(* Title: HOL/Library/Sublist_Order.thy
Authors: Peter Lammich, Uni Muenster <peter.lammich@uni-muenster.de>
Florian Haftmann, Tobias Nipkow, TU Muenchen
*)
section \<open>Sublist Ordering\<close>
theory Sublist_Order
imports Sublist
begin
text \<open>
This theory defines sublist ordering on lists.
A list \<open>ys\<close> is a sublist of a list \<open>xs\<close>,
iff one obtains \<open>ys\<close> by erasing some elements from \<open>xs\<close>.
\<close>
subsection \<open>Definitions and basic lemmas\<close>
instantiation list :: (type) ord
begin
definition
"(xs :: 'a list) \<le> ys \<longleftrightarrow> sublisteq xs ys"
definition
"(xs :: 'a list) < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs"
instance ..
end
instance list :: (type) order
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 (simp add: less_eq_list_def)
next
fix xs ys :: "'a list"
assume "xs <= ys" and "ys <= xs"
thus "xs = ys" by (unfold less_eq_list_def) (rule sublisteq_antisym)
next
fix xs ys zs :: "'a list"
assume "xs <= ys" and "ys <= zs"
thus "xs <= zs" by (unfold less_eq_list_def) (rule sublisteq_trans)
qed
lemmas less_eq_list_induct [consumes 1, case_names empty drop take] =
list_emb.induct [of "op =", folded less_eq_list_def]
lemmas less_eq_list_drop = list_emb.list_emb_Cons [of "op =", folded less_eq_list_def]
lemmas le_list_Cons2_iff [simp, code] = sublisteq_Cons2_iff [folded less_eq_list_def]
lemmas le_list_map = sublisteq_map [folded less_eq_list_def]
lemmas le_list_filter = sublisteq_filter [folded less_eq_list_def]
lemmas le_list_length = list_emb_length [of "op =", folded less_eq_list_def]
lemma less_list_length: "xs < ys \<Longrightarrow> length xs < length ys"
by (metis list_emb_length sublisteq_same_length le_neq_implies_less less_list_def less_eq_list_def)
lemma less_list_empty [simp]: "[] < xs \<longleftrightarrow> xs \<noteq> []"
by (metis less_eq_list_def list_emb_Nil order_less_le)
lemma less_list_below_empty [simp]: "xs < [] \<longleftrightarrow> False"
by (metis list_emb_Nil less_eq_list_def less_list_def)
lemma less_list_drop: "xs < ys \<Longrightarrow> xs < x # ys"
by (unfold less_le less_eq_list_def) (auto)
lemma less_list_take_iff: "x # xs < x # ys \<longleftrightarrow> xs < ys"
by (metis sublisteq_Cons2_iff less_list_def less_eq_list_def)
lemma less_list_drop_many: "xs < ys \<Longrightarrow> xs < zs @ ys"
by (metis sublisteq_append_le_same_iff sublisteq_drop_many order_less_le self_append_conv2 less_eq_list_def)
lemma less_list_take_many_iff: "zs @ xs < zs @ ys \<longleftrightarrow> xs < ys"
by (metis less_list_def less_eq_list_def sublisteq_append')
lemma less_list_rev_take: "xs @ zs < ys @ zs \<longleftrightarrow> xs < ys"
by (unfold less_le less_eq_list_def) auto
end