src/HOL/Library/Order_Relation.thy
 author Christian Sternagel Thu Dec 13 13:11:38 2012 +0100 (2012-12-13) changeset 50516 ed6b40d15d1c parent 48750 a151db85a62b child 52182 57b4fdc59d3b permissions -rw-r--r--
renamed "emb" to "list_hembeq";
make "list_hembeq" reflexive independent of the base order;
renamed "sub" to "sublisteq";
dropped "transp_on" (state transitivity explicitly instead);
no need to hide "sub" after renaming;
replaced some ASCII symbols by proper Isabelle symbols;
NEWS
```     1 (* Author: Tobias Nipkow *)
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```     2
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```     3 header {* Orders as Relations *}
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```     4
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```     5 theory Order_Relation
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```     6 imports Main
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```     7 begin
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```     8
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```     9 subsection{* Orders on a set *}
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```    10
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```    11 definition "preorder_on A r \<equiv> refl_on A r \<and> trans r"
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```    12
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```    13 definition "partial_order_on A r \<equiv> preorder_on A r \<and> antisym r"
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```    14
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```    15 definition "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
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```    16
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```    17 definition "strict_linear_order_on A r \<equiv> trans r \<and> irrefl r \<and> total_on A r"
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```    18
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```    19 definition "well_order_on A r \<equiv> linear_order_on A r \<and> wf(r - Id)"
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```    20
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```    21 lemmas order_on_defs =
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```    22   preorder_on_def partial_order_on_def linear_order_on_def
```
```    23   strict_linear_order_on_def well_order_on_def
```
```    24
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```    25
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```    26 lemma preorder_on_empty[simp]: "preorder_on {} {}"
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```    27 by(simp add:preorder_on_def trans_def)
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```    28
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```    29 lemma partial_order_on_empty[simp]: "partial_order_on {} {}"
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```    30 by(simp add:partial_order_on_def)
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```    31
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```    32 lemma lnear_order_on_empty[simp]: "linear_order_on {} {}"
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```    33 by(simp add:linear_order_on_def)
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```    34
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```    35 lemma well_order_on_empty[simp]: "well_order_on {} {}"
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```    36 by(simp add:well_order_on_def)
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```    37
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```    38
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```    39 lemma preorder_on_converse[simp]: "preorder_on A (r^-1) = preorder_on A r"
```
```    40 by (simp add:preorder_on_def)
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```    41
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```    42 lemma partial_order_on_converse[simp]:
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```    43   "partial_order_on A (r^-1) = partial_order_on A r"
```
```    44 by (simp add: partial_order_on_def)
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```    45
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```    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)
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```    49
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```    50
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```    51 lemma strict_linear_order_on_diff_Id:
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```    52   "linear_order_on A r \<Longrightarrow> strict_linear_order_on A (r-Id)"
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```    53 by(simp add: order_on_defs trans_diff_Id)
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```    54
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```    55
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```    56 subsection{* Orders on the field *}
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```    57
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```    58 abbreviation "Refl r \<equiv> refl_on (Field r) r"
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```    59
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```    60 abbreviation "Preorder r \<equiv> preorder_on (Field r) r"
```
```    61
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```    62 abbreviation "Partial_order r \<equiv> partial_order_on (Field r) r"
```
```    63
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```    64 abbreviation "Total r \<equiv> total_on (Field r) r"
```
```    65
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```    66 abbreviation "Linear_order r \<equiv> linear_order_on (Field r) r"
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```    67
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```    68 abbreviation "Well_order r \<equiv> well_order_on (Field r) r"
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```    69
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```    70
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```    71 lemma subset_Image_Image_iff:
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```    72   "\<lbrakk> Preorder r; A \<subseteq> Field r; B \<subseteq> Field r\<rbrakk> \<Longrightarrow>
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```    73    r `` A \<subseteq> r `` B \<longleftrightarrow> (\<forall>a\<in>A.\<exists>b\<in>B. (b,a):r)"
```
```    74 unfolding preorder_on_def refl_on_def Image_def
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```    75 apply (simp add: subset_eq)
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```    76 unfolding trans_def by fast
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```    77
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```    78 lemma subset_Image1_Image1_iff:
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```    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)
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```    81
```
```    82 lemma Refl_antisym_eq_Image1_Image1_iff:
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```    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
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```    86 lemma Partial_order_eq_Image1_Image1_iff:
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```    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)
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```    89
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```    90
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```    91 subsection{* Orders on a type *}
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```    92
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```    93 abbreviation "strict_linear_order \<equiv> strict_linear_order_on UNIV"
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```    94
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```    95 abbreviation "linear_order \<equiv> linear_order_on UNIV"
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```    96
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```    97 abbreviation "well_order r \<equiv> well_order_on UNIV"
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```    98
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```    99 end
```