(* Title: ZF/Cardinal_AC.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
These results help justify infinite-branching datatypes
*)
header{*Cardinal Arithmetic Using AC*}
theory Cardinal_AC = CardinalArith + Zorn:
subsection{*Strengthened Forms of Existing Theorems on Cardinals*}
lemma cardinal_eqpoll: "|A| eqpoll A"
apply (rule AC_well_ord [THEN exE])
apply (erule well_ord_cardinal_eqpoll)
done
text{*The theorem @{term "||A|| = |A|"} *}
lemmas cardinal_idem = cardinal_eqpoll [THEN cardinal_cong, standard, simp]
lemma cardinal_eqE: "|X| = |Y| ==> X eqpoll Y"
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
apply (rule well_ord_cardinal_eqE, assumption+)
done
lemma cardinal_eqpoll_iff: "|X| = |Y| <-> X eqpoll Y"
by (blast intro: cardinal_cong cardinal_eqE)
lemma cardinal_disjoint_Un:
"[| |A|=|B|; |C|=|D|; A Int C = 0; B Int D = 0 |]
==> |A Un C| = |B Un D|"
by (simp add: cardinal_eqpoll_iff eqpoll_disjoint_Un)
lemma lepoll_imp_Card_le: "A lepoll B ==> |A| le |B|"
apply (rule AC_well_ord [THEN exE])
apply (erule well_ord_lepoll_imp_Card_le, assumption)
done
lemma cadd_assoc: "(i |+| j) |+| k = i |+| (j |+| k)"
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
apply (rule well_ord_cadd_assoc, assumption+)
done
lemma cmult_assoc: "(i |*| j) |*| k = i |*| (j |*| k)"
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
apply (rule well_ord_cmult_assoc, assumption+)
done
lemma cadd_cmult_distrib: "(i |+| j) |*| k = (i |*| k) |+| (j |*| k)"
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
apply (rule AC_well_ord [THEN exE])
apply (rule well_ord_cadd_cmult_distrib, assumption+)
done
lemma InfCard_square_eq: "InfCard(|A|) ==> A*A eqpoll A"
apply (rule AC_well_ord [THEN exE])
apply (erule well_ord_InfCard_square_eq, assumption)
done
subsection {*The relationship between cardinality and le-pollence*}
lemma Card_le_imp_lepoll: "|A| le |B| ==> A lepoll B"
apply (rule cardinal_eqpoll
[THEN eqpoll_sym, THEN eqpoll_imp_lepoll, THEN lepoll_trans])
apply (erule le_imp_subset [THEN subset_imp_lepoll, THEN lepoll_trans])
apply (rule cardinal_eqpoll [THEN eqpoll_imp_lepoll])
done
lemma le_Card_iff: "Card(K) ==> |A| le K <-> A lepoll K"
apply (erule Card_cardinal_eq [THEN subst], rule iffI,
erule Card_le_imp_lepoll)
apply (erule lepoll_imp_Card_le)
done
lemma cardinal_0_iff_0 [simp]: "|A| = 0 <-> A = 0";
apply auto
apply (drule cardinal_0 [THEN ssubst])
apply (blast intro: eqpoll_0_iff [THEN iffD1] cardinal_eqpoll_iff [THEN iffD1])
done
lemma cardinal_lt_iff_lesspoll: "Ord(i) ==> i < |A| <-> i lesspoll A"
apply (cut_tac A = "A" in cardinal_eqpoll)
apply (auto simp add: eqpoll_iff)
apply (blast intro: lesspoll_trans2 lt_Card_imp_lesspoll Card_cardinal)
apply (force intro: cardinal_lt_imp_lt lesspoll_cardinal_lt lesspoll_trans2
simp add: cardinal_idem)
done
lemma cardinal_le_imp_lepoll: " i \<le> |A| ==> i \<lesssim> A"
apply (blast intro: lt_Ord Card_le_imp_lepoll Ord_cardinal_le le_trans)
done
subsection{*Other Applications of AC*}
lemma surj_implies_inj: "f: surj(X,Y) ==> EX g. g: inj(Y,X)"
apply (unfold surj_def)
apply (erule CollectE)
apply (rule_tac A1 = Y and B1 = "%y. f-``{y}" in AC_Pi [THEN exE])
apply (fast elim!: apply_Pair)
