(*  Title:      HOL/Imperative_HOL/ex/Sorted_List.thy

     Author:     Lukas Bulwahn, TU Muenchen

 *)

 header {* Sorted lists as representation of finite sets *}

 theory Sorted_List

 imports Main

 begin

 text {* Merge function for two distinct sorted lists to get compound distinct sorted list *}

 fun merge :: "('a::linorder) list \<Rightarrow> 'a list \<Rightarrow> 'a list"

 where

   "merge (x#xs) (y#ys) =

   (if x < y then x # merge xs (y#ys) else (if x > y then y # merge (x#xs) ys else x # merge xs ys))"

 | "merge xs [] = xs"

 | "merge [] ys = ys"

 text {* The function package does not derive automatically the more general rewrite rule as follows: *}

 lemma merge_Nil[simp]: "merge [] ys = ys"

 by (cases ys) auto

 lemma set_merge[simp]: "set (merge xs ys) = set xs \<union> set ys"

 by (induct xs ys rule: merge.induct, auto)

 lemma sorted_merge[simp]:

      "List.sorted (merge xs ys) = (List.sorted xs \<and> List.sorted ys)"

 by (induct xs ys rule: merge.induct, auto simp add: sorted_Cons)

 lemma distinct_merge[simp]: "\<lbrakk> distinct xs; distinct ys; List.sorted xs; List.sorted ys \<rbrakk> \<Longrightarrow> distinct (merge xs ys)"

 by (induct xs ys rule: merge.induct, auto simp add: sorted_Cons)

 text {* The remove function removes an element from a sorted list *}

 primrec remove :: "('a :: linorder) \<Rightarrow> 'a list \<Rightarrow> 'a list"

 where

   "remove a [] = []"

   |  "remove a (x#xs) = (if a > x then (x # remove a xs) else (if a = x then xs else x#xs))" 

 lemma remove': "sorted xs \<and> distinct xs \<Longrightarrow> sorted (remove a xs) \<and> distinct (remove a xs) \<and> set (remove a xs) = set xs - {a}"

 apply (induct xs)

 apply (auto simp add: sorted_Cons)

 done

 lemma set_remove[simp]: "\<lbrakk> sorted xs; distinct xs \<rbrakk> \<Longrightarrow> set (remove a xs) = set xs - {a}"

 using remove' by auto

 lemma sorted_remove[simp]: "\<lbrakk> sorted xs; distinct xs \<rbrakk> \<Longrightarrow> sorted (remove a xs)" 

 using remove' by auto

 lemma distinct_remove[simp]: "\<lbrakk> sorted xs; distinct xs \<rbrakk> \<Longrightarrow> distinct (remove a xs)" 

 using remove' by auto

 lemma remove_insort_cancel: "remove a (insort a xs) = xs"

 apply (induct xs)

 apply simp

 apply auto

 done

 lemma remove_insort_commute: "\<lbrakk> a \<noteq> b; sorted xs \<rbrakk> \<Longrightarrow> remove b (insort a xs) = insort a (remove b xs)"

 apply (induct xs)

 apply auto

 apply (auto simp add: sorted_Cons)

 apply (case_tac xs)

 apply auto

 done

 lemma notinset_remove: "x \<notin> set xs \<Longrightarrow> remove x xs = xs"

 apply (induct xs)

 apply auto

 done

 lemma remove1_eq_remove:

   "sorted xs \<Longrightarrow> distinct xs \<Longrightarrow> remove1 x xs = remove x xs"

 apply (induct xs)

 apply (auto simp add: sorted_Cons)

 apply (subgoal_tac "x \<notin> set xs")

 apply (simp add: notinset_remove)

 apply fastsimp

 done

 lemma sorted_remove1:

   "sorted xs \<Longrightarrow> sorted (remove1 x xs)"

 apply (induct xs)

 apply (auto simp add: sorted_Cons)

 done

 subsection {* Efficient member function for sorted lists *}

 primrec smember :: "'a list \<Rightarrow> 'a::linorder \<Rightarrow> bool" where

   "smember [] x \<longleftrightarrow> False"

 | "smember (y#ys) x \<longleftrightarrow> x = y \<or> (x > y \<and> smember ys x)"

 lemma "sorted xs \<Longrightarrow> smember xs x \<longleftrightarrow> (x \<in> set xs)" 

   by (induct xs) (auto simp add: sorted_Cons)

 end