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
1.12 -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"
1.25 -
1.26 -lemma refl_on_converse[simp]: "refl A (r^-1) = refl A r"
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)"
1.34 -
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"
1.42 -
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 {} {}"
1.63 -
1.64 -lemma partial_order_on_empty[simp]: "partial_order_on {} {}"
1.66 -
1.67 -lemma lnear_order_on_empty[simp]: "linear_order_on {} {}"
1.69 -
1.70 -lemma well_order_on_empty[simp]: "well_order_on {} {}"
1.72 -
1.73 -
1.74 -lemma preorder_on_converse[simp]: "preorder_on A (r^-1) = preorder_on A r"
1.76 -
1.77 -lemma partial_order_on_converse[simp]:
1.78 -  "partial_order_on A (r^-1) = partial_order_on A r"
1.80 -
1.81 -lemma linear_order_on_converse[simp]:
1.82 -  "linear_order_on A (r^-1) = linear_order_on A r"
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)"
1.89 -
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"
1.116 -
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
```