nipkow@15737: (*  Title:      HOL/Library/Product_ord.thy
nipkow@15737:     Author:     Norbert Voelker
nipkow@15737: *)
nipkow@15737: 
wenzelm@17200: header {* Order on product types *}
nipkow@15737: 
nipkow@15737: theory Product_ord
haftmann@30738: imports Main
nipkow@15737: begin
nipkow@15737: 
haftmann@37678: instantiation prod :: (ord, ord) ord
haftmann@25571: begin
haftmann@25571: 
haftmann@25571: definition
haftmann@37765:   prod_le_def: "x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x \<le> fst y \<and> snd x \<le> snd y"
haftmann@25571: 
haftmann@25571: definition
haftmann@37765:   prod_less_def: "x < y \<longleftrightarrow> fst x < fst y \<or> fst x \<le> fst y \<and> snd x < snd y"
haftmann@25571: 
haftmann@25571: instance ..
haftmann@25571: 
haftmann@25571: end
nipkow@15737: 
haftmann@28562: lemma [code]:
wenzelm@47961:   "(x1::'a::{ord, equal}, y1) \<le> (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 \<le> y2"
wenzelm@47961:   "(x1::'a::{ord, equal}, y1) < (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 < y2"
haftmann@25502:   unfolding prod_le_def prod_less_def by simp_all
haftmann@22177: 
wenzelm@47961: instance prod :: (preorder, preorder) preorder
wenzelm@47961:   by default (auto simp: prod_le_def prod_less_def less_le_not_le intro: order_trans)
nipkow@15737: 
wenzelm@47961: instance prod :: (order, order) order
wenzelm@47961:   by default (auto simp add: prod_le_def)
haftmann@31040: 
wenzelm@47961: instance prod :: (linorder, linorder) linorder
wenzelm@47961:   by default (auto simp: prod_le_def)
nipkow@15737: 
haftmann@37678: instantiation prod :: (linorder, linorder) distrib_lattice
haftmann@25571: begin
haftmann@25571: 
haftmann@25571: definition
wenzelm@47961:   inf_prod_def: "(inf :: 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = min"
haftmann@25571: 
haftmann@25571: definition
wenzelm@47961:   sup_prod_def: "(sup :: 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = max"
haftmann@25571: 
wenzelm@47961: instance
wenzelm@47961:   by default (auto simp add: inf_prod_def sup_prod_def min_max.sup_inf_distrib1)
haftmann@31040: 
haftmann@31040: end
haftmann@31040: 
haftmann@37678: instantiation prod :: (bot, bot) bot
haftmann@31040: begin
haftmann@31040: 
haftmann@31040: definition
haftmann@31040:   bot_prod_def: "bot = (bot, bot)"
haftmann@31040: 
wenzelm@47961: instance
wenzelm@47961:   by default (auto simp add: bot_prod_def prod_le_def)
haftmann@31040: 
haftmann@31040: end
haftmann@31040: 
haftmann@37678: instantiation prod :: (top, top) top
haftmann@31040: begin
haftmann@31040: 
haftmann@31040: definition
haftmann@31040:   top_prod_def: "top = (top, top)"
haftmann@31040: 
wenzelm@47961: instance
wenzelm@47961:   by default (auto simp add: top_prod_def prod_le_def)
haftmann@22483: 
wenzelm@19736: end
haftmann@25571: 
huffman@44063: text {* A stronger version of the definition holds for partial orders. *}
huffman@44063: 
huffman@44063: lemma prod_less_eq:
huffman@44063:   fixes x y :: "'a::order \<times> 'b::ord"
huffman@44063:   shows "x < y \<longleftrightarrow> fst x < fst y \<or> (fst x = fst y \<and> snd x < snd y)"
huffman@44063:   unfolding prod_less_def fst_conv snd_conv le_less by auto
huffman@44063: 
huffman@44063: instance prod :: (wellorder, wellorder) wellorder
huffman@44063: proof
huffman@44063:   fix P :: "'a \<times> 'b \<Rightarrow> bool" and z :: "'a \<times> 'b"
huffman@44063:   assume P: "\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x"
huffman@44063:   show "P z"
huffman@44063:   proof (induct z)
huffman@44063:     case (Pair a b)
huffman@44063:     show "P (a, b)"
wenzelm@47961:     proof (induct a arbitrary: b rule: less_induct)
wenzelm@47961:       case (less a\<^isub>1) note a\<^isub>1 = this
wenzelm@47961:       show "P (a\<^isub>1, b)"
wenzelm@47961:       proof (induct b rule: less_induct)
wenzelm@47961:         case (less b\<^isub>1) note b\<^isub>1 = this
wenzelm@47961:         show "P (a\<^isub>1, b\<^isub>1)"
wenzelm@47961:         proof (rule P)
wenzelm@47961:           fix p assume p: "p < (a\<^isub>1, b\<^isub>1)"
wenzelm@47961:           show "P p"
wenzelm@47961:           proof (cases "fst p < a\<^isub>1")
wenzelm@47961:             case True
wenzelm@47961:             then have "P (fst p, snd p)" by (rule a\<^isub>1)
wenzelm@47961:             then show ?thesis by simp
wenzelm@47961:           next
wenzelm@47961:             case False
wenzelm@47961:             with p have 1: "a\<^isub>1 = fst p" and 2: "snd p < b\<^isub>1"
wenzelm@47961:               by (simp_all add: prod_less_eq)
wenzelm@47961:             from 2 have "P (a\<^isub>1, snd p)" by (rule b\<^isub>1)
wenzelm@47961:             with 1 show ?thesis by simp
wenzelm@47961:           qed
wenzelm@47961:         qed
wenzelm@47961:       qed
wenzelm@47961:     qed
huffman@44063:   qed
huffman@44063: qed
huffman@44063: 
haftmann@25571: end