author | wenzelm |
Sun, 02 Nov 2014 18:21:45 +0100 | |
changeset 58889 | 5b7a9633cfa8 |
parent 55102 | 761e40ce91bc |
child 60585 | 48fdff264eb2 |
permissions | -rw-r--r-- |
49310 | 1 |
(* Title: HOL/Cardinals/Order_Relation_More.thy |
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2 |
Author: Andrei Popescu, TU Muenchen |
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3 |
Copyright 2012 |
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|
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5 |
Basics on order-like relations. |
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6 |
*) |
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|
58889 | 8 |
section {* Basics on Order-Like Relations *} |
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|
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10 |
theory Order_Relation_More |
55026
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11 |
imports Main |
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12 |
begin |
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|
55102 | 14 |
subsection {* The upper and lower bounds operators *} |
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15 |
|
55023
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16 |
lemma aboveS_subset_above: "aboveS r a \<le> above r a" |
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17 |
by(auto simp add: aboveS_def above_def) |
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|
55023
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19 |
lemma AboveS_subset_Above: "AboveS r A \<le> Above r A" |
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20 |
by(auto simp add: AboveS_def Above_def) |
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21 |
|
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lemma UnderS_disjoint: "A Int (UnderS r A) = {}" |
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by(auto simp add: UnderS_def) |
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|
55023
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lemma aboveS_notIn: "a \<notin> aboveS r a" |
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by(auto simp add: aboveS_def) |
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|
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lemma Refl_above_in: "\<lbrakk>Refl r; a \<in> Field r\<rbrakk> \<Longrightarrow> a \<in> above r a" |
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by(auto simp add: refl_on_def above_def) |
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|
55023
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lemma in_Above_under: "a \<in> Field r \<Longrightarrow> a \<in> Above r (under r a)" |
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by(auto simp add: Above_def under_def) |
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|
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lemma in_Under_above: "a \<in> Field r \<Longrightarrow> a \<in> Under r (above r a)" |
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by(auto simp add: Under_def above_def) |
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36 |
|
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37 |
lemma in_UnderS_aboveS: "a \<in> Field r \<Longrightarrow> a \<in> UnderS r (aboveS r a)" |
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38 |
by(auto simp add: UnderS_def aboveS_def) |
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39 |
|
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40 |
lemma UnderS_subset_Under: "UnderS r A \<le> Under r A" |
54479 | 41 |
by(auto simp add: UnderS_def Under_def) |
42 |
||
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lemma subset_Above_Under: "B \<le> Field r \<Longrightarrow> B \<le> Above r (Under r B)" |
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by(auto simp add: Above_def Under_def) |
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|
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lemma subset_Under_Above: "B \<le> Field r \<Longrightarrow> B \<le> Under r (Above r B)" |
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by(auto simp add: Under_def Above_def) |
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48 |
|
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49 |
lemma subset_AboveS_UnderS: "B \<le> Field r \<Longrightarrow> B \<le> AboveS r (UnderS r B)" |
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by(auto simp add: AboveS_def UnderS_def) |
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|
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52 |
lemma subset_UnderS_AboveS: "B \<le> Field r \<Longrightarrow> B \<le> UnderS r (AboveS r B)" |
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by(auto simp add: UnderS_def AboveS_def) |
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54 |
|
54479 | 55 |
lemma Under_Above_Galois: |
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56 |
"\<lbrakk>B \<le> Field r; C \<le> Field r\<rbrakk> \<Longrightarrow> (B \<le> Above r C) = (C \<le> Under r B)" |
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57 |
by(unfold Above_def Under_def, blast) |
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58 |
|
54479 | 59 |
lemma UnderS_AboveS_Galois: |
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60 |
"\<lbrakk>B \<le> Field r; C \<le> Field r\<rbrakk> \<Longrightarrow> (B \<le> AboveS r C) = (C \<le> UnderS r B)" |
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61 |
by(unfold AboveS_def UnderS_def, blast) |
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62 |
|
54479 | 63 |
lemma Refl_above_aboveS: |
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64 |
assumes REFL: "Refl r" and IN: "a \<in> Field r" |
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65 |
shows "above r a = aboveS r a \<union> {a}" |
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66 |
proof(unfold above_def aboveS_def, auto) |
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67 |
show "(a,a) \<in> r" using REFL IN refl_on_def[of _ r] by blast |
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68 |
qed |
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69 |
|
54479 | 70 |
lemma Linear_order_under_aboveS_Field: |
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71 |
assumes LIN: "Linear_order r" and IN: "a \<in> Field r" |
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parents:
54552
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72 |
shows "Field r = under r a \<union> aboveS r a" |
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73 |
