1.1 --- a/src/HOL/Library/Order_Relation.thy	Thu Dec 05 17:52:12 2013 +0100
     1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.3 @@ -1,116 +0,0 @@
     1.4 -(* Author: Tobias Nipkow *)
     1.5 -
     1.6 -header {* Orders as Relations *}
     1.7 -
     1.8 -theory Order_Relation
     1.9 -imports Main
    1.10 -begin
    1.11 -
    1.12 -subsection{* Orders on a set *}
    1.13 -
    1.14 -definition "preorder_on A r \<equiv> refl_on A r \<and> trans r"
    1.15 -
    1.16 -definition "partial_order_on A r \<equiv> preorder_on A r \<and> antisym r"
    1.17 -
    1.18 -definition "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
    1.19 -
    1.20 -definition "strict_linear_order_on A r \<equiv> trans r \<and> irrefl r \<and> total_on A r"
    1.21 -
    1.22 -definition "well_order_on A r \<equiv> linear_order_on A r \<and> wf(r - Id)"
    1.23 -
    1.24 -lemmas order_on_defs =
    1.25 -  preorder_on_def partial_order_on_def linear_order_on_def
    1.26 -  strict_linear_order_on_def well_order_on_def
    1.27 -
    1.28 -
    1.29 -lemma preorder_on_empty[simp]: "preorder_on {} {}"
    1.30 -by(simp add:preorder_on_def trans_def)
    1.31 -
    1.32 -lemma partial_order_on_empty[simp]: "partial_order_on {} {}"
    1.33 -by(simp add:partial_order_on_def)
    1.34 -
    1.35 -lemma lnear_order_on_empty[simp]: "linear_order_on {} {}"
    1.36 -by(simp add:linear_order_on_def)
    1.37 -
    1.38 -lemma well_order_on_empty[simp]: "well_order_on {} {}"
    1.39 -by(simp add:well_order_on_def)
    1.40 -
    1.41 -
    1.42 -lemma preorder_on_converse[simp]: "preorder_on A (r^-1) = preorder_on A r"
    1.43 -by (simp add:preorder_on_def)
    1.44 -
    1.45 -lemma partial_order_on_converse[simp]:
    1.46 -  "partial_order_on A (r^-1) = partial_order_on A r"
    1.47 -by (simp add: partial_order_on_def)
    1.48 -
    1.49 -lemma linear_order_on_converse[simp]:
    1.50 -  "linear_order_on A (r^-1) = linear_order_on A r"
    1.51 -by (simp add: linear_order_on_def)
    1.52 -
    1.53 -
    1.54 -lemma strict_linear_order_on_diff_Id:
    1.55 -  "linear_order_on A r \<Longrightarrow> strict_linear_order_on A (r-Id)"
    1.56 -by(simp add: order_on_defs trans_diff_Id)
    1.57 -
    1.58 -
    1.59 -subsection{* Orders on the field *}
    1.60 -
    1.61 -abbreviation "Refl r \<equiv> refl_on (Field r) r"
    1.62 -
    1.63 -abbreviation "Preorder r \<equiv> preorder_on (Field r) r"
    1.64 -
    1.65 -abbreviation "Partial_order r \<equiv> partial_order_on (Field r) r"
    1.66 -
    1.67 -abbreviation "Total r \<equiv> total_on (Field r) r"
    1.68 -
    1.69 -abbreviation "Linear_order r \<equiv> linear_order_on (Field r) r"
    1.70 -
    1.71 -abbreviation "Well_order r \<equiv> well_order_on (Field r) r"
    1.72 -
    1.73 -
    1.74 -lemma subset_Image_Image_iff:
    1.75 -  "\<lbrakk> Preorder r; A \<subseteq> Field r; B \<subseteq> Field r\<rbrakk> \<Longrightarrow>
    1.76 -   r `` A \<subseteq> r `` B \<longleftrightarrow> (\<forall>a\<in>A.\<exists>b\<in>B. (b,a):r)"
    1.77 -unfolding preorder_on_def refl_on_def Image_def
    1.78 -apply (simp add: subset_eq)
    1.79 -unfolding trans_def by fast
    1.80 -
    1.81 -lemma subset_Image1_Image1_iff:
    1.82 -  "\<lbrakk> Preorder r; a : Field r; b : Field r\<rbrakk> \<Longrightarrow> r `` {a} \<subseteq> r `` {b} \<longleftrightarrow> (b,a):r"
    1.83 -by(simp add:subset_Image_Image_iff)
    1.84 -
    1.85 -lemma Refl_antisym_eq_Image1_Image1_iff:
    1.86 -  "\<lbrakk>Refl r; antisym r; a:Field r; b:Field r\<rbrakk> \<Longrightarrow> r `` {a} = r `` {b} \<longleftrightarrow> a=b"
    1.87 -by(simp add: set_eq_iff antisym_def refl_on_def) metis
    1.88 -
    1.89 -lemma Partial_order_eq_Image1_Image1_iff:
    1.90 -  "\<lbrakk>Partial_order r; a:Field r; b:Field r\<rbrakk> \<Longrightarrow> r `` {a} = r `` {b} \<longleftrightarrow> a=b"
    1.91 -by(auto simp:order_on_defs Refl_antisym_eq_Image1_Image1_iff)
    1.92 -
    1.93 -lemma Total_Id_Field:
    1.94 -assumes TOT: "Total r" and NID: "\<not> (r <= Id)"
    1.95 -shows "Field r = Field(r - Id)"
    1.96 -using mono_Field[of "r - Id" r] Diff_subset[of r Id]
    1.97 -proof(auto)
    1.98 -  have "r \<noteq> {}" using NID by fast
    1.99 -  then obtain b and c where "b \<noteq> c \<and> (b,c) \<in> r" using NID by fast
   1.100 -  hence 1: "b \<noteq> c \<and> {b,c} \<le> Field r" by (auto simp: Field_def)
   1.101 -  (*  *)
   1.102 -  fix a assume *: "a \<in> Field r"
   1.103 -  obtain d where 2: "d \<in> Field r" and 3: "d \<noteq> a"
   1.104 -  using * 1 by auto
   1.105 -  hence "(a,d) \<in> r \<or> (d,a) \<in> r" using * TOT
   1.106 -  by (simp add: total_on_def)
   1.107 -  thus "a \<in> Field(r - Id)" using 3 unfolding Field_def by blast
   1.108 -qed
   1.109 -
   1.110 -
   1.111 -subsection{* Orders on a type *}
   1.112 -
   1.113 -abbreviation "strict_linear_order \<equiv> strict_linear_order_on UNIV"
   1.114 -
   1.115 -abbreviation "linear_order \<equiv> linear_order_on UNIV"
   1.116 -
   1.117 -abbreviation "well_order r \<equiv> well_order_on UNIV"
   1.118 -
   1.119 -end