src/HOL/Library/Product_ord.thy
 changeset 31040 996ae76c9eda parent 30738 0842e906300c child 37678 0040bafffdef
```--- a/src/HOL/Library/Product_ord.thy	Mon May 04 23:45:58 2009 +0200
+++ b/src/HOL/Library/Product_ord.thy	Wed May 06 09:08:47 2009 +0200
@@ -12,25 +12,28 @@
begin

definition
-  prod_le_def [code del]: "x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x \<le> snd y"
+  prod_le_def [code del]: "x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x \<le> fst y \<and> snd x \<le> snd y"

definition
-  prod_less_def [code del]: "x < y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x < snd y"
+  prod_less_def [code del]: "x < y \<longleftrightarrow> fst x < fst y \<or> fst x \<le> fst y \<and> snd x < snd y"

instance ..

end

lemma [code]:
-  "(x1\<Colon>'a\<Colon>{ord, eq}, y1) \<le> (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
-  "(x1\<Colon>'a\<Colon>{ord, eq}, y1) < (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
+  "(x1\<Colon>'a\<Colon>{ord, eq}, y1) \<le> (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 \<le> y2"
+  "(x1\<Colon>'a\<Colon>{ord, eq}, y1) < (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 < y2"
unfolding prod_le_def prod_less_def by simp_all

-instance * :: (order, order) order
-  by default (auto simp: prod_le_def prod_less_def intro: order_less_trans)
+instance * :: (preorder, preorder) preorder proof
+qed (auto simp: prod_le_def prod_less_def less_le_not_le intro: order_trans)

-instance * :: (linorder, linorder) linorder
-  by default (auto simp: prod_le_def)
+instance * :: (order, order) order proof
+
+instance * :: (linorder, linorder) linorder proof
+qed (auto simp: prod_le_def)

instantiation * :: (linorder, linorder) distrib_lattice
begin
@@ -41,9 +44,30 @@
definition
sup_prod_def: "(sup \<Colon> 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = max"

-instance
-  by intro_classes
-    (auto simp add: inf_prod_def sup_prod_def min_max.sup_inf_distrib1)
+instance proof
+qed (auto simp add: inf_prod_def sup_prod_def min_max.sup_inf_distrib1)
+
+end
+
+instantiation * :: (bot, bot) bot
+begin
+
+definition
+  bot_prod_def: "bot = (bot, bot)"
+
+instance proof
+qed (auto simp add: bot_prod_def prod_le_def)
+
+end
+
+instantiation * :: (top, top) top
+begin
+
+definition
+  top_prod_def: "top = (top, top)"
+
+instance proof
+qed (auto simp add: top_prod_def prod_le_def)

end
```