proof(unfold under_def aboveS_def, auto) |
7f79f94a432c
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74 |
assume "a \<in> Field r" "(a, a) \<notin> r" |
7f79f94a432c
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parents:
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75 |
with LIN IN order_on_defs[of _ r] refl_on_def[of _ r] |
7f79f94a432c
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parents:
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76 |
show False by auto |
7f79f94a432c
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77 |
next |
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78 |
fix b assume "b \<in> Field r" "(b, a) \<notin> r" |
7f79f94a432c
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parents:
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79 |
with LIN IN order_on_defs[of "Field r" r] total_on_def[of "Field r" r] |
7f79f94a432c
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parents:
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80 |
have "(a,b) \<in> r \<or> a = b" by blast |
7f79f94a432c
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parents:
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81 |
thus "(a,b) \<in> r" |
7f79f94a432c
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parents:
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82 |
using LIN IN order_on_defs[of _ r] refl_on_def[of _ r] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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83 |
next |
7f79f94a432c
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parents:
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84 |
fix b assume "(b, a) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
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85 |
thus "b \<in> Field r" |
7f79f94a432c
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blanchet
parents:
diff
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86 |
using LIN order_on_defs[of _ r] refl_on_def[of _ r] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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87 |
next |
7f79f94a432c
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parents:
diff
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88 |
fix b assume "b \<noteq> a" "(a, b) \<in> r" |
7f79f94a432c
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parents:
diff
changeset
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89 |
thus "b \<in> Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
90 |
using LIN order_on_defs[of "Field r" r] refl_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:
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91 |
qed |
7f79f94a432c
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parents:
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92 |
|
54479 | 93 |
lemma Linear_order_underS_above_Field: |
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94 |
assumes LIN: "Linear_order r" and IN: "a \<in> Field r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
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parents:
54552
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changeset
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95 |
shows "Field r = underS r a \<union> above r a" |
48975
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parents:
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96 |
proof(unfold underS_def above_def, auto) |
7f79f94a432c
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parents:
diff
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97 |
assume "a \<in> Field r" "(a, a) \<notin> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
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98 |
with LIN IN order_on_defs[of _ r] refl_on_def[of _ r] |
51764 | 99 |
show False by metis |
48975
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parents:
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100 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
101 |
fix b assume "b \<in> Field r" "(a, b) \<notin> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
102 |
with LIN IN order_on_defs[of "Field r" r] total_on_def[of "Field r" r] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
103 |
have "(b,a) \<in> r \<or> b = a" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
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changeset
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104 |
thus "(b,a) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
105 |
using LIN IN order_on_defs[of _ r] refl_on_def[of _ r] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
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|
106 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
107 |
fix b assume "b \<noteq> a" "(b, a) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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parents:
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108 |
thus "b \<in> Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
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|
109 |
using LIN order_on_defs[of _ r] refl_on_def[of _ r] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
110 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
111 |
fix b assume "(a, b) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
112 |
thus "b \<in> Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
113 |
using LIN order_on_defs[of "Field r" r] refl_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
|
114 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
115 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
116 |
lemma under_empty: "a \<notin> Field r \<Longrightarrow> under r a = {}" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
117 |
unfolding Field_def under_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
118 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
119 |
lemma Under_Field: "Under r A \<le> Field r" |
54479 | 120 |
by(unfold Under_def Field_def, auto) |
121 |
||
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
122 |
lemma UnderS_Field: "UnderS r A \<le> Field r" |
54479 | 123 |
by(unfold UnderS_def Field_def, auto) |
124 |
||
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
125 |
lemma above_Field: "above r a \<le> Field r" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
126 |
by(unfold above_def Field_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
127 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
128 |
lemma aboveS_Field: "aboveS r a \<le> Field r" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
