(* Title: HOL/Library/Product_ord.thy
ID: $Id$
Author: Norbert Voelker
*)
header {* Order on product types *}
theory Product_ord
imports Main
begin
instance "*" :: (ord, ord) ord
prod_le_def: "(x \<le> y) \<equiv> (fst x < fst y) \<or> (fst x = fst y \<and> snd x \<le> snd y)"
prod_less_def: "(x < y) \<equiv> (fst x < fst y) \<or> (fst x = fst y \<and> snd x < snd y)" ..
lemmas prod_ord_defs [code func del] = prod_less_def prod_le_def
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_ord_defs by simp_all
lemma [code]:
"(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_ord_defs by simp_all
instance * :: (order, order) order
by default (auto simp: prod_ord_defs intro: order_less_trans)
instance * :: (linorder, linorder) linorder
by default (auto simp: prod_le_def)
instance * :: (linorder, linorder) distrib_lattice
inf_prod_def: "inf \<equiv> min"
sup_prod_def: "sup \<equiv> max"
by intro_classes
(auto simp add: inf_prod_def sup_prod_def min_max.sup_inf_distrib1)
end