author | blanchet |
Mon, 18 Nov 2013 18:04:45 +0100 | |
changeset 54475 | b4d644be252c |
parent 54473 | 8bee5ca99e63 |
child 54481 | 5c9819d7713b |
permissions | -rw-r--r-- |
49310 | 1 |
(* Title: HOL/Cardinals/Cardinal_Order_Relation.thy |
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Author: Andrei Popescu, TU Muenchen |
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3 |
Copyright 2012 |
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Cardinal-order relations. |
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*) |
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|
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header {* Cardinal-Order Relations *} |
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|
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theory Cardinal_Order_Relation |
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imports Cardinal_Order_Relation_LFP Constructions_on_Wellorders |
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begin |
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declare |
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card_order_on_well_order_on[simp] |
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card_of_card_order_on[simp] |
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card_of_well_order_on[simp] |
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Field_card_of[simp] |
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card_of_Card_order[simp] |
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card_of_Well_order[simp] |
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card_of_least[simp] |
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card_of_unique[simp] |
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card_of_mono1[simp] |
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card_of_mono2[simp] |
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card_of_cong[simp] |
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card_of_Field_ordLess[simp] |
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card_of_Field_ordIso[simp] |
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card_of_underS[simp] |
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ordLess_Field[simp] |
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card_of_empty[simp] |
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card_of_empty1[simp] |
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card_of_image[simp] |
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card_of_singl_ordLeq[simp] |
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Card_order_singl_ordLeq[simp] |
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card_of_Pow[simp] |
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Card_order_Pow[simp] |
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card_of_Plus1[simp] |
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Card_order_Plus1[simp] |
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card_of_Plus2[simp] |
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Card_order_Plus2[simp] |
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card_of_Plus_mono1[simp] |
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card_of_Plus_mono2[simp] |
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card_of_Plus_mono[simp] |
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card_of_Plus_cong2[simp] |
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card_of_Plus_cong[simp] |
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card_of_Un_Plus_ordLeq[simp] |
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card_of_Times1[simp] |
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card_of_Times2[simp] |
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card_of_Times3[simp] |
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card_of_Times_mono1[simp] |
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card_of_Times_mono2[simp] |
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card_of_ordIso_finite[simp] |
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card_of_Times_same_infinite[simp] |
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card_of_Times_infinite_simps[simp] |
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card_of_Plus_infinite1[simp] |
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card_of_Plus_infinite2[simp] |
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card_of_Plus_ordLess_infinite[simp] |
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card_of_Plus_ordLess_infinite_Field[simp] |
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infinite_cartesian_product[simp] |
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cardSuc_Card_order[simp] |
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cardSuc_greater[simp] |
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cardSuc_ordLeq[simp] |
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cardSuc_ordLeq_ordLess[simp] |
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cardSuc_mono_ordLeq[simp] |
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cardSuc_invar_ordIso[simp] |
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card_of_cardSuc_finite[simp] |
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cardSuc_finite[simp] |
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card_of_Plus_ordLeq_infinite_Field[simp] |
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curr_in[intro, simp] |
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Func_empty[simp] |
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Func_is_emp[simp] |
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subsection {* Cardinal of a set *} |
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lemma card_of_inj_rel: assumes INJ: "!! x y y'. \<lbrakk>(x,y) : R; (x,y') : R\<rbrakk> \<Longrightarrow> y = y'" |
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shows "|{y. EX x. (x,y) : R}| <=o |{x. EX y. (x,y) : R}|" |
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proof- |
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let ?Y = "{y. EX x. (x,y) : R}" let ?X = "{x. EX y. (x,y) : R}" |
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let ?f = "% y. SOME x. (x,y) : R" |
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have "?f ` ?Y <= ?X" by (auto dest: someI) |
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moreover have "inj_on ?f ?Y" |
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unfolding inj_on_def proof(auto) |
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fix y1 x1 y2 x2 |
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assume *: "(x1, y1) \<in> R" "(x2, y2) \<in> R" and **: "?f y1 = ?f y2" |
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hence "(?f y1,y1) : R" using someI[of "% x. (x,y1) : R"] by auto |
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moreover have "(?f y2,y2) : R" using * someI[of "% x. (x,y2) : R"] by auto |
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ultimately show "y1 = y2" using ** INJ by auto |
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qed |
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ultimately show "|?Y| <=o |?X|" using card_of_ordLeq by blast |
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qed |
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|
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lemma card_of_unique2: "\<lbrakk>card_order_on B r; bij_betw f A B\<rbrakk> \<Longrightarrow> r =o |A|" |
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using card_of_ordIso card_of_unique ordIso_equivalence by blast |
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|
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lemma internalize_card_of_ordLess: |
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"( |A| <o r) = (\<exists>B < Field r. |A| =o |B| \<and> |B| <o r)" |
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98 |
proof |
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assume "|A| <o r" |
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100 |
then obtain p where 1: "Field p < Field r \<and> |A| =o p \<and> p <o r" |
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101 |
using internalize_ordLess[of "|A|" r] by blast |
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hence "Card_order p" using card_of_Card_order Card_order_ordIso2 by blast |
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103 |
hence "|Field p| =o p" using card_of_Field_ordIso by blast |
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hence "|A| =o |Field p| \<and> |Field p| <o r" |
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105 |
using 1 ordIso_equivalence ordIso_ordLess_trans by blast |
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106 |
thus "\<exists>B < Field r. |A| =o |B| \<and> |B| <o r" using 1 by blast |
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107 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
108 |
assume "\<exists>B < Field r. |A| =o |B| \<and> |B| <o r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
109 |
thus "|A| <o r" using ordIso_ordLess_trans by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
110 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
111 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
112 |
lemma internalize_card_of_ordLess2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
113 |
"( |A| <o |C| ) = (\<exists>B < C. |A| =o |B| \<and> |B| <o |C| )" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
114 |
using internalize_card_of_ordLess[of "A" "|C|"] Field_card_of[of C] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
115 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
116 |
lemma Card_order_omax: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
117 |
assumes "finite R" and "R \<noteq> {}" and "\<forall>r\<in>R. Card_order r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
118 |
shows "Card_order (omax R)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
119 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
120 |
have "\<forall>r\<in>R. Well_order r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
121 |
using assms unfolding card_order_on_def by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
122 |
thus ?thesis using assms apply - apply(drule omax_in) by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
123 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
124 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
125 |
lemma Card_order_omax2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
126 |
assumes "finite I" and "I \<noteq> {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
127 |
shows "Card_order (omax {|A i| | i. i \<in> I})" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
128 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
129 |
let ?R = "{|A i| | i. i \<in> I}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
130 |
have "finite ?R" and "?R \<noteq> {}" using assms by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
131 |
moreover have "\<forall>r\<in>?R. Card_order r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
132 |
using card_of_Card_order by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
133 |
ultimately show ?thesis by(rule Card_order_omax) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
134 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
135 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
136 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
137 |
subsection {* Cardinals versus set operations on arbitrary sets *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
138 |
|
54475 | 139 |
lemma infinite_Pow: |
140 |
assumes "infinite A" |
|
141 |
shows "infinite (Pow A)" |
|
142 |
proof- |
|
143 |
have "|A| \<le>o |Pow A|" by (metis card_of_Pow ordLess_imp_ordLeq) |
|
144 |
thus ?thesis by (metis assms finite_Pow_iff) |
|
145 |
qed |
|
146 |
||
147 |
corollary card_of_set_type[simp]: "|UNIV::'a set| <o |UNIV::'a set set|" |
|
148 |
using card_of_Pow[of "UNIV::'a set"] by simp |
|
149 |
||
150 |
lemma card_of_Un1[simp]: |
|
151 |
shows "|A| \<le>o |A \<union> B| " |
|
152 |
using inj_on_id[of A] card_of_ordLeq[of A _] by fastforce |
|
153 |
||
154 |
lemma card_of_diff[simp]: |
|
155 |
shows "|A - B| \<le>o |A|" |
|
156 |
using inj_on_id[of "A - B"] card_of_ordLeq[of "A - B" _] by fastforce |
|
157 |
||
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
158 |
lemma subset_ordLeq_strict: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
159 |
assumes "A \<le> B" and "|A| <o |B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
160 |
shows "A < B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
161 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
162 |
{assume "\<not>(A < B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
163 |
hence "A = B" using assms(1) by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
164 |
hence False using assms(2) not_ordLess_ordIso card_of_refl by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
165 |
} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
166 |
thus ?thesis by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
167 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
168 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
169 |
corollary subset_ordLeq_diff: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
170 |
assumes "A \<le> B" and "|A| <o |B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
171 |
shows "B - A \<noteq> {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
172 |
using assms subset_ordLeq_strict by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
173 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
174 |
lemma card_of_empty4: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
175 |
"|{}::'b set| <o |A::'a set| = (A \<noteq> {})" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
176 |
proof(intro iffI notI) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
177 |
assume *: "|{}::'b set| <o |A|" and "A = {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
178 |
hence "|A| =o |{}::'b set|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
179 |
using card_of_ordIso unfolding bij_betw_def inj_on_def by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
180 |
hence "|{}::'b set| =o |A|" using ordIso_symmetric by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
181 |
with * show False using not_ordLess_ordIso[of "|{}::'b set|" "|A|"] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
