src/HOL/Library/Order_Relation.thy
author Christian Sternagel
Wed Aug 29 12:23:14 2012 +0900 (2012-08-29)
changeset 49083 01081bca31b6
parent 48750 a151db85a62b
child 52182 57b4fdc59d3b
permissions -rw-r--r--
dropped ord and bot instance for list prefixes (use locale interpretation instead, which allows users to decide what order to use on lists)
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(* Author: Tobias Nipkow *)
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header {* Orders as Relations *}
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theory Order_Relation
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imports Main
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begin
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subsection{* Orders on a set *}
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definition "preorder_on A r \<equiv> refl_on A r \<and> trans r"
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definition "partial_order_on A r \<equiv> preorder_on A r \<and> antisym r"
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definition "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
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definition "strict_linear_order_on A r \<equiv> trans r \<and> irrefl r \<and> total_on A r"
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definition "well_order_on A r \<equiv> linear_order_on A r \<and> wf(r - Id)"
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lemmas order_on_defs =
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  preorder_on_def partial_order_on_def linear_order_on_def
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  strict_linear_order_on_def well_order_on_def
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lemma preorder_on_empty[simp]: "preorder_on {} {}"
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by(simp add:preorder_on_def trans_def)
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lemma partial_order_on_empty[simp]: "partial_order_on {} {}"
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by(simp add:partial_order_on_def)
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lemma lnear_order_on_empty[simp]: "linear_order_on {} {}"
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by(simp add:linear_order_on_def)
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lemma well_order_on_empty[simp]: "well_order_on {} {}"
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by(simp add:well_order_on_def)
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lemma preorder_on_converse[simp]: "preorder_on A (r^-1) = preorder_on A r"
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by (simp add:preorder_on_def)
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lemma partial_order_on_converse[simp]:
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  "partial_order_on A (r^-1) = partial_order_on A r"
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by (simp add: partial_order_on_def)
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lemma linear_order_on_converse[simp]:
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  "linear_order_on A (r^-1) = linear_order_on A r"
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by (simp add: linear_order_on_def)
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lemma strict_linear_order_on_diff_Id:
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  "linear_order_on A r \<Longrightarrow> strict_linear_order_on A (r-Id)"
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by(simp add: order_on_defs trans_diff_Id)
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subsection{* Orders on the field *}
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abbreviation "Refl r \<equiv> refl_on (Field r) r"
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abbreviation "Preorder r \<equiv> preorder_on (Field r) r"
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abbreviation "Partial_order r \<equiv> partial_order_on (Field r) r"
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abbreviation "Total r \<equiv> total_on (Field r) r"
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abbreviation "Linear_order r \<equiv> linear_order_on (Field r) r"
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abbreviation "Well_order r \<equiv> well_order_on (Field r) r"
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lemma subset_Image_Image_iff:
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  "\<lbrakk> Preorder r; A \<subseteq> Field r; B \<subseteq> Field r\<rbrakk> \<Longrightarrow>
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   r `` A \<subseteq> r `` B \<longleftrightarrow> (\<forall>a\<in>A.\<exists>b\<in>B. (b,a):r)"
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unfolding preorder_on_def refl_on_def Image_def
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apply (simp add: subset_eq)
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unfolding trans_def by fast
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lemma subset_Image1_Image1_iff:
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  "\<lbrakk> Preorder r; a : Field r; b : Field r\<rbrakk> \<Longrightarrow> r `` {a} \<subseteq> r `` {b} \<longleftrightarrow> (b,a):r"
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by(simp add:subset_Image_Image_iff)
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lemma Refl_antisym_eq_Image1_Image1_iff:
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  "\<lbrakk>Refl r; antisym r; a:Field r; b:Field r\<rbrakk> \<Longrightarrow> r `` {a} = r `` {b} \<longleftrightarrow> a=b"
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by(simp add: set_eq_iff antisym_def refl_on_def) metis
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lemma Partial_order_eq_Image1_Image1_iff:
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  "\<lbrakk>Partial_order r; a:Field r; b:Field r\<rbrakk> \<Longrightarrow> r `` {a} = r `` {b} \<longleftrightarrow> a=b"
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by(auto simp:order_on_defs Refl_antisym_eq_Image1_Image1_iff)
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subsection{* Orders on a type *}
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abbreviation "strict_linear_order \<equiv> strict_linear_order_on UNIV"
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abbreviation "linear_order \<equiv> linear_order_on UNIV"
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abbreviation "well_order r \<equiv> well_order_on UNIV"
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end