
diff -r 984191be0357 -r 33bff35f1335 src/HOL/Library/Order_Relation.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Order_Relation.thy	Wed Feb 11 10:51:07 2009 +0100
@@ -0,0 +1,101 @@
+(*  ID          : $Id$
+    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 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 (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)"
+apply(auto simp add: subset_eq preorder_on_def refl_def Image_def)
+apply metis
+by(metis trans_def)
+
+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: expand_set_eq antisym_def refl_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)
+
+
+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




