--- a/src/HOL/Library/Fun_Lexorder.thy Mon May 28 23:15:30 2018 +0100
+++ b/src/HOL/Library/Fun_Lexorder.thy Tue May 29 14:05:59 2018 +0200
@@ -1,6 +1,6 @@
(* Author: Florian Haftmann, TU Muenchen *)
-section \<open>Lexical order on functions\<close>
+section \<open>Lexicographic order on functions\<close>
theory Fun_Lexorder
imports Main
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/List_Lexorder.thy Tue May 29 14:05:59 2018 +0200
@@ -0,0 +1,121 @@
+(* Title: HOL/Library/List_Lexorder.thy
+ Author: Norbert Voelker
+*)
+
+section \<open>Lexicographic order on lists\<close>
+
+theory List_Lexorder
+imports Main
+begin
+
+instantiation list :: (ord) ord
+begin
+
+definition
+ list_less_def: "xs < ys \<longleftrightarrow> (xs, ys) \<in> lexord {(u, v). u < v}"
+
+definition
+ list_le_def: "(xs :: _ list) \<le> ys \<longleftrightarrow> xs < ys \<or> xs = ys"
+
+instance ..
+
+end
+
+instance list :: (order) order
+proof
+ fix xs :: "'a list"
+ show "xs \<le> xs" by (simp add: list_le_def)
+next
+ fix xs ys zs :: "'a list"
+ assume "xs \<le> ys" and "ys \<le> zs"
+ then show "xs \<le> zs"
+ apply (auto simp add: list_le_def list_less_def)
+ apply (rule lexord_trans)
+ apply (auto intro: transI)
+ done
+next
+ fix xs ys :: "'a list"
+ assume "xs \<le> ys" and "ys \<le> xs"
+ then show "xs = ys"
+ apply (auto simp add: list_le_def list_less_def)
+ apply (rule lexord_irreflexive [THEN notE])
+ defer
+ apply (rule lexord_trans)
+ apply (auto intro: transI)
+ done
+next
+ fix xs ys :: "'a list"
+ show "xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs"
+ apply (auto simp add: list_less_def list_le_def)
+ defer
+ apply (rule lexord_irreflexive [THEN notE])
+ apply auto
+ apply (rule lexord_irreflexive [THEN notE])
+ defer
+ apply (rule lexord_trans)
+ apply (auto intro: transI)
+ done
+qed
+
+instance list :: (linorder) linorder
+proof
+ fix xs ys :: "'a list"
+ have "(xs, ys) \<in> lexord {(u, v). u < v} \<or> xs = ys \<or> (ys, xs) \<in> lexord {(u, v). u < v}"
+ by (rule lexord_linear) auto
+ then show "xs \<le> ys \<or> ys \<le> xs"
+ by (auto simp add: list_le_def list_less_def)
+qed
+
+instantiation list :: (linorder) distrib_lattice
+begin
+
+definition "(inf :: 'a list \<Rightarrow> _) = min"
+
+definition "(sup :: 'a list \<Rightarrow> _) = max"
+
+instance
+ by standard (auto simp add: inf_list_def sup_list_def max_min_distrib2)
+
+end
+
+lemma not_less_Nil [simp]: "\<not> x < []"
+ by (simp add: list_less_def)
+
+lemma Nil_less_Cons [simp]: "[] < a # x"
+ by (simp add: list_less_def)
+
+lemma Cons_less_Cons [simp]: "a # x < b # y \<longleftrightarrow> a < b \<or> a = b \<and> x < y"
+ by (simp add: list_less_def)
+
+lemma le_Nil [simp]: "x \<le> [] \<longleftrightarrow> x = []"
+ unfolding list_le_def by (cases x) auto
+
+lemma Nil_le_Cons [simp]: "[] \<le> x"
+ unfolding list_le_def by (cases x) auto
+
+lemma Cons_le_Cons [simp]: "a # x \<le> b # y \<longleftrightarrow> a < b \<or> a = b \<and> x \<le> y"
+ unfolding list_le_def by auto
+
+instantiation list :: (order) order_bot
+begin
+
+definition "bot = []"
+
+instance
+ by standard (simp add: bot_list_def)
+
+end
+
+lemma less_list_code [code]:
+ "xs < ([]::'a::{equal, order} list) \<longleftrightarrow> False"
+ "[] < (x::'a::{equal, order}) # xs \<longleftrightarrow> True"
+ "(x::'a::{equal, order}) # xs < y # ys \<longleftrightarrow> x < y \<or> x = y \<and> xs < ys"
+ by simp_all
+
+lemma less_eq_list_code [code]:
+ "x # xs \<le> ([]::'a::{equal, order} list) \<longleftrightarrow> False"
+ "[] \<le> (xs::'a::{equal, order} list) \<longleftrightarrow> True"
+ "(x::'a::{equal, order}) # xs \<le> y # ys \<longleftrightarrow> x < y \<or> x = y \<and> xs \<le> ys"
+ by simp_all
+
+end
--- a/src/HOL/Library/List_lexord.thy Mon May 28 23:15:30 2018 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,121 +0,0 @@
-(* Title: HOL/Library/List_lexord.thy
- Author: Norbert Voelker
-*)
-
-section \<open>Lexicographic order on lists\<close>
-
-theory List_lexord
-imports Main
-begin
-
-instantiation list :: (ord) ord
-begin
-
-definition
- list_less_def: "xs < ys \<longleftrightarrow> (xs, ys) \<in> lexord {(u, v). u < v}"
-
-definition
- list_le_def: "(xs :: _ list) \<le> ys \<longleftrightarrow> xs < ys \<or> xs = ys"
-
-instance ..
