(* 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, equal}, y1) \<le> (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 \<le> y2"
"(x1\<Colon>'a\<Colon>{ord, equal}, 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
text {* A stronger version of the definition holds for partial orders. *}
lemma prod_less_eq:
fixes x y :: "'a::order \<times> 'b::ord"
shows "x < y \<longleftrightarrow> fst x < fst y \<or> (fst x = fst y \<and> snd x < snd y)"
unfolding prod_less_def fst_conv snd_conv le_less by auto
instance prod :: (wellorder, wellorder) wellorder
proof
fix P :: "'a \<times> 'b \<Rightarrow> bool" and z :: "'a \<times> 'b"
assume P: "\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x"
show "P z"
proof (induct z)
case (Pair a b)
show "P (a, b)"
apply (induct a arbitrary: b rule: less_induct)
apply (rule less_induct [where 'a='b])
apply (rule P)
apply (auto simp add: prod_less_eq)
done
qed
qed
end