182 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
183 |
assume "A \<noteq> {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
184 |
hence "(\<not> (\<exists>f. inj_on f A \<and> f ` A \<subseteq> {}))" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
185 |
unfolding inj_on_def by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
186 |
thus "| {} | <o | A |" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
187 |
using card_of_ordLess by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
188 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
189 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
190 |
lemma card_of_empty5: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
191 |
"|A| <o |B| \<Longrightarrow> B \<noteq> {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
192 |
using card_of_empty not_ordLess_ordLeq by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
193 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
194 |
lemma Well_order_card_of_empty: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
195 |
"Well_order r \<Longrightarrow> |{}| \<le>o r" by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
196 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
197 |
lemma card_of_UNIV[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
198 |
"|A :: 'a set| \<le>o |UNIV :: 'a set|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
199 |
using card_of_mono1[of A] by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
200 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
201 |
lemma card_of_UNIV2[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
202 |
"Card_order r \<Longrightarrow> (r :: 'a rel) \<le>o |UNIV :: 'a set|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
203 |
using card_of_UNIV[of "Field r"] card_of_Field_ordIso |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
204 |
ordIso_symmetric ordIso_ordLeq_trans by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
205 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
206 |
lemma card_of_Pow_mono[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
207 |
assumes "|A| \<le>o |B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
208 |
shows "|Pow A| \<le>o |Pow B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
209 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
210 |
obtain f where "inj_on f A \<and> f ` A \<le> B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
211 |
using assms card_of_ordLeq[of A B] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
212 |
hence "inj_on (image f) (Pow A) \<and> (image f) ` (Pow A) \<le> (Pow B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
213 |
by (auto simp add: inj_on_image_Pow image_Pow_mono) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
214 |
thus ?thesis using card_of_ordLeq[of "Pow A"] by metis |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
215 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
216 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
217 |
lemma ordIso_Pow_mono[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
218 |
assumes "r \<le>o r'" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
219 |
shows "|Pow(Field r)| \<le>o |Pow(Field r')|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
220 |
using assms card_of_mono2 card_of_Pow_mono by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
221 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
222 |
lemma card_of_Pow_cong[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
223 |
assumes "|A| =o |B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
224 |
shows "|Pow A| =o |Pow B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
225 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
226 |
obtain f where "bij_betw f A B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
227 |
using assms card_of_ordIso[of A B] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
228 |
hence "bij_betw (image f) (Pow A) (Pow B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
229 |
by (auto simp add: bij_betw_image_Pow) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
230 |
thus ?thesis using card_of_ordIso[of "Pow A"] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
231 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
232 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
233 |
lemma ordIso_Pow_cong[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
234 |
assumes "r =o r'" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
235 |
shows "|Pow(Field r)| =o |Pow(Field r')|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
236 |
using assms card_of_cong card_of_Pow_cong by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
237 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
238 |
corollary Card_order_Plus_empty1: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
239 |
"Card_order r \<Longrightarrow> r =o |(Field r) <+> {}|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
240 |
using card_of_Plus_empty1 card_of_Field_ordIso ordIso_equivalence by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
241 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
242 |
corollary Card_order_Plus_empty2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
243 |
"Card_order r \<Longrightarrow> r =o |{} <+> (Field r)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
244 |
using card_of_Plus_empty2 card_of_Field_ordIso ordIso_equivalence by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
245 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
246 |
lemma Card_order_Un1: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
247 |
shows "Card_order r \<Longrightarrow> |Field r| \<le>o |(Field r) \<union> B| " |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
248 |
using card_of_Un1 card_of_Field_ordIso ordIso_symmetric ordIso_ordLeq_trans by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
249 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
250 |
lemma card_of_Un2[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
251 |
shows "|A| \<le>o |B \<union> A|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
252 |
using inj_on_id[of A] card_of_ordLeq[of A _] by fastforce |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
253 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
254 |
lemma Card_order_Un2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
255 |
shows "Card_order r \<Longrightarrow> |Field r| \<le>o |A \<union> (Field r)| " |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
256 |
using card_of_Un2 card_of_Field_ordIso ordIso_symmetric ordIso_ordLeq_trans by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
257 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
258 |
lemma Un_Plus_bij_betw: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
259 |
assumes "A Int B = {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
260 |
shows "\<exists>f. bij_betw f (A \<union> B) (A <+> B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
261 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
262 |
let ?f = "\<lambda> c. if c \<in> A then Inl c else Inr c" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
263 |
have "bij_betw ?f (A \<union> B) (A <+> B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
264 |
using assms by(unfold bij_betw_def inj_on_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
265 |
thus ?thesis by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
266 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
267 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
268 |
lemma card_of_Un_Plus_ordIso: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
269 |
assumes "A Int B = {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
270 |
shows "|A \<union> B| =o |A <+> B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
271 |
using assms card_of_ordIso[of "A \<union> B"] Un_Plus_bij_betw[of A B] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
272 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
273 |
lemma card_of_Un_Plus_ordIso1: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
274 |
"|A \<union> B| =o |A <+> (B - A)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
275 |
using card_of_Un_Plus_ordIso[of A "B - A"] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
276 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
277 |
lemma card_of_Un_Plus_ordIso2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
278 |
"|A \<union> B| =o |(A - B) <+> B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
279 |
using card_of_Un_Plus_ordIso[of "A - B" B] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
280 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
281 |
lemma card_of_Times_singl1: "|A| =o |A \<times> {b}|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
282 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
283 |
have "bij_betw fst (A \<times> {b}) A" unfolding bij_betw_def inj_on_def by force |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
284 |
thus ?thesis using card_of_ordIso ordIso_symmetric by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
285 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
286 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
287 |
corollary Card_order_Times_singl1: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
288 |
"Card_order r \<Longrightarrow> r =o |(Field r) \<times> {b}|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
289 |
using card_of_Times_singl1[of _ b] card_of_Field_ordIso ordIso_equivalence by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
290 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
291 |
lemma card_of_Times_singl2: "|A| =o |{b} \<times> A|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
292 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
293 |
have "bij_betw snd ({b} \<times> A) A" unfolding bij_betw_def inj_on_def by force |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
294 |
thus ?thesis using card_of_ordIso ordIso_symmetric by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
295 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
296 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
297 |
corollary Card_order_Times_singl2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
298 |
"Card_order r \<Longrightarrow> r =o |{a} \<times> (Field r)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
299 |
using card_of_Times_singl2[of _ a] card_of_Field_ordIso ordIso_equivalence by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
300 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
301 |
lemma card_of_Times_assoc: "|(A \<times> B) \<times> C| =o |A \<times> B \<times> C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
302 |
proof - |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
303 |
let ?f = "\<lambda>((a,b),c). (a,(b,c))" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
304 |
have "A \<times> B \<times> C \<subseteq> ?f ` ((A \<times> B) \<times> C)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
305 |
proof |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
306 |
fix x assume "x \<in> A \<times> B \<times> C" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
307 |
then obtain a b c where *: "a \<in> A" "b \<in> B" "c \<in> C" "x = (a, b, c)" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
308 |
let ?x = "((a, b), c)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
309 |
from * have "?x \<in> (A \<times> B) \<times> C" "x = ?f ?x" by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
310 |
thus "x \<in> ?f ` ((A \<times> B) \<times> C)" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
311 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
312 |
hence "bij_betw ?f ((A \<times> B) \<times> C) (A \<times> B \<times> C)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
313 |
unfolding bij_betw_def inj_on_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
314 |
thus ?thesis using card_of_ordIso by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
315 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
316 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
317 |
corollary Card_order_Times3: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
318 |
"Card_order r \<Longrightarrow> |Field r| \<le>o |(Field r) \<times> (Field r)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
319 |
using card_of_Times3 card_of_Field_ordIso |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
320 |
ordIso_ordLeq_trans ordIso_symmetric by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
321 |
|
54475 | 322 |
lemma card_of_Times_cong1[simp]: |
323 |
assumes "|A| =o |B|" |
|
324 |
shows "|A \<times> C| =o |B \<times> C|" |
|
325 |
using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono1) |
|
326 |
||
327 |
lemma card_of_Times_cong2[simp]: |
|
328 |
assumes "|A| =o |B|" |
|
329 |
shows "|C \<times> A| =o |C \<times> B|" |
|
330 |
using assms by (simp add: ordIso_iff_ordLeq card_of_Times_mono2) |
|
331 |
||
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
332 |
lemma card_of_Times_mono[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
333 |
assumes "|A| \<le>o |B|" and "|C| \<le>o |D|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
334 |
shows "|A \<times> C| \<le>o |B \<times> D|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
335 |
using assms card_of_Times_mono1[of A B C] card_of_Times_mono2[of C D B] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
336 |
ordLeq_transitive[of "|A \<times> C|"] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
337 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
338 |
corollary ordLeq_Times_mono: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
339 |
assumes "r \<le>o r'" and "p \<le>o p'" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
340 |
shows "|(Field r) \<times> (Field p)| \<le>o |(Field r') \<times> (Field p')|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
341 |
using assms card_of_mono2[of r r'] card_of_mono2[of p p'] card_of_Times_mono by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
342 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
343 |
corollary ordIso_Times_cong1[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
344 |
assumes "r =o r'" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
345 |
shows "|(Field r) \<times> C| =o |(Field r') \<times> C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
346 |
using assms card_of_cong card_of_Times_cong1 by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
347 |
|
54475 | 348 |
corollary ordIso_Times_cong2: |
