src/HOL/Library/Sublist_Order.thy
author wenzelm
Mon Dec 28 01:28:28 2015 +0100 (2015-12-28)
changeset 61945 1135b8de26c3
parent 61585 a9599d3d7610
child 63465 d7610beb98bc
permissions -rw-r--r--
more symbols;
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(*  Title:      HOL/Library/Sublist_Order.thy
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    Authors:    Peter Lammich, Uni Muenster <peter.lammich@uni-muenster.de>
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                Florian Haftmann, Tobias Nipkow, TU Muenchen
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*)
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section \<open>Sublist Ordering\<close>
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theory Sublist_Order
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imports Sublist
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begin
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text \<open>
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  This theory defines sublist ordering on lists.
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  A list \<open>ys\<close> is a sublist of a list \<open>xs\<close>,
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  iff one obtains \<open>ys\<close> by erasing some elements from \<open>xs\<close>.
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\<close>
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subsection \<open>Definitions and basic lemmas\<close>
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instantiation list :: (type) ord
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begin
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definition
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  "(xs :: 'a list) \<le> ys \<longleftrightarrow> sublisteq xs ys"
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definition
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  "(xs :: 'a list) < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs"
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instance ..
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end
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instance list :: (type) order
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proof
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  fix xs ys :: "'a list"
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  show "xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs" unfolding less_list_def .. 
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next
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  fix xs :: "'a list"
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  show "xs \<le> xs" by (simp add: less_eq_list_def)
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next
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  fix xs ys :: "'a list"
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  assume "xs <= ys" and "ys <= xs"
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  thus "xs = ys" by (unfold less_eq_list_def) (rule sublisteq_antisym)
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next
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  fix xs ys zs :: "'a list"
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  assume "xs <= ys" and "ys <= zs"
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  thus "xs <= zs" by (unfold less_eq_list_def) (rule sublisteq_trans)
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qed
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lemmas less_eq_list_induct [consumes 1, case_names empty drop take] =
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  list_emb.induct [of "op =", folded less_eq_list_def]
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lemmas less_eq_list_drop = list_emb.list_emb_Cons [of "op =", folded less_eq_list_def]
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lemmas le_list_Cons2_iff [simp, code] = sublisteq_Cons2_iff [folded less_eq_list_def]
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lemmas le_list_map = sublisteq_map [folded less_eq_list_def]
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lemmas le_list_filter = sublisteq_filter [folded less_eq_list_def]
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lemmas le_list_length = list_emb_length [of "op =", folded less_eq_list_def]
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lemma less_list_length: "xs < ys \<Longrightarrow> length xs < length ys"
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  by (metis list_emb_length sublisteq_same_length le_neq_implies_less less_list_def less_eq_list_def)
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lemma less_list_empty [simp]: "[] < xs \<longleftrightarrow> xs \<noteq> []"
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  by (metis less_eq_list_def list_emb_Nil order_less_le)
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lemma less_list_below_empty [simp]: "xs < [] \<longleftrightarrow> False"
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  by (metis list_emb_Nil less_eq_list_def less_list_def)
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lemma less_list_drop: "xs < ys \<Longrightarrow> xs < x # ys"
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  by (unfold less_le less_eq_list_def) (auto)
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lemma less_list_take_iff: "x # xs < x # ys \<longleftrightarrow> xs < ys"
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  by (metis sublisteq_Cons2_iff less_list_def less_eq_list_def)
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lemma less_list_drop_many: "xs < ys \<Longrightarrow> xs < zs @ ys"
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  by (metis sublisteq_append_le_same_iff sublisteq_drop_many order_less_le self_append_conv2 less_eq_list_def)
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lemma less_list_take_many_iff: "zs @ xs < zs @ ys \<longleftrightarrow> xs < ys"
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  by (metis less_list_def less_eq_list_def sublisteq_append')
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lemma less_list_rev_take: "xs @ zs < ys @ zs \<longleftrightarrow> xs < ys"
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  by (unfold less_le less_eq_list_def) auto
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end