(* Author: Tobias Nipkow *)

header {* Orders as Relations *}

theory Order_Relation

imports Main

begin

subsection{* Orders on a set *}

definition "preorder_on A r \<equiv> refl_on A r \<and> trans r"

definition "partial_order_on A r \<equiv> preorder_on A r \<and> antisym r"

definition "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"

definition "strict_linear_order_on A r \<equiv> trans r \<and> irrefl r \<and> total_on A r"

definition "well_order_on A r \<equiv> linear_order_on A r \<and> wf(r - Id)"

lemmas order_on_defs =

  preorder_on_def partial_order_on_def linear_order_on_def

  strict_linear_order_on_def well_order_on_def

lemma preorder_on_empty[simp]: "preorder_on {} {}"

by(simp add:preorder_on_def trans_def)

lemma partial_order_on_empty[simp]: "partial_order_on {} {}"

by(simp add:partial_order_on_def)

lemma lnear_order_on_empty[simp]: "linear_order_on {} {}"

by(simp add:linear_order_on_def)

lemma well_order_on_empty[simp]: "well_order_on {} {}"

by(simp add:well_order_on_def)

lemma preorder_on_converse[simp]: "preorder_on A (r^-1) = preorder_on A r"

by (simp add:preorder_on_def)

lemma partial_order_on_converse[simp]:

  "partial_order_on A (r^-1) = partial_order_on A r"

by (simp add: partial_order_on_def)

lemma linear_order_on_converse[simp]:

  "linear_order_on A (r^-1) = linear_order_on A r"

by (simp add: linear_order_on_def)

lemma strict_linear_order_on_diff_Id:

  "linear_order_on A r \<Longrightarrow> strict_linear_order_on A (r-Id)"

by(simp add: order_on_defs trans_diff_Id)

subsection{* Orders on the field *}

abbreviation "Refl r \<equiv> refl_on (Field r) r"

abbreviation "Preorder r \<equiv> preorder_on (Field r) r"

abbreviation "Partial_order r \<equiv> partial_order_on (Field r) r"

abbreviation "Total r \<equiv> total_on (Field r) r"

abbreviation "Linear_order r \<equiv> linear_order_on (Field r) r"

abbreviation "Well_order r \<equiv> well_order_on (Field r) r"

lemma subset_Image_Image_iff:

  "\<lbrakk> Preorder r; A \<subseteq> Field r; B \<subseteq> Field r\<rbrakk> \<Longrightarrow>

   r `` A \<subseteq> r `` B \<longleftrightarrow> (\<forall>a\<in>A.\<exists>b\<in>B. (b,a):r)"

unfolding preorder_on_def refl_on_def Image_def

apply (simp add: subset_eq)

unfolding trans_def by fast

lemma subset_Image1_Image1_iff:

  "\<lbrakk> Preorder r; a : Field r; b : Field r\<rbrakk> \<Longrightarrow> r `` {a} \<subseteq> r `` {b} \<longleftrightarrow> (b,a):r"

by(simp add:subset_Image_Image_iff)

lemma Refl_antisym_eq_Image1_Image1_iff:

  "\<lbrakk>Refl r; antisym r; a:Field r; b:Field r\<rbrakk> \<Longrightarrow> r `` {a} = r `` {b} \<longleftrightarrow> a=b"

by(simp add: set_eq_iff antisym_def refl_on_def) metis

lemma Partial_order_eq_Image1_Image1_iff:

  "\<lbrakk>Partial_order r; a:Field r; b:Field r\<rbrakk> \<Longrightarrow> r `` {a} = r `` {b} \<longleftrightarrow> a=b"

by(auto simp:order_on_defs Refl_antisym_eq_Image1_Image1_iff)

lemma Total_Id_Field:

assumes TOT: "Total r" and NID: "\<not> (r <= Id)"

shows "Field r = Field(r - Id)"

using mono_Field[of "r - Id" r] Diff_subset[of r Id]

proof(auto)

  have "r \<noteq> {}" using NID by fast

  then obtain b and c where "b \<noteq> c \<and> (b,c) \<in> r" using NID by auto

  hence 1: "b \<noteq> c \<and> {b,c} \<le> Field r" by (auto simp: Field_def)

  (*  *)

  fix a assume *: "a \<in> Field r"

  obtain d where 2: "d \<in> Field r" and 3: "d \<noteq> a"

  using * 1 by auto

  hence "(a,d) \<in> r \<or> (d,a) \<in> r" using * TOT

  by (simp add: total_on_def)

  thus "a \<in> Field(r - Id)" using 3 unfolding Field_def by blast

qed

subsection{* Orders on a type *}

abbreviation "strict_linear_order \<equiv> strict_linear_order_on UNIV"

abbreviation "linear_order \<equiv> linear_order_on UNIV"

abbreviation "well_order r \<equiv> well_order_on UNIV"

end