apply (blast dest: apply_type Pi_memberD
intro: apply_equality Pi_type f_imp_injective)
done
(*Kunen's Lemma 10.20*)
lemma surj_implies_cardinal_le: "f: surj(X,Y) ==> |Y| le |X|"
apply (rule lepoll_imp_Card_le)
apply (erule surj_implies_inj [THEN exE])
apply (unfold lepoll_def)
apply (erule exI)
done
(*Kunen's Lemma 10.21*)
lemma cardinal_UN_le:
"[| InfCard(K); ALL i:K. |X(i)| le K |] ==> |\<Union>i\<in>K. X(i)| le K"
apply (simp add: InfCard_is_Card le_Card_iff)
apply (rule lepoll_trans)
prefer 2
apply (rule InfCard_square_eq [THEN eqpoll_imp_lepoll])
apply (simp add: InfCard_is_Card Card_cardinal_eq)
apply (unfold lepoll_def)
apply (frule InfCard_is_Card [THEN Card_is_Ord])
apply (erule AC_ball_Pi [THEN exE])
apply (rule exI)
(*Lemma needed in both subgoals, for a fixed z*)
apply (subgoal_tac "ALL z: (\<Union>i\<in>K. X (i)). z: X (LEAST i. z:X (i)) &
(LEAST i. z:X (i)) : K")
prefer 2
apply (fast intro!: Least_le [THEN lt_trans1, THEN ltD] ltI
elim!: LeastI Ord_in_Ord)
apply (rule_tac c = "%z. <LEAST i. z:X (i), f ` (LEAST i. z:X (i)) ` z>"
and d = "%<i,j>. converse (f`i) ` j" in lam_injective)
(*Instantiate the lemma proved above*)
by (blast intro: inj_is_fun [THEN apply_type] dest: apply_type, force)
(*The same again, using csucc*)
lemma cardinal_UN_lt_csucc:
"[| InfCard(K); ALL i:K. |X(i)| < csucc(K) |]
==> |\<Union>i\<in>K. X(i)| < csucc(K)"
by (simp add: Card_lt_csucc_iff cardinal_UN_le InfCard_is_Card Card_cardinal)
(*The same again, for a union of ordinals. In use, j(i) is a bit like rank(i),
the least ordinal j such that i:Vfrom(A,j). *)
lemma cardinal_UN_Ord_lt_csucc:
"[| InfCard(K); ALL i:K. j(i) < csucc(K) |]
==> (\<Union>i\<in>K. j(i)) < csucc(K)"
apply (rule cardinal_UN_lt_csucc [THEN Card_lt_imp_lt], assumption)
apply (blast intro: Ord_cardinal_le [THEN lt_trans1] elim: ltE)
apply (blast intro!: Ord_UN elim: ltE)
apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN Card_csucc])
done
(** Main result for infinite-branching datatypes. As above, but the index
set need not be a cardinal. Surprisingly complicated proof!
**)
(*Work backwards along the injection from W into K, given that W~=0.*)
lemma inj_UN_subset:
"[| f: inj(A,B); a:A |] ==>
(\<Union>x\<in>A. C(x)) <= (\<Union>y\<in>B. C(if y: range(f) then converse(f)`y else a))"
apply (rule UN_least)
apply (rule_tac x1= "f`x" in subset_trans [OF _ UN_upper])
apply (simp add: inj_is_fun [THEN apply_rangeI])
apply (blast intro: inj_is_fun [THEN apply_type])
done
(*Simpler to require |W|=K; we'd have a bijection; but the theorem would
be weaker.*)
lemma le_UN_Ord_lt_csucc:
"[| InfCard(K); |W| le K; ALL w:W. j(w) < csucc(K) |]
==> (\<Union>w\<in>W. j(w)) < csucc(K)"
apply (case_tac "W=0")
(*solve the easy 0 case*)
apply (simp add: InfCard_is_Card Card_is_Ord [THEN Card_csucc]
Card_is_Ord Ord_0_lt_csucc)
apply (simp add: InfCard_is_Card le_Card_iff lepoll_def)
apply (safe intro!: equalityI)
apply (erule swap)
apply (rule lt_subset_trans [OF inj_UN_subset cardinal_UN_Ord_lt_csucc], assumption+)
apply (simp add: inj_converse_fun [THEN apply_type])
apply (blast intro!: Ord_UN elim: ltE)
done
ML
{*
val cardinal_0_iff_0 = thm "cardinal_0_iff_0";
val cardinal_lt_iff_lesspoll = thm "cardinal_lt_iff_lesspoll";
*}
end