349 |
assumes "r =o r'" |
|
350 |
shows "|A \<times> (Field r)| =o |A \<times> (Field r')|" |
|
351 |
using assms card_of_cong card_of_Times_cong2 by blast |
|
352 |
||
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
353 |
lemma card_of_Times_cong[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
354 |
assumes "|A| =o |B|" and "|C| =o |D|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
355 |
shows "|A \<times> C| =o |B \<times> D|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
356 |
using assms |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
357 |
by (auto simp add: ordIso_iff_ordLeq) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
358 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
359 |
corollary ordIso_Times_cong: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
360 |
assumes "r =o r'" and "p =o p'" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
361 |
shows "|(Field r) \<times> (Field p)| =o |(Field r') \<times> (Field p')|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
362 |
using assms card_of_cong[of r r'] card_of_cong[of p p'] card_of_Times_cong by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
363 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
364 |
lemma card_of_Sigma_mono2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
365 |
assumes "inj_on f (I::'i set)" and "f ` I \<le> (J::'j set)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
366 |
shows "|SIGMA i : I. (A::'j \<Rightarrow> 'a set) (f i)| \<le>o |SIGMA j : J. A j|" |
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 |
let ?LEFT = "SIGMA i : I. A (f i)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
369 |
let ?RIGHT = "SIGMA j : J. A j" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
370 |
obtain u where u_def: "u = (\<lambda>(i::'i,a::'a). (f i,a))" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
371 |
have "inj_on u ?LEFT \<and> u `?LEFT \<le> ?RIGHT" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
372 |
using assms unfolding u_def inj_on_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
373 |
thus ?thesis using card_of_ordLeq by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
374 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
375 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
376 |
lemma card_of_Sigma_mono: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
377 |
assumes INJ: "inj_on f I" and IM: "f ` I \<le> J" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
378 |
LEQ: "\<forall>j \<in> J. |A j| \<le>o |B j|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
379 |
shows "|SIGMA i : I. A (f i)| \<le>o |SIGMA j : J. B j|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
380 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
381 |
have "\<forall>i \<in> I. |A(f i)| \<le>o |B(f i)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
382 |
using IM LEQ by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
383 |
hence "|SIGMA i : I. A (f i)| \<le>o |SIGMA i : I. B (f i)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
384 |
using card_of_Sigma_mono1[of I] by metis |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
385 |
moreover have "|SIGMA i : I. B (f i)| \<le>o |SIGMA j : J. B j|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
386 |
using INJ IM card_of_Sigma_mono2 by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
387 |
ultimately show ?thesis using ordLeq_transitive by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
388 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
389 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
390 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
391 |
lemma ordLeq_Sigma_mono1: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
392 |
assumes "\<forall>i \<in> I. p i \<le>o r i" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
393 |
shows "|SIGMA i : I. Field(p i)| \<le>o |SIGMA i : I. Field(r i)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
394 |
using assms by (auto simp add: card_of_Sigma_mono1) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
395 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
396 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
397 |
lemma ordLeq_Sigma_mono: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
398 |
assumes "inj_on f I" and "f ` I \<le> J" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
399 |
"\<forall>j \<in> J. p j \<le>o r j" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
400 |
shows "|SIGMA i : I. Field(p(f i))| \<le>o |SIGMA j : J. Field(r j)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
401 |
using assms card_of_mono2 card_of_Sigma_mono |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
402 |
[of f I J "\<lambda> i. Field(p i)" "\<lambda> j. Field(r j)"] by metis |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
403 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
404 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
405 |
lemma card_of_Sigma_cong1: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
406 |
assumes "\<forall>i \<in> I. |A i| =o |B i|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
407 |
shows "|SIGMA i : I. A i| =o |SIGMA i : I. B i|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
408 |
using assms by (auto simp add: card_of_Sigma_mono1 ordIso_iff_ordLeq) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
409 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
410 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
411 |
lemma card_of_Sigma_cong2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
412 |
assumes "bij_betw f (I::'i set) (J::'j set)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
413 |
shows "|SIGMA i : I. (A::'j \<Rightarrow> 'a set) (f i)| =o |SIGMA j : J. A j|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
414 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
415 |
let ?LEFT = "SIGMA i : I. A (f i)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
416 |
let ?RIGHT = "SIGMA j : J. A j" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
417 |
obtain u where u_def: "u = (\<lambda>(i::'i,a::'a). (f i,a))" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
418 |
have "bij_betw u ?LEFT ?RIGHT" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
419 |
using assms unfolding u_def bij_betw_def inj_on_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
420 |
thus ?thesis using card_of_ordIso by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
421 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
422 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
423 |
lemma card_of_Sigma_cong: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
424 |
assumes BIJ: "bij_betw f I J" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
425 |
ISO: "\<forall>j \<in> J. |A j| =o |B j|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
426 |
shows "|SIGMA i : I. A (f i)| =o |SIGMA j : J. B j|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
427 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
428 |
have "\<forall>i \<in> I. |A(f i)| =o |B(f i)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
429 |
using ISO BIJ unfolding bij_betw_def by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
430 |
hence "|SIGMA i : I. A (f i)| =o |SIGMA i : I. B (f i)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
431 |
using card_of_Sigma_cong1 by metis |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
432 |
moreover have "|SIGMA i : I. B (f i)| =o |SIGMA j : J. B j|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
433 |
using BIJ card_of_Sigma_cong2 by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
434 |
ultimately show ?thesis using ordIso_transitive by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
435 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
436 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
437 |
lemma ordIso_Sigma_cong1: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
438 |
assumes "\<forall>i \<in> I. p i =o r i" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
439 |
shows "|SIGMA i : I. Field(p i)| =o |SIGMA i : I. Field(r i)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
440 |
using assms by (auto simp add: card_of_Sigma_cong1) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
441 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
442 |
lemma ordLeq_Sigma_cong: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
443 |
assumes "bij_betw f I J" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
444 |
"\<forall>j \<in> J. p j =o r j" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
445 |
shows "|SIGMA i : I. Field(p(f i))| =o |SIGMA j : J. Field(r j)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
446 |
using assms card_of_cong card_of_Sigma_cong |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
447 |
[of f I J "\<lambda> j. Field(p j)" "\<lambda> j. Field(r j)"] by blast |
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 |
corollary ordLeq_Sigma_Times: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
450 |
"\<forall>i \<in> I. p i \<le>o r \<Longrightarrow> |SIGMA i : I. Field (p i)| \<le>o |I \<times> (Field r)|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
451 |
by (auto simp add: card_of_Sigma_Times) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
452 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
453 |
lemma card_of_UNION_Sigma2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
454 |
assumes |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
455 |
"!! i j. \<lbrakk>{i,j} <= I; i ~= j\<rbrakk> \<Longrightarrow> A i Int A j = {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
456 |
shows |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
457 |
"|\<Union>i\<in>I. A i| =o |Sigma I A|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
458 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
459 |
let ?L = "\<Union>i\<in>I. A i" let ?R = "Sigma I A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
460 |
have "|?L| <=o |?R|" using card_of_UNION_Sigma . |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
461 |
moreover have "|?R| <=o |?L|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
462 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
463 |
have "inj_on snd ?R" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
464 |
unfolding inj_on_def using assms by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
465 |
moreover have "snd ` ?R <= ?L" by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
466 |
ultimately show ?thesis using card_of_ordLeq by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
467 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
468 |
ultimately show ?thesis by(simp add: ordIso_iff_ordLeq) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
469 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
470 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
471 |
corollary Plus_into_Times: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
472 |
assumes A2: "a1 \<noteq> a2 \<and> {a1,a2} \<le> A" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
473 |
B2: "b1 \<noteq> b2 \<and> {b1,b2} \<le> B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
474 |
shows "\<exists>f. inj_on f (A <+> B) \<and> f ` (A <+> B) \<le> A \<times> B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
475 |
using assms by (auto simp add: card_of_Plus_Times card_of_ordLeq) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
476 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
477 |
corollary Plus_into_Times_types: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
478 |
assumes A2: "(a1::'a) \<noteq> a2" and B2: "(b1::'b) \<noteq> b2" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
479 |
shows "\<exists>(f::'a + 'b \<Rightarrow> 'a * 'b). inj f" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
480 |
using assms Plus_into_Times[of a1 a2 UNIV b1 b2 UNIV] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
481 |
by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
482 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
483 |
corollary Times_same_infinite_bij_betw: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
484 |
assumes "infinite A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
485 |
shows "\<exists>f. bij_betw f (A \<times> A) A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
486 |
using assms by (auto simp add: card_of_ordIso) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
487 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
488 |
corollary Times_same_infinite_bij_betw_types: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
489 |
assumes INF: "infinite(UNIV::'a set)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
490 |
shows "\<exists>(f::('a * 'a) => 'a). bij f" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
491 |
using assms Times_same_infinite_bij_betw[of "UNIV::'a set"] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
492 |
by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
493 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
494 |
corollary Times_infinite_bij_betw: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
495 |
assumes INF: "infinite A" and NE: "B \<noteq> {}" and INJ: "inj_on g B \<and> g ` B \<le> A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
496 |
shows "(\<exists>f. bij_betw f (A \<times> B) A) \<and> (\<exists>h. bij_betw h (B \<times> A) A)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
497 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
498 |
have "|B| \<le>o |A|" using INJ card_of_ordLeq by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
499 |
thus ?thesis using INF NE |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
500 |
by (auto simp add: card_of_ordIso card_of_Times_infinite) |
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 |
corollary Times_infinite_bij_betw_types: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
504 |
assumes INF: "infinite(UNIV::'a set)" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
505 |
BIJ: "inj(g::'b \<Rightarrow> 'a)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
506 |
shows "(\<exists>(f::('b * 'a) => 'a). bij f) \<and> (\<exists>(h::('a * 'b) => 'a). bij h)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
507 |
using assms Times_infinite_bij_betw[of "UNIV::'a set" "UNIV::'b set" g] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
508 |
by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
509 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
