--- a/src/HOL/Library/List_lexord.thy Mon Aug 26 22:01:39 2013 +0200
+++ b/src/HOL/Library/List_lexord.thy Mon Aug 26 23:39:53 2013 +0200
@@ -5,7 +5,7 @@
header {* Lexicographic order on lists *}
theory List_lexord
-imports List Main
+imports Main
begin
instantiation list :: (ord) ord
@@ -28,25 +28,33 @@
next
fix xs ys zs :: "'a list"
assume "xs \<le> ys" and "ys \<le> zs"
- then show "xs \<le> zs" by (auto simp add: list_le_def list_less_def)
- (rule lexord_trans, auto intro: transI)
+ 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
+ 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
+ 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
@@ -54,51 +62,47 @@
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"
+ 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 \<Colon> 'a list \<Rightarrow> _) = min"
+definition "(inf \<Colon> 'a list \<Rightarrow> _) = min"
-definition
- "(sup \<Colon> 'a list \<Rightarrow> _) = max"
+definition "(sup \<Colon> 'a list \<Rightarrow> _) = max"
instance
- by intro_classes
- (auto simp add: inf_list_def sup_list_def min_max.sup_inf_distrib1)
+ by default (auto simp add: inf_list_def sup_list_def min_max.sup_inf_distrib1)
end
-lemma not_less_Nil [simp]: "\<not> (x < [])"
- by (unfold list_less_def) simp
+lemma not_less_Nil [simp]: "\<not> x < []"
+ by (simp add: list_less_def)
lemma Nil_less_Cons [simp]: "[] < a # x"
- by (unfold list_less_def) simp
+ 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 (unfold list_less_def) simp
+ by (simp add: list_less_def)
lemma le_Nil [simp]: "x \<le> [] \<longleftrightarrow> x = []"
- by (unfold list_le_def, cases x) auto
+ unfolding list_le_def by (cases x) auto
lemma Nil_le_Cons [simp]: "[] \<le> x"
- by (unfold list_le_def, cases x) auto
+ 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"
- by (unfold list_le_def) auto
+ unfolding list_le_def by auto
instantiation list :: (order) order_bot
begin
-definition
- "bot = []"
+definition "bot = []"
-instance proof
-qed (simp add: bot_list_def)
+instance
+ by default (simp add: bot_list_def)
end