(* Title: HOL/Library/Product_ord.thy
Author: Norbert Voelker
*)
header {* Order on product types *}
theory Product_ord
imports Main
begin
instantiation prod :: (ord, ord) ord
begin
definition
prod_le_def: "x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x \<le> fst y \<and> snd x \<le> snd y"
definition
prod_less_def: "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 \<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 prod :: (preorder, preorder) preorder proof
qed (auto simp: prod_le_def prod_less_def less_le_not_le intro: order_trans)
instance prod :: (order, order) order proof
qed (auto simp add: prod_le_def)
instance prod :: (linorder, linorder) linorder proof
qed (auto simp: prod_le_def)
instantiation prod :: (linorder, linorder) distrib_lattice
begin
definition
inf_prod_def: "(inf \<Colon> 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = min"
definition
sup_prod_def: "(sup \<Colon> 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = max"
instance proof
qed (auto simp add: inf_prod_def sup_prod_def min_max.sup_inf_distrib1)
end
instantiation prod :: (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 prod :: (top, top) top
begin
definition
top_prod_def: "top = (top, top)"
instance proof
qed (auto simp add: top_prod_def prod_le_def)
end
end