src/HOL/Library/Product_Lexorder.thy
 author wenzelm Wed, 08 Mar 2017 10:50:59 +0100 changeset 65151 a7394aa4d21c parent 60679 ade12ef2773c permissions -rw-r--r--
tuned proofs;
```
(*  Title:      HOL/Library/Product_Lexorder.thy
Author:     Norbert Voelker
*)

section \<open>Lexicographic order on product types\<close>

theory Product_Lexorder
imports Main
begin

instantiation prod :: (ord, ord) ord
begin

definition
"x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x \<le> fst y \<and> snd x \<le> snd y"

definition
"x < y \<longleftrightarrow> fst x < fst y \<or> fst x \<le> fst y \<and> snd x < snd y"

instance ..

end

lemma less_eq_prod_simp [simp, code]:
"(x1, y1) \<le> (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 \<le> y2"

lemma less_prod_simp [simp, code]:
"(x1, y1) < (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 < y2"

text \<open>A stronger version for partial orders.\<close>

lemma less_prod_def':
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"
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 \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = min"

definition
"(sup :: 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = 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 \<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\<^sub>1) note a\<^sub>1 = this
show "P (a\<^sub>1, b)"
proof (induct b rule: less_induct)
case (less b\<^sub>1) note b\<^sub>1 = this
show "P (a\<^sub>1, b\<^sub>1)"
proof (rule P)
fix p assume p: "p < (a\<^sub>1, b\<^sub>1)"
show "P p"
proof (cases "fst p < a\<^sub>1")
case True
then have "P (fst p, snd p)" by (rule a\<^sub>1)
then show ?thesis by simp
next
case False
with p have 1: "a\<^sub>1 = fst p" and 2: "snd p < b\<^sub>1"
from 2 have "P (a\<^sub>1, snd p)" by (rule b\<^sub>1)
with 1 show ?thesis by simp
qed
qed
qed
qed
qed
qed

text \<open>Legacy lemma bindings\<close>

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

end

```