"
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   467
  shows "inv2 (rbt_ins f k x t)" "bheight (rbt_ins f k x t) = bheight t" 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   468
  "color_of t = B \<Longrightarrow> inv1 (rbt_ins f k x t)" "inv1l (rbt_ins f k x t)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   469
  using assms
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   470
  by (induct f k x t rule: rbt_ins.induct) (auto simp: balance_inv1 balance_inv2 balance_bheight)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   471
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   472
end
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   473
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   474
context linorder begin
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   475
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   476
lemma ins_rbt_greater[simp]: "(v \<guillemotleft>| rbt_ins f (k :: 'a) x t) = (v \<guillemotleft>| t \<and> k > v)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   477
  by (induct f k x t rule: rbt_ins.induct) auto
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   478
lemma ins_rbt_less[simp]: "(rbt_ins f k x t |\<guillemotleft> v) = (t |\<guillemotleft> v \<and> k < v)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   479
  by (induct f k x t rule: rbt_ins.induct) auto
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   480
lemma ins_rbt_sorted[simp]: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_ins f k x t)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   481
  by (induct f k x t rule: rbt_ins.induct) (auto simp: balance_rbt_sorted)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   482
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   483
lemma keys_ins: "set (keys (rbt_ins f k v t)) = { k } \<union> set (keys t)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   484
  by (induct f k v t rule: rbt_ins.induct) auto
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   485
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   486
lemma rbt_lookup_ins: 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   487
  fixes k :: "'a"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   488
  assumes "rbt_sorted t"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   489
  shows "rbt_lookup (rbt_ins f k v t) x = ((rbt_lookup t)(k |-> case rbt_lookup t k of None \<Rightarrow> v 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   490
                                                                | Some w \<Rightarrow> f k w v)) x"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   491
using assms by (induct f k v t rule: rbt_ins.induct) auto
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   492
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   493
end
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   494
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   495
context ord begin
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   496
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   497
definition rbt_insert_with_key :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   498
where "rbt_insert_with_key f k v t = paint B (rbt_ins f k v t)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   499
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   500
definition rbt_insertw_def: "rbt_insert_with f = rbt_insert_with_key (\<lambda>_. f)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   501
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   502
definition rbt_insert :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   503
  "rbt_insert = rbt_insert_with_key (\<lambda>_ _ nv. nv)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   504
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   505
end
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   506
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   507
context linorder begin
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   508
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   509
lemma rbt_insertwk_rbt_sorted: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_insert_with_key f (k :: 'a) x t)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   510
  by (auto simp: rbt_insert_with_key_def)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   511
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   512
theorem rbt_insertwk_is_rbt: 
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   513
  assumes inv: "is_rbt t" 
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   514
  shows "is_rbt (rbt_insert_with_key f k x t)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   515
using assms
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   516
unfolding rbt_insert_with_key_def is_rbt_def
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   517
by (auto simp: ins_inv1_inv2)
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   518
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   519
lemma rbt_lookup_rbt_insertwk: 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   520
  assumes "rbt_sorted t"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   521
  shows "rbt_lookup (rbt_insert_with_key f k v t) x = ((rbt_lookup t)(k |-> case rbt_lookup t k of None \<Rightarrow> v 
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   522
                                                       | Some w \<Rightarrow> f k w v)) x"
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   523
unfolding rbt_insert_with_key_def using assms
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   524
by (simp add:rbt_lookup_ins)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   525
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   526
lemma rbt_insertw_rbt_sorted: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_insert_with f k v t)" 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   527
  by (simp add: rbt_insertwk_rbt_sorted rbt_insertw_def)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   528
theorem rbt_insertw_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (rbt_insert_with f k v t)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   529
  by (simp add: rbt_insertwk_is_rbt rbt_insertw_def)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   530
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   531
lemma rbt_lookup_rbt_insertw:
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   532
  assumes "is_rbt t"
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   533
  shows "rbt_lookup (rbt_insert_with f k v t) = (rbt_lookup t)(k \<mapsto> (if k:dom (rbt_lookup t) then f (the (rbt_lookup t k)) v else v))"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   534
using assms
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   535
unfolding rbt_insertw_def
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   536
by (rule_tac ext) (cases "rbt_lookup t k", auto simp:rbt_lookup_rbt_insertwk dom_def)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   537
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   538
lemma rbt_insert_rbt_sorted: "rbt_sorted t \<Longrightarrow> rbt_sorted (rbt_insert k v t)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   539
  by (simp add: rbt_insertwk_rbt_sorted rbt_insert_def)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   540
theorem rbt_insert_is_rbt [simp]: "is_rbt t \<Longrightarrow> is_rbt (rbt_insert k v t)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   541
  by (simp add: rbt_insertwk_is_rbt rbt_insert_def)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   542
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   543
lemma rbt_lookup_rbt_insert: 
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   544
  assumes "is_rbt t"
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   545
  shows "rbt_lookup (rbt_insert k v t) = (rbt_lookup t)(k\<mapsto>v)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   546
unfolding rbt_insert_def
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   547
using assms
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   548
by (rule_tac ext) (simp add: rbt_lookup_rbt_insertwk split:option.split)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   549
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   550
end
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   551
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
   552
subsection \<open>Deletion\<close>
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   553
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   554
lemma bheight_paintR'[simp]: "color_of t = B \<Longrightarrow> bheight (paint R t) = bheight t - 1"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   555
by (cases t rule: rbt_cases) auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   556
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   557
fun
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   558
  balance_left :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   559
where
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   560
  "balance_left (Branch R a k x b) s y c = Branch R (Branch B a k x b) s y c" |
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   561
  "balance_left bl k x (Branch B a s y b) = balance bl k x (Branch R a s y b)" |
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   562
  "balance_left bl k x (Branch R (Branch B a s y b) t z c) = Branch R (Branch B bl k x a) s y (balance b t z (paint R c))" |
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   563
  "balance_left t k x s = Empty"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   564
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   565
lemma balance_left_inv2_with_inv1:
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   566
  assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "inv1 rt"
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   567
  shows "bheight (balance_left lt k v rt) = bheight lt + 1"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   568
  and   "inv2 (balance_left lt k v rt)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   569
using assms 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   570
by (induct lt k v rt rule: balance_left.induct) (auto simp: balance_inv2 balance_bheight)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   571
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   572
lemma balance_left_inv2_app: 
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   573
  assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "color_of rt = B"
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   574
  shows "inv2 (balance_left lt k v rt)" 
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   575
        "bheight (balance_left lt k v rt) = bheight rt"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   576
using assms 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   577
by (induct lt k v rt rule: balance_left.induct) (auto simp add: balance_inv2 balance_bheight)+ 
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   578
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   579
lemma balance_left_inv1: "\<lbrakk>inv1l a; inv1 b; color_of b = B\<rbrakk> \<Longrightarrow> inv1 (balance_left a k x b)"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   580
  by (induct a k x b rule: balance_left.induct) (simp add: balance_inv1)+
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   581
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   582
lemma balance_left_inv1l: "\<lbrakk> inv1l lt; inv1 rt \<rbrakk> \<Longrightarrow> inv1l (balance_left lt k x rt)"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   583
by (induct lt k x rt rule: balance_left.induct) (auto simp: balance_inv1)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   584
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   585
lemma (in linorder) balance_left_rbt_sorted: 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   586
  "\<lbrakk> rbt_sorted l; rbt_sorted r; rbt_less k l; k \<guillemotleft>| r \<rbrakk> \<Longrightarrow> rbt_sorted (balance_left l k v r)"
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   587
apply (induct l k v r rule: balance_left.induct)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   588
apply (auto simp: balance_rbt_sorted)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   589
apply (unfold rbt_greater_prop rbt_less_prop)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   590
by force+
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   591
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   592
context order begin
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   593
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   594
lemma balance_left_rbt_greater: 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   595
  fixes k :: "'a"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   596
  assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x" 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   597
  shows "k \<guillemotleft>| balance_left a x t b"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   598
using assms 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   599
by (induct a x t b rule: balance_left.induct) auto
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   600
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   601
lemma balance_left_rbt_less: 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   602
  fixes k :: "'a"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   603
  assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k" 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   604
  shows "balance_left a x t b |\<guillemotleft> k"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   605
using assms
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   606
by (induct a x t b rule: balance_left.induct) auto
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   607
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   608
end
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   609
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   610
lemma balance_left_in_tree: 
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   611
  assumes "inv1l l" "inv1 r" "bheight l + 1 = bheight r"
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   612
  shows "entry_in_tree k v (balance_left l a b r) = (entry_in_tree k v l \<or> k = a \<and> v = b \<or> entry_in_tree k v r)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   613
using assms 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   614
by (induct l k v r rule: balance_left.induct) (auto simp: balance_in_tree)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   615
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   616
fun
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   617
  balance_right :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   618
where
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   619
  "balance_right a k x (Branch R b s y c) = Branch R a k x (Branch B b s y c)" |
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   620
  "balance_right (Branch B a k x b) s y bl = balance (Branch R a k x b) s y bl" |
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   621
  "balance_right (Branch R a k x (Branch B b s y c)) t z bl = Branch R (balance (paint R a) k x b) s y (Branch B c t z bl)" |
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   622
  "balance_right t k x s = Empty"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   623
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   624
lemma balance_right_inv2_with_inv1:
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   625
  assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt + 1" "inv1 lt"
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   626
  shows "inv2 (balance_right lt k v rt) \<and> bheight (balance_right lt k v rt) = bheight lt"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   627
using assms
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   628
by (induct lt k v rt rule: balance_right.induct) (auto simp: balance_inv2 balance_bheight)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   629
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   630
lemma balance_right_inv1: "\<lbrakk>inv1 a; inv1l b; color_of a = B\<rbrakk> \<Longrightarrow> inv1 (balance_right a k x b)"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   631
by (induct a k x b rule: balance_right.induct) (simp add: balance_inv1)+
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   632
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   633
lemma balance_right_inv1l: "\<lbrakk> inv1 lt; inv1l rt \<rbrakk> \<Longrightarrow>inv1l (balance_right lt k x rt)"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   634
by (induct lt k x rt rule: balance_right.induct) (auto simp: balance_inv1)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   635
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   636
lemma (in linorder) balance_right_rbt_sorted:
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   637
  "\<lbrakk> rbt_sorted l; rbt_sorted r; rbt_less k l; k \<guillemotleft>| r \<rbrakk> \<Longrightarrow> rbt_sorted (balance_right l k v r)"
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   638
apply (induct l k v r rule: balance_right.induct)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   639
apply (auto simp:balance_rbt_sorted)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   640
apply (unfold rbt_less_prop rbt_greater_prop)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   641
by force+
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   642
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   643
context order begin
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   644
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   645
lemma balance_right_rbt_greater: 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   646
  fixes k :: "'a"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   647
  assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x" 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   648
  shows "k \<guillemotleft>| balance_right a x t b"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   649
using assms by (induct a x t b rule: balance_right.induct) auto
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   650
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   651
lemma balance_right_rbt_less: 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   652
  fixes k :: "'a"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   653
  assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k" 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   654
  shows "balance_right a x t b |\<guillemotleft> k"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   655
using assms by (induct a x t b rule: balance_right.induct) auto
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   656
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   657
end
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   658
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   659
lemma balance_right_in_tree:
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   660
  assumes "inv1 l" "inv1l r" "bheight l = bheight r + 1" "inv2 l" "inv2 r"
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   661
  shows "entry_in_tree x y (balance_right l k v r) = (entry_in_tree x y l \<or> x = k \<and> y = v \<or> entry_in_tree x y r)"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   662
using assms by (induct l k v r rule: balance_right.induct) (auto simp: balance_in_tree)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   663
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   664
fun
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   665
  combine :: "('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   666
where
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   667
  "combine Empty x = x" 
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   668
| "combine x Empty = x" 
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   669
| "combine (Branch R a k x b) (Branch R c s y d) = (case (combine b c) of
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   670
                                    Branch R b2 t z c2 \<Rightarrow> (Branch R (Branch R a k x b2) t z (Branch R c2 s y d)) |
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   671
                                    bc \<Rightarrow> Branch R a k x (Branch R bc s y d))" 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   672
| "combine (Branch B a k x b) (Branch B c s y d) = (case (combine b c) of
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   673
                                    Branch R b2 t z c2 \<Rightarrow> Branch R (Branch B a k x b2) t z (Branch B c2 s y d) |
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   674
                                    bc \<Rightarrow> balance_left a k x (Branch B bc s y d))" 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   675