129 |
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
|
130 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
131 |
lemma Above_Field: "Above r A \<le> Field r" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
132 |
by(unfold Above_def Field_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
133 |
|
54479 | 134 |
lemma Refl_under_Under: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
135 |
assumes REFL: "Refl r" and NE: "A \<noteq> {}" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
136 |
shows "Under r A = (\<Inter> a \<in> A. under r a)" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
137 |
proof |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
138 |
show "Under r A \<subseteq> (\<Inter> a \<in> A. under r a)" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
139 |
by(unfold Under_def under_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
140 |
next |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
141 |
show "(\<Inter> a \<in> A. under r a) \<subseteq> Under r A" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
142 |
proof(auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
143 |
fix x |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
144 |
assume *: "\<forall>xa \<in> A. x \<in> under r xa" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
145 |
hence "\<forall>xa \<in> A. (x,xa) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
146 |
by (simp add: under_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
147 |
moreover |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
148 |
{from NE obtain a where "a \<in> A" by blast |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
149 |
with * have "x \<in> under r a" by simp |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
150 |
hence "x \<in> Field r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
151 |
using under_Field[of r a] by auto |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
152 |
} |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
153 |
ultimately show "x \<in> Under r A" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
154 |
unfolding Under_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
155 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
156 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
157 |
|
54479 | 158 |
lemma Refl_underS_UnderS: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
159 |
assumes REFL: "Refl r" and NE: "A \<noteq> {}" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
160 |
shows "UnderS r A = (\<Inter> a \<in> A. underS r a)" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
161 |
proof |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
162 |
show "UnderS r A \<subseteq> (\<Inter> a \<in> A. underS r a)" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
163 |
by(unfold UnderS_def underS_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
164 |
next |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
165 |
show "(\<Inter> a \<in> A. underS r a) \<subseteq> UnderS r A" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
166 |
proof(auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
167 |
fix x |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
168 |
assume *: "\<forall>xa \<in> A. x \<in> underS r xa" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
169 |
hence "\<forall>xa \<in> A. x \<noteq> xa \<and> (x,xa) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
170 |
by (auto simp add: underS_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
171 |
moreover |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
172 |
{from NE obtain a where "a \<in> A" by blast |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
173 |
with * have "x \<in> underS r a" by simp |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
174 |
hence "x \<in> Field r" |
55065 | 175 |
using underS_Field[of _ r a] by auto |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
176 |
} |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
177 |
ultimately show "x \<in> UnderS r A" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
178 |
unfolding UnderS_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
179 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
180 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
181 |
|
54479 | 182 |
lemma Refl_above_Above: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
183 |
assumes REFL: "Refl r" and NE: "A \<noteq> {}" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
184 |
shows "Above r A = (\<Inter> a \<in> A. above r a)" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
185 |
proof |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
186 |
show "Above r A \<subseteq> (\<Inter> a \<in> A. above r a)" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
187 |
by(unfold Above_def above_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
188 |
next |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
189 |
show "(\<Inter> a \<in> A. above r a) \<subseteq> Above r A" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
190 |
proof(auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
191 |
fix x |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
192 |
assume *: "\<forall>xa \<in> A. x \<in> above r xa" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
193 |
hence "\<forall>xa \<in> A. (xa,x) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
194 |
by (simp add: above_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
195 |
moreover |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
196 |
{from NE obtain a where "a \<in> A" by blast |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
197 |
with * have "x \<in> above r a" by simp |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
198 |
hence "x \<in> Field r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
199 |
using above_Field[of r a] by auto |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
200 |
} |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
201 |
ultimately show "x \<in> Above r A" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
202 |
unfolding Above_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
203 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
204 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
205 |
|
54479 | 206 |
lemma Refl_aboveS_AboveS: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
207 |
assumes REFL: "Refl r" and NE: "A \<noteq> {}" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
208 |
shows "AboveS r A = (\<Inter> a \<in> A. aboveS r a)" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
