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(* Title: HOL/Library/List_lexord.thy
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ID: $Id$
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Author: Norbert Voelker
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*)
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header {* Lexicographic order on lists *}
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theory List_lexord
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imports Plain "~~/src/HOL/List"
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begin
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instantiation list :: (ord) ord
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begin
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definition
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list_less_def [code del]: "(xs::('a::ord) list) < ys \<longleftrightarrow> (xs, ys) \<in> lexord {(u,v). u < v}"
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definition
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list_le_def [code del]: "(xs::('a::ord) list) \<le> ys \<longleftrightarrow> (xs < ys \<or> xs = ys)"
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instance ..
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end
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instance list :: (order) order
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proof
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fix xs :: "'a list"
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show "xs \<le> xs" by (simp add: list_le_def)
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next
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fix xs ys zs :: "'a list"
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assume "xs \<le> ys" and "ys \<le> zs"
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then show "xs \<le> zs" by (auto simp add: list_le_def list_less_def)
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(rule lexord_trans, auto intro: transI)
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next
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fix xs ys :: "'a list"
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assume "xs \<le> ys" and "ys \<le> xs"
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then show "xs = ys" apply (auto simp add: list_le_def list_less_def)
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apply (rule lexord_irreflexive [THEN notE])
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defer
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apply (rule lexord_trans) apply (auto intro: transI) done
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next
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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"
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apply (auto simp add: list_less_def list_le_def)
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defer
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apply (rule lexord_irreflexive [THEN notE])
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apply auto
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apply (rule lexord_irreflexive [THEN notE])
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defer
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apply (rule lexord_trans) apply (auto intro: transI) done
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qed
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instance list :: (linorder) linorder
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proof
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fix xs ys :: "'a list"
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have "(xs, ys) \<in> lexord {(u, v). u < v} \<or> xs = ys \<or> (ys, xs) \<in> lexord {(u, v). u < v}"
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by (rule lexord_linear) auto
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then show "xs \<le> ys \<or> ys \<le> xs"
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by (auto simp add: list_le_def list_less_def)
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qed
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instantiation list :: (linorder) distrib_lattice
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begin
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definition
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[code del]: "(inf \<Colon> 'a list \<Rightarrow> _) = min"
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definition
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[code del]: "(sup \<Colon> 'a list \<Rightarrow> _) = max"
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instance
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by intro_classes
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(auto simp add: inf_list_def sup_list_def min_max.sup_inf_distrib1)
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end
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lemma not_less_Nil [simp]: "\<not> (x < [])"
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by (unfold list_less_def) simp
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lemma Nil_less_Cons [simp]: "[] < a # x"
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by (unfold list_less_def) simp
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lemma Cons_less_Cons [simp]: "a # x < b # y \<longleftrightarrow> a < b \<or> a = b \<and> x < y"
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by (unfold list_less_def) simp
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lemma le_Nil [simp]: "x \<le> [] \<longleftrightarrow> x = []"
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by (unfold list_le_def, cases x) auto
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lemma Nil_le_Cons [simp]: "[] \<le> x"
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by (unfold list_le_def, cases x) auto
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lemma Cons_le_Cons [simp]: "a # x \<le> b # y \<longleftrightarrow> a < b \<or> a = b \<and> x \<le> y"
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by (unfold list_le_def) auto
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lemma less_code [code]:
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"xs < ([]\<Colon>'a\<Colon>{eq, order} list) \<longleftrightarrow> False"
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"[] < (x\<Colon>'a\<Colon>{eq, order}) # xs \<longleftrightarrow> True"
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"(x\<Colon>'a\<Colon>{eq, order}) # xs < y # ys \<longleftrightarrow> x < y \<or> x = y \<and> xs < ys"
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by simp_all
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lemma less_eq_code [code]:
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"x # xs \<le> ([]\<Colon>'a\<Colon>{eq, order} list) \<longleftrightarrow> False"
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"[] \<le> (xs\<Colon>'a\<Colon>{eq, order} list) \<longleftrightarrow> True"
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"(x\<Colon>'a\<Colon>{eq, order}) # xs \<le> y # ys \<longleftrightarrow> x < y \<or> x = y \<and> xs \<le> ys"
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by simp_all
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end
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