| "combine a (Branch R b k x c) = Branch R (combine a b) k x c" 
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   676
| "combine (Branch R a k x b) c = Branch R a k x (combine b c)" 
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   677
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   678
lemma combine_inv2:
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   679
  assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt"
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   680
  shows "bheight (combine lt rt) = bheight lt" "inv2 (combine lt rt)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   681
using assms 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   682
by (induct lt rt rule: combine.induct) 
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   683
   (auto simp: balance_left_inv2_app split: rbt.splits color.splits)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   684
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   685
lemma combine_inv1: 
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   686
  assumes "inv1 lt" "inv1 rt"
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   687
  shows "color_of lt = B \<Longrightarrow> color_of rt = B \<Longrightarrow> inv1 (combine lt rt)"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   688
         "inv1l (combine lt rt)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   689
using assms 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   690
by (induct lt rt rule: combine.induct)
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   691
   (auto simp: balance_left_inv1 split: rbt.splits color.splits)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   692
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   693
context linorder begin
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   694
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   695
lemma combine_rbt_greater[simp]: 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   696
  fixes k :: "'a"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   697
  assumes "k \<guillemotleft>| l" "k \<guillemotleft>| r" 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   698
  shows "k \<guillemotleft>| combine l r"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   699
using assms 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   700
by (induct l r rule: combine.induct)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   701
   (auto simp: balance_left_rbt_greater split:rbt.splits color.splits)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   702
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   703
lemma combine_rbt_less[simp]: 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   704
  fixes k :: "'a"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   705
  assumes "l |\<guillemotleft> k" "r |\<guillemotleft> k" 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   706
  shows "combine l r |\<guillemotleft> k"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   707
using assms 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   708
by (induct l r rule: combine.induct)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   709
   (auto simp: balance_left_rbt_less split:rbt.splits color.splits)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   710
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   711
lemma combine_rbt_sorted: 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   712
  fixes k :: "'a"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   713
  assumes "rbt_sorted l" "rbt_sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   714
  shows "rbt_sorted (combine l r)"
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   715
using assms proof (induct l r rule: combine.induct)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   716
  case (3 a x v b c y w d)
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   717
  hence ineqs: "a |\<guillemotleft> x" "x \<guillemotleft>| b" "b |\<guillemotleft> k" "k \<guillemotleft>| c" "c |\<guillemotleft> y" "y \<guillemotleft>| d"
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   718
    by auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   719
  with 3
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   720
  show ?case
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   721
    by (cases "combine b c" rule: rbt_cases)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   722
      (auto, (metis combine_rbt_greater combine_rbt_less ineqs ineqs rbt_less_simps(2) rbt_greater_simps(2) rbt_greater_trans rbt_less_trans)+)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   723
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   724
  case (4 a x v b c y w d)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   725
  hence "x < k \<and> rbt_greater k c" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   726
  hence "rbt_greater x c" by (blast dest: rbt_greater_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   727
  with 4 have 2: "rbt_greater x (combine b c)" by (simp add: combine_rbt_greater)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   728
  from 4 have "k < y \<and> rbt_less k b" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   729
  hence "rbt_less y b" by (blast dest: rbt_less_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   730
  with 4 have 3: "rbt_less y (combine b c)" by (simp add: combine_rbt_less)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   731
  show ?case
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   732
  proof (cases "combine b c" rule: rbt_cases)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   733
    case Empty
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   734
    from 4 have "x < y \<and> rbt_greater y d" by auto
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   735
    hence "rbt_greater x d" by (blast dest: rbt_greater_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   736
    with 4 Empty have "rbt_sorted a" and "rbt_sorted (Branch B Empty y w d)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   737
      and "rbt_less x a" and "rbt_greater x (Branch B Empty y w d)" by auto
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   738
    with Empty show ?thesis by (simp add: balance_left_rbt_sorted)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   739
  next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   740
    case (Red lta va ka rta)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   741
    with 2 4 have "x < va \<and> rbt_less x a" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   742
    hence 5: "rbt_less va a" by (blast dest: rbt_less_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   743
    from Red 3 4 have "va < y \<and> rbt_greater y d" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   744
    hence "rbt_greater va d" by (blast dest: rbt_greater_trans)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   745
    with Red 2 3 4 5 show ?thesis by simp
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   746
  next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   747
    case (Black lta va ka rta)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   748
    from 4 have "x < y \<and> rbt_greater y d" by auto
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   749
    hence "rbt_greater x d" by (blast dest: rbt_greater_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   750
    with Black 2 3 4 have "rbt_sorted a" and "rbt_sorted (Branch B (combine b c) y w d)" 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   751
      and "rbt_less x a" and "rbt_greater x (Branch B (combine b c) y w d)" by auto
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   752
    with Black show ?thesis by (simp add: balance_left_rbt_sorted)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   753
  qed
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   754
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   755
  case (5 va vb vd vc b x w c)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   756
  hence "k < x \<and> rbt_less k (Branch B va vb vd vc)" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   757
  hence "rbt_less x (Branch B va vb vd vc)" by (blast dest: rbt_less_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   758
  with 5 show ?case by (simp add: combine_rbt_less)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   759
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   760
  case (6 a x v b va vb vd vc)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   761
  hence "x < k \<and> rbt_greater k (Branch B va vb vd vc)" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   762
  hence "rbt_greater x (Branch B va vb vd vc)" by (blast dest: rbt_greater_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   763
  with 6 show ?case by (simp add: combine_rbt_greater)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   764
qed simp+
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   765
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   766
end
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   767
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   768
lemma combine_in_tree: 
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   769
  assumes "inv2 l" "inv2 r" "bheight l = bheight r" "inv1 l" "inv1 r"
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   770
  shows "entry_in_tree k v (combine l r) = (entry_in_tree k v l \<or> entry_in_tree k v r)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   771
using assms 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   772
proof (induct l r rule: combine.induct)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   773
  case (4 _ _ _ b c)
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   774
  hence a: "bheight (combine b c) = bheight b" by (simp add: combine_inv2)
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   775
  from 4 have b: "inv1l (combine b c)" by (simp add: combine_inv1)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   776
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   777
  show ?case
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   778
  proof (cases "combine b c" rule: rbt_cases)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   779
    case Empty
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   780
    with 4 a show ?thesis by (auto simp: balance_left_in_tree)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   781