209 |
proof |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
210 |
show "AboveS r A \<subseteq> (\<Inter> a \<in> A. aboveS r a)" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
211 |
by(unfold AboveS_def aboveS_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
212 |
next |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
213 |
show "(\<Inter> a \<in> A. aboveS r a) \<subseteq> AboveS r A" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
214 |
proof(auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
215 |
fix x |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
216 |
assume *: "\<forall>xa \<in> A. x \<in> aboveS r xa" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
217 |
hence "\<forall>xa \<in> A. xa \<noteq> x \<and> (xa,x) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
218 |
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
|
219 |
moreover |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
220 |
{from NE obtain a where "a \<in> A" by blast |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
221 |
with * have "x \<in> aboveS r a" by simp |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
222 |
hence "x \<in> Field r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
223 |
using aboveS_Field[of r a] by auto |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
224 |
} |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
225 |
ultimately show "x \<in> AboveS r A" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
226 |
unfolding AboveS_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
227 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
228 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
229 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
230 |
lemma under_Under_singl: "under r a = Under r {a}" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
231 |
by(unfold Under_def under_def, auto simp add: Field_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
232 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
233 |
lemma underS_UnderS_singl: "underS r a = UnderS r {a}" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
234 |
by(unfold UnderS_def underS_def, auto simp add: Field_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
235 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
236 |
lemma above_Above_singl: "above r a = Above r {a}" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
237 |
by(unfold Above_def above_def, auto simp add: Field_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
238 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
239 |
lemma aboveS_AboveS_singl: "aboveS r a = AboveS r {a}" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
240 |
by(unfold AboveS_def aboveS_def, auto simp add: Field_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
241 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
242 |
lemma Under_decr: "A \<le> B \<Longrightarrow> Under r B \<le> Under r A" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
243 |
by(unfold Under_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
244 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
245 |
lemma UnderS_decr: "A \<le> B \<Longrightarrow> UnderS r B \<le> UnderS r A" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
246 |
by(unfold UnderS_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
247 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
248 |
lemma Above_decr: "A \<le> B \<Longrightarrow> Above r B \<le> Above r A" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
249 |
by(unfold Above_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
250 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
251 |
lemma AboveS_decr: "A \<le> B \<Longrightarrow> AboveS r B \<le> AboveS r A" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
252 |
by(unfold AboveS_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
253 |
|
54479 | 254 |
lemma under_incl_iff: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
255 |
assumes TRANS: "trans r" and REFL: "Refl r" and IN: "a \<in> Field r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
256 |
shows "(under r a \<le> under r b) = ((a,b) \<in> r)" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
257 |
proof |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
258 |
assume "(a,b) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
259 |
thus "under r a \<le> under r b" using TRANS |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
260 |
by (auto simp add: under_incr) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
261 |
next |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
262 |
assume "under r a \<le> under r b" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
263 |
moreover |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
264 |
have "a \<in> under r a" using REFL IN |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
265 |
by (auto simp add: Refl_under_in) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
266 |
ultimately show "(a,b) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
267 |
by (auto simp add: under_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
268 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
269 |
|
54479 | 270 |
lemma above_decr: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
271 |
assumes TRANS: "trans r" and REL: "(a,b) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
272 |
shows "above r b \<le> above r a" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
273 |
proof(unfold above_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
274 |
fix x assume "(b,x) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
275 |
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
|
276 |
show "(a,x) \<in> r" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
277 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
278 |
|
54479 | 279 |
lemma aboveS_decr: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
280 |
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
|
281 |
REL: "(a,b) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
282 |
shows "aboveS r b \<le> aboveS r a" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
283 |
proof(unfold aboveS_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
284 |
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
|
285 |
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
|
286 |
show False by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
287 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
288 |
