(*  Title:      HOL/Library/Sublist_Order.thy

     ID:         $Id$

     Authors:    Peter Lammich, Uni Muenster <peter.lammich@uni-muenster.de>

                 Florian Haftmann, TU München

 *)

 header {* Sublist Ordering *}

 theory Sublist_Order

 imports Plain "~~/src/HOL/List"

 begin

 text {*

   This theory defines sublist ordering on lists.

   A list @{text ys} is a sublist of a list @{text xs},

   iff one obtains @{text ys} by erasing some elements from @{text xs}.

 *}

 subsection {* Definitions and basic lemmas *}

 instantiation list :: (type) order

 begin

 inductive less_eq_list where

   empty [simp, intro!]: "[] \<le> xs"

   | drop: "ys \<le> xs \<Longrightarrow> ys \<le> x # xs"

   | take: "ys \<le> xs \<Longrightarrow> x # ys \<le> x # xs"

 lemmas ileq_empty = empty

 lemmas ileq_drop = drop

 lemmas ileq_take = take

 lemma ileq_cases [cases set, case_names empty drop take]:

   assumes "xs \<le> ys"

     and "xs = [] \<Longrightarrow> P"

     and "\<And>z zs. ys = z # zs \<Longrightarrow> xs \<le> zs \<Longrightarrow> P"

     and "\<And>x zs ws. xs = x # zs \<Longrightarrow> ys = x # ws \<Longrightarrow> zs \<le> ws \<Longrightarrow> P"

   shows P

   using assms by (blast elim: less_eq_list.cases)

 lemma ileq_induct [induct set, case_names empty drop take]:

   assumes "xs \<le> ys"

     and "\<And>zs. P [] zs"

     and "\<And>z zs ws. ws \<le> zs \<Longrightarrow>  P ws zs \<Longrightarrow> P ws (z # zs)"

     and "\<And>z zs ws. ws \<le> zs \<Longrightarrow> P ws zs \<Longrightarrow> P (z # ws) (z # zs)"

   shows "P xs ys" 

   using assms by (induct rule: less_eq_list.induct) blast+

 definition

   [code del]: "(xs \<Colon> 'a list) < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs"

 lemma ileq_length: "xs \<le> ys \<Longrightarrow> length xs \<le> length ys"

   by (induct rule: ileq_induct) auto

 lemma ileq_below_empty [simp]: "xs \<le> [] \<longleftrightarrow> xs = []"

   by (auto dest: ileq_length)

 instance proof

   fix xs ys :: "'a list"

   show "xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs" unfolding less_list_def .. 

 next

   fix xs :: "'a list"

   show "xs \<le> xs" by (induct xs) (auto intro!: ileq_empty ileq_drop ileq_take)

 next

   fix xs ys :: "'a list"

   (* TODO: Is there a simpler proof ? *)

   { fix n

     have "!!l l'. \<lbrakk>l\<le>l'; l'\<le>l; n=length l + length l'\<rbrakk> \<Longrightarrow> l=l'"

     proof (induct n rule: nat_less_induct)

       case (1 n l l') from "1.prems"(1) show ?case proof (cases rule: ileq_cases)

         case empty with "1.prems"(2) show ?thesis by auto 

       next

         case (drop a l2') with "1.prems"(2) have "length l'\<le>length l" "length l \<le> length l2'" "1+length l2' = length l'" by (auto dest: ileq_length)

         hence False by simp thus ?thesis ..

       next

         case (take a l1' l2') hence LEN': "length l1' + length l2' < length l + length l'" by simp

         from "1.prems" have LEN: "length l' = length l" by (auto dest!: ileq_length)

         from "1.prems"(2) show ?thesis proof (cases rule: ileq_cases[case_names empty' drop' take'])

           case empty' with take LEN show ?thesis by simp 

         next

           case (drop' ah l2h) with take LEN have "length l1' \<le> length l2h" "1+length l2h = length l2'" "length l2' = length l1'" by (auto dest: ileq_length)

           hence False by simp thus ?thesis ..

         next

           case (take' ah l1h l2h)

           with take have 2: "ah=a" "l1h=l2'" "l2h=l1'" "l1' \<le> l2'" "l2' \<le> l1'" by auto

           with LEN' "1.hyps" "1.prems"(3) have "l1'=l2'" by blast

           with take 2 show ?thesis by simp

         qed

       qed

     qed

   }

   moreover assume "xs \<le> ys" "ys \<le> xs"

   ultimately show "xs = ys" by blast

 next

   fix xs ys zs :: "'a list"

