(* Title: HOL/Library/Product_ord.thy
ID: $Id$
Author: Norbert Voelker
*)
header {* Order on product types *}
theory Product_ord
imports Plain
begin
instantiation "*" :: (ord, ord) ord
begin
definition
prod_le_def [code func del]: "x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x \<le> snd y"
definition
prod_less_def [code func del]: "x < y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x < snd y"
instance ..
end
lemma [code, code func del]:
"(x1, y1) \<le> (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
"(x1, y1) < (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
unfolding prod_le_def prod_less_def by simp_all
lemma [code func]:
"(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"
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 * :: (linorder, linorder) linorder
by default (auto simp: prod_le_def)
instantiation * :: (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
by intro_classes
(auto simp add: inf_prod_def sup_prod_def min_max.sup_inf_distrib1)
end
end