fix x assume "x \<noteq> b" "(b,x) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
289 |
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
|
290 |
show "(a,x) \<in> r" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
291 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
292 |
|
54479 | 293 |
lemma under_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
294 |
assumes TRANS: "trans r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
295 |
IN1: "a \<in> under r b" and IN2: "b \<in> under r c" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
296 |
shows "a \<in> under r c" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
297 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
298 |
have "(a,b) \<in> r \<and> (b,c) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
299 |
using IN1 IN2 under_def by fastforce |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
300 |
hence "(a,c) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
301 |
using TRANS trans_def[of r] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
302 |
thus ?thesis unfolding under_def by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
303 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
304 |
|
54479 | 305 |
lemma under_underS_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
306 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
307 |
IN1: "a \<in> under r b" and IN2: "b \<in> underS r c" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
308 |
shows "a \<in> underS r c" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
309 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
310 |
have 0: "(a,b) \<in> r \<and> (b,c) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
311 |
using IN1 IN2 under_def underS_def by fastforce |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
312 |
hence 1: "(a,c) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
313 |
using TRANS trans_def[of r] by blast |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
314 |
have 2: "b \<noteq> c" using IN2 underS_def by force |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
315 |
have 3: "a \<noteq> c" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
316 |
proof |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
317 |
assume "a = c" with 0 2 ANTISYM antisym_def[of r] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
318 |
show False by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
319 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
320 |
from 1 3 show ?thesis unfolding underS_def by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
321 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
322 |
|
54479 | 323 |
lemma underS_under_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
324 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
325 |
IN1: "a \<in> underS r b" and IN2: "b \<in> under r c" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
326 |
shows "a \<in> underS r c" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
327 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
328 |
have 0: "(a,b) \<in> r \<and> (b,c) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
329 |
using IN1 IN2 under_def underS_def by fast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
330 |
hence 1: "(a,c) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
331 |
using TRANS trans_def[of r] by fast |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
332 |
have 2: "a \<noteq> b" using IN1 underS_def by force |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
333 |
have 3: "a \<noteq> c" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
334 |
proof |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
335 |
assume "a = c" with 0 2 ANTISYM antisym_def[of r] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
336 |
show False by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
337 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
338 |
from 1 3 show ?thesis unfolding underS_def by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
339 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
340 |
|
54479 | 341 |
lemma underS_underS_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
342 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
343 |
IN1: "a \<in> underS r b" and IN2: "b \<in> underS r c" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
344 |
shows "a \<in> underS r c" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
345 |
proof- |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
346 |
have "a \<in> under r b" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
347 |
using IN1 underS_subset_under by fast |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
348 |
with assms under_underS_trans show ?thesis by fast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
349 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
350 |
|
54479 | 351 |
lemma above_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
352 |
assumes TRANS: "trans r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
353 |
IN1: "b \<in> above r a" and IN2: "c \<in> above r b" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
354 |
shows "c \<in> above r a" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
355 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
356 |
have "(a,b) \<in> r \<and> (b,c) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
357 |
using IN1 IN2 above_def by fast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
358 |
hence "(a,c) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
359 |
using TRANS trans_def[of r] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
360 |
thus ?thesis unfolding above_def by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
361 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
362 |
|
54479 | 363 |
lemma above_aboveS_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
364 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
365 |
IN1: "b \<in> above r a" and IN2: "c \<in> aboveS r b" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
366 |
shows "c \<in> aboveS r a" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
367 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
368 |
have 0: "(a,b) \<in> r \<and> (b,c) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
369 |
using IN1 IN2 above_def aboveS_def by fast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
370 |
hence 1: "(a,c) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
371 |
using TRANS trans_def[of r] by blast |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
372 |
