nipkow@15737: (*  Title:      HOL/Library/List_lexord.thy
nipkow@15737:     Author:     Norbert Voelker
nipkow@15737: *)
nipkow@15737: 
wenzelm@17200: header {* Lexicographic order on lists *}
nipkow@15737: 
nipkow@15737: theory List_lexord
haftmann@30663: imports List Main
nipkow@15737: begin
nipkow@15737: 
haftmann@25764: instantiation list :: (ord) ord
haftmann@25764: begin
haftmann@25764: 
haftmann@25764: definition
haftmann@37474:   list_less_def: "xs < ys \<longleftrightarrow> (xs, ys) \<in> lexord {(u, v). u < v}"
haftmann@25764: 
haftmann@25764: definition
haftmann@37474:   list_le_def: "(xs :: _ list) \<le> ys \<longleftrightarrow> xs < ys \<or> xs = ys"
haftmann@25764: 
haftmann@25764: instance ..
haftmann@25764: 
haftmann@25764: end
nipkow@15737: 
wenzelm@17200: instance list :: (order) order
haftmann@27682: proof
haftmann@27682:   fix xs :: "'a list"
haftmann@27682:   show "xs \<le> xs" by (simp add: list_le_def)
haftmann@27682: next
haftmann@27682:   fix xs ys zs :: "'a list"
haftmann@27682:   assume "xs \<le> ys" and "ys \<le> zs"
haftmann@27682:   then show "xs \<le> zs" by (auto simp add: list_le_def list_less_def)
haftmann@27682:     (rule lexord_trans, auto intro: transI)
haftmann@27682: next
haftmann@27682:   fix xs ys :: "'a list"
haftmann@27682:   assume "xs \<le> ys" and "ys \<le> xs"
haftmann@27682:   then show "xs = ys" apply (auto simp add: list_le_def list_less_def)
haftmann@27682:   apply (rule lexord_irreflexive [THEN notE])
haftmann@27682:   defer
haftmann@27682:   apply (rule lexord_trans) apply (auto intro: transI) done
haftmann@27682: next
haftmann@27682:   fix xs ys :: "'a list"
haftmann@27682:   show "xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs" 
haftmann@27682:   apply (auto simp add: list_less_def list_le_def)
haftmann@27682:   defer
haftmann@27682:   apply (rule lexord_irreflexive [THEN notE])
haftmann@27682:   apply auto
haftmann@27682:   apply (rule lexord_irreflexive [THEN notE])
haftmann@27682:   defer
haftmann@27682:   apply (rule lexord_trans) apply (auto intro: transI) done
haftmann@27682: qed
nipkow@15737: 
haftmann@21458: instance list :: (linorder) linorder
haftmann@27682: proof
haftmann@27682:   fix xs ys :: "'a list"
haftmann@27682:   have "(xs, ys) \<in> lexord {(u, v). u < v} \<or> xs = ys \<or> (ys, xs) \<in> lexord {(u, v). u < v}"
haftmann@27682:     by (rule lexord_linear) auto
haftmann@27682:   then show "xs \<le> ys \<or> ys \<le> xs" 
haftmann@27682:     by (auto simp add: list_le_def list_less_def)
haftmann@27682: qed
nipkow@15737: 
haftmann@25764: instantiation list :: (linorder) distrib_lattice
haftmann@25764: begin
haftmann@25764: 
haftmann@25764: definition
haftmann@37765:   "(inf \<Colon> 'a list \<Rightarrow> _) = min"
haftmann@25764: 
haftmann@25764: definition
haftmann@37765:   "(sup \<Colon> 'a list \<Rightarrow> _) = max"
haftmann@25764: 
haftmann@25764: instance
haftmann@22483:   by intro_classes
haftmann@22483:     (auto simp add: inf_list_def sup_list_def min_max.sup_inf_distrib1)
haftmann@22483: 
haftmann@25764: end
haftmann@25764: 
haftmann@22177: lemma not_less_Nil [simp]: "\<not> (x < [])"
wenzelm@17200:   by (unfold list_less_def) simp
nipkow@15737: 
haftmann@22177: lemma Nil_less_Cons [simp]: "[] < a # x"
wenzelm@17200:   by (unfold list_less_def) simp
nipkow@15737: 
haftmann@22177: lemma Cons_less_Cons [simp]: "a # x < b # y \<longleftrightarrow> a < b \<or> a = b \<and> x < y"
wenzelm@17200:   by (unfold list_less_def) simp
nipkow@15737: 
haftmann@22177: lemma le_Nil [simp]: "x \<le> [] \<longleftrightarrow> x = []"
haftmann@25502:   by (unfold list_le_def, cases x) auto
haftmann@22177: 
haftmann@22177: lemma Nil_le_Cons [simp]: "[] \<le> x"
haftmann@25502:   by (unfold list_le_def, cases x) auto
nipkow@15737: 
haftmann@22177: lemma Cons_le_Cons [simp]: "a # x \<le> b # y \<longleftrightarrow> a < b \<or> a = b \<and> x \<le> y"
haftmann@25502:   by (unfold list_le_def) auto
nipkow@15737: 
haftmann@37474: instantiation list :: (order) bot
haftmann@37474: begin
haftmann@37474: 
haftmann@37474: definition
haftmann@37474:   "bot = []"
haftmann@37474: 
haftmann@37474: instance proof
haftmann@37474: qed (simp add: bot_list_def)
haftmann@37474: 
haftmann@37474: end
haftmann@37474: 
haftmann@37474: lemma less_list_code [code]:
haftmann@38857:   "xs < ([]\<Colon>'a\<Colon>{equal, order} list) \<longleftrightarrow> False"
haftmann@38857:   "[] < (x\<Colon>'a\<Colon>{equal, order}) # xs \<longleftrightarrow> True"
haftmann@38857:   "(x\<Colon>'a\<Colon>{equal, order}) # xs < y # ys \<longleftrightarrow> x < y \<or> x = y \<and> xs < ys"
haftmann@22177:   by simp_all
haftmann@22177: 
haftmann@37474: lemma less_eq_list_code [code]:
haftmann@38857:   "x # xs \<le> ([]\<Colon>'a\<Colon>{equal, order} list) \<longleftrightarrow> False"
haftmann@38857:   "[] \<le> (xs\<Colon>'a\<Colon>{equal, order} list) \<longleftrightarrow> True"
haftmann@38857:   "(x\<Colon>'a\<Colon>{equal, order}) # xs \<le> y # ys \<longleftrightarrow> x < y \<or> x = y \<and> xs \<le> ys"
haftmann@22177:   by simp_all
haftmann@22177: 
wenzelm@17200: end