author | blanchet |
Thu, 16 Jan 2014 16:33:19 +0100 | |
changeset 55018 | 2a526bd279ed |
parent 54552 | 5d57cbec0f0f |
child 55023 | 38db7814481d |
permissions | -rw-r--r-- |
54481 | 1 |
(* Title: HOL/Cardinals/Order_Relation_More_FP.thy |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
2 |
Author: Andrei Popescu, TU Muenchen |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
3 |
Copyright 2012 |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
4 |
|
54481 | 5 |
Basics on order-like relations (FP). |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
6 |
*) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
7 |
|
54481 | 8 |
header {* Basics on Order-Like Relations (FP) *} |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
9 |
|
54481 | 10 |
theory Order_Relation_More_FP |
54552
5d57cbec0f0f
moving 'Order_Relation' to 'HOL' (since it's a BNF dependency)
blanchet
parents:
54481
diff
changeset
|
11 |
imports Order_Relation |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
12 |
begin |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
13 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
14 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
15 |
text{* In this section, we develop basic concepts and results pertaining |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
16 |
to order-like relations, i.e., to reflexive and/or transitive and/or symmetric and/or |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
17 |
total relations. The development is placed on top of the definitions |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
18 |
from the theory @{text "Order_Relation"}. We also |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
19 |
further define upper and lower bounds operators. *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
20 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
21 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
22 |
locale rel = fixes r :: "'a rel" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
23 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
24 |
text{* The following context encompasses all this section, except |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
25 |
for its last subsection. In other words, for the rest of this section except its last |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
26 |
subsection, we consider a fixed relation @{text "r"}. *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
27 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
28 |
context rel |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
29 |
begin |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
30 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
31 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
32 |
subsection {* Auxiliaries *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
33 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
34 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
35 |
lemma refl_on_domain: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
36 |
"\<lbrakk>refl_on A r; (a,b) : r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
37 |
by(auto simp add: refl_on_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
38 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
39 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
40 |
corollary well_order_on_domain: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
41 |
"\<lbrakk>well_order_on A r; (a,b) \<in> r\<rbrakk> \<Longrightarrow> a \<in> A \<and> b \<in> A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
42 |
by(simp add: refl_on_domain order_on_defs) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
43 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
44 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
45 |
lemma well_order_on_Field: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
46 |
"well_order_on A r \<Longrightarrow> A = Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
47 |
by(auto simp add: refl_on_def Field_def order_on_defs) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
48 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
49 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
50 |
lemma well_order_on_Well_order: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
51 |
"well_order_on A r \<Longrightarrow> A = Field r \<and> Well_order r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
52 |
using well_order_on_Field by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
53 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
54 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
55 |
lemma Total_subset_Id: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
56 |
assumes TOT: "Total r" and SUB: "r \<le> Id" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
57 |
shows "r = {} \<or> (\<exists>a. r = {(a,a)})" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
58 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
59 |
{assume "r \<noteq> {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
60 |
then obtain a b where 1: "(a,b) \<in> r" by fast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
61 |
hence "a = b" using SUB by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
62 |
hence 2: "(a,a) \<in> r" using 1 by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
63 |
{fix c d assume "(c,d) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
64 |
hence "{a,c,d} \<le> Field r" using 1 unfolding Field_def by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
65 |
hence "((a,c) \<in> r \<or> (c,a) \<in> r \<or> a = c) \<and> |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
66 |
((a,d) \<in> r \<or> (d,a) \<in> r \<or> a = d)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
67 |
using TOT unfolding total_on_def by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
68 |
hence "a = c \<and> a = d" using SUB by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
69 |
} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
70 |
hence "r \<le> {(a,a)}" by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
71 |
with 2 have "\<exists>a. r = {(a,a)}" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
72 |
} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
73 |
thus ?thesis by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
74 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
75 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
76 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
77 |
lemma Linear_order_in_diff_Id: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
78 |
assumes LI: "Linear_order r" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
79 |
IN1: "a \<in> Field r" and IN2: "b \<in> Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
80 |
shows "((a,b) \<in> r) = ((b,a) \<notin> r - Id)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
81 |
using assms unfolding order_on_defs total_on_def antisym_def Id_def refl_on_def by force |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
82 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
83 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
84 |
subsection {* The upper and lower bounds operators *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
85 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
86 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
87 |
text{* Here we define upper (``above") and lower (``below") bounds operators. |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
88 |
We think of @{text "r"} as a {\em non-strict} relation. The suffix ``S" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
89 |
at the names of some operators indicates that the bounds are strict -- e.g., |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
90 |
@{text "underS a"} is the set of all strict lower bounds of @{text "a"} (w.r.t. @{text "r"}). |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
91 |
Capitalization of the first letter in the name reminds that the operator acts on sets, rather |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
92 |
than on individual elements. *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
93 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
94 |
definition under::"'a \<Rightarrow> 'a set" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
95 |
where "under a \<equiv> {b. (b,a) \<in> r}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
96 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
97 |
definition underS::"'a \<Rightarrow> 'a set" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
98 |
where "underS a \<equiv> {b. b \<noteq> a \<and> (b,a) \<in> r}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
99 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
100 |
definition Under::"'a set \<Rightarrow> 'a set" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
101 |
where "Under A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (b,a) \<in> r}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
102 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
103 |
definition UnderS::"'a set \<Rightarrow> 'a set" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
104 |
where "UnderS A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (b,a) \<in> r}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
105 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
106 |
definition above::"'a \<Rightarrow> 'a set" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
107 |
where "above a \<equiv> {b. (a,b) \<in> r}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
108 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
109 |
definition aboveS::"'a \<Rightarrow> 'a set" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
110 |
where "aboveS a \<equiv> {b. b \<noteq> a \<and> (a,b) \<in> r}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
