# Theory Product_ord

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theory Product_ord
imports Main
begin

    (*  Title:      HOL/Library/Product_ord.thy
        ID:         $Id: Product_ord.thy,v 1.2 2005/08/31 13:46:37 wenzelm Exp $
        Author:     Norbert Voelker
    *)

    header {* Order on product types *}

    theory Product_ord
    imports Main
    begin

    instance "*" :: (ord,ord) ord ..

    defs (overloaded)
      prod_le_def: "(x ≤ y) ≡ (fst x < fst y) | (fst x = fst y & snd x ≤ snd y)"
      prod_less_def: "(x < y) ≡ (fst x < fst y) | (fst x = fst y & snd x < snd y)"

    lemmas prod_ord_defs = prod_less_def prod_le_def

    instance "*" :: (order,order) order
      apply (intro_classes, unfold prod_ord_defs)
      by (auto intro: order_less_trans)

    instance "*":: (linorder,linorder)linorder
      by (intro_classes, unfold prod_le_def, auto)

    end

lemmas prod_ord_defs:

      x < y == fst x < fst y ∨ fst x = fst y ∧ snd x < snd y

      x ≤ y == fst x < fst y ∨ fst x = fst y ∧ snd x ≤ snd y

lemmas prod_ord_defs:

      x < y == fst x < fst y ∨ fst x = fst y ∧ snd x < snd y

      x ≤ y == fst x < fst y ∨ fst x = fst y ∧ snd x ≤ snd y