Theory NatOrder_ZF

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    Copyright (C) 2008  Seo Sanghyeon

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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