   {

     fix n

     have "!!x y z. \<lbrakk>x \<le> y; y \<le> z; n=length x + length y + length z\<rbrakk> \<Longrightarrow> x \<le> z" 

     proof (induct rule: nat_less_induct[case_names I])

       case (I n x y z)

       from I.prems(2) show ?case proof (cases rule: ileq_cases)

         case empty with I.prems(1) show ?thesis by auto

       next

         case (drop a z') hence "length x + length y + length z' < length x + length y + length z" by simp

         with I.hyps I.prems(3,1) drop(2) have "x\<le>z'" by blast

         with drop(1) show ?thesis by (auto intro: ileq_drop)

       next

         case (take a y' z') from I.prems(1) show ?thesis proof (cases rule: ileq_cases[case_names empty' drop' take'])

           case empty' thus ?thesis by auto

         next

           case (drop' ah y'h) with take have "x\<le>y'" "y'\<le>z'" "length x + length y' + length z' < length x + length y + length z" by auto

           with I.hyps I.prems(3) have "x\<le>z'" by (blast)

           with take(2) show ?thesis  by (auto intro: ileq_drop)

         next

           case (take' ah x' y'h) with take have 2: "x=a#x'" "x'\<le>y'" "y'\<le>z'" "length x' + length y' + length z' < length x + length y + length z" by auto

           with I.hyps I.prems(3) have "x'\<le>z'" by blast

           with 2 take(2) show ?thesis by (auto intro: ileq_take)

         qed

       qed

     qed

   }

   moreover assume "xs \<le> ys" "ys \<le> zs"

   ultimately show "xs \<le> zs" by blast

 qed

 end

 lemmas ileq_intros = ileq_empty ileq_drop ileq_take

 lemma ileq_drop_many: "xs \<le> ys \<Longrightarrow> xs \<le> zs @ ys"

   by (induct zs) (auto intro: ileq_drop)

 lemma ileq_take_many: "xs \<le> ys \<Longrightarrow> zs @ xs \<le> zs @ ys"

   by (induct zs) (auto intro: ileq_take)

 lemma ileq_same_length: "xs \<le> ys \<Longrightarrow> length xs = length ys \<Longrightarrow> xs = ys"

   by (induct rule: ileq_induct) (auto dest: ileq_length)

 lemma ileq_same_append [simp]: "x # xs \<le> xs \<longleftrightarrow> False"

   by (auto dest: ileq_length)

 lemma ilt_length [intro]:

   assumes "xs < ys"

   shows "length xs < length ys"

 proof -

   from assms have "xs \<le> ys" and "xs \<noteq> ys" by (simp_all add: less_le)

   moreover with ileq_length have "length xs \<le> length ys" by auto

   ultimately show ?thesis by (auto intro: ileq_same_length)

 qed

 lemma ilt_empty [simp]: "[] < xs \<longleftrightarrow> xs \<noteq> []"

   by (unfold less_list_def, auto)

 lemma ilt_emptyI: "xs \<noteq> [] \<Longrightarrow> [] < xs"

   by (unfold less_list_def, auto)

 lemma ilt_emptyD: "[] < xs \<Longrightarrow> xs \<noteq> []"

   by (unfold less_list_def, auto)

 lemma ilt_below_empty[simp]: "xs < [] \<Longrightarrow> False"

   by (auto dest: ilt_length)

 lemma ilt_drop: "xs < ys \<Longrightarrow> xs < x # ys"

   by (unfold less_le) (auto intro: ileq_intros)

 lemma ilt_take: "xs < ys \<Longrightarrow> x # xs < x # ys"

   by (unfold less_le) (auto intro: ileq_intros)

 lemma ilt_drop_many: "xs < ys \<Longrightarrow> xs < zs @ ys"

   by (induct zs) (auto intro: ilt_drop)

 lemma ilt_take_many: "xs < ys \<Longrightarrow> zs @ xs < zs @ ys"

   by (induct zs) (auto intro: ilt_take)

 subsection {* Appending elements *}

 lemma ileq_rev_take: "xs \<le> ys \<Longrightarrow> xs @ [x] \<le> ys @ [x]"

   by (induct rule: ileq_induct) (auto intro: ileq_intros ileq_drop_many)

 lemma ilt_rev_take: "xs < ys \<Longrightarrow> xs @ [x] < ys @ [x]"

   by (unfold less_le) (auto dest: ileq_rev_take)

 lemma ileq_rev_drop: "xs \<le> ys \<Longrightarrow> xs \<le> ys @ [x]"

   by (induct rule: ileq_induct) (auto intro: ileq_intros)

 lemma ileq_rev_drop_many: "xs \<le> ys \<Longrightarrow> xs \<le> ys @ zs"

   by (induct zs rule: rev_induct) (auto dest: ileq_rev_drop)

 subsection {* Relation to standard list operations *}

 lemma ileq_map: "xs \<le> ys \<Longrightarrow> map f xs \<le> map f ys"

   by (induct rule: ileq_induct) (auto intro: ileq_intros)

 lemma ileq_filter_left[simp]: "filter f xs \<le> xs"

   by (induct xs) (auto intro: ileq_intros)

 lemma ileq_filter: "xs \<le> ys \<Longrightarrow> filter f xs \<le> filter f ys"

   by (induct rule: ileq_induct) (auto intro: ileq_intros) 

 end