header {* Order on product types *}
theory Product_ord
imports Main
begin
instantiation prod :: (ord, ord) ord
begin
definition
prod_le_def: "x ≤ y <-> fst x < fst y ∨ fst x ≤ fst y ∧ snd x ≤ snd y"
definition
prod_less_def: "x < y <-> fst x < fst y ∨ fst x ≤ fst y ∧ snd x < snd y"
instance ..
end
lemma [code]:
"(x1::'a::{ord, equal}, y1) ≤ (x2, y2) <-> x1 < x2 ∨ x1 ≤ x2 ∧ y1 ≤ y2"
"(x1::'a::{ord, equal}, y1) < (x2, y2) <-> x1 < x2 ∨ x1 ≤ x2 ∧ 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 × 'b => _ => _) = min"
definition
sup_prod_def: "(sup :: 'a × 'b => _ => _) = 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 × 'b::ord"
shows "x < y <-> fst x < fst y ∨ (fst x = fst y ∧ 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 × '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: prod_less_eq)
from 2 have "P (a⇣1, snd p)" by (rule b⇣1)
with 1 show ?thesis by simp
qed
qed
qed
qed
qed
qed
end