510 |
lemma card_of_Times_ordLeq_infinite: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
511 |
"\<lbrakk>infinite C; |A| \<le>o |C|; |B| \<le>o |C|\<rbrakk> |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
512 |
\<Longrightarrow> |A <*> B| \<le>o |C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
513 |
by(simp add: card_of_Sigma_ordLeq_infinite) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
514 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
515 |
corollary Plus_infinite_bij_betw: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
516 |
assumes INF: "infinite A" and INJ: "inj_on g B \<and> g ` B \<le> A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
517 |
shows "(\<exists>f. bij_betw f (A <+> B) A) \<and> (\<exists>h. bij_betw h (B <+> A) A)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
518 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
519 |
have "|B| \<le>o |A|" using INJ card_of_ordLeq by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
520 |
thus ?thesis using INF |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
521 |
by (auto simp add: card_of_ordIso) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
522 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
523 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
524 |
corollary Plus_infinite_bij_betw_types: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
525 |
assumes INF: "infinite(UNIV::'a set)" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
526 |
BIJ: "inj(g::'b \<Rightarrow> 'a)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
527 |
shows "(\<exists>(f::('b + 'a) => 'a). bij f) \<and> (\<exists>(h::('a + 'b) => 'a). bij h)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
528 |
using assms Plus_infinite_bij_betw[of "UNIV::'a set" g "UNIV::'b set"] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
529 |
by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
530 |
|
54475 | 531 |
lemma card_of_Un_infinite: |
532 |
assumes INF: "infinite A" and LEQ: "|B| \<le>o |A|" |
|
533 |
shows "|A \<union> B| =o |A| \<and> |B \<union> A| =o |A|" |
|
534 |
proof- |
|
535 |
have "|A \<union> B| \<le>o |A <+> B|" by (rule card_of_Un_Plus_ordLeq) |
|
536 |
moreover have "|A <+> B| =o |A|" |
|
537 |
using assms by (metis card_of_Plus_infinite) |
|
538 |
ultimately have "|A \<union> B| \<le>o |A|" using ordLeq_ordIso_trans by blast |
|
539 |
hence "|A \<union> B| =o |A|" using card_of_Un1 ordIso_iff_ordLeq by blast |
|
540 |
thus ?thesis using Un_commute[of B A] by auto |
|
541 |
qed |
|
542 |
||
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
543 |
lemma card_of_Un_infinite_simps[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
544 |
"\<lbrakk>infinite A; |B| \<le>o |A| \<rbrakk> \<Longrightarrow> |A \<union> B| =o |A|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
545 |
"\<lbrakk>infinite A; |B| \<le>o |A| \<rbrakk> \<Longrightarrow> |B \<union> A| =o |A|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
546 |
using card_of_Un_infinite by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
547 |
|
54475 | 548 |
lemma card_of_Un_diff_infinite: |
549 |
assumes INF: "infinite A" and LESS: "|B| <o |A|" |
|
550 |
shows "|A - B| =o |A|" |
|
551 |
proof- |
|
552 |
obtain C where C_def: "C = A - B" by blast |
|
553 |
have "|A \<union> B| =o |A|" |
|
554 |
using assms ordLeq_iff_ordLess_or_ordIso card_of_Un_infinite by blast |
|
555 |
moreover have "C \<union> B = A \<union> B" unfolding C_def by auto |
|
556 |
ultimately have 1: "|C \<union> B| =o |A|" by auto |
|
557 |
(* *) |
|
558 |
{assume *: "|C| \<le>o |B|" |
|
559 |
moreover |
|
560 |
{assume **: "finite B" |
|
561 |
hence "finite C" |
|
562 |
using card_of_ordLeq_finite * by blast |
|
563 |
hence False using ** INF card_of_ordIso_finite 1 by blast |
|
564 |
} |
|
565 |
hence "infinite B" by auto |
|
566 |
ultimately have False |
|
567 |
using card_of_Un_infinite 1 ordIso_equivalence(1,3) LESS not_ordLess_ordIso by metis |
|
568 |
} |
|
569 |
hence 2: "|B| \<le>o |C|" using card_of_Well_order ordLeq_total by blast |
|
570 |
{assume *: "finite C" |
|
571 |
hence "finite B" using card_of_ordLeq_finite 2 by blast |
|
572 |
hence False using * INF card_of_ordIso_finite 1 by blast |
|
573 |
} |
|
574 |
hence "infinite C" by auto |
|
575 |
hence "|C| =o |A|" |
|
576 |
using card_of_Un_infinite 1 2 ordIso_equivalence(1,3) by metis |
|
577 |
thus ?thesis unfolding C_def . |
|
578 |
qed |
|
579 |
||
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
580 |
corollary Card_order_Un_infinite: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
581 |
assumes INF: "infinite(Field r)" and CARD: "Card_order r" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
582 |
LEQ: "p \<le>o r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
583 |
shows "| (Field r) \<union> (Field p) | =o r \<and> | (Field p) \<union> (Field r) | =o r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
584 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
585 |
have "| Field r \<union> Field p | =o | Field r | \<and> |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
586 |
| Field p \<union> Field r | =o | Field r |" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
587 |
using assms by (auto simp add: card_of_Un_infinite) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
588 |
thus ?thesis |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
589 |
using assms card_of_Field_ordIso[of r] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
590 |
ordIso_transitive[of "|Field r \<union> Field p|"] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
591 |
ordIso_transitive[of _ "|Field r|"] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
592 |
qed |
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 |
corollary subset_ordLeq_diff_infinite: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
595 |
assumes INF: "infinite B" and SUB: "A \<le> B" and LESS: "|A| <o |B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
596 |
shows "infinite (B - A)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
597 |
using assms card_of_Un_diff_infinite card_of_ordIso_finite by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
598 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
599 |
lemma card_of_Times_ordLess_infinite[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
600 |
assumes INF: "infinite C" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
601 |
LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
602 |
shows "|A \<times> B| <o |C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
603 |
proof(cases "A = {} \<or> B = {}") |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
604 |
assume Case1: "A = {} \<or> B = {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
605 |
hence "A \<times> B = {}" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
606 |
moreover have "C \<noteq> {}" using |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
607 |
LESS1 card_of_empty5 by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
608 |
ultimately show ?thesis by(auto simp add: card_of_empty4) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
609 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
610 |
assume Case2: "\<not>(A = {} \<or> B = {})" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
611 |
{assume *: "|C| \<le>o |A \<times> B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
612 |
hence "infinite (A \<times> B)" using INF card_of_ordLeq_finite by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
613 |
hence 1: "infinite A \<or> infinite B" using finite_cartesian_product by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
614 |
{assume Case21: "|A| \<le>o |B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
615 |
hence "infinite B" using 1 card_of_ordLeq_finite by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
616 |
hence "|A \<times> B| =o |B|" using Case2 Case21 |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
617 |
by (auto simp add: card_of_Times_infinite) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
618 |
hence False using LESS2 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast |
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 |
moreover |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
621 |
{assume Case22: "|B| \<le>o |A|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
622 |
hence "infinite A" using 1 card_of_ordLeq_finite by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
623 |
hence "|A \<times> B| =o |A|" using Case2 Case22 |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
624 |
by (auto simp add: card_of_Times_infinite) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
625 |
hence False using LESS1 not_ordLess_ordLeq * ordLeq_ordIso_trans by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
626 |
} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
627 |
ultimately have False using ordLeq_total card_of_Well_order[of A] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
628 |
card_of_Well_order[of B] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
629 |
} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
630 |
thus ?thesis using ordLess_or_ordLeq[of "|A \<times> B|" "|C|"] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
631 |
card_of_Well_order[of "A \<times> B"] card_of_Well_order[of "C"] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
632 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
633 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
634 |
lemma card_of_Times_ordLess_infinite_Field[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
635 |
assumes INF: "infinite (Field r)" and r: "Card_order r" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
636 |
LESS1: "|A| <o r" and LESS2: "|B| <o r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
637 |
shows "|A \<times> B| <o r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
638 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
639 |
let ?C = "Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
640 |
have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
641 |
ordIso_symmetric by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
642 |
hence "|A| <o |?C|" "|B| <o |?C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
643 |
using LESS1 LESS2 ordLess_ordIso_trans by blast+ |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
644 |
hence "|A <*> B| <o |?C|" using INF |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
645 |
card_of_Times_ordLess_infinite by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
646 |
thus ?thesis using 1 ordLess_ordIso_trans by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
647 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
648 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
649 |
lemma card_of_Un_ordLess_infinite[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
650 |
assumes INF: "infinite C" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
651 |
LESS1: "|A| <o |C|" and LESS2: "|B| <o |C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
652 |
shows "|A \<union> B| <o |C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
653 |
using assms card_of_Plus_ordLess_infinite card_of_Un_Plus_ordLeq |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
654 |
ordLeq_ordLess_trans by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
655 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
656 |
lemma card_of_Un_ordLess_infinite_Field[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
657 |
assumes INF: "infinite (Field r)" and r: "Card_order r" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
658 |
LESS1: "|A| <o r" and LESS2: "|B| <o r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
659 |
shows "|A Un B| <o r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
660 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
661 |
let ?C = "Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
662 |
have 1: "r =o |?C| \<and> |?C| =o r" using r card_of_Field_ordIso |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
663 |
ordIso_symmetric by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
664 |
hence "|A| <o |?C|" "|B| <o |?C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
665 |
using LESS1 LESS2 ordLess_ordIso_trans by blast+ |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
666 |
hence "|A Un B| <o |?C|" using INF |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
667 |
card_of_Un_ordLess_infinite by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
668 |
thus ?thesis using 1 ordLess_ordIso_trans by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
669 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
670 |
|
54475 | 671 |
|
672 |
subsection {* Cardinals versus set operations involving infinite sets *} |
|
673 |
||
674 |
lemma finite_iff_cardOf_nat: |
|
675 |
"finite A = ( |A| <o |UNIV :: nat set| )" |
|
676 |
using infinite_iff_card_of_nat[of A] |
|
677 |
not_ordLeq_iff_ordLess[of "|A|" "|UNIV :: nat set|"] |
|
678 |
by (fastforce simp: card_of_Well_order) |
|
679 |
||
680 |
lemma finite_ordLess_infinite2[simp]: |
|
681 |
assumes "finite A" and "infinite B" |
|
682 |
shows "|A| <o |B|" |
|
683 |
using assms |
|
684 |
finite_ordLess_infinite[of "|A|" "|B|"] |
|
685 |
card_of_Well_order[of A] card_of_Well_order[of B] |
|
686 |
Field_card_of[of A] Field_card_of[of B] by auto |
|
687 |
||
688 |
lemma infinite_card_of_insert: |
|
689 |
assumes "infinite A" |
|
690 |
shows "|insert a A| =o |A|" |
|
691 |
proof- |
|
692 |
have iA: "insert a A = A \<union> {a}" by simp |
|
693 |
show ?thesis |
|
694 |
using infinite_imp_bij_betw2[OF assms] unfolding iA |
|
695 |
by (metis bij_betw_inv card_of_ordIso) |
|
696 |
qed |
|
697 |
||
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
698 |
lemma card_of_Un_singl_ordLess_infinite1: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
699 |
assumes "infinite B" and "|A| <o |B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
700 |
shows "|{a} Un A| <o |B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
701 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
702 |
have "|{a}| <o |B|" using assms by auto |
51764 | 703 |
thus ?thesis using assms card_of_Un_ordLess_infinite[of B] by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
704 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
705 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
706 |
lemma card_of_Un_singl_ordLess_infinite: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
707 |