have 2: "b \<noteq> c" using IN2 aboveS_def by force |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
373 |
have 3: "a \<noteq> c" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
374 |
proof |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
375 |
assume "a = c" with 0 2 ANTISYM antisym_def[of r] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
376 |
show False by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
377 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
378 |
from 1 3 show ?thesis unfolding aboveS_def by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
379 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
380 |
|
54479 | 381 |
lemma aboveS_above_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
382 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
383 |
IN1: "b \<in> aboveS r a" and IN2: "c \<in> above r b" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
384 |
shows "c \<in> aboveS r a" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
385 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
386 |
have 0: "(a,b) \<in> r \<and> (b,c) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
387 |
using IN1 IN2 above_def aboveS_def by fast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
388 |
hence 1: "(a,c) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
389 |
using TRANS trans_def[of r] by blast |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
390 |
have 2: "a \<noteq> b" using IN1 aboveS_def by force |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
391 |
have 3: "a \<noteq> c" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
392 |
proof |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
393 |
assume "a = c" with 0 2 ANTISYM antisym_def[of r] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
394 |
show False by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
395 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
396 |
from 1 3 show ?thesis unfolding aboveS_def by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
397 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
398 |
|
54479 | 399 |
lemma aboveS_aboveS_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
400 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
401 |
IN1: "b \<in> aboveS r a" and IN2: "c \<in> aboveS r b" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
402 |
shows "c \<in> aboveS r a" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
403 |
proof- |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
404 |
have "b \<in> above r a" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
405 |
using IN1 aboveS_subset_above by fast |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
406 |
with assms above_aboveS_trans show ?thesis by fast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
407 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
408 |
|
54479 | 409 |
lemma under_Under_trans: |
410 |
assumes TRANS: "trans r" and |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
411 |
IN1: "a \<in> under r b" and IN2: "b \<in> Under r C" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
412 |
shows "a \<in> Under r C" |
54479 | 413 |
proof- |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
414 |
have "b \<in> {u \<in> Field r. \<forall>x. x \<in> C \<longrightarrow> (u, x) \<in> r}" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
415 |
using IN2 Under_def by force |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
416 |
hence "(a,b) \<in> r \<and> (\<forall>c \<in> C. (b,c) \<in> r)" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
417 |
using IN1 under_def by fast |
54479 | 418 |
hence "\<forall>c \<in> C. (a,c) \<in> r" |
419 |
using TRANS trans_def[of r] by blast |
|
420 |
moreover |
|
421 |
have "a \<in> Field r" using IN1 unfolding Field_def under_def by blast |
|
422 |
ultimately |
|
423 |
show ?thesis unfolding Under_def by blast |
|
424 |
qed |
|
425 |
||
426 |
lemma underS_Under_trans: |
|
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
427 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
428 |
IN1: "a \<in> underS r b" and IN2: "b \<in> Under r C" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
429 |
shows "a \<in> UnderS r C" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
430 |
proof- |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
431 |
from IN1 have "a \<in> under r b" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
432 |
using underS_subset_under[of r b] by fast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
433 |
with assms under_Under_trans |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
434 |
have "a \<in> Under r C" by fast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
435 |
(* *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
436 |
moreover |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
437 |
have "a \<notin> C" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
438 |
proof |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
439 |
assume *: "a \<in> C" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
440 |
have 1: "b \<noteq> a \<and> (a,b) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
441 |
using IN1 underS_def[of r b] by auto |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
442 |
have "\<forall>c \<in> C. (b,c) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
443 |
using IN2 Under_def[of r C] by auto |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
444 |
with * have "(b,a) \<in> r" by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
445 |
with 1 ANTISYM antisym_def[of r] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
446 |
show False by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
447 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
448 |
(* *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
449 |
ultimately |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
450 |
show ?thesis unfolding UnderS_def |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
451 |
using Under_def by force |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
452 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
453 |
|
54479 | 454 |
lemma underS_UnderS_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
455 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
456 |
IN1: "a \<in> underS r b" and IN2: "b \<in> UnderS r C" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
457 |
shows "a \<in> UnderS r C" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
458 |
proof- |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
459 |
