src/HOL/Library/Order_Relation.thy
author bulwahn
Fri Apr 08 16:31:14 2011 +0200 (2011-04-08)
changeset 42316 12635bb655fd
parent 39302 d7728f65b353
child 48750 a151db85a62b
permissions -rw-r--r--
deactivating other compilations in quickcheck_exhaustive momentarily that only interesting for my benchmarks and experiments
     1 (* Author: Tobias Nipkow *)
     2 
     3 header {* Orders as Relations *}
     4 
     5 theory Order_Relation
     6 imports Main
     7 begin
     8 
     9 subsection{* Orders on a set *}
    10 
    11 definition "preorder_on A r \<equiv> refl_on A r \<and> trans r"
    12 
    13 definition "partial_order_on A r \<equiv> preorder_on A r \<and> antisym r"
    14 
    15 definition "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
    16 
    17 definition "strict_linear_order_on A r \<equiv> trans r \<and> irrefl r \<and> total_on A r"
    18 
    19 definition "well_order_on A r \<equiv> linear_order_on A r \<and> wf(r - Id)"
    20 
    21 lemmas order_on_defs =
    22   preorder_on_def partial_order_on_def linear_order_on_def
    23   strict_linear_order_on_def well_order_on_def
    24 
    25 
    26 lemma preorder_on_empty[simp]: "preorder_on {} {}"
    27 by(simp add:preorder_on_def trans_def)
    28 
    29 lemma partial_order_on_empty[simp]: "partial_order_on {} {}"
    30 by(simp add:partial_order_on_def)
    31 
    32 lemma lnear_order_on_empty[simp]: "linear_order_on {} {}"
    33 by(simp add:linear_order_on_def)
    34 
    35 lemma well_order_on_empty[simp]: "well_order_on {} {}"
    36 by(simp add:well_order_on_def)
    37 
    38 
    39 lemma preorder_on_converse[simp]: "preorder_on A (r^-1) = preorder_on A r"
    40 by (simp add:preorder_on_def)
    41 
    42 lemma partial_order_on_converse[simp]:
    43   "partial_order_on A (r^-1) = partial_order_on A r"
    44 by (simp add: partial_order_on_def)
    45 
    46 lemma linear_order_on_converse[simp]:
    47   "linear_order_on A (r^-1) = linear_order_on A r"
    48 by (simp add: linear_order_on_def)
    49 
    50 
    51 lemma strict_linear_order_on_diff_Id:
    52   "linear_order_on A r \<Longrightarrow> strict_linear_order_on A (r-Id)"
    53 by(simp add: order_on_defs trans_diff_Id)
    54 
    55 
    56 subsection{* Orders on the field *}
    57 
    58 abbreviation "Refl r \<equiv> refl_on (Field r) r"
    59 
    60 abbreviation "Preorder r \<equiv> preorder_on (Field r) r"
    61 
    62 abbreviation "Partial_order r \<equiv> partial_order_on (Field r) r"
    63 
    64 abbreviation "Total r \<equiv> total_on (Field r) r"
    65 
    66 abbreviation "Linear_order r \<equiv> linear_order_on (Field r) r"
    67 
    68 abbreviation "Well_order r \<equiv> well_order_on (Field r) r"
    69 
    70 
    71 lemma subset_Image_Image_iff:
    72   "\<lbrakk> Preorder r; A \<subseteq> Field r; B \<subseteq> Field r\<rbrakk> \<Longrightarrow>
    73    r `` A \<subseteq> r `` B \<longleftrightarrow> (\<forall>a\<in>A.\<exists>b\<in>B. (b,a):r)"
    74 apply(auto simp add: subset_eq preorder_on_def refl_on_def Image_def)
    75 apply metis
    76 by(metis trans_def)
    77 
    78 lemma subset_Image1_Image1_iff:
    79   "\<lbrakk> Preorder r; a : Field r; b : Field r\<rbrakk> \<Longrightarrow> r `` {a} \<subseteq> r `` {b} \<longleftrightarrow> (b,a):r"
    80 by(simp add:subset_Image_Image_iff)
    81 
    82 lemma Refl_antisym_eq_Image1_Image1_iff:
    83   "\<lbrakk>Refl r; antisym r; a:Field r; b:Field r\<rbrakk> \<Longrightarrow> r `` {a} = r `` {b} \<longleftrightarrow> a=b"
    84 by(simp add: set_eq_iff antisym_def refl_on_def) metis
    85 
    86 lemma Partial_order_eq_Image1_Image1_iff:
    87   "\<lbrakk>Partial_order r; a:Field r; b:Field r\<rbrakk> \<Longrightarrow> r `` {a} = r `` {b} \<longleftrightarrow> a=b"
    88 by(auto simp:order_on_defs Refl_antisym_eq_Image1_Image1_iff)
    89 
    90 
    91 subsection{* Orders on a type *}
    92 
    93 abbreviation "strict_linear_order \<equiv> strict_linear_order_on UNIV"
    94 
    95 abbreviation "linear_order \<equiv> linear_order_on UNIV"
    96 
    97 abbreviation "well_order r \<equiv> well_order_on UNIV"
    98 
    99 end