assumes "infinite B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
708 |
shows "( |A| <o |B| ) = ( |{a} Un A| <o |B| )" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
709 |
using assms card_of_Un_singl_ordLess_infinite1[of B A] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
710 |
proof(auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
711 |
assume "|insert a A| <o |B|" |
51764 | 712 |
moreover have "|A| <=o |insert a A|" using card_of_mono1[of A "insert a A"] by blast |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
713 |
ultimately show "|A| <o |B|" using ordLeq_ordLess_trans by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
714 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
715 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
716 |
|
54475 | 717 |
subsection {* Cardinals versus lists *} |
718 |
||
719 |
text{* The next is an auxiliary operator, which shall be used for inductive |
|
720 |
proofs of facts concerning the cardinality of @{text "List"} : *} |
|
721 |
||
722 |
definition nlists :: "'a set \<Rightarrow> nat \<Rightarrow> 'a list set" |
|
723 |
where "nlists A n \<equiv> {l. set l \<le> A \<and> length l = n}" |
|
724 |
||
725 |
lemma lists_def2: "lists A = {l. set l \<le> A}" |
|
726 |
using in_listsI by blast |
|
727 |
||
728 |
lemma lists_UNION_nlists: "lists A = (\<Union> n. nlists A n)" |
|
729 |
unfolding lists_def2 nlists_def by blast |
|
730 |
||
731 |
lemma card_of_lists: "|A| \<le>o |lists A|" |
|
732 |
proof- |
|
733 |
let ?h = "\<lambda> a. [a]" |
|
734 |
have "inj_on ?h A \<and> ?h ` A \<le> lists A" |
|
735 |
unfolding inj_on_def lists_def2 by auto |
|
736 |
thus ?thesis by (metis card_of_ordLeq) |
|
737 |
qed |
|
738 |
||
739 |
lemma nlists_0: "nlists A 0 = {[]}" |
|
740 |
unfolding nlists_def by auto |
|
741 |
||
742 |
lemma nlists_not_empty: |
|
743 |
assumes "A \<noteq> {}" |
|
744 |
shows "nlists A n \<noteq> {}" |
|
745 |
proof(induct n, simp add: nlists_0) |
|
746 |
fix n assume "nlists A n \<noteq> {}" |
|
747 |
then obtain a and l where "a \<in> A \<and> l \<in> nlists A n" using assms by auto |
|
748 |
hence "a # l \<in> nlists A (Suc n)" unfolding nlists_def by auto |
|
749 |
thus "nlists A (Suc n) \<noteq> {}" by auto |
|
750 |
qed |
|
751 |
||
752 |
lemma Nil_in_lists: "[] \<in> lists A" |
|
753 |
unfolding lists_def2 by auto |
|
754 |
||
755 |
lemma lists_not_empty: "lists A \<noteq> {}" |
|
756 |
using Nil_in_lists by blast |
|
757 |
||
758 |
lemma card_of_nlists_Succ: "|nlists A (Suc n)| =o |A \<times> (nlists A n)|" |
|
759 |
proof- |
|
760 |
let ?B = "A \<times> (nlists A n)" let ?h = "\<lambda>(a,l). a # l" |
|
761 |
have "inj_on ?h ?B \<and> ?h ` ?B \<le> nlists A (Suc n)" |
|
762 |
unfolding inj_on_def nlists_def by auto |
|
763 |
moreover have "nlists A (Suc n) \<le> ?h ` ?B" |
|
764 |
proof(auto) |
|
765 |
fix l assume "l \<in> nlists A (Suc n)" |
|
766 |
hence 1: "length l = Suc n \<and> set l \<le> A" unfolding nlists_def by auto |
|
767 |
then obtain a and l' where 2: "l = a # l'" by (auto simp: length_Suc_conv) |
|
768 |
hence "a \<in> A \<and> set l' \<le> A \<and> length l' = n" using 1 by auto |
|
769 |
thus "l \<in> ?h ` ?B" using 2 unfolding nlists_def by auto |
|
770 |
qed |
|
771 |
ultimately have "bij_betw ?h ?B (nlists A (Suc n))" |
|
772 |
unfolding bij_betw_def by auto |
|
773 |
thus ?thesis using card_of_ordIso ordIso_symmetric by blast |
|
774 |
qed |
|
775 |
||
776 |
lemma card_of_nlists_infinite: |
|
777 |
assumes "infinite A" |
|
778 |
shows "|nlists A n| \<le>o |A|" |
|
779 |
proof(induct n) |
|
780 |
have "A \<noteq> {}" using assms by auto |
|
781 |
thus "|nlists A 0| \<le>o |A|" by (simp add: nlists_0 card_of_singl_ordLeq) |
|
782 |
next |
|
783 |
fix n assume IH: "|nlists A n| \<le>o |A|" |
|
784 |
have "|nlists A (Suc n)| =o |A \<times> (nlists A n)|" |
|
785 |
using card_of_nlists_Succ by blast |
|
786 |
moreover |
|
787 |
{have "nlists A n \<noteq> {}" using assms nlists_not_empty[of A] by blast |
|
788 |
hence "|A \<times> (nlists A n)| =o |A|" |
|
789 |
using assms IH by (auto simp add: card_of_Times_infinite) |
|
790 |
} |
|
791 |
ultimately show "|nlists A (Suc n)| \<le>o |A|" |
|
792 |
using ordIso_transitive ordIso_iff_ordLeq by blast |
|
793 |
qed |
|
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
794 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
795 |
lemma Card_order_lists: "Card_order r \<Longrightarrow> r \<le>o |lists(Field r) |" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
796 |
using card_of_lists card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
797 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
798 |
lemma Union_set_lists: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
799 |
"Union(set ` (lists A)) = A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
800 |
unfolding lists_def2 proof(auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
801 |
fix a assume "a \<in> A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
802 |
hence "set [a] \<le> A \<and> a \<in> set [a]" by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
803 |
thus "\<exists>l. set l \<le> A \<and> a \<in> set l" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
804 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
805 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
806 |
lemma inj_on_map_lists: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
807 |
assumes "inj_on f A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
808 |
shows "inj_on (map f) (lists A)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
809 |
using assms Union_set_lists[of A] inj_on_mapI[of f "lists A"] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
810 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
811 |
lemma map_lists_mono: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
812 |
assumes "f ` A \<le> B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
813 |
shows "(map f) ` (lists A) \<le> lists B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
814 |
using assms unfolding lists_def2 by (auto, blast) (* lethal combination of methods :) *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
815 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
816 |
lemma map_lists_surjective: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
817 |
assumes "f ` A = B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
818 |
shows "(map f) ` (lists A) = lists B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
819 |
using assms unfolding lists_def2 |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
820 |
proof (auto, blast) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
821 |
fix l' assume *: "set l' \<le> f ` A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
822 |
have "set l' \<le> f ` A \<longrightarrow> l' \<in> map f ` {l. set l \<le> A}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
823 |
proof(induct l', auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
824 |
fix l a |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
825 |
assume "a \<in> A" and "set l \<le> A" and |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
826 |
IH: "f ` (set l) \<le> f ` A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
827 |
hence "set (a # l) \<le> A" by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
828 |
hence "map f (a # l) \<in> map f ` {l. set l \<le> A}" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
829 |
thus "f a # map f l \<in> map f ` {l. set l \<le> A}" by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
830 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
831 |
thus "l' \<in> map f ` {l. set l \<le> A}" using * by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
832 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
833 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
834 |
lemma bij_betw_map_lists: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
835 |
assumes "bij_betw f A B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
836 |
shows "bij_betw (map f) (lists A) (lists B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
837 |
using assms unfolding bij_betw_def |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
838 |
by(auto simp add: inj_on_map_lists map_lists_surjective) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
839 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
840 |
lemma card_of_lists_mono[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
841 |
assumes "|A| \<le>o |B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
842 |
shows "|lists A| \<le>o |lists B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
843 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
844 |
obtain f where "inj_on f A \<and> f ` A \<le> B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
845 |
using assms card_of_ordLeq[of A B] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
846 |
hence "inj_on (map f) (lists A) \<and> (map f) ` (lists A) \<le> (lists B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
847 |
by (auto simp add: inj_on_map_lists map_lists_mono) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
848 |
thus ?thesis using card_of_ordLeq[of "lists A"] by metis |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
849 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
850 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
851 |
lemma ordIso_lists_mono: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
852 |
assumes "r \<le>o r'" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
853 |
shows "|lists(Field r)| \<le>o |lists(Field r')|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
854 |
using assms card_of_mono2 card_of_lists_mono by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
855 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
856 |
lemma card_of_lists_cong[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
857 |
assumes "|A| =o |B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
858 |
shows "|lists A| =o |lists B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
859 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
860 |
obtain f where "bij_betw f A B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
861 |
using assms card_of_ordIso[of A B] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
862 |
hence "bij_betw (map f) (lists A) (lists B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
863 |
by (auto simp add: bij_betw_map_lists) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
864 |
thus ?thesis using card_of_ordIso[of "lists A"] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
865 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
866 |
|
54475 | 867 |
lemma card_of_lists_infinite[simp]: |
868 |
assumes "infinite A" |
|
869 |
shows "|lists A| =o |A|" |
|
870 |
proof- |
|
871 |
have "|lists A| \<le>o |A|" |
|
872 |
using assms |
|
873 |
by (auto simp add: lists_UNION_nlists card_of_UNION_ordLeq_infinite |
|
874 |
infinite_iff_card_of_nat card_of_nlists_infinite) |
|
875 |
thus ?thesis using card_of_lists ordIso_iff_ordLeq by blast |
|
876 |
qed |
|
877 |
||
878 |
lemma Card_order_lists_infinite: |
|
879 |
assumes "Card_order r" and "infinite(Field r)" |
|
880 |
shows "|lists(Field r)| =o r" |
|
881 |
using assms card_of_lists_infinite card_of_Field_ordIso ordIso_transitive by blast |
|
882 |
||
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
883 |
lemma ordIso_lists_cong: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
884 |
assumes "r =o r'" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
885 |
shows "|lists(Field r)| =o |lists(Field r')|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
886 |
using assms card_of_cong card_of_lists_cong by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
887 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
888 |
corollary lists_infinite_bij_betw: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
889 |
assumes "infinite A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
890 |
shows "\<exists>f. bij_betw f (lists A) A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
891 |
using assms card_of_lists_infinite card_of_ordIso by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
892 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
893 |
corollary lists_infinite_bij_betw_types: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
894 |
assumes "infinite(UNIV :: 'a set)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
895 |
shows "\<exists>(f::'a list \<Rightarrow> 'a). bij f" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
896 |
using assms assms lists_infinite_bij_betw[of "UNIV::'a set"] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
897 |
using lists_UNIV by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
898 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
899 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
900 |
subsection {* Cardinals versus the set-of-finite-sets operator *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
901 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
902 |
definition Fpow :: "'a set \<Rightarrow> 'a set set" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
903 |
where "Fpow A \<equiv> {X. X \<le> A \<and> finite X}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
904 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
905 |
lemma Fpow_mono: "A \<le> B \<Longrightarrow> Fpow A \<le> Fpow B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
906 |
unfolding Fpow_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
907 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
908 |
lemma empty_in_Fpow: "{} \<in> Fpow A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
909 |
unfolding Fpow_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
910 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
911 |
lemma Fpow_not_empty: "Fpow A \<noteq> {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
912 |
using empty_in_Fpow by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
913 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
914 |
lemma Fpow_subset_Pow: "Fpow A \<le> Pow A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
915 |
unfolding Fpow_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
916 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
917 |
lemma card_of_Fpow[simp]: "|A| \<le>o |Fpow A|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
918 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
919 |
let ?h = "\<lambda> a. {a}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
920 |
have "inj_on ?h A \<and> ?h ` A \<le> Fpow A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
