src/HOL/Library/Product_ord.thy
author haftmann
Thu, 29 Nov 2007 17:08:26 +0100
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(*  Title:      HOL/Library/Product_ord.thy
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    ID:         $Id$
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    Author:     Norbert Voelker
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*)
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header {* Order on product types *}
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theory Product_ord
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imports Main
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begin
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instance "*" :: (ord, ord) ord
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  prod_le_def [code func del]: "x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x \<le> snd y"
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  prod_less_def [code func del]: "x < y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x < snd y" ..
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lemma [code func]:
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  "(x1\<Colon>'a\<Colon>{ord, eq}, y1) \<le> (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
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  "(x1\<Colon>'a\<Colon>{ord, eq}, y1) < (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
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  unfolding prod_le_def prod_less_def by simp_all
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lemma [code]:
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  "(x1, y1) \<le> (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 = x2 \<and> y1 \<le> y2"
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  "(x1, y1) < (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 = x2 \<and> y1 < y2"
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  unfolding prod_le_def prod_less_def by simp_all
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instance * :: (order, order) order
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  by default (auto simp: prod_le_def prod_less_def intro: order_less_trans)
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instance * :: (linorder, linorder) linorder
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  by default (auto simp: prod_le_def)
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instance * :: (linorder, linorder) distrib_lattice
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  inf_prod_def: "(inf \<Colon> 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = min"
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  sup_prod_def: "(sup \<Colon> 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = max"
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  by intro_classes
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    (auto simp add: inf_prod_def sup_prod_def min_max.sup_inf_distrib1)
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end