--- a/src/HOL/Library/List_Lexorder.thy Thu Aug 20 10:39:26 2020 +0100
+++ b/src/HOL/Library/List_Lexorder.thy Fri Aug 21 12:42:57 2020 +0100
@@ -26,20 +26,17 @@
let ?r = "{(u, v::'a). u < v}"
have tr: "trans ?r"
using trans_def by fastforce
- have *: "antisym {(u, v::'a). u < v}"
- using antisym_def by fastforce
have \<section>: False
if "(xs,ys) \<in> lexord ?r" "(ys,xs) \<in> lexord ?r" for xs ys :: "'a list"
proof -
have "(xs,xs) \<in> lexord ?r"
- using lexord_trans that tr * by blast
+ using that transD [OF lexord_transI [OF tr]] by blast
then show False
by (meson case_prodD lexord_irreflexive less_irrefl mem_Collect_eq)
qed
show "xs \<le> xs" for xs :: "'a list" by (simp add: list_le_def)
show "xs \<le> zs" if "xs \<le> ys" and "ys \<le> zs" for xs ys zs :: "'a list"
- using that transD [OF lexord_transI [OF tr]] *
- by (auto simp add: list_le_def list_less_def)
+ using that transD [OF lexord_transI [OF tr]] by (auto simp add: list_le_def list_less_def)
show "xs = ys" if "xs \<le> ys" "ys \<le> xs" for xs ys :: "'a list"
using \<section> that list_le_def list_less_def by blast
show "xs < ys \<longleftrightarrow> xs \<le> ys \<and> \<not> ys \<le> xs" for xs ys :: "'a list"
@@ -73,7 +70,7 @@
lemma Nil_less_Cons [simp]: "[] < a # x"
by (simp add: list_less_def)
-lemma Cons_less_Cons [simp]: "a # x < b # y \<longleftrightarrow> (if a = b then x < y else a < b)"
+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 = []"
@@ -82,7 +79,7 @@
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> (if a = b then x \<le> y else a < b)"
+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