921 |
unfolding inj_on_def Fpow_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
922 |
thus ?thesis using card_of_ordLeq by metis |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
923 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
924 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
925 |
lemma Card_order_Fpow: "Card_order r \<Longrightarrow> r \<le>o |Fpow(Field r) |" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
926 |
using card_of_Fpow card_of_Field_ordIso ordIso_ordLeq_trans ordIso_symmetric by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
927 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
928 |
lemma Fpow_Pow_finite: "Fpow A = Pow A Int {A. finite A}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
929 |
unfolding Fpow_def Pow_def by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
930 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
931 |
lemma inj_on_image_Fpow: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
932 |
assumes "inj_on f A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
933 |
shows "inj_on (image f) (Fpow A)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
934 |
using assms Fpow_subset_Pow[of A] subset_inj_on[of "image f" "Pow A"] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
935 |
inj_on_image_Pow by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
936 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
937 |
lemma image_Fpow_mono: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
938 |
assumes "f ` A \<le> B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
939 |
shows "(image f) ` (Fpow A) \<le> Fpow B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
940 |
using assms by(unfold Fpow_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
941 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
942 |
lemma image_Fpow_surjective: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
943 |
assumes "f ` A = B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
944 |
shows "(image f) ` (Fpow A) = Fpow B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
945 |
using assms proof(unfold Fpow_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
946 |
fix Y assume *: "Y \<le> f ` A" and **: "finite Y" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
947 |
hence "\<forall>b \<in> Y. \<exists>a. a \<in> A \<and> f a = b" by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
948 |
with bchoice[of Y "\<lambda>b a. a \<in> A \<and> f a = b"] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
949 |
obtain g where 1: "\<forall>b \<in> Y. g b \<in> A \<and> f(g b) = b" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
950 |
obtain X where X_def: "X = g ` Y" by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
951 |
have "f ` X = Y \<and> X \<le> A \<and> finite X" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
952 |
by(unfold X_def, force simp add: ** 1) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
953 |
thus "Y \<in> (image f) ` {X. X \<le> A \<and> finite X}" by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
954 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
955 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
956 |
lemma bij_betw_image_Fpow: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
957 |
assumes "bij_betw f A B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
958 |
shows "bij_betw (image f) (Fpow A) (Fpow B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
959 |
using assms unfolding bij_betw_def |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
960 |
by (auto simp add: inj_on_image_Fpow image_Fpow_surjective) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
961 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
962 |
lemma card_of_Fpow_mono[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
963 |
assumes "|A| \<le>o |B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
964 |
shows "|Fpow A| \<le>o |Fpow B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
965 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
966 |
obtain f where "inj_on f A \<and> f ` A \<le> B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
967 |
using assms card_of_ordLeq[of A B] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
968 |
hence "inj_on (image f) (Fpow A) \<and> (image f) ` (Fpow A) \<le> (Fpow B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
969 |
by (auto simp add: inj_on_image_Fpow image_Fpow_mono) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
970 |
thus ?thesis using card_of_ordLeq[of "Fpow A"] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
971 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
972 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
973 |
lemma ordIso_Fpow_mono: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
974 |
assumes "r \<le>o r'" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
975 |
shows "|Fpow(Field r)| \<le>o |Fpow(Field r')|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
976 |
using assms card_of_mono2 card_of_Fpow_mono by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
977 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
978 |
lemma card_of_Fpow_cong[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
979 |
assumes "|A| =o |B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
980 |
shows "|Fpow A| =o |Fpow B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
981 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
982 |
obtain f where "bij_betw f A B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
983 |
using assms card_of_ordIso[of A B] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
984 |
hence "bij_betw (image f) (Fpow A) (Fpow B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
985 |
by (auto simp add: bij_betw_image_Fpow) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
986 |
thus ?thesis using card_of_ordIso[of "Fpow A"] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
987 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
988 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
989 |
lemma ordIso_Fpow_cong: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
990 |
assumes "r =o r'" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
991 |
shows "|Fpow(Field r)| =o |Fpow(Field r')|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
992 |
using assms card_of_cong card_of_Fpow_cong by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
993 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
994 |
lemma card_of_Fpow_lists: "|Fpow A| \<le>o |lists A|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
995 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
996 |
have "set ` (lists A) = Fpow A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
997 |
unfolding lists_def2 Fpow_def using finite_list finite_set by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
998 |
thus ?thesis using card_of_ordLeq2[of "Fpow A"] Fpow_not_empty[of A] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
999 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1000 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1001 |
lemma card_of_Fpow_infinite[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1002 |
assumes "infinite A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1003 |
shows "|Fpow A| =o |A|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1004 |
using assms card_of_Fpow_lists card_of_lists_infinite card_of_Fpow |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1005 |
ordLeq_ordIso_trans ordIso_iff_ordLeq by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1006 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1007 |
corollary Fpow_infinite_bij_betw: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1008 |
assumes "infinite A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1009 |
shows "\<exists>f. bij_betw f (Fpow A) A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1010 |
using assms card_of_Fpow_infinite card_of_ordIso by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1011 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1012 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1013 |
subsection {* The cardinal $\omega$ and the finite cardinals *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1014 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1015 |
subsubsection {* First as well-orders *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1016 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1017 |
lemma Field_natLess: "Field natLess = (UNIV::nat set)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1018 |
by(unfold Field_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1019 |
|
54475 | 1020 |
lemma natLeq_well_order_on: "well_order_on UNIV natLeq" |
1021 |
using natLeq_Well_order Field_natLeq by auto |
|
1022 |
||
1023 |
lemma natLeq_wo_rel: "wo_rel natLeq" |
|
1024 |
unfolding wo_rel_def using natLeq_Well_order . |
|
1025 |
||
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1026 |
lemma natLeq_ofilter_less: "ofilter natLeq {0 ..< n}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1027 |
by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def, |
54475 | 1028 |
simp add: Field_natLeq, unfold rel.under_def, auto) |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1029 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1030 |
lemma natLeq_ofilter_leq: "ofilter natLeq {0 .. n}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1031 |
by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def, |
54475 | 1032 |
simp add: Field_natLeq, unfold rel.under_def, auto) |
1033 |
||
1034 |
lemma natLeq_UNIV_ofilter: "wo_rel.ofilter natLeq UNIV" |
|
1035 |
using natLeq_wo_rel Field_natLeq wo_rel.Field_ofilter[of natLeq] by auto |
|
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1036 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1037 |
lemma natLeq_ofilter_iff: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1038 |
"ofilter natLeq A = (A = UNIV \<or> (\<exists>n. A = {0 ..< n}))" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1039 |
proof(rule iffI) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1040 |
assume "ofilter natLeq A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1041 |
hence "\<forall>m n. n \<in> A \<and> m \<le> n \<longrightarrow> m \<in> A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1042 |
by(auto simp add: natLeq_wo_rel wo_rel.ofilter_def rel.under_def) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1043 |
thus "A = UNIV \<or> (\<exists>n. A = {0 ..< n})" using closed_nat_set_iff by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1044 |
next |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1045 |
assume "A = UNIV \<or> (\<exists>n. A = {0 ..< n})" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1046 |
thus "ofilter natLeq A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1047 |
by(auto simp add: natLeq_ofilter_less natLeq_UNIV_ofilter) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1048 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1049 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1050 |
lemma natLeq_under_leq: "under natLeq n = {0 .. n}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1051 |
unfolding rel.under_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1052 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1053 |
corollary natLeq_on_ofilter: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1054 |
"ofilter(natLeq_on n) {0 ..< n}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1055 |
by (auto simp add: natLeq_on_ofilter_less_eq) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1056 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1057 |
lemma natLeq_on_ofilter_less: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1058 |
"n < m \<Longrightarrow> ofilter (natLeq_on m) {0 .. n}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1059 |
by(auto simp add: natLeq_on_wo_rel wo_rel.ofilter_def, |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1060 |
simp add: Field_natLeq_on, unfold rel.under_def, auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1061 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1062 |
lemma natLeq_on_ordLess_natLeq: "natLeq_on n <o natLeq" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1063 |
using Field_natLeq Field_natLeq_on[of n] nat_infinite |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1064 |
finite_ordLess_infinite[of "natLeq_on n" natLeq] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1065 |
natLeq_Well_order natLeq_on_Well_order[of n] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1066 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1067 |
lemma natLeq_on_injective: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1068 |
"natLeq_on m = natLeq_on n \<Longrightarrow> m = n" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1069 |
using Field_natLeq_on[of m] Field_natLeq_on[of n] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1070 |
atLeastLessThan_injective[of m n] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1071 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1072 |
lemma natLeq_on_injective_ordIso: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1073 |
"(natLeq_on m =o natLeq_on n) = (m = n)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1074 |
proof(auto simp add: natLeq_on_Well_order ordIso_reflexive) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1075 |
assume "natLeq_on m =o natLeq_on n" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1076 |
then obtain f where "bij_betw f {0..<m} {0..<n}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1077 |
using Field_natLeq_on assms unfolding ordIso_def iso_def[abs_def] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1078 |
thus "m = n" using atLeastLessThan_injective2 by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1079 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1080 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1081 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1082 |
subsubsection {* Then as cardinals *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1083 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1084 |
lemma ordIso_natLeq_infinite1: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1085 |
"|A| =o natLeq \<Longrightarrow> infinite A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1086 |
using ordIso_symmetric ordIso_imp_ordLeq infinite_iff_natLeq_ordLeq by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1087 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1088 |