from IN2 have "b \<in> Under r C" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
460 |
using UnderS_subset_Under[of r C] by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
461 |
with underS_Under_trans assms |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
462 |
show ?thesis by force |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
463 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
464 |
|
54479 | 465 |
lemma above_Above_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
466 |
assumes TRANS: "trans r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
467 |
IN1: "a \<in> above r b" and IN2: "b \<in> Above r C" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
468 |
shows "a \<in> Above r C" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
469 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
470 |
have "(b,a) \<in> r \<and> (\<forall>c \<in> C. (c,b) \<in> r)" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
471 |
using IN1[unfolded above_def] IN2[unfolded Above_def] by simp |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
472 |
hence "\<forall>c \<in> C. (c,a) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
473 |
using TRANS trans_def[of r] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
474 |
moreover |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
475 |
have "a \<in> Field r" using IN1[unfolded above_def] Field_def by fast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
476 |
ultimately |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
477 |
show ?thesis unfolding Above_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
478 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
479 |
|
54479 | 480 |
lemma aboveS_Above_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
481 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
482 |
IN1: "a \<in> aboveS r b" and IN2: "b \<in> Above r C" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
483 |
shows "a \<in> AboveS r C" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
484 |
proof- |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
485 |
from IN1 have "a \<in> above r b" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
486 |
using aboveS_subset_above[of r b] by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
487 |
with assms above_Above_trans |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
488 |
have "a \<in> Above r C" by fast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
489 |
(* *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
490 |
moreover |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
491 |
have "a \<notin> C" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
492 |
proof |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
493 |
assume *: "a \<in> C" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
494 |
have 1: "b \<noteq> a \<and> (b,a) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
495 |
using IN1 aboveS_def[of r b] by auto |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
496 |
have "\<forall>c \<in> C. (c,b) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
497 |
using IN2 Above_def[of r C] by auto |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
498 |
with * have "(a,b) \<in> r" by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
499 |
with 1 ANTISYM antisym_def[of r] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
500 |
show False by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
501 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
502 |
(* *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
503 |
ultimately |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
504 |
show ?thesis unfolding AboveS_def |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
505 |
using Above_def by force |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
506 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
507 |
|
54479 | 508 |
lemma above_AboveS_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
509 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
510 |
IN1: "a \<in> above r b" and IN2: "b \<in> AboveS r C" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
511 |
shows "a \<in> AboveS r C" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
512 |
proof- |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
513 |
from IN2 have "b \<in> Above r C" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
514 |
using AboveS_subset_Above[of r C] by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
515 |
with assms above_Above_trans |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
516 |
have "a \<in> Above r C" by force |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
517 |
(* *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
518 |
moreover |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
519 |
have "a \<notin> C" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
520 |
proof |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
521 |
assume *: "a \<in> C" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
522 |
have 1: "(b,a) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
523 |
using IN1 above_def[of r b] by auto |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
524 |
have "\<forall>c \<in> C. b \<noteq> c \<and> (c,b) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
525 |
using IN2 AboveS_def[of r C] by auto |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
526 |
with * have "b \<noteq> a \<and> (a,b) \<in> r" by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
527 |
with 1 ANTISYM antisym_def[of r] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
528 |
show False by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
529 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
530 |
(* *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
531 |
ultimately |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
532 |
show ?thesis unfolding AboveS_def |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
533 |
using Above_def by force |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
534 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
535 |
|
54479 | 536 |
lemma aboveS_AboveS_trans: |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
537 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
538 |
IN1: "a \<in> aboveS r b" and IN2: "b \<in> AboveS r C" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
539 |
shows "a \<in> AboveS r C" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
540 |
proof- |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
541 |