  next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   782
    case (Red lta ka va rta)
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   783
    with 4 show ?thesis by auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   784
  next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   785
    case (Black lta ka va rta)
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   786
    with a b 4  show ?thesis by (auto simp: balance_left_in_tree)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   787
  qed 
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   788
qed (auto split: rbt.splits color.splits)
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   789
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   790
context ord begin
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   791
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   792
fun
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   793
  rbt_del_from_left :: "'a \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   794
  rbt_del_from_right :: "'a \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt" and
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   795
  rbt_del :: "'a\<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   796
where
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   797
  "rbt_del x Empty = Empty" |
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   798
  "rbt_del x (Branch c a y s b) = 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   799
   (if x < y then rbt_del_from_left x a y s b 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   800
    else (if x > y then rbt_del_from_right x a y s b else combine a b))" |
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   801
  "rbt_del_from_left x (Branch B lt z v rt) y s b = balance_left (rbt_del x (Branch B lt z v rt)) y s b" |
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   802
  "rbt_del_from_left x a y s b = Branch R (rbt_del x a) y s b" |
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   803
  "rbt_del_from_right x a y s (Branch B lt z v rt) = balance_right a y s (rbt_del x (Branch B lt z v rt))" | 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   804
  "rbt_del_from_right x a y s b = Branch R a y s (rbt_del x b)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   805
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   806
end
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   807
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   808
context linorder begin
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   809
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   810
lemma 
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   811
  assumes "inv2 lt" "inv1 lt"
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   812
  shows
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   813
  "\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   814
   inv2 (rbt_del_from_left x lt k v rt) \<and> 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   815
   bheight (rbt_del_from_left x lt k v rt) = bheight lt \<and> 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   816
   (color_of lt = B \<and> color_of rt = B \<and> inv1 (rbt_del_from_left x lt k v rt) \<or> 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   817
    (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (rbt_del_from_left x lt k v rt))"
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   818
  and "\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   819
  inv2 (rbt_del_from_right x lt k v rt) \<and> 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   820
  bheight (rbt_del_from_right x lt k v rt) = bheight lt \<and> 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   821
  (color_of lt = B \<and> color_of rt = B \<and> inv1 (rbt_del_from_right x lt k v rt) \<or> 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   822
   (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (rbt_del_from_right x lt k v rt))"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   823
  and rbt_del_inv1_inv2: "inv2 (rbt_del x lt) \<and> (color_of lt = R \<and> bheight (rbt_del x lt) = bheight lt \<and> inv1 (rbt_del x lt) 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   824
  \<or> color_of lt = B \<and> bheight (rbt_del x lt) = bheight lt - 1 \<and> inv1l (rbt_del x lt))"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   825
using assms
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   826
proof (induct x lt k v rt and x lt k v rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   827
case (2 y c _ y')
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   828
  have "y = y' \<or> y < y' \<or> y > y'" by auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   829
  thus ?case proof (elim disjE)
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   830
    assume "y = y'"
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   831
    with 2 show ?thesis by (cases c) (simp add: combine_inv2 combine_inv1)+
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   832
  next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   833
    assume "y < y'"
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   834
    with 2 show ?thesis by (cases c) auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   835
  next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   836
    assume "y' < y"
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   837
    with 2 show ?thesis by (cases c) auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   838
  qed
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   839
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   840
  case (3 y lt z v rta y' ss bb) 
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   841
  thus ?case by (cases "color_of (Branch B lt z v rta) = B \<and> color_of bb = B") (simp add: balance_left_inv2_with_inv1 balance_left_inv1 balance_left_inv1l)+
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   842
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   843
  case (5 y a y' ss lt z v rta)
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   844
  thus ?case by (cases "color_of a = B \<and> color_of (Branch B lt z v rta) = B") (simp add: balance_right_inv2_with_inv1 balance_right_inv1 balance_right_inv1l)+
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   845
next
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   846
  case ("6_1" y a y' ss) thus ?case by (cases "color_of a = B \<and> color_of Empty = B") simp+
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   847
qed auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   848
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   849
lemma 
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   850
  rbt_del_from_left_rbt_less: "\<lbrakk> lt |\<guillemotleft> v; rt |\<guillemotleft> v; k < v\<rbrakk> \<Longrightarrow> rbt_del_from_left x lt k y rt |\<guillemotleft> v"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   851
  and rbt_del_from_right_rbt_less: "\<lbrakk>lt |\<guillemotleft> v; rt |\<guillemotleft> v; k < v\<rbrakk> \<Longrightarrow> rbt_del_from_right x lt k y rt |\<guillemotleft> v"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   852
  and rbt_del_rbt_less: "lt |\<guillemotleft> v \<Longrightarrow> rbt_del x lt |\<guillemotleft> v"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   853
by (induct x lt k y rt and x lt k y rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct) 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   854
   (auto simp: balance_left_rbt_less balance_right_rbt_less)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   855
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   856
lemma rbt_del_from_left_rbt_greater: "\<lbrakk>v \<guillemotleft>| lt; v \<guillemotleft>| rt; k > v\<rbrakk> \<Longrightarrow> v \<guillemotleft>| rbt_del_from_left x lt k y rt"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   857
  and rbt_del_from_right_rbt_greater: "\<lbrakk>v \<guillemotleft>| lt; v \<guillemotleft>| rt; k > v\<rbrakk> \<Longrightarrow> v \<guillemotleft>| rbt_del_from_right x lt k y rt"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   858
  and rbt_del_rbt_greater: "v \<guillemotleft>| lt \<Longrightarrow> v \<guillemotleft>| rbt_del x lt"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   859
by (induct x lt k y rt and x lt k y rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   860
   (auto simp: balance_left_rbt_greater balance_right_rbt_greater)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   861
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   862
lemma "\<lbrakk>rbt_sorted lt; rbt_sorted rt; lt |\<guillemotleft> k; k \<guillemotleft>| rt\<rbrakk> \<Longrightarrow> rbt_sorted (rbt_del_from_left x lt k y rt)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   863
  and "\<lbrakk>rbt_sorted lt; rbt_sorted rt; lt |\<guillemotleft> k; k \<guillemotleft>| rt\<rbrakk> \<Longrightarrow> rbt_sorted (rbt_del_from_right x lt k y rt)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   864
  and rbt_del_rbt_sorted: "rbt_sorted lt \<Longrightarrow> rbt_sorted (rbt_del x lt)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   865
proof (induct x lt k y rt and x lt k y rt and x lt rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   866
  case (3 x lta zz v rta yy ss bb)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   867
  from 3 have "Branch B lta zz v rta |\<guillemotleft> yy" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   868
  hence "rbt_del x (Branch B lta zz v rta) |\<guillemotleft> yy" by (rule rbt_del_rbt_less)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   869
  with 3 show ?case by (simp add: balance_left_rbt_sorted)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   870
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   871
  case ("4_2" x vaa vbb vdd vc yy ss bb)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   872
  hence "Branch R vaa vbb vdd vc |\<guillemotleft> yy" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   873
  hence "rbt_del x (Branch R vaa vbb vdd vc) |\<guillemotleft> yy" by (rule rbt_del_rbt_less)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   874
  with "4_2" show ?case by simp
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   875
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   876
  case (5 x aa yy ss lta zz v rta) 
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   877
