src/HOL/Order_Relation.thy
changeset 29860 f735e4027656
parent 29858 c8cee17d7e50
parent 29859 33bff35f1335
child 29861 3c348f5873f3
     1.1 --- a/src/HOL/Order_Relation.thy	Wed Feb 11 16:03:10 2009 +1100
     1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.3 @@ -1,131 +0,0 @@
     1.4 -(*  ID          : $Id$
     1.5 -    Author      : Tobias Nipkow
     1.6 -*)
     1.7 -
     1.8 -header {* Orders as Relations *}
     1.9 -
    1.10 -theory Order_Relation
    1.11 -imports Plain "~~/src/HOL/Hilbert_Choice" "~~/src/HOL/ATP_Linkup"
    1.12 -begin
    1.13 -
    1.14 -text{* This prelude could be moved to theory Relation: *}
    1.15 -
    1.16 -definition "irrefl r \<equiv> \<forall>x. (x,x) \<notin> r"
    1.17 -
    1.18 -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"
    1.19 -
    1.20 -abbreviation "total \<equiv> total_on UNIV"
    1.21 -
    1.22 -
    1.23 -lemma total_on_empty[simp]: "total_on {} r"
    1.24 -by(simp add:total_on_def)
    1.25 -
    1.26 -lemma refl_on_converse[simp]: "refl A (r^-1) = refl A r"
    1.27 -by(auto simp add:refl_def)
    1.28 -
    1.29 -lemma total_on_converse[simp]: "total_on A (r^-1) = total_on A r"
    1.30 -by (auto simp: total_on_def)
    1.31 -
    1.32 -lemma irrefl_diff_Id[simp]: "irrefl(r-Id)"
    1.33 -by(simp add:irrefl_def)
    1.34 -
    1.35 -declare [[simp_depth_limit = 2]]
    1.36 -lemma trans_diff_Id: " trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r-Id)"
    1.37 -by(simp add: antisym_def trans_def) blast
    1.38 -declare [[simp_depth_limit = 50]]
    1.39 -
    1.40 -lemma total_on_diff_Id[simp]: "total_on A (r-Id) = total_on A r"
    1.41 -by(simp add: total_on_def)
    1.42 -
    1.43 -
    1.44 -subsection{* Orders on a set *}
    1.45 -
    1.46 -definition "preorder_on A r \<equiv> refl A r \<and> trans r"
    1.47 -
    1.48 -definition "partial_order_on A r \<equiv> preorder_on A r \<and> antisym r"
    1.49 -
    1.50 -definition "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
    1.51 -
    1.52 -definition "strict_linear_order_on A r \<equiv> trans r \<and> irrefl r \<and> total_on A r"
    1.53 -
    1.54 -definition "well_order_on A r \<equiv> linear_order_on A r \<and> wf(r - Id)"
    1.55 -
    1.56 -lemmas order_on_defs =
    1.57 -  preorder_on_def partial_order_on_def linear_order_on_def
    1.58 -  strict_linear_order_on_def well_order_on_def
    1.59 -
    1.60 -
    1.61 -lemma preorder_on_empty[simp]: "preorder_on {} {}"
    1.62 -by(simp add:preorder_on_def trans_def)
    1.63 -
    1.64 -lemma partial_order_on_empty[simp]: "partial_order_on {} {}"
    1.65 -by(simp add:partial_order_on_def)
    1.66 -
    1.67 -lemma lnear_order_on_empty[simp]: "linear_order_on {} {}"
    1.68 -by(simp add:linear_order_on_def)
    1.69 -
    1.70 -lemma well_order_on_empty[simp]: "well_order_on {} {}"
    1.71 -by(simp add:well_order_on_def)
    1.72 -
    1.73 -
    1.74 -lemma preorder_on_converse[simp]: "preorder_on A (r^-1) = preorder_on A r"
    1.75 -by (simp add:preorder_on_def)
    1.76 -
    1.77 -lemma partial_order_on_converse[simp]:
    1.78 -  "partial_order_on A (r^-1) = partial_order_on A r"
    1.79 -by (simp add: partial_order_on_def)
    1.80 -
    1.81 -lemma linear_order_on_converse[simp]:
    1.82 -  "linear_order_on A (r^-1) = linear_order_on A r"
    1.83 -by (simp add: linear_order_on_def)
    1.84 -
    1.85 -
    1.86 -lemma strict_linear_order_on_diff_Id:
    1.87 -  "linear_order_on A r \<Longrightarrow> strict_linear_order_on A (r-Id)"
    1.88 -by(simp add: order_on_defs trans_diff_Id)
    1.89 -
    1.90 -
    1.91 -subsection{* Orders on the field *}
    1.92 -
    1.93 -abbreviation "Refl r \<equiv> refl (Field r) r"
    1.94 -
    1.95 -abbreviation "Preorder r \<equiv> preorder_on (Field r) r"
    1.96 -
    1.97 -abbreviation "Partial_order r \<equiv> partial_order_on (Field r) r"
    1.98 -
    1.99 -abbreviation "Total r \<equiv> total_on (Field r) r"
   1.100 -
   1.101 -abbreviation "Linear_order r \<equiv> linear_order_on (Field r) r"
   1.102 -
   1.103 -abbreviation "Well_order r \<equiv> well_order_on (Field r) r"
   1.104 -
   1.105 -
   1.106 -lemma subset_Image_Image_iff:
   1.107 -  "\<lbrakk> Preorder r; A \<subseteq> Field r; B \<subseteq> Field r\<rbrakk> \<Longrightarrow>
   1.108 -   r `` A \<subseteq> r `` B \<longleftrightarrow> (\<forall>a\<in>A.\<exists>b\<in>B. (b,a):r)"
   1.109 -apply(auto simp add: subset_eq preorder_on_def refl_def Image_def)
   1.110 -apply metis
   1.111 -by(metis trans_def)
   1.112 -
   1.113 -lemma subset_Image1_Image1_iff:
   1.114 -  "\<lbrakk> Preorder r; a : Field r; b : Field r\<rbrakk> \<Longrightarrow> r `` {a} \<subseteq> r `` {b} \<longleftrightarrow> (b,a):r"
   1.115 -by(simp add:subset_Image_Image_iff)
   1.116 -
   1.117 -lemma Refl_antisym_eq_Image1_Image1_iff:
   1.118 -  "\<lbrakk>Refl r; antisym r; a:Field r; b:Field r\<rbrakk> \<Longrightarrow> r `` {a} = r `` {b} \<longleftrightarrow> a=b"
   1.119 -by(simp add: expand_set_eq antisym_def refl_def) metis
   1.120 -
   1.121 -lemma Partial_order_eq_Image1_Image1_iff:
   1.122 -  "\<lbrakk>Partial_order r; a:Field r; b:Field r\<rbrakk> \<Longrightarrow> r `` {a} = r `` {b} \<longleftrightarrow> a=b"
   1.123 -by(auto simp:order_on_defs Refl_antisym_eq_Image1_Image1_iff)
   1.124 -
   1.125 -
   1.126 -subsection{* Orders on a type *}
   1.127 -
   1.128 -abbreviation "strict_linear_order \<equiv> strict_linear_order_on UNIV"
   1.129 -
   1.130 -abbreviation "linear_order \<equiv> linear_order_on UNIV"
   1.131 -
   1.132 -abbreviation "well_order r \<equiv> well_order_on UNIV"
   1.133 -
   1.134 -end