1 (* Title: ZF/Cardinal_AC.thy
2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
3 Copyright 1994 University of Cambridge
5 These results help justify infinite-branching datatypes
8 header{*Cardinal Arithmetic Using AC*}
10 theory Cardinal_AC imports CardinalArith Zorn begin
12 subsection{*Strengthened Forms of Existing Theorems on Cardinals*}
14 lemma cardinal_eqpoll: "|A| eqpoll A"
15 apply (rule AC_well_ord [THEN exE])
16 apply (erule well_ord_cardinal_eqpoll)
19 text{*The theorem @{term "||A|| = |A|"} *}
20 lemmas cardinal_idem = cardinal_eqpoll [THEN cardinal_cong, standard, simp]
22 lemma cardinal_eqE: "|X| = |Y| ==> X eqpoll Y"
23 apply (rule AC_well_ord [THEN exE])
24 apply (rule AC_well_ord [THEN exE])
25 apply (rule well_ord_cardinal_eqE, assumption+)
28 lemma cardinal_eqpoll_iff: "|X| = |Y| <-> X eqpoll Y"
29 by (blast intro: cardinal_cong cardinal_eqE)
31 lemma cardinal_disjoint_Un:
32 "[| |A|=|B|; |C|=|D|; A Int C = 0; B Int D = 0 |]
33 ==> |A Un C| = |B Un D|"
34 by (simp add: cardinal_eqpoll_iff eqpoll_disjoint_Un)
36 lemma lepoll_imp_Card_le: "A lepoll B ==> |A| le |B|"
37 apply (rule AC_well_ord [THEN exE])
38 apply (erule well_ord_lepoll_imp_Card_le, assumption)
41 lemma cadd_assoc: "(i |+| j) |+| k = i |+| (j |+| k)"
42 apply (rule AC_well_ord [THEN exE])
43 apply (rule AC_well_ord [THEN exE])
44 apply (rule AC_well_ord [THEN exE])
45 apply (rule well_ord_cadd_assoc, assumption+)
48 lemma cmult_assoc: "(i |*| j) |*| k = i |*| (j |*| k)"
49 apply (rule AC_well_ord [THEN exE])
50 apply (rule AC_well_ord [THEN exE])
51 apply (rule AC_well_ord [THEN exE])
52 apply (rule well_ord_cmult_assoc, assumption+)
55 lemma cadd_cmult_distrib: "(i |+| j) |*| k = (i |*| k) |+| (j |*| k)"
56 apply (rule AC_well_ord [THEN exE])
57 apply (rule AC_well_ord [THEN exE])
58 apply (rule AC_well_ord [THEN exE])
59 apply (rule well_ord_cadd_cmult_distrib, assumption+)
62 lemma InfCard_square_eq: "InfCard(|A|) ==> A*A eqpoll A"
63 apply (rule AC_well_ord [THEN exE])
64 apply (erule well_ord_InfCard_square_eq, assumption)
68 subsection {*The relationship between cardinality and le-pollence*}
70 lemma Card_le_imp_lepoll: "|A| le |B| ==> A lepoll B"
71 apply (rule cardinal_eqpoll
72 [THEN eqpoll_sym, THEN eqpoll_imp_lepoll, THEN lepoll_trans])
73 apply (erule le_imp_subset [THEN subset_imp_lepoll, THEN lepoll_trans])
74 apply (rule cardinal_eqpoll [THEN eqpoll_imp_lepoll])
77 lemma le_Card_iff: "Card(K) ==> |A| le K <-> A lepoll K"
78 apply (erule Card_cardinal_eq [THEN subst], rule iffI,
79 erule Card_le_imp_lepoll)
80 apply (erule lepoll_imp_Card_le)
83 lemma cardinal_0_iff_0 [simp]: "|A| = 0 <-> A = 0";
85 apply (drule cardinal_0 [THEN ssubst])
86 apply (blast intro: eqpoll_0_iff [THEN iffD1] cardinal_eqpoll_iff [THEN iffD1])
89 lemma cardinal_lt_iff_lesspoll: "Ord(i) ==> i < |A| <-> i lesspoll A"
90 apply (cut_tac A = "A" in cardinal_eqpoll)
91 apply (auto simp add: eqpoll_iff)
92 apply (blast intro: lesspoll_trans2 lt_Card_imp_lesspoll Card_cardinal)
93 apply (force intro: cardinal_lt_imp_lt lesspoll_cardinal_lt lesspoll_trans2
94 simp add: cardinal_idem)
97 lemma cardinal_le_imp_lepoll: " i \<le> |A| ==> i \<lesssim> A"
98 apply (blast intro: lt_Ord Card_le_imp_lepoll Ord_cardinal_le le_trans)
102 subsection{*Other Applications of AC*}
104 lemma surj_implies_inj: "f: surj(X,Y) ==> EX g. g: inj(Y,X)"
105 apply (unfold surj_def)
106 apply (erule CollectE)
