(* 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::'a::{ord, equal}, y1) \<le> (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 \<le> y2"
"(x1::'a::{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
by default (auto simp: prod_le_def prod_less_def less_le_not_le intro: order_trans)
instance prod :: (order, order) order
by default (auto simp add: prod_le_def)
instance prod :: (linorder, linorder) linorder
by default (auto simp: prod_le_def)
instantiation prod :: (linorder, linorder) distrib_lattice
begin
definition
inf_prod_def: "(inf :: 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = min"
definition
sup_prod_def: "(sup :: 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = max"
instance
by default (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
by default (auto simp add: bot_prod_def prod_le_def)
end
instantiation prod :: (top, top) top
begin
definition
top_prod_def: "top = (top, top)"
instance
by default (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)"
proof (induct a arbitrary: b rule: less_induct)
case (less a\<^isub>1) note a\<^isub>1 = this
show "P (a\<^isub>1, b)"
proof (induct b rule: less_induct)
case (less b\<^isub>1) note b\<^isub>1 = this
show "P (a\<^isub>1, b\<^isub>1)"
proof (rule P)
fix p assume p: "p < (a\<^isub>1, b\<^isub>1)"
show "P p"
proof (cases "fst p < a\<^isub>1")
case True
then have "P (fst p, snd p)" by (rule a\<^isub>1)
then show ?thesis by simp
next
case False
with p have 1: "a\<^isub>1 = fst p" and 2: "snd p < b\<^isub>1"
by (simp_all add: prod_less_eq)
from 2 have "P (a\<^isub>1, snd p)" by (rule b\<^isub>1)
with 1 show ?thesis by simp
qed
qed
qed
qed
qed
qed
end