-
-end
-
-instance list :: (order) order
-proof
- fix xs :: "'a list"
- show "xs \<le> xs" by (simp add: list_le_def)
-next
- fix xs ys zs :: "'a list"
- assume "xs \<le> ys" and "ys \<le> zs"
- then show "xs \<le> zs"
- apply (auto simp add: list_le_def list_less_def)
- apply (rule lexord_trans)
- apply (auto intro: transI)
- done
-next
- fix xs ys :: "'a list"
- assume "xs \<le> ys" and "ys \<le> xs"
- then show "xs = ys"
- apply (auto simp add: list_le_def list_less_def)
- apply (rule lexord_irreflexive [THEN notE])
- defer
- apply (rule lexord_trans)
- apply (auto intro: transI)
- done
-next
- fix xs ys :: "'a list"
- show "xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs"
- apply (auto simp add: list_less_def list_le_def)
- defer
- apply (rule lexord_irreflexive [THEN notE])
- apply auto
- apply (rule lexord_irreflexive [THEN notE])
- defer
- apply (rule lexord_trans)
- apply (auto intro: transI)
- done
-qed
-
-instance list :: (linorder) linorder
-proof
- fix xs ys :: "'a list"
- have "(xs, ys) \<in> lexord {(u, v). u < v} \<or> xs = ys \<or> (ys, xs) \<in> lexord {(u, v). u < v}"
- by (rule lexord_linear) auto
- then show "xs \<le> ys \<or> ys \<le> xs"
- by (auto simp add: list_le_def list_less_def)
-qed
-
-instantiation list :: (linorder) distrib_lattice
-begin
-
-definition "(inf :: 'a list \<Rightarrow> _) = min"
-
-definition "(sup :: 'a list \<Rightarrow> _) = max"
-
-instance
- by standard (auto simp add: inf_list_def sup_list_def max_min_distrib2)
-
-end
-
-lemma not_less_Nil [simp]: "\<not> x < []"
- by (simp add: list_less_def)
-
-lemma Nil_less_Cons [simp]: "[] < a # x"
- by (simp add: list_less_def)
-
-lemma Cons_less_Cons [simp]: "a # x < b # y \<longleftrightarrow> a < b \<or> a = b \<and> x < y"
- by (simp add: list_less_def)
-
-lemma le_Nil [simp]: "x \<le> [] \<longleftrightarrow> x = []"
- unfolding list_le_def by (cases x) auto
-
-lemma Nil_le_Cons [simp]: "[] \<le> x"
- unfolding list_le_def by (cases x) auto
-
-lemma Cons_le_Cons [simp]: "a # x \<le> b # y \<longleftrightarrow> a < b \<or> a = b \<and> x \<le> y"
- unfolding list_le_def by auto
-
-instantiation list :: (order) order_bot
-begin
-
-definition "bot = []"
-
-instance
- by standard (simp add: bot_list_def)
-
-end
-
-lemma less_list_code [code]:
- "xs < ([]::'a::{equal, order} list) \<longleftrightarrow> False"
- "[] < (x::'a::{equal, order}) # xs \<longleftrightarrow> True"
- "(x::'a::{equal, order}) # xs < y # ys \<longleftrightarrow> x < y \<or> x = y \<and> xs < ys"
- by simp_all
-
-lemma less_eq_list_code [code]:
- "x # xs \<le> ([]::'a::{equal, order} list) \<longleftrightarrow> False"
- "[] \<le> (xs::'a::{equal, order} list) \<longleftrightarrow> True"
- "(x::'a::{equal, order}) # xs \<le> y # ys \<longleftrightarrow> x < y \<or> x = y \<and> xs \<le> ys"
- by simp_all
-
-end
--- a/src/HOL/ROOT Mon May 28 23:15:30 2018 +0100
+++ b/src/HOL/ROOT Tue May 29 14:05:59 2018 +0200
@@ -29,7 +29,7 @@
Library
(*conflicting type class instantiations and dependent applications*)
Finite_Lattice
- List_lexord
+ List_Lexorder
Prefix_Order
Product_Lexorder
Product_Order
--- a/src/HOL/ex/Radix_Sort.thy Mon May 28 23:15:30 2018 +0100
+++ b/src/HOL/ex/Radix_Sort.thy Tue May 29 14:05:59 2018 +0200
@@ -2,7 +2,7 @@
theory Radix_Sort
imports
- "HOL-Library.List_lexord"
+ "HOL-Library.List_Lexorder"
"HOL-Library.Sublist"
"HOL-Library.Multiset"
begin