Theory Product_ord

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theory Product_ord
imports Main

(*  Title:      HOL/Library/Product_ord.thy
Author: Norbert Voelker
*)


header {* Order on product types *}

theory Product_ord
imports Main
begin


instantiation "*" :: (ord, ord) ord
begin


definition
prod_le_def [code del]: "x ≤ y <-> fst x < fst y ∨ fst x ≤ fst y ∧ snd x ≤ snd y"


definition
prod_less_def [code del]: "x < y <-> fst x < fst y ∨ fst x ≤ fst y ∧ snd x < snd y"


instance ..

end

lemma [code]:
"(x1::'a::{ord, eq}, y1) ≤ (x2, y2) <-> x1 < x2 ∨ x1 ≤ x2 ∧ y1 ≤ y2"
"(x1::'a::{ord, eq}, y1) < (x2, y2) <-> x1 < x2 ∨ x1 ≤ x2 ∧ y1 < y2"

unfolding prod_le_def prod_less_def by simp_all

instance * :: (preorder, preorder) preorder proof
qed (auto simp: prod_le_def prod_less_def less_le_not_le intro: order_trans)

instance * :: (order, order) order proof
qed (auto simp add: prod_le_def)

instance * :: (linorder, linorder) linorder proof
qed (auto simp: prod_le_def)

instantiation * :: (linorder, linorder) distrib_lattice
begin


definition
inf_prod_def: "(inf :: 'a × 'b => _ => _) = min"


definition
sup_prod_def: "(sup :: 'a × 'b => _ => _) = max"


instance proof
qed (auto simp add: inf_prod_def sup_prod_def min_max.sup_inf_distrib1)

end

instantiation * :: (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 * :: (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