  hence "yy \<guillemotleft>| Branch B lta zz v rta" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   878
  hence "yy \<guillemotleft>| rbt_del x (Branch B lta zz v rta)" by (rule rbt_del_rbt_greater)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   879
  with 5 show ?case by (simp add: balance_right_rbt_sorted)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   880
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   881
  case ("6_2" x aa yy ss vaa vbb vdd vc)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   882
  hence "yy \<guillemotleft>| Branch R vaa vbb vdd vc" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   883
  hence "yy \<guillemotleft>| rbt_del x (Branch R vaa vbb vdd vc)" by (rule rbt_del_rbt_greater)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   884
  with "6_2" show ?case by simp
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   885
qed (auto simp: combine_rbt_sorted)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   886
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   887
lemma "\<lbrakk>rbt_sorted lt; rbt_sorted rt; lt |\<guillemotleft> kt; kt \<guillemotleft>| rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x < kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (rbt_del_from_left x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   888
  and "\<lbrakk>rbt_sorted lt; rbt_sorted rt; lt |\<guillemotleft> kt; kt \<guillemotleft>| rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x > kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (rbt_del_from_right x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   889
  and rbt_del_in_tree: "\<lbrakk>rbt_sorted t; inv1 t; inv2 t\<rbrakk> \<Longrightarrow> entry_in_tree k v (rbt_del x t) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v t))"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   890
proof (induct x lt kt y rt and x lt kt y rt and x t rule: rbt_del_from_left_rbt_del_from_right_rbt_del.induct)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   891
  case (2 xx c aa yy ss bb)
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   892
  have "xx = yy \<or> xx < yy \<or> xx > yy" by auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   893
  from this 2 show ?case proof (elim disjE)
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   894
    assume "xx = yy"
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   895
    with 2 show ?thesis proof (cases "xx = k")
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   896
      case True
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
   897
      from 2 \<open>xx = yy\<close> \<open>xx = k\<close> have "rbt_sorted (Branch c aa yy ss bb) \<and> k = yy" by simp
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   898
      hence "\<not> entry_in_tree k v aa" "\<not> entry_in_tree k v bb" by (auto simp: rbt_less_nit rbt_greater_prop)
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
   899
      with \<open>xx = yy\<close> 2 \<open>xx = k\<close> show ?thesis by (simp add: combine_in_tree)
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   900
    qed (simp add: combine_in_tree)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   901
  qed simp+
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   902
next    
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   903
  case (3 xx lta zz vv rta yy ss bb)
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   904
  def mt[simp]: mt == "Branch B lta zz vv rta"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   905
  from 3 have "inv2 mt \<and> inv1 mt" by simp
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   906
  hence "inv2 (rbt_del xx mt) \<and> (color_of mt = R \<and> bheight (rbt_del xx mt) = bheight mt \<and> inv1 (rbt_del xx mt) \<or> color_of mt = B \<and> bheight (rbt_del xx mt) = bheight mt - 1 \<and> inv1l (rbt_del xx mt))" by (blast dest: rbt_del_inv1_inv2)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   907
  with 3 have 4: "entry_in_tree k v (rbt_del_from_left xx mt yy ss bb) = (False \<or> xx \<noteq> k \<and> entry_in_tree k v mt \<or> (k = yy \<and> v = ss) \<or> entry_in_tree k v bb)" by (simp add: balance_left_in_tree)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   908
  thus ?case proof (cases "xx = k")
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   909
    case True
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   910
    from 3 True have "yy \<guillemotleft>| bb \<and> yy > k" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   911
    hence "k \<guillemotleft>| bb" by (blast dest: rbt_greater_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   912
    with 3 4 True show ?thesis by (auto simp: rbt_greater_nit)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   913
  qed auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   914
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   915
  case ("4_1" xx yy ss bb)
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   916
  show ?case proof (cases "xx = k")
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   917
    case True
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   918
    with "4_1" have "yy \<guillemotleft>| bb \<and> k < yy" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   919
    hence "k \<guillemotleft>| bb" by (blast dest: rbt_greater_trans)
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
   920
    with "4_1" \<open>xx = k\<close> 
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   921
   have "entry_in_tree k v (Branch R Empty yy ss bb) = entry_in_tree k v Empty" by (auto simp: rbt_greater_nit)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   922
    thus ?thesis by auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   923
  qed simp+
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   924
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   925
  case ("4_2" xx vaa vbb vdd vc yy ss bb)
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   926
  thus ?case proof (cases "xx = k")
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   927
    case True
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   928
    with "4_2" have "k < yy \<and> yy \<guillemotleft>| bb" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   929
    hence "k \<guillemotleft>| bb" by (blast dest: rbt_greater_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   930
    with True "4_2" show ?thesis by (auto simp: rbt_greater_nit)
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   931
  qed auto
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   932
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   933
  case (5 xx aa yy ss lta zz vv rta)
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   934
  def mt[simp]: mt == "Branch B lta zz vv rta"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   935
  from 5 have "inv2 mt \<and> inv1 mt" by simp
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   936
  hence "inv2 (rbt_del xx mt) \<and> (color_of mt = R \<and> bheight (rbt_del xx mt) = bheight mt \<and> inv1 (rbt_del xx mt) \<or> color_of mt = B \<and> bheight (rbt_del xx mt) = bheight mt - 1 \<and> inv1l (rbt_del xx mt))" by (blast dest: rbt_del_inv1_inv2)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   937
  with 5 have 3: "entry_in_tree k v (rbt_del_from_right xx aa yy ss mt) = (entry_in_tree k v aa \<or> (k = yy \<and> v = ss) \<or> False \<or> xx \<noteq> k \<and> entry_in_tree k v mt)" by (simp add: balance_right_in_tree)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   938
  thus ?case proof (cases "xx = k")
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   939
    case True
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   940
    from 5 True have "aa |\<guillemotleft> yy \<and> yy < k" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   941
    hence "aa |\<guillemotleft> k" by (blast dest: rbt_less_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   942
    with 3 5 True show ?thesis by (auto simp: rbt_less_nit)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   943
  qed auto
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   944
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   945
  case ("6_1" xx aa yy ss)
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   946
  show ?case proof (cases "xx = k")
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   947
    case True
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   948
    with "6_1" have "aa |\<guillemotleft> yy \<and> k > yy" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   949
    hence "aa |\<guillemotleft> k" by (blast dest: rbt_less_trans)
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
   950
    with "6_1" \<open>xx = k\<close> show ?thesis by (auto simp: rbt_less_nit)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   951
  qed simp
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   952
next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   953
  case ("6_2" xx aa yy ss vaa vbb vdd vc)
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   954
  thus ?case proof (cases "xx = k")
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   955
    case True
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   956
    with "6_2" have "k > yy \<and> aa |\<guillemotleft> yy" by simp
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   957
    hence "aa |\<guillemotleft> k" by (blast dest: rbt_less_trans)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   958
    with True "6_2" show ?thesis by (auto simp: rbt_less_nit)
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   959
  qed auto
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   960
qed simp
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   961
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   962
definition (in ord) rbt_delete where
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   963
  "rbt_delete k t = paint B (rbt_del k t)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   964
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   965
theorem rbt_delete_is_rbt [simp]: assumes "is_rbt t" shows "is_rbt (rbt_delete k t)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   966
proof -
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   967
  from assms have "inv2 t" and "inv1 t" unfolding is_rbt_def by auto 
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   968
  hence "inv2 (rbt_del k t) \<and> (color_of t = R \<and> bheight (rbt_del k t) = bheight t \<and> inv1 (rbt_del k t) \<or> color_of t = B \<and> bheight (rbt_del k t) = bheight t - 1 \<and> inv1l (rbt_del k t))" by (rule rbt_del_inv1_inv2)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   969