lemma ordIso_natLeq_infinite2: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1089 |
"natLeq =o |A| \<Longrightarrow> infinite A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1090 |
using ordIso_imp_ordLeq infinite_iff_natLeq_ordLeq by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1091 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1092 |
lemma ordLeq_natLeq_on_imp_finite: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1093 |
assumes "|A| \<le>o natLeq_on n" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1094 |
shows "finite A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1095 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1096 |
have "|A| \<le>o |{0 ..< n}|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1097 |
using assms card_of_less ordIso_symmetric ordLeq_ordIso_trans by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1098 |
thus ?thesis by (auto simp add: card_of_ordLeq_finite) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1099 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1100 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1101 |
|
54475 | 1102 |
subsubsection {* "Backward compatibility" with the numeric cardinal operator for finite sets *} |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1103 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1104 |
lemma finite_card_of_iff_card: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1105 |
assumes FIN: "finite A" and FIN': "finite B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1106 |
shows "( |A| =o |B| ) = (card A = card B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1107 |
using assms card_of_ordIso[of A B] bij_betw_iff_card[of A B] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1108 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1109 |
lemma finite_card_of_iff_card3: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1110 |
assumes FIN: "finite A" and FIN': "finite B" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1111 |
shows "( |A| <o |B| ) = (card A < card B)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1112 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1113 |
have "( |A| <o |B| ) = (~ ( |B| \<le>o |A| ))" by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1114 |
also have "... = (~ (card B \<le> card A))" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1115 |
using assms by(simp add: finite_card_of_iff_card2) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1116 |
also have "... = (card A < card B)" by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1117 |
finally show ?thesis . |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1118 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1119 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1120 |
lemma card_Field_natLeq_on: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1121 |
"card(Field(natLeq_on n)) = n" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1122 |
using Field_natLeq_on card_atLeastLessThan by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1123 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1124 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1125 |
subsection {* The successor of a cardinal *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1126 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1127 |
lemma embed_implies_ordIso_Restr: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1128 |
assumes WELL: "Well_order r" and WELL': "Well_order r'" and EMB: "embed r' r f" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1129 |
shows "r' =o Restr r (f ` (Field r'))" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1130 |
using assms embed_implies_iso_Restr Well_order_Restr unfolding ordIso_def by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1131 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1132 |
lemma cardSuc_Well_order[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1133 |
"Card_order r \<Longrightarrow> Well_order(cardSuc r)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1134 |
using cardSuc_Card_order unfolding card_order_on_def by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1135 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1136 |
lemma Field_cardSuc_not_empty: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1137 |
assumes "Card_order r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1138 |
shows "Field (cardSuc r) \<noteq> {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1139 |
proof |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1140 |
assume "Field(cardSuc r) = {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1141 |
hence "|Field(cardSuc r)| \<le>o r" using assms Card_order_empty[of r] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1142 |
hence "cardSuc r \<le>o r" using assms card_of_Field_ordIso |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1143 |
cardSuc_Card_order ordIso_symmetric ordIso_ordLeq_trans by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1144 |
thus False using cardSuc_greater not_ordLess_ordLeq assms by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1145 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1146 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1147 |
lemma cardSuc_mono_ordLess[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1148 |
assumes CARD: "Card_order r" and CARD': "Card_order r'" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1149 |
shows "(cardSuc r <o cardSuc r') = (r <o r')" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1150 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1151 |
have 0: "Well_order r \<and> Well_order r' \<and> Well_order(cardSuc r) \<and> Well_order(cardSuc r')" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1152 |
using assms by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1153 |
thus ?thesis |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1154 |
using not_ordLeq_iff_ordLess not_ordLeq_iff_ordLess[of r r'] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1155 |
using cardSuc_mono_ordLeq[of r' r] assms by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1156 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1157 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1158 |
lemma card_of_Plus_ordLeq_infinite[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1159 |
assumes C: "infinite C" and A: "|A| \<le>o |C|" and B: "|B| \<le>o |C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1160 |
shows "|A <+> B| \<le>o |C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1161 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1162 |
let ?r = "cardSuc |C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1163 |
have "Card_order ?r \<and> infinite (Field ?r)" using assms by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1164 |
moreover have "|A| <o ?r" and "|B| <o ?r" using A B by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1165 |
ultimately have "|A <+> B| <o ?r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1166 |
using card_of_Plus_ordLess_infinite_Field by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1167 |
thus ?thesis using C by simp |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1168 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1169 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1170 |
lemma card_of_Un_ordLeq_infinite[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1171 |
assumes C: "infinite C" and A: "|A| \<le>o |C|" and B: "|B| \<le>o |C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1172 |
shows "|A Un B| \<le>o |C|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1173 |
using assms card_of_Plus_ordLeq_infinite card_of_Un_Plus_ordLeq |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1174 |
ordLeq_transitive by metis |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1175 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1176 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1177 |
subsection {* Others *} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1178 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1179 |
lemma under_mono[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1180 |
assumes "Well_order r" and "(i,j) \<in> r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1181 |
shows "under r i \<subseteq> under r j" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1182 |
using assms unfolding rel.under_def order_on_defs |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1183 |
trans_def by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1184 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1185 |
lemma underS_under: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1186 |
assumes "i \<in> Field r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1187 |
shows "underS r i = under r i - {i}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1188 |
using assms unfolding rel.underS_def rel.under_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1189 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1190 |
lemma relChain_under: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1191 |
assumes "Well_order r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1192 |
shows "relChain r (\<lambda> i. under r i)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1193 |
using assms unfolding relChain_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1194 |
|
54475 | 1195 |
lemma card_of_infinite_diff_finite: |
1196 |
assumes "infinite A" and "finite B" |
|
1197 |
shows "|A - B| =o |A|" |
|
1198 |
by (metis assms card_of_Un_diff_infinite finite_ordLess_infinite2) |
|
1199 |
||
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1200 |
lemma infinite_card_of_diff_singl: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1201 |
assumes "infinite A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1202 |
shows "|A - {a}| =o |A|" |
52544 | 1203 |
by (metis assms card_of_infinite_diff_finite finite.emptyI finite_insert) |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1204 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1205 |
lemma card_of_vimage: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1206 |
assumes "B \<subseteq> range f" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1207 |
shows "|B| \<le>o |f -` B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1208 |
apply(rule surj_imp_ordLeq[of _ f]) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1209 |
using assms by (metis Int_absorb2 image_vimage_eq order_refl) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1210 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1211 |
lemma surj_card_of_vimage: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1212 |
assumes "surj f" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1213 |
shows "|B| \<le>o |f -` B|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1214 |
by (metis assms card_of_vimage subset_UNIV) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1215 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1216 |
(* bounded powerset *) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1217 |
definition Bpow where |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1218 |
"Bpow r A \<equiv> {X . X \<subseteq> A \<and> |X| \<le>o r}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1219 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1220 |
lemma Bpow_empty[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1221 |
assumes "Card_order r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1222 |
shows "Bpow r {} = {{}}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1223 |
using assms unfolding Bpow_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1224 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1225 |
lemma singl_in_Bpow: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1226 |
assumes rc: "Card_order r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1227 |
and r: "Field r \<noteq> {}" and a: "a \<in> A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1228 |
shows "{a} \<in> Bpow r A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1229 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1230 |
have "|{a}| \<le>o r" using r rc by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1231 |
thus ?thesis unfolding Bpow_def using a by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1232 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1233 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1234 |
lemma ordLeq_card_Bpow: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1235 |
assumes rc: "Card_order r" and r: "Field r \<noteq> {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1236 |
shows "|A| \<le>o |Bpow r A|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1237 |
proof- |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1238 |
have "inj_on (\<lambda> a. {a}) A" unfolding inj_on_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1239 |
moreover have "(\<lambda> a. {a}) ` A \<subseteq> Bpow r A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1240 |
using singl_in_Bpow[OF assms] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1241 |
ultimately show ?thesis unfolding card_of_ordLeq[symmetric] by blast |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1242 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1243 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1244 |
lemma infinite_Bpow: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1245 |
assumes rc: "Card_order r" and r: "Field r \<noteq> {}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1246 |
and A: "infinite A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1247 |
shows "infinite (Bpow r A)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1248 |
using ordLeq_card_Bpow[OF rc r] |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1249 |
by (metis A card_of_ordLeq_infinite) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1250 |
|
52545
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1251 |
definition Func_option where |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1252 |
"Func_option A B \<equiv> |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1253 |
{f. (\<forall> a. f a \<noteq> None \<longleftrightarrow> a \<in> A) \<and> (\<forall> a \<in> A. case f a of Some b \<Rightarrow> b \<in> B |None \<Rightarrow> True)}" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1254 |
|
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1255 |
