(* This file is a part of IsarMathLib -

a library of formalized mathematics for Isabelle/Isar.

Copyright (C) 2008 Seo Sanghyeon

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header{*\isaheader{NatOrder\_ZF.thy}*}

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 @{text "Nat"} theory, and

linear order is defined in @{text "Order_ZF"} theory.

Contributed by Seo Sanghyeon.*}

section{*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