  hence "inv2 (rbt_del k t) \<and> inv1l (rbt_del k t)" by (cases "color_of t") auto
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   970
  with assms show ?thesis
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   971
    unfolding is_rbt_def rbt_delete_def
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   972
    by (auto intro: paint_rbt_sorted rbt_del_rbt_sorted)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   973
qed
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   974
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   975
lemma rbt_delete_in_tree: 
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   976
  assumes "is_rbt t" 
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   977
  shows "entry_in_tree k v (rbt_delete x t) = (x \<noteq> k \<and> entry_in_tree k v t)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   978
  using assms unfolding is_rbt_def rbt_delete_def
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   979
  by (auto simp: rbt_del_in_tree)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   980
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   981
lemma rbt_lookup_rbt_delete:
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   982
  assumes is_rbt: "is_rbt t"
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   983
  shows "rbt_lookup (rbt_delete k t) = (rbt_lookup t)|`(-{k})"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   984
proof
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   985
  fix x
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   986
  show "rbt_lookup (rbt_delete k t) x = (rbt_lookup t |` (-{k})) x" 
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   987
  proof (cases "x = k")
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   988
    assume "x = k" 
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
   989
    with is_rbt show ?thesis
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   990
      by (cases "rbt_lookup (rbt_delete k t) k") (auto simp: rbt_lookup_in_tree rbt_delete_in_tree)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   991
  next
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   992
    assume "x \<noteq> k"
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   993
    thus ?thesis
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   994
      by auto (metis is_rbt rbt_delete_is_rbt rbt_delete_in_tree is_rbt_rbt_sorted rbt_lookup_from_in_tree)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   995
  qed
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   996
qed
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
   997
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
   998
end
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
   999
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
  1000
subsection \<open>Modifying existing entries\<close>
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
  1001
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1002
context ord begin
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1003
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
  1004
primrec
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1005
  rbt_map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
  1006
where
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1007
  "rbt_map_entry k f Empty = Empty"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1008
| "rbt_map_entry k f (Branch c lt x v rt) =
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1009
    (if k < x then Branch c (rbt_map_entry k f lt) x v rt
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1010
    else if k > x then (Branch c lt x v (rbt_map_entry k f rt))
35602
e814157560e8 various refinements
haftmann
parents: 35550
diff changeset
  1011
    else Branch c lt x (f v) rt)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
  1012
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1013
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1014
lemma rbt_map_entry_color_of: "color_of (rbt_map_entry k f t) = color_of t" by (induct t) simp+
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1015
lemma rbt_map_entry_inv1: "inv1 (rbt_map_entry k f t) = inv1 t" by (induct t) (simp add: rbt_map_entry_color_of)+
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1016
lemma rbt_map_entry_inv2: "inv2 (rbt_map_entry k f t) = inv2 t" "bheight (rbt_map_entry k f t) = bheight t" by (induct t) simp+
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1017
lemma rbt_map_entry_rbt_greater: "rbt_greater a (rbt_map_entry k f t) = rbt_greater a t" by (induct t) simp+
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1018
lemma rbt_map_entry_rbt_less: "rbt_less a (rbt_map_entry k f t) = rbt_less a t" by (induct t) simp+
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1019
lemma rbt_map_entry_rbt_sorted: "rbt_sorted (rbt_map_entry k f t) = rbt_sorted t"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1020
  by (induct t) (simp_all add: rbt_map_entry_rbt_less rbt_map_entry_rbt_greater)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
  1021
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1022
theorem rbt_map_entry_is_rbt [simp]: "is_rbt (rbt_map_entry k f t) = is_rbt t" 
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1023
unfolding is_rbt_def by (simp add: rbt_map_entry_inv2 rbt_map_entry_color_of rbt_map_entry_rbt_sorted rbt_map_entry_inv1 )
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
  1024
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1025
end
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1026
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1027
theorem (in linorder) rbt_lookup_rbt_map_entry:
55466
786edc984c98 merged 'Option.map' and 'Option.map_option'
blanchet
parents: 55417
diff changeset
  1028
  "rbt_lookup (rbt_map_entry k f t) = (rbt_lookup t)(k := map_option f (rbt_lookup t k))"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  1029
  by (induct t) (auto split: option.splits simp add: fun_eq_iff)
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
  1030
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
  1031
subsection \<open>Mapping all entries\<close>
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
  1032
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
  1033
primrec
35602
e814157560e8 various refinements
haftmann
parents: 35550
diff changeset
  1034
  map :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'c) rbt"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
  1035
where
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1036
  "map f Empty = Empty"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1037
| "map f (Branch c lt k v rt) = Branch c (map f lt) k (f k v) (map f rt)"
32237
cdc76a42fed4 added missing proof of RBT.map_of_alist_of (contributed by Peter Lammich)
krauss
parents: 30738
diff changeset
  1038
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1039
lemma map_entries [simp]: "entries (map f t) = List.map (\<lambda>(k, v). (k, f k v)) (entries t)"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1040
  by (induct t) auto
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1041
lemma map_keys [simp]: "keys (map f t) = keys t" by (simp add: keys_def split_def)
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1042
lemma map_color_of: "color_of (map f t) = color_of t" by (induct t) simp+
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1043
lemma map_inv1: "inv1 (map f t) = inv1 t" by (induct t) (simp add: map_color_of)+
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1044
lemma map_inv2: "inv2 (map f t) = inv2 t" "bheight (map f t) = bheight t" by (induct t) simp+
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1045
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1046
context ord begin
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1047
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1048
lemma map_rbt_greater: "rbt_greater k (map f t) = rbt_greater k t" by (induct t) simp+
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1049
lemma map_rbt_less: "rbt_less k (map f t) = rbt_less k t" by (induct t) simp+
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1050
lemma map_rbt_sorted: "rbt_sorted (map f t) = rbt_sorted t"  by (induct t) (simp add: map_rbt_less map_rbt_greater)+
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1051
theorem map_is_rbt [simp]: "is_rbt (map f t) = is_rbt t" 
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1052
unfolding is_rbt_def by (simp add: map_inv1 map_inv2 map_rbt_sorted map_color_of)
32237
cdc76a42fed4 added missing proof of RBT.map_of_alist_of (contributed by Peter Lammich)
krauss
parents: 30738
diff changeset
  1053
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1054
end
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
  1055
55466
786edc984c98 merged 'Option.map' and 'Option.map_option'
blanchet
parents: 55417
diff changeset
  1056
theorem (in linorder) rbt_lookup_map: "rbt_lookup (map f t) x = map_option (f x) (rbt_lookup t x)"
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1057
  apply(induct t)
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1058
  apply auto
58257
0662f35534fe half-ported Imperative HOL to new datatypes
blanchet
parents: 58249
diff changeset
  1059
  apply(rename_tac a b c, subgoal_tac "x = a")
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1060
  apply auto
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1061
  done
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1062
 (* FIXME: simproc "antisym less" does not work for linorder context, only for linorder type class
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1063
    by (induct t) auto *)
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1064
49770
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1065
hide_const (open) map
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1066
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
  1067
subsection \<open>Folding over entries\<close>
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1068
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1069
definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" where
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55412
diff changeset
  1070
  "fold f t = List.fold (case_prod f) (entries t)"
26192
52617dca8386 new theory of red-black trees, an efficient implementation of finite maps.