lemma card_of_Func_option_Func: |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1256 |
"|Func_option A B| =o |Func A B|" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1257 |
proof (rule ordIso_symmetric, unfold card_of_ordIso[symmetric], intro exI) |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1258 |
let ?F = "\<lambda> f a. if a \<in> A then Some (f a) else None" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1259 |
show "bij_betw ?F (Func A B) (Func_option A B)" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1260 |
unfolding bij_betw_def unfolding inj_on_def proof(intro conjI ballI impI) |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1261 |
fix f g assume f: "f \<in> Func A B" and g: "g \<in> Func A B" and eq: "?F f = ?F g" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1262 |
show "f = g" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1263 |
proof(rule ext) |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1264 |
fix a |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1265 |
show "f a = g a" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1266 |
proof(cases "a \<in> A") |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1267 |
case True |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1268 |
have "Some (f a) = ?F f a" using True by auto |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1269 |
also have "... = ?F g a" using eq unfolding fun_eq_iff by(rule allE) |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1270 |
also have "... = Some (g a)" using True by auto |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1271 |
finally have "Some (f a) = Some (g a)" . |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1272 |
thus ?thesis by simp |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1273 |
qed(insert f g, unfold Func_def, auto) |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1274 |
qed |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1275 |
next |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1276 |
show "?F ` Func A B = Func_option A B" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1277 |
proof safe |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1278 |
fix f assume f: "f \<in> Func_option A B" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1279 |
def g \<equiv> "\<lambda> a. case f a of Some b \<Rightarrow> b | None \<Rightarrow> undefined" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1280 |
have "g \<in> Func A B" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1281 |
using f unfolding g_def Func_def Func_option_def by force+ |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1282 |
moreover have "f = ?F g" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1283 |
proof(rule ext) |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1284 |
fix a show "f a = ?F g a" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1285 |
using f unfolding Func_option_def g_def by (cases "a \<in> A") force+ |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1286 |
qed |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1287 |
ultimately show "f \<in> ?F ` (Func A B)" by blast |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1288 |
qed(unfold Func_def Func_option_def, auto) |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1289 |
qed |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1290 |
qed |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1291 |
|
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1292 |
(* partial-function space: *) |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1293 |
definition Pfunc where |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1294 |
"Pfunc A B \<equiv> |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1295 |
{f. (\<forall>a. f a \<noteq> None \<longrightarrow> a \<in> A) \<and> |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1296 |
(\<forall>a. case f a of None \<Rightarrow> True | Some b \<Rightarrow> b \<in> B)}" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1297 |
|
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1298 |
lemma Func_Pfunc: |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1299 |
"Func_option A B \<subseteq> Pfunc A B" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1300 |
unfolding Func_option_def Pfunc_def by auto |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1301 |
|
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1302 |
lemma Pfunc_Func_option: |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1303 |
"Pfunc A B = (\<Union> A' \<in> Pow A. Func_option A' B)" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1304 |
proof safe |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1305 |
fix f assume f: "f \<in> Pfunc A B" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1306 |
show "f \<in> (\<Union>A'\<in>Pow A. Func_option A' B)" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1307 |
proof (intro UN_I) |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1308 |
let ?A' = "{a. f a \<noteq> None}" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1309 |
show "?A' \<in> Pow A" using f unfolding Pow_def Pfunc_def by auto |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1310 |
show "f \<in> Func_option ?A' B" using f unfolding Func_option_def Pfunc_def by auto |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1311 |
qed |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1312 |
next |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1313 |
fix f A' assume "f \<in> Func_option A' B" and "A' \<subseteq> A" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1314 |
thus "f \<in> Pfunc A B" unfolding Func_option_def Pfunc_def by auto |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1315 |
qed |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1316 |
|
54475 | 1317 |
lemma card_of_Func_mono: |
1318 |
fixes A1 A2 :: "'a set" and B :: "'b set" |
|
1319 |
assumes A12: "A1 \<subseteq> A2" and B: "B \<noteq> {}" |
|
1320 |
shows "|Func A1 B| \<le>o |Func A2 B|" |
|
1321 |
proof- |
|
1322 |
obtain bb where bb: "bb \<in> B" using B by auto |
|
1323 |
def F \<equiv> "\<lambda> (f1::'a \<Rightarrow> 'b) a. if a \<in> A2 then (if a \<in> A1 then f1 a else bb) |
|
1324 |
else undefined" |
|
1325 |
show ?thesis unfolding card_of_ordLeq[symmetric] proof(intro exI[of _ F] conjI) |
|
1326 |
show "inj_on F (Func A1 B)" unfolding inj_on_def proof safe |
|
1327 |
fix f g assume f: "f \<in> Func A1 B" and g: "g \<in> Func A1 B" and eq: "F f = F g" |
|
1328 |
show "f = g" |
|
1329 |
proof(rule ext) |
|
1330 |
fix a show "f a = g a" |
|
1331 |
proof(cases "a \<in> A1") |
|
1332 |
case True |
|
1333 |
thus ?thesis using eq A12 unfolding F_def fun_eq_iff |
|
1334 |
by (elim allE[of _ a]) auto |
|
1335 |
qed(insert f g, unfold Func_def, fastforce) |
|
1336 |
qed |
|
1337 |
qed |
|
1338 |
qed(insert bb, unfold Func_def F_def, force) |
|
1339 |
qed |
|
1340 |
||
52545
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1341 |
lemma card_of_Func_option_mono: |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1342 |
fixes A1 A2 :: "'a set" and B :: "'b set" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1343 |
assumes A12: "A1 \<subseteq> A2" and B: "B \<noteq> {}" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1344 |
shows "|Func_option A1 B| \<le>o |Func_option A2 B|" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1345 |
by (metis card_of_Func_mono[OF A12 B] card_of_Func_option_Func |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1346 |
ordIso_ordLeq_trans ordLeq_ordIso_trans ordIso_symmetric) |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1347 |
|
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1348 |
lemma card_of_Pfunc_Pow_Func_option: |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1349 |
assumes "B \<noteq> {}" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1350 |
shows "|Pfunc A B| \<le>o |Pow A <*> Func_option A B|" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1351 |
proof- |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1352 |
have "|Pfunc A B| =o |\<Union> A' \<in> Pow A. Func_option A' B|" (is "_ =o ?K") |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1353 |
unfolding Pfunc_Func_option by(rule card_of_refl) |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1354 |
also have "?K \<le>o |Sigma (Pow A) (\<lambda> A'. Func_option A' B)|" using card_of_UNION_Sigma . |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1355 |
also have "|Sigma (Pow A) (\<lambda> A'. Func_option A' B)| \<le>o |Pow A <*> Func_option A B|" |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1356 |
apply(rule card_of_Sigma_mono1) using card_of_Func_option_mono[OF _ assms] by auto |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1357 |
finally show ?thesis . |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1358 |
qed |
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1359 |
|
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1360 |
lemma Bpow_ordLeq_Func_Field: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1361 |
assumes rc: "Card_order r" and r: "Field r \<noteq> {}" and A: "infinite A" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1362 |
shows "|Bpow r A| \<le>o |Func (Field r) A|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1363 |
proof- |
52545
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1364 |
let ?F = "\<lambda> f. {x | x a. f a = x \<and> a \<in> Field r}" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1365 |
{fix X assume "X \<in> Bpow r A - {{}}" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1366 |
hence XA: "X \<subseteq> A" and "|X| \<le>o r" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1367 |
and X: "X \<noteq> {}" unfolding Bpow_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1368 |
hence "|X| \<le>o |Field r|" by (metis Field_card_of card_of_mono2) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1369 |
then obtain F where 1: "X = F ` (Field r)" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1370 |
using card_of_ordLeq2[OF X] by metis |
52545
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1371 |
def f \<equiv> "\<lambda> i. if i \<in> Field r then F i else undefined" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1372 |
have "\<exists> f \<in> Func (Field r) A. X = ?F f" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1373 |
apply (intro bexI[of _ f]) using 1 XA unfolding Func_def f_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1374 |
} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1375 |
hence "Bpow r A - {{}} \<subseteq> ?F ` (Func (Field r) A)" by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1376 |
hence "|Bpow r A - {{}}| \<le>o |Func (Field r) A|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1377 |
by (rule surj_imp_ordLeq) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1378 |
moreover |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1379 |
{have 2: "infinite (Bpow r A)" using infinite_Bpow[OF rc r A] . |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1380 |
have "|Bpow r A| =o |Bpow r A - {{}}|" |
52544 | 1381 |
using card_of_infinite_diff_finite |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1382 |
by (metis Pow_empty 2 finite_Pow_iff infinite_imp_nonempty ordIso_symmetric) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1383 |
} |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1384 |
ultimately show ?thesis by (metis ordIso_ordLeq_trans) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1385 |
qed |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1386 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1387 |
lemma Func_emp2[simp]: "A \<noteq> {} \<Longrightarrow> Func A {} = {}" by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1388 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1389 |
lemma empty_in_Func[simp]: |
52545
d2ad6eae514f
Func -> Func_option, Ffunc -> Func (avoids dependence of codatatypes on the option type)
traytel
parents:
52544
diff
changeset
|
1390 |
"B \<noteq> {} \<Longrightarrow> (\<lambda>x. undefined) \<in> Func {} B" |
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1391 |
unfolding Func_def by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1392 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1393 |
lemma Func_mono[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1394 |
assumes "B1 \<subseteq> B2" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1395 |
shows "Func A B1 \<subseteq> Func A B2" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1396 |
using assms unfolding Func_def by force |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1397 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1398 |
lemma Pfunc_mono[simp]: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1399 |
assumes "A1 \<subseteq> A2" and "B1 \<subseteq> B2" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1400 |
shows "Pfunc A B1 \<subseteq> Pfunc A B2" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1401 |
using assms in_mono unfolding Pfunc_def apply safe |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1402 |
apply(case_tac "x a", auto) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1403 |
by (metis in_mono option.simps(5)) |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1404 |
|
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1405 |
lemma card_of_Func_UNIV_UNIV: |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1406 |
"|Func (UNIV::'a set) (UNIV::'b set)| =o |UNIV::('a \<Rightarrow> 'b) set|" |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1407 |
using card_of_Func_UNIV[of "UNIV::'b set"] by auto |
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1408 |
|
54475 | 1409 |
lemma ordLeq_Func: |
1410 |
assumes "{b1,b2} \<subseteq> B" "b1 \<noteq> b2" |
|
1411 |
shows "|A| \<le>o |Func A B|" |
|
1412 |
unfolding card_of_ordLeq[symmetric] proof(intro exI conjI) |
|
1413 |
let ?F = "\<lambda> aa a. if a \<in> A then (if a = aa then b1 else b2) else undefined" |
|
1414 |
show "inj_on ?F A" using assms unfolding inj_on_def fun_eq_iff by auto |
|
1415 |
show "?F ` A \<subseteq> Func A B" using assms unfolding Func_def by auto |
|
1416 |
qed |
|
1417 |
||
1418 |
lemma infinite_Func: |
|
1419 |
assumes A: "infinite A" and B: "{b1,b2} \<subseteq> B" "b1 \<noteq> b2" |
|
1420 |
shows "infinite (Func A B)" |
|
1421 |
using ordLeq_Func[OF B] by (metis A card_of_ordLeq_finite) |
|
1422 |
||
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
blanchet
parents:
diff
changeset
|
1423 |
end |