src/HOL/Library/Product_Lexorder.thy
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(*  Title:      HOL/Library/Product_Lexorder.thy
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    Author:     Norbert Voelker
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*)
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header {* Lexicographic order on product types *}
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theory Product_Lexorder
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imports Main
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begin
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instantiation prod :: (ord, ord) ord
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begin
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definition
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  "x \<le> y \<longleftrightarrow> fst x < fst y \<or> fst x \<le> fst y \<and> snd x \<le> snd y"
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definition
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  "x < y \<longleftrightarrow> fst x < fst y \<or> fst x \<le> fst y \<and> snd x < snd y"
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instance ..
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end
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lemma less_eq_prod_simp [simp, code]:
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  "(x1, y1) \<le> (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 \<le> y2"
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  by (simp add: less_eq_prod_def)
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lemma less_prod_simp [simp, code]:
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  "(x1, y1) < (x2, y2) \<longleftrightarrow> x1 < x2 \<or> x1 \<le> x2 \<and> y1 < y2"
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  by (simp add: less_prod_def)
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text {* A stronger version for partial orders. *}
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lemma less_prod_def':
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  fixes x y :: "'a::order \<times> 'b::ord"
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  shows "x < y \<longleftrightarrow> fst x < fst y \<or> fst x = fst y \<and> snd x < snd y"
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  by (auto simp add: less_prod_def le_less)
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instance prod :: (preorder, preorder) preorder
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  by default (auto simp: less_eq_prod_def less_prod_def less_le_not_le intro: order_trans)
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instance prod :: (order, order) order
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  by default (auto simp add: less_eq_prod_def)
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instance prod :: (linorder, linorder) linorder
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  by default (auto simp: less_eq_prod_def)
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instantiation prod :: (linorder, linorder) distrib_lattice
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begin
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definition
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  "(inf :: 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = min"
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definition
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  "(sup :: 'a \<times> 'b \<Rightarrow> _ \<Rightarrow> _) = max"
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instance
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  by default (auto simp add: inf_prod_def sup_prod_def min_max.sup_inf_distrib1)
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end
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instantiation prod :: (bot, bot) bot
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begin
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definition
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  "bot = (bot, bot)"
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instance ..
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end
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instance prod :: (order_bot, order_bot) order_bot
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  by default (auto simp add: bot_prod_def)
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instantiation prod :: (top, top) top
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begin
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definition
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  "top = (top, top)"
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instance ..
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end
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instance prod :: (order_top, order_top) order_top
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  by default (auto simp add: top_prod_def)
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instance prod :: (wellorder, wellorder) wellorder
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proof
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  fix P :: "'a \<times> 'b \<Rightarrow> bool" and z :: "'a \<times> 'b"
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  assume P: "\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x"
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  show "P z"
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  proof (induct z)
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    case (Pair a b)
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    show "P (a, b)"
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    proof (induct a arbitrary: b rule: less_induct)
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      case (less a\<^sub>1) note a\<^sub>1 = this
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      show "P (a\<^sub>1, b)"
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      proof (induct b rule: less_induct)
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        case (less b\<^sub>1) note b\<^sub>1 = this
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        show "P (a\<^sub>1, b\<^sub>1)"
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        proof (rule P)
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          fix p assume p: "p < (a\<^sub>1, b\<^sub>1)"
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          show "P p"
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          proof (cases "fst p < a\<^sub>1")
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            case True
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            then have "P (fst p, snd p)" by (rule a\<^sub>1)
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            then show ?thesis by simp
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          next
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            case False
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            with p have 1: "a\<^sub>1 = fst p" and 2: "snd p < b\<^sub>1"
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              by (simp_all add: less_prod_def')
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            from 2 have "P (a\<^sub>1, snd p)" by (rule b\<^sub>1)
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            with 1 show ?thesis by simp
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          qed
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        qed
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      qed
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    qed
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  qed
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qed
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text {* Legacy lemma bindings *}
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lemmas prod_le_def = less_eq_prod_def
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lemmas prod_less_def = less_prod_def
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lemmas prod_less_eq = less_prod_def'
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end
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