krauss
parents:
diff changeset
  1071
49770
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1072
lemma fold_simps [simp]:
35550
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1073
  "fold f Empty = id"
e2bc7f8d8d51 restructured RBT theory
haftmann
parents: 35534
diff changeset
  1074
  "fold f (Branch c lt k v rt) = fold f rt \<circ> f k v \<circ> fold f lt"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  1075
  by (simp_all add: fold_def fun_eq_iff)
35534
14d8d72f8b1f more explicit naming scheme
haftmann
parents: 32245
diff changeset
  1076
49770
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1077
lemma fold_code [code]:
49810
53f14f62cca2 fix code equation for RBT_Impl.fold
Andreas Lochbihler
parents: 49807
diff changeset
  1078
  "fold f Empty x = x"
53f14f62cca2 fix code equation for RBT_Impl.fold
Andreas Lochbihler
parents: 49807
diff changeset
  1079
  "fold f (Branch c lt k v rt) x = fold f rt (f k v (fold f lt x))"
49770
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1080
by(simp_all)
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1081
48621
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1082
(* fold with continuation predicate *)
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1083
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1084
fun foldi :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a :: linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" 
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1085
  where
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1086
  "foldi c f Empty s = s" |
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1087
  "foldi c f (Branch col l k v r) s = (
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1088
    if (c s) then
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1089
      let s' = foldi c f l s in
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1090
        if (c s') then
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1091
          foldi c f r (f k v s')
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1092
        else s'
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1093
    else 
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1094
      s
877df57629e3 a couple of additions to RBT formalization to allow us to implement RBT_Set
kuncar
parents: 47455
diff changeset
  1095
  )"
35606
7c5b40c7e8c4 added bulkload; tuned document
haftmann
parents: 35603
diff changeset
  1096
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
  1097
subsection \<open>Bulkloading a tree\<close>
35606
7c5b40c7e8c4 added bulkload; tuned document
haftmann
parents: 35603
diff changeset
  1098
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1099
definition (in ord) rbt_bulkload :: "('a \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" where
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1100
  "rbt_bulkload xs = foldr (\<lambda>(k, v). rbt_insert k v) xs Empty"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1101
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1102
context linorder begin
35606
7c5b40c7e8c4 added bulkload; tuned document
haftmann
parents: 35603
diff changeset
  1103
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1104
lemma rbt_bulkload_is_rbt [simp, intro]:
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1105
  "is_rbt (rbt_bulkload xs)"
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1106
  unfolding rbt_bulkload_def by (induct xs) auto
35606
7c5b40c7e8c4 added bulkload; tuned document
haftmann
parents: 35603
diff changeset
  1107
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1108
lemma rbt_lookup_rbt_bulkload:
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1109
  "rbt_lookup (rbt_bulkload xs) = map_of xs"
35606
7c5b40c7e8c4 added bulkload; tuned document
haftmann
parents: 35603
diff changeset
  1110
proof -
7c5b40c7e8c4 added bulkload; tuned document
haftmann
parents: 35603
diff changeset
  1111
  obtain ys where "ys = rev xs" by simp
7c5b40c7e8c4 added bulkload; tuned document
haftmann
parents: 35603
diff changeset
  1112
  have "\<And>t. is_rbt t \<Longrightarrow>
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55412
diff changeset
  1113
    rbt_lookup (List.fold (case_prod rbt_insert) ys t) = rbt_lookup t ++ map_of (rev ys)"
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55412
diff changeset
  1114
      by (induct ys) (simp_all add: rbt_bulkload_def rbt_lookup_rbt_insert case_prod_beta)
35606
7c5b40c7e8c4 added bulkload; tuned document
haftmann
parents: 35603
diff changeset
  1115
  from this Empty_is_rbt have
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55412
diff changeset
  1116
    "rbt_lookup (List.fold (case_prod rbt_insert) (rev xs) Empty) = rbt_lookup Empty ++ map_of xs"
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
  1117
     by (simp add: \<open>ys = rev xs\<close>)
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1118
  then show ?thesis by (simp add: rbt_bulkload_def rbt_lookup_Empty foldr_conv_fold)
35606
7c5b40c7e8c4 added bulkload; tuned document
haftmann
parents: 35603
diff changeset
  1119
qed
7c5b40c7e8c4 added bulkload; tuned document
haftmann
parents: 35603
diff changeset
  1120
47450
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1121
end
2ada2be850cb move RBT implementation into type class contexts
Andreas Lochbihler
parents: 47397
diff changeset
  1122
49770
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1123
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1124
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
  1125
subsection \<open>Building a RBT from a sorted list\<close>
49770
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1126
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
  1127
text \<open>
49770
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1128
  These functions have been adapted from 
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1129
  Andrew W. Appel, Efficient Verified Red-Black Trees (September 2011) 
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59575
diff changeset
  1130
\<close>
49770
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1131
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1132
fun rbtreeify_f :: "nat \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a, 'b) rbt \<times> ('a \<times> 'b) list"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1133
  and rbtreeify_g :: "nat \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a, 'b) rbt \<times> ('a \<times> 'b) list"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1134
where
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1135
  "rbtreeify_f n kvs =
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1136
   (if n = 0 then (Empty, kvs)
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1137
    else if n = 1 then
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1138
      case kvs of (k, v) # kvs' \<Rightarrow> (Branch R Empty k v Empty, kvs')
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1139
    else if (n mod 2 = 0) then
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1140
      case rbtreeify_f (n div 2) kvs of (t1, (k, v) # kvs') \<Rightarrow>
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1141
        apfst (Branch B t1 k v) (rbtreeify_g (n div 2) kvs')
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1142
    else case rbtreeify_f (n div 2) kvs of (t1, (k, v) # kvs') \<Rightarrow>
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1143
        apfst (Branch B t1 k v) (rbtreeify_f (n div 2) kvs'))"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1144
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1145
| "rbtreeify_g n kvs =
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1146
