added theory Sublist_Order
authorhaftmann
Tue, 22 Apr 2008 08:33:21 +0200
changeset 26735 39be3c7e643a
parent 26734 a92057c1ee21
child 26736 e6091328718f
added theory Sublist_Order
src/HOL/IsaMakefile
src/HOL/Library/Library/ROOT.ML
src/HOL/Library/Sublist_Order.thy
--- a/src/HOL/IsaMakefile	Tue Apr 22 08:33:20 2008 +0200
+++ b/src/HOL/IsaMakefile	Tue Apr 22 08:33:21 2008 +0200
@@ -226,7 +226,7 @@
   Library/Library/ROOT.ML Library/Library/document/root.tex \
   Library/Library/document/root.bib Library/While_Combinator.thy \
   Library/Product_ord.thy Library/Char_nat.thy Library/Char_ord.thy \
-  Library/Option_ord.thy \
+  Library/Option_ord.thy Library/Sublist_Order.thy \
   Library/List_lexord.thy Library/Commutative_Ring.thy Library/comm_ring.ML \
   Library/Coinductive_List.thy Library/AssocList.thy \
   Library/Parity.thy Library/GCD.thy Library/Binomial.thy \
--- a/src/HOL/Library/Library/ROOT.ML	Tue Apr 22 08:33:20 2008 +0200
+++ b/src/HOL/Library/Library/ROOT.ML	Tue Apr 22 08:33:21 2008 +0200
@@ -1,3 +1,3 @@
 (* $Id$ *)
 
-use_thys ["Library", "List_Prefix", "List_lexord"];
+use_thys ["Library", "List_Prefix", "List_lexord", "Sublist_Order"];
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Sublist_Order.thy	Tue Apr 22 08:33:21 2008 +0200
@@ -0,0 +1,187 @@
+(*  Title:      HOL/Library/Sublist_Order.thy
+    ID:         $Id$
+    Authors:    Peter Lammich, Uni Muenster <peter.lammich@uni-muenster.de>
+                Florian Haftmann, TU M√ľnchen
+*)
+
+header {* Sublist Ordering *}
+
+theory Sublist_Order
+imports Main
+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 func del]: "(xs \<Colon> 'a list) < ys \<longleftrightarrow> xs \<le> ys \<and> xs \<noteq> ys"
+
+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> xs \<noteq> ys" 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_list_def)
+  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_list_def) (auto intro: ileq_intros)
+lemma ilt_take: "xs < ys \<Longrightarrow> x # xs < x # ys"
+  by (unfold less_list_def) (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_list_def) (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