Moved Order_Relation into Library and moved some of it into Relation.

--- a/src/HOL/IsaMakefile	Tue Feb 10 17:53:51 2009 -0800
+++ b/src/HOL/IsaMakefile	Wed Feb 11 10:51:07 2009 +0100
@@ -285,7 +285,6 @@
   Taylor.thy \
   Transcendental.thy \
   GCD.thy \
-  Order_Relation.thy \
   Parity.thy \
   Lubs.thy \
   Polynomial.thy \
@@ -322,7 +321,7 @@
   Library/Multiset.thy Library/Permutation.thy	\
   Library/Primes.thy Library/Pocklington.thy Library/Quotient.thy	\
   Library/Quicksort.thy Library/Nat_Infinity.thy Library/Word.thy	\
-  Library/README.html Library/Continuity.thy				\
+  Library/README.html Library/Continuity.thy Library/Order_Relation.thy \
   Library/Nested_Environment.thy Library/Ramsey.thy Library/Zorn.thy	\
   Library/Library/ROOT.ML Library/Library/document/root.tex		\
   Library/Library/document/root.bib Library/While_Combinator.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

--- a/src/HOL/Order_Relation.thy	Tue Feb 10 17:53:51 2009 -0800
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,131 +0,0 @@
-(*  ID          : $Id$
-    Author      : Tobias Nipkow
-*)
-
-header {* Orders as Relations *}
-
-theory Order_Relation
-imports Plain "~~/src/HOL/Hilbert_Choice" "~~/src/HOL/ATP_Linkup"
-begin
-
-text{* This prelude could be moved to theory Relation: *}
-
-definition "irrefl r \<equiv> \<forall>x. (x,x) \<notin> r"
-
-definition "total_on A r \<equiv> \<forall>x\<in>A.\<forall>y\<in>A. x\<noteq>y \<longrightarrow> (x,y)\<in>r \<or> (y,x)\<in>r"
-
-abbreviation "total \<equiv> total_on UNIV"
-
-
-lemma total_on_empty[simp]: "total_on {} r"
-by(simp add:total_on_def)
-
-lemma refl_on_converse[simp]: "refl A (r^-1) = refl A r"
-by(auto simp add:refl_def)
-
-lemma total_on_converse[simp]: "total_on A (r^-1) = total_on A r"
-by (auto simp: total_on_def)
-
-lemma irrefl_diff_Id[simp]: "irrefl(r-Id)"
-by(simp add:irrefl_def)
-
-declare [[simp_depth_limit = 2]]
-lemma trans_diff_Id: " trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r-Id)"
-by(simp add: antisym_def trans_def) blast
-declare [[simp_depth_limit = 50]]
-
-lemma total_on_diff_Id[simp]: "total_on A (r-Id) = total_on A r"
-by(simp add: total_on_def)
-
-
-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

--- a/src/HOL/Relation.thy	Tue Feb 10 17:53:51 2009 -0800
+++ b/src/HOL/Relation.thy	Wed Feb 11 10:51:07 2009 +0100
@@ -70,6 +70,16 @@
   "trans r == (ALL x y z. (x,y):r --> (y,z):r --> (x,z):r)"
 
 definition
+irrefl :: "('a * 'a) set => bool" where
+"irrefl r \<equiv> \<forall>x. (x,x) \<notin> r"
+
+definition
+total_on :: "'a set => ('a * 'a) set => bool" where
+"total_on A r \<equiv> \<forall>x\<in>A.\<forall>y\<in>A. x\<noteq>y \<longrightarrow> (x,y)\<in>r \<or> (y,x)\<in>r"
+
+abbreviation "total \<equiv> total_on UNIV"
+
+definition
   single_valued :: "('a * 'b) set => bool" where
   "single_valued r == ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z)"
 
@@ -268,6 +278,21 @@
 lemma trans_diag [simp]: "trans (diag A)"
 by (fast intro: transI elim: transD)
 
+lemma trans_diff_Id: " trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r-Id)"
+unfolding antisym_def trans_def by blast
+
+subsection {* Irreflexivity *}
+
+lemma irrefl_diff_Id[simp]: "irrefl(r-Id)"
+by(simp add:irrefl_def)
+
+subsection {* Totality *}
+
+lemma total_on_empty[simp]: "total_on {} r"
+by(simp add:total_on_def)
+
+lemma total_on_diff_Id[simp]: "total_on A (r-Id) = total_on A r"
+by(simp add: total_on_def)
 
 subsection {* Converse *}
 
@@ -330,6 +355,9 @@
 lemma sym_Int_converse: "sym (r \<inter> r^-1)"
 by (unfold sym_def) blast
 
+lemma total_on_converse[simp]: "total_on A (r^-1) = total_on A r"
+by (auto simp: total_on_def)
+
 
 subsection {* Domain *}