(*  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 imports CardinalArith Zorn begin

 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