# Theory HOL-Library.Product_Lexorder

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

section ‹Lexicographic order on product types›

theory Product_Lexorder
imports Main
begin

instantiation prod :: (ord, ord) ord
begin

definition
"x ≤ y ⟷ fst x < fst y ∨ fst x ≤ fst y ∧ snd x ≤ snd y"

definition
"x < y ⟷ fst x < fst y ∨ fst x ≤ fst y ∧ snd x < snd y"

instance ..

end

lemma less_eq_prod_simp [simp, code]:
"(x1, y1) ≤ (x2, y2) ⟷ x1 < x2 ∨ x1 ≤ x2 ∧ y1 ≤ y2"
by (simp add: less_eq_prod_def)

lemma less_prod_simp [simp, code]:
"(x1, y1) < (x2, y2) ⟷ x1 < x2 ∨ x1 ≤ x2 ∧ y1 < y2"
by (simp add: less_prod_def)

text ‹A stronger version for partial orders.›

lemma less_prod_def':
fixes x y :: "'a::order × 'b::ord"
shows "x < y ⟷ fst x < fst y ∨ fst x = fst y ∧ snd x < snd y"
by (auto simp add: less_prod_def le_less)

instance prod :: (preorder, preorder) preorder
by standard (auto simp: less_eq_prod_def less_prod_def less_le_not_le intro: order_trans)

instance prod :: (order, order) order
by standard (auto simp add: less_eq_prod_def)

instance prod :: (linorder, linorder) linorder
by standard (auto simp: less_eq_prod_def)

instantiation prod :: (linorder, linorder) distrib_lattice
begin

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

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

instance
by standard (auto simp add: inf_prod_def sup_prod_def max_min_distrib2)

end

instantiation prod :: (bot, bot) bot
begin

definition
"bot = (bot, bot)"

instance ..

end

instance prod :: (order_bot, order_bot) order_bot
by standard (auto simp add: bot_prod_def)

instantiation prod :: (top, top) top
begin

definition
"top = (top, top)"

instance ..

end

instance prod :: (order_top, order_top) order_top
by standard (auto simp add: top_prod_def)

instance prod :: (wellorder, wellorder) wellorder
proof
fix P :: "'a × 'b ⇒ bool" and z :: "'a × 'b"
assume P: "⋀x. (⋀y. y < x ⟹ P y) ⟹ 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⇩1) note a⇩1 = this
show "P (a⇩1, b)"
proof (induct b rule: less_induct)
case (less b⇩1) note b⇩1 = this
show "P (a⇩1, b⇩1)"
proof (rule P)
fix p assume p: "p < (a⇩1, b⇩1)"
show "P p"
proof (cases "fst p < a⇩1")
case True
then have "P (fst p, snd p)" by (rule a⇩1)
then show ?thesis by simp
next
case False
with p have 1: "a⇩1 = fst p" and 2: "snd p < b⇩1"
by (simp_all add: less_prod_def')
from 2 have "P (a⇩1, snd p)" by (rule b⇩1)
with 1 show ?thesis by simp
qed
qed
qed
qed
qed
qed

text ‹Legacy lemma bindings›

lemmas prod_le_def = less_eq_prod_def
lemmas prod_less_def = less_prod_def
lemmas prod_less_eq = less_prod_def'

end

```