111 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
112 |
definition Above::"'a set \<Rightarrow> 'a set" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
113 |
where "Above A \<equiv> {b \<in> Field r. \<forall>a \<in> A. (a,b) \<in> r}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
114 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
115 |
definition AboveS::"'a set \<Rightarrow> 'a set" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
116 |
where "AboveS A \<equiv> {b \<in> Field r. \<forall>a \<in> A. b \<noteq> a \<and> (a,b) \<in> r}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
117 |
(* *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
118 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
119 |
text{* Note: In the definitions of @{text "Above[S]"} and @{text "Under[S]"}, |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
120 |
we bounded comprehension by @{text "Field r"} in order to properly cover |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
121 |
the case of @{text "A"} being empty. *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
122 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
123 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
124 |
lemma underS_subset_under: "underS a \<le> under a" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
125 |
by(auto simp add: underS_def under_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
126 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
127 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
128 |
lemma underS_notIn: "a \<notin> underS a" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
129 |
by(simp add: underS_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
130 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
131 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
132 |
lemma Refl_under_in: "\<lbrakk>Refl r; a \<in> Field r\<rbrakk> \<Longrightarrow> a \<in> under a" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
133 |
by(simp add: refl_on_def under_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
134 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
135 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
136 |
lemma AboveS_disjoint: "A Int (AboveS A) = {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
137 |
by(auto simp add: AboveS_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
138 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
139 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
140 |
lemma in_AboveS_underS: "a \<in> Field r \<Longrightarrow> a \<in> AboveS (underS a)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
141 |
by(auto simp add: AboveS_def underS_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
142 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
143 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
144 |
lemma Refl_under_underS: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
145 |
assumes "Refl r" "a \<in> Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
146 |
shows "under a = underS a \<union> {a}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
147 |
unfolding under_def underS_def |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
148 |
using assms refl_on_def[of _ r] by fastforce |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
149 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
150 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
151 |
lemma underS_empty: "a \<notin> Field r \<Longrightarrow> underS a = {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
152 |
by (auto simp: Field_def underS_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
153 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
154 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
155 |
lemma under_Field: "under a \<le> Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
156 |
by(unfold under_def Field_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
157 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
158 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
159 |
lemma underS_Field: "underS a \<le> Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
160 |
by(unfold underS_def Field_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
161 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
162 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
163 |
lemma underS_Field2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
164 |
"a \<in> Field r \<Longrightarrow> underS a < Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
165 |
using assms underS_notIn underS_Field by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
166 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
167 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
168 |
lemma underS_Field3: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
169 |
"Field r \<noteq> {} \<Longrightarrow> underS a < Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
170 |
by(cases "a \<in> Field r", simp add: underS_Field2, auto simp add: underS_empty) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
171 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
172 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
173 |
lemma AboveS_Field: "AboveS A \<le> Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
174 |
by(unfold AboveS_def Field_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
175 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
176 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
177 |
lemma under_incr: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
178 |
assumes TRANS: "trans r" and REL: "(a,b) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
179 |
shows "under a \<le> under b" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
180 |
proof(unfold under_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
181 |
fix x assume "(x,a) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
182 |
with REL TRANS trans_def[of r] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
183 |
show "(x,b) \<in> r" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
184 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
185 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
186 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
187 |
lemma underS_incr: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
188 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
189 |
REL: "(a,b) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
190 |
shows "underS a \<le> underS b" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
191 |
proof(unfold underS_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
192 |
assume *: "b \<noteq> a" and **: "(b,a) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
193 |
with ANTISYM antisym_def[of r] REL |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
194 |
show False by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
195 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
196 |
fix x assume "x \<noteq> a" "(x,a) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
197 |
with REL TRANS trans_def[of r] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
198 |
show "(x,b) \<in> r" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
199 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
200 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
201 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
202 |
lemma underS_incl_iff: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
203 |
assumes LO: "Linear_order r" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
204 |
INa: "a \<in> Field r" and INb: "b \<in> Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
205 |
shows "(underS a \<le> underS b) = ((a,b) \<in> r)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
206 |
proof |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
207 |
assume "(a,b) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
208 |
thus "underS a \<le> underS b" using LO |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
209 |
by (simp add: order_on_defs underS_incr) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
210 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
211 |
assume *: "underS a \<le> underS b" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
212 |
{assume "a = b" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
213 |
hence "(a,b) \<in> r" using assms |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
214 |
by (simp add: order_on_defs refl_on_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
215 |
} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
216 |
moreover |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
217 |
{assume "a \<noteq> b \<and> (b,a) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
218 |
hence "b \<in> underS a" unfolding underS_def by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
219 |
hence "b \<in> underS b" using * by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
220 |
hence False by (simp add: underS_notIn) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
221 |
} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
222 |
ultimately |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
223 |
show "(a,b) \<in> r" using assms |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
224 |
order_on_defs[of "Field r" r] total_on_def[of "Field r" r] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
225 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
226 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
227 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
228 |
end (* context rel *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
229 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
230 |
end |