   (if n = 0 \<or> n = 1 then (Empty, kvs)
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1147
    else if n mod 2 = 0 then
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1148
      case rbtreeify_g (n div 2) kvs of (t1, (k, v) # kvs') \<Rightarrow>
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1149
        apfst (Branch B t1 k v) (rbtreeify_g (n div 2) kvs')
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1150
    else case rbtreeify_f (n div 2) kvs of (t1, (k, v) # kvs') \<Rightarrow>
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1151
        apfst (Branch B t1 k v) (rbtreeify_g (n div 2) kvs'))"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1152
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1153
definition rbtreeify :: "('a \<times> 'b) list \<Rightarrow> ('a, 'b) rbt"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1154
where "rbtreeify kvs = fst (rbtreeify_g (Suc (length kvs)) kvs)"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1155
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1156
declare rbtreeify_f.simps [simp del] rbtreeify_g.simps [simp del]
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1157
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1158
lemma rbtreeify_f_code [code]:
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1159
  "rbtreeify_f n kvs =
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1160
   (if n = 0 then (Empty, kvs)
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1161
    else if n = 1 then
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1162
      case kvs of (k, v) # kvs' \<Rightarrow> 
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1163
        (Branch R Empty k v Empty, kvs')
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1164
    else let (n', r) = divmod_nat n 2 in
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1165
      if r = 0 then
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1166
        case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1167
          apfst (Branch B t1 k v) (rbtreeify_g n' kvs')
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1168
      else case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1169
          apfst (Branch B t1 k v) (rbtreeify_f n' kvs'))"
55412
eb2caacf3ba4 avoid old 'prod.simps' -- better be more specific
blanchet
parents: 53374
diff changeset
  1170
by (subst rbtreeify_f.simps) (simp only: Let_def divmod_nat_div_mod prod.case)
49770
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1171
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1172
lemma rbtreeify_g_code [code]:
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1173
  "rbtreeify_g n kvs =
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1174
   (if n = 0 \<or> n = 1 then (Empty, kvs)
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1175
    else let (n', r) = divmod_nat n 2 in
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1176
      if r = 0 then
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1177
        case rbtreeify_g n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1178
          apfst (Branch B t1 k v) (rbtreeify_g n' kvs')
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1179
      else case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow>
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1180
          apfst (Branch B t1 k v) (rbtreeify_g n' kvs'))"
55412
eb2caacf3ba4 avoid old 'prod.simps' -- better be more specific
blanchet
parents: 53374
diff changeset
  1181
by(subst rbtreeify_g.simps)(simp only: Let_def divmod_nat_div_mod prod.case)
49770
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1182
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1183
lemma Suc_double_half: "Suc (2 * n) div 2 = n"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1184
by simp
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1185
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1186
lemma div2_plus_div2: "n div 2 + n div 2 = (n :: nat) - n mod 2"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1187
by arith
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1188
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1189
lemma rbtreeify_f_rec_aux_lemma:
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1190
  "\<lbrakk>k - n div 2 = Suc k'; n \<le> k; n mod 2 = Suc 0\<rbrakk>
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1191
  \<Longrightarrow> k' - n div 2 = k - n"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1192
apply(rule add_right_imp_eq[where a = "n - n div 2"])
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1193
apply(subst add_diff_assoc2, arith)
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1194
apply(simp add: div2_plus_div2)
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1195
done
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1196
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1197
lemma rbtreeify_f_simps:
59575
55f5e1cbf2a7 removed needless (and inconsistent) qualifier that messes up with Mirabelle
blanchet
parents: 59554
diff changeset
  1198
  "rbtreeify_f 0 kvs = (Empty, kvs)"
49770
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1199
  "rbtreeify_f (Suc 0) ((k, v) # kvs) = 
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1200
  (Branch R Empty k v Empty, kvs)"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1201
  "0 < n \<Longrightarrow> rbtreeify_f (2 * n) kvs =
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1202
   (case rbtreeify_f n kvs of (t1, (k, v) # kvs') \<Rightarrow>
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1203
     apfst (Branch B t1 k v) (rbtreeify_g n kvs'))"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1204
  "0 < n \<Longrightarrow> rbtreeify_f (Suc (2 * n)) kvs =
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1205
   (case rbtreeify_f n kvs of (t1, (k, v) # kvs') \<Rightarrow> 
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1206
     apfst (Branch B t1 k v) (rbtreeify_f n kvs'))"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1207
by(subst (1) rbtreeify_f.simps, simp add: Suc_double_half)+
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1208
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1209
lemma rbtreeify_g_simps:
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1210
  "rbtreeify_g 0 kvs = (Empty, kvs)"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1211
  "rbtreeify_g (Suc 0) kvs = (Empty, kvs)"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1212
  "0 < n \<Longrightarrow> rbtreeify_g (2 * n) kvs =
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1213
   (case rbtreeify_g n kvs of (t1, (k, v) # kvs') \<Rightarrow> 
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1214
     apfst (Branch B t1 k v) (rbtreeify_g n kvs'))"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1215
  "0 < n \<Longrightarrow> rbtreeify_g (Suc (2 * n)) kvs =
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1216
   (case rbtreeify_f n kvs of (t1, (k, v) # kvs') \<Rightarrow> 
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1217
     apfst (Branch B t1 k v) (rbtreeify_g n kvs'))"
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1218
by(subst (1) rbtreeify_g.simps, simp add: Suc_double_half)+
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1219
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1220
declare rbtreeify_f_simps[simp] rbtreeify_g_simps[simp]
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1221
cf6a78acf445 efficient construction of red black trees from sorted associative lists
Andreas Lochbihler
parents: 49480
diff changeset
  1222
lemma length_rbtreeify_f: "n \<le> length kvs
cf6a78acf445 efficient construction of red black trees