from IN2 have "b \<in> Above r C" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
542 |
using AboveS_subset_Above[of r C] by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
543 |
with aboveS_Above_trans assms |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
544 |
show ?thesis by force |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
545 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
546 |
|
54479 | 547 |
lemma under_UnderS_trans: |
548 |
assumes TRANS: "trans r" and ANTISYM: "antisym r" and |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
549 |
IN1: "a \<in> under r b" and IN2: "b \<in> UnderS r C" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
550 |
shows "a \<in> UnderS r C" |
54479 | 551 |
proof- |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
552 |
from IN2 have "b \<in> Under r C" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
553 |
using UnderS_subset_Under[of r C] by blast |
54479 | 554 |
with assms under_Under_trans |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
555 |
have "a \<in> Under r C" by force |
54479 | 556 |
(* *) |
557 |
moreover |
|
558 |
have "a \<notin> C" |
|
559 |
proof |
|
560 |
assume *: "a \<in> C" |
|
561 |
have 1: "(a,b) \<in> r" |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
562 |
using IN1 under_def[of r b] by auto |
54479 | 563 |
have "\<forall>c \<in> C. b \<noteq> c \<and> (b,c) \<in> r" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
564 |
using IN2 UnderS_def[of r C] by blast |
54479 | 565 |
with * have "b \<noteq> a \<and> (b,a) \<in> r" by blast |
566 |
with 1 ANTISYM antisym_def[of r] |
|
567 |
show False by blast |
|
568 |
qed |
|
569 |
(* *) |
|
570 |
ultimately |
|
571 |
show ?thesis unfolding UnderS_def Under_def by fast |
|
572 |
qed |
|
573 |
||
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
574 |
|
55102 | 575 |
subsection {* Properties depending on more than one relation *} |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
576 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
577 |
lemma under_incr2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
578 |
"r \<le> r' \<Longrightarrow> under r a \<le> under r' a" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
579 |
unfolding under_def by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
580 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
581 |
lemma underS_incr2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
582 |
"r \<le> r' \<Longrightarrow> underS r a \<le> underS r' a" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
583 |
unfolding underS_def by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
584 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
585 |
(* FIXME: r \<leadsto> r' |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
586 |
lemma Under_incr: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
587 |
"r \<le> r' \<Longrightarrow> Under r A \<le> Under r A" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
588 |
unfolding Under_def by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
589 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
590 |
lemma UnderS_incr: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
591 |
"r \<le> r' \<Longrightarrow> UnderS r A \<le> UnderS r A" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
592 |
unfolding UnderS_def by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
593 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
594 |
lemma Under_incr_decr: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
595 |
"\<lbrakk>r \<le> r'; A' \<le> A\<rbrakk> \<Longrightarrow> Under r A \<le> Under r A'" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
596 |
unfolding Under_def by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
597 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
598 |
lemma UnderS_incr_decr: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
599 |
"\<lbrakk>r \<le> r'; A' \<le> A\<rbrakk> \<Longrightarrow> UnderS r A \<le> UnderS r A'" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
600 |
unfolding UnderS_def by blast |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
601 |
*) |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
602 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
603 |
lemma above_incr2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
604 |
"r \<le> r' \<Longrightarrow> above r a \<le> above r' a" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
605 |
unfolding above_def by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
606 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
607 |
lemma aboveS_incr2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
608 |
"r \<le> r' \<Longrightarrow> aboveS r a \<le> aboveS r' a" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
609 |
unfolding aboveS_def by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
610 |
|
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
611 |
(* FIXME |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
612 |
lemma Above_incr: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
613 |
"r \<le> r' \<Longrightarrow> Above r A \<le> Above r A" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
614 |
unfolding Above_def by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
615 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
616 |
lemma AboveS_incr: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
617 |
"r \<le> r' \<Longrightarrow> AboveS r A \<le> AboveS r A" |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
618 |
unfolding AboveS_def by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
619 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
620 |
lemma Above_incr_decr: |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
621 |
"\<lbrakk>r \<le> r'; A' \<le> A\<rbrakk> \<Longrightarrow> Above r A \<le> Above r A'" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
622 |
unfolding Above_def by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
623 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
624 |
lemma AboveS_incr_decr: |
55023
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
625 |
"\<lbrakk>r \<le> r'; A' \<le> A\<rbrakk> \<Longrightarrow> AboveS r A \<le> AboveS r A'" |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
626 |
unfolding AboveS_def by blast |
38db7814481d
get rid of 'rel' locale, to facilitate inclusion of 'Order_Relation_More_FP' into 'Order_Relation'
blanchet
parents:
54552
diff
changeset
|
627 |
*) |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
628 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
629 |
end |