107 apply (rule_tac A1 = Y and B1 = "%y. f-``{y}" in AC_Pi [THEN exE])
108 apply (fast elim!: apply_Pair)
109 apply (blast dest: apply_type Pi_memberD
110 intro: apply_equality Pi_type f_imp_injective)
113 (*Kunen's Lemma 10.20*)
114 lemma surj_implies_cardinal_le: "f: surj(X,Y) ==> |Y| le |X|"
115 apply (rule lepoll_imp_Card_le)
116 apply (erule surj_implies_inj [THEN exE])
117 apply (unfold lepoll_def)
121 (*Kunen's Lemma 10.21*)
122 lemma cardinal_UN_le:
123 "[| InfCard(K); ALL i:K. |X(i)| le K |] ==> |\<Union>i\<in>K. X(i)| le K"
124 apply (simp add: InfCard_is_Card le_Card_iff)
125 apply (rule lepoll_trans)
127 apply (rule InfCard_square_eq [THEN eqpoll_imp_lepoll])
128 apply (simp add: InfCard_is_Card Card_cardinal_eq)
129 apply (unfold lepoll_def)
130 apply (frule InfCard_is_Card [THEN Card_is_Ord])
131 apply (erule AC_ball_Pi [THEN exE])
133 (*Lemma needed in both subgoals, for a fixed z*)
134 apply (subgoal_tac "ALL z: (\<Union>i\<in>K. X (i)). z: X (LEAST i. z:X (i)) &
135 (LEAST i. z:X (i)) : K")
137 apply (fast intro!: Least_le [THEN lt_trans1, THEN ltD] ltI
138 elim!: LeastI Ord_in_Ord)
139 apply (rule_tac c = "%z. <LEAST i. z:X (i), f ` (LEAST i. z:X (i)) ` z>"
140 and d = "%<i,j>. converse (f`i) ` j" in lam_injective)
141 (*Instantiate the lemma proved above*)
142 by (blast intro: inj_is_fun [THEN apply_type] dest: apply_type, force)
145 (*The same again, using csucc*)
146 lemma cardinal_UN_lt_csucc:
147 "[| InfCard(K); ALL i:K. |X(i)| < csucc(K) |]
148 ==> |\<Union>i\<in>K. X(i)| < csucc(K)"
149 by (simp add: Card_lt_csucc_iff cardinal_UN_le InfCard_is_Card Card_cardinal)
151 (*The same again, for a union of ordinals. In use, j(i) is a bit like rank(i),
152 the least ordinal j such that i:Vfrom(A,j). *)
153 lemma cardinal_UN_Ord_lt_csucc:
154 "[| InfCard(K); ALL i:K. j(i) < csucc(K) |]
155 ==> (\<Union>i\<in>K. j(i)) < csucc(K)"
156 apply (rule cardinal_UN_lt_csucc [THEN Card_lt_imp_lt], assumption)
157 apply (blast intro: Ord_cardinal_le [THEN lt_trans1] elim: ltE)
158 apply (blast intro!: Ord_UN elim: ltE)
159 apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN Card_csucc])
163 (** Main result for infinite-branching datatypes. As above, but the index
164 set need not be a cardinal. Surprisingly complicated proof!
167 (*Work backwards along the injection from W into K, given that W~=0.*)
169 "[| f: inj(A,B); a:A |] ==>
170 (\<Union>x\<in>A. C(x)) <= (\<Union>y\<in>B. C(if y: range(f) then converse(f)`y else a))"
171 apply (rule UN_least)
172 apply (rule_tac x1= "f`x" in subset_trans [OF _ UN_upper])
173 apply (simp add: inj_is_fun [THEN apply_rangeI])
174 apply (blast intro: inj_is_fun [THEN apply_type])
177 (*Simpler to require |W|=K; we'd have a bijection; but the theorem would
179 lemma le_UN_Ord_lt_csucc:
180 "[| InfCard(K); |W| le K; ALL w:W. j(w) < csucc(K) |]
181 ==> (\<Union>w\<in>W. j(w)) < csucc(K)"
182 apply (case_tac "W=0")
183 (*solve the easy 0 case*)
184 apply (simp add: InfCard_is_Card Card_is_Ord [THEN Card_csucc]
185 Card_is_Ord Ord_0_lt_csucc)
186 apply (simp add: InfCard_is_Card le_Card_iff lepoll_def)
187 apply (safe intro!: equalityI)
189 apply (rule lt_subset_trans [OF inj_UN_subset cardinal_UN_Ord_lt_csucc], assumption+)
190 apply (simp add: inj_converse_fun [THEN apply_type])
191 apply (blast intro!: Ord_UN elim: ltE)
196 val cardinal_0_iff_0 = thm "cardinal_0_iff_0";
197 val cardinal_lt_iff_lesspoll = thm "cardinal_lt_iff_lesspoll";