(* This file is a part of IsarMathLib - a library of formalized mathematics for Isabelle/Isar. Copyright (C) 2008 Seo Sanghyeon This program is free software; Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The name of the author may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. *) section ‹Order on natural numbers› theory NatOrder_ZF imports Nat_ZF_IML Order_ZF begin text‹This theory proves that $\leq$ is a linear order on $\mathbb{N}$. $\leq$ is defined in Isabelle's ‹Nat› theory, and linear order is defined in ‹Order_ZF› theory. Contributed by Seo Sanghyeon.› subsection‹Order on natural numbers› text‹This is the only section in this theory.› text‹To prove that $\leq$ is a total order, we use a result on ordinals.› lemma NatOrder_ZF_1_L1: assumes "a∈nat" and "b∈nat" shows "a ≤ b ∨ b ≤ a" proof - from assms have I: "Ord(a) ∧ Ord(b)" using nat_into_Ord by auto then have "a ∈ b ∨ a = b ∨ b ∈ a" using Ord_linear by simp with I have "a < b ∨ a = b ∨ b < a" using ltI by auto with I show "a ≤ b ∨ b ≤ a" using le_iff by auto qed text‹$\leq$ is antisymmetric, transitive, total, and linear. Proofs by rewrite using definitions.› lemma NatOrder_ZF_1_L2: shows "antisym(Le)" "trans(Le)" "Le {is total on} nat" "IsLinOrder(nat,Le)" proof - show "antisym(Le)" using antisym_def Le_def le_anti_sym by auto moreover show "trans(Le)" using trans_def Le_def le_trans by blast moreover show "Le {is total on} nat" using IsTotal_def Le_def NatOrder_ZF_1_L1 by simp ultimately show "IsLinOrder(nat,Le)" using IsLinOrder_def by simp qed text‹The order on natural numbers is linear on every natural number. Recall that each natural number is a subset of the set of all natural numbers (as well as a member).› lemma natord_lin_on_each_nat: assumes A1: "n ∈ nat" shows "IsLinOrder(n,Le)" proof - from A1 have "n ⊆ nat" using nat_subset_nat by simp then show ?thesis using NatOrder_ZF_1